MTENSOR

<!DOCTYPE html PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN" "http://www.w3.org/TR/html4/loose.dtd"> <html> <head> <meta name="author" content="Bilbao Crystallographic Server" /> <meta name="description" content="provides the most general form of any tensor adapted to the symmetry of a magnetic point group"/> <meta name="keywords" content="tensor magnetic point groups symmetry-adapted form" /> <meta http-equiv="Content-Type" content="text/html;charset=ISO-8859-1" > <title> MTENSOR </title> <link rev="MADE" href="samuel.vidal@ehu.es"> <link href="/html/css/bcs_index.css" rel="stylesheet" title="BCS Style" type="text/css"> <script language="javascript"> function showNonconv() { var obj = document.getElementById('nonconvDiv'); if (document.form1.nonconv.checked) { obj.style.display = "block"; document.form1.input_type.value = "standard"; document.getElementById('byhandDiv').style.display = "none"; } else { obj.style.display = "none"; } block_state(); } function showbyhand() { var obj = document.getElementById('byhandDiv'); 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block_state(); } function carga() { //var obj = document.getElementById('nonconvDiv'); //obj.style.display = 'none'; showNonconv(); showbyhandno(); var choose_tensor = document.getElementById('choose_tensor'); //choose_tensor.selectedIndex = 0; document.form1.choose_tensor.value = "polarization"; document.form1.input_type.value = "standard"; //document.form1.choose_tensor.selectedIndex = 0; //document.getElementById('choose_order').options[0].onChange(); //document.getElementById('order1').style.display = "inline"; //for (loop = 2; loop < 9; loop++) { //var loopword = 'order' + loop; //document.getElementById(loopword).style.display = "none"; //} document.form1.domain.checked = false; document.form1.tensor_type.value = "standard"; block_state(); } function selectos() { var choose_order = document.getElementById('choose_order'); //var V_options = document.getElementById('V_options'); var index = document.form1.choose_order.selectedIndex; //document.write(index); index++; for (loop = 1; loop < 9; loop++) { var loopword = 'order' + loop; //document.write(loopword); document.getElementById(loopword).style.display = "none"; if (index == loop) { document.getElementById(loopword).style.display = "inline"; document.getElementById(loopword).options[0].selected = true; } } } </script> </head> <body onLoad="carga()"> <table class="signature"> <tr> <td> <a class="blue" href="/">Bilbao Crystallographic Server</a> <img alt="arrow" src="/html/gif/a.gif">MTENSOR </td> <td> <a class="blue" href="/html/cryst/mtensor_help.html">Help</a> </td> </tr> </table> <center> <h2 class="blue" align="center"> MTENSOR: Tensor calculation for Magnetic Point Groups </h2> </center>  <input type="hidden" name="type" value="user"> <center><h4>For the symmetry-adapted form of non-magnetic crystal tensors see <a href="/cgi-bin/cryst/programs/tensor.pl" target="_blank">TENSOR</a></h4></center> <table width="100%"> <tr valign="top"> <td class="blue" width="35%"> <table class="darkblue" width="100%"> <tr> <td> Tensor calculation for Magnetic Point Groups </td> </tr> </table> <table width="100%"> <tr> <td>  MTENSOR provides the symmetry-adapted form of tensor properties for any magnetic point (or space) group. On the one hand, a point or space group must be selected. On the other hand, a tensor must be defined by the user or selected from the lists of known equilibrium, optical, nonlinear optical susceptibility and transport tensors, gathered from scientific literature. If a magnetic point or space group is defined and a known tensor is selected from the lists the program will obtain the required tensor from an internal database; otherwise, the tensor is calculated live. Live calculation of tensors may take too much time and even exceed the time limit, giving an empty result, if high-rank tensors, and/or a lot of symmetry elements are introduced. <hr> Tutorial of MTENSOR: <a href="/html/cryst/tutorials/Tutorial_magnetic_section_BCS_1.pdf">download</a> <hr> Further information can be found <a class="blue" href="/html/cryst/mtensor_help.html" target="_blank">here</a> If you are using this program in the preparation of an article, please cite this reference: <a class="blue" href="http://scripts.iucr.org/cgi-bin/paper?S2053273319001748"> Gallego et al. "Automatic calculation of symmetry-adapted tensors in magnetic and non-magnetic materials: a new tool of the Bilbao Crystallographic Server" Acta Cryst. A (2019) 75, 438-447.</a> If you are interested in other publications related to Bilbao Crystallographic Server, click <a class="blue" href="http://www.cryst.ehu.es/wiki/index.php/Articles">here</a> <table cellpading="0" cellspacing="0" style="height: 0px"><tr><td><div id='block1' style="display:none"> </div><div id='block2' style="display:none"> </div><div id='block3' style="display:none"> </div><div id='block4' style="display:none"> </div><div id='block5' style="display:none"> </div><div id='legend_1' style="display:none"> <center><table style="width:500px" bgcolor="#CCEEFF"><tr><td><center><big>Information about the selected tensor</big></center> • 1 st rank Electric polarization vector Pi • Polar tensor invariant under time-reversal symmetry operation • Intrinsic symmetry symbol: V </td></tr></table></center> </div> <div id='legend_2' style="display:none"> <center><table style="width:500px" bgcolor="#CCEEFF"><tr><td><center><big>Information about the selected tensor</big></center> • 1 st rank Electrocaloric effect tensor pi • Polar tensor invariant under time-reversal symmetry operation • Defining equation: ΔS=piEi • Relates Electric field E with Entropy variation ΔS • Intrinsic symmetry symbol: V </td></tr></table></center> </div> <div id='legend_3' style="display:none"> <center><table style="width:500px" bgcolor="#CCEEFF"><tr><td><center><big>Information about the selected tensor</big></center> • 1 st rank Electrothermal effect tensor ti • Polar tensor invariant under time-reversal symmetry operation • Defining equation: Ei=-tiΔT • Relates temperature variation ΔT with Electric field E • Intrinsic symmetry symbol: V </td></tr></table></center> </div> <div id='legend_4' style="display:none"> <center><table style="width:500px" bgcolor="#CCEEFF"><tr><td><center><big>Information about the selected tensor</big></center> • 1 st rank Heat of polarization tensor ti • Polar tensor invariant under time-reversal symmetry operation • Defining equation: ΔS=tiΔPi • Relates Polarization vector P variation with Entropy variation ΔS • Intrinsic symmetry symbol: V </td></tr></table></center> </div> <div id='legend_5' style="display:none"> <center><table style="width:500px" bgcolor="#CCEEFF"><tr><td><center><big>Information about the selected tensor</big></center> • 1 st rank Piezoelectric polarization tensor under hydrostatic pressure dijj • Polar tensor invariant under time-reversal symmetry operation • Defining equation: Pi=-dijjp • Relates Hydrostatic pressure p with Polarization vector P. • This tensor of rank 2 is obtained from the contraction of the last two indices of the elastic compliance tensor sijkl. • Intrinsic symmetry symbol: V </td></tr></table></center> </div> <div id='legend_6' style="display:none"> <center><table style="width:500px" bgcolor="#CCEEFF"><tr><td><center><big>Information about the selected tensor</big></center> • 1 st rank Pyroelectric tensor pi • Polar tensor invariant under time-reversal symmetry operation • Defining equation: ΔPi=piΔT • Relates Temperature variation ΔT with Polarization vector P variation • Intrinsic symmetry symbol: V </td></tr></table></center> </div> <div id='legend_7' style="display:none"> <center><table style="width:500px" bgcolor="#CCEEFF"><tr><td><center><big>Information about the selected tensor</big></center> • 1 st rank Axial toroidal moment Ai • Axial tensor invariant under time-reversal symmetry operation • Intrinsic symmetry symbol: eV </td></tr></table></center> </div> <div id='legend_8' style="display:none"> <center><table style="width:500px" bgcolor="#CCEEFF"><tr><td><center><big>Information about the selected tensor</big></center> • 1 st rank Polar Toroidal moment Ti • Polar tensor which inverts under time-reversal symmetry operation • Intrinsic symmetry symbol: aV </td></tr></table></center> </div> <div id='legend_9' style="display:none"> <center><table style="width:500px" bgcolor="#CCEEFF"><tr><td><center><big>Information about the selected tensor</big></center> • 1 st rank Pyrotoroidic tensor ri • Polar tensor which inverts under time-reversal symmetry operation • Defining equation: Ti=riΔT • Relates Temperature variation ΔT with Toroidal moment T • Intrinsic symmetry symbol: aV </td></tr></table></center> </div> <div id='legend_10' style="display:none"> <center><table style="width:500px" bgcolor="#CCEEFF"><tr><td><center><big>Information about the selected tensor</big></center> • 1 st rank Toroidalcaloric tensor rTi • Polar tensor which inverts under time-reversal symmetry operation • Defining equation: ΔS=rTiSi • Relates Toroidal field S with Entropy variation ΔS • Intrinsic symmetry symbol: aV </td></tr></table></center> </div> <div id='legend_11' style="display:none"> <center><table style="width:500px" bgcolor="#CCEEFF"><tr><td><center><big>Information about the selected tensor</big></center> • 1 st rank Heat of Magnetization tensor ti • Axial tensor which inverts under time-reversal symmetry operation • Defining equation: ΔS=tiMi • Relates Magnetization M with Entropy variation ΔS • Intrinsic symmetry symbol: aeV </td></tr></table></center> </div> <div id='legend_12' style="display:none"> <center><table style="width:500px" bgcolor="#CCEEFF"><tr><td><center><big>Information about the selected tensor</big></center> • 1 st rank Magnetization vector Mi • Axial tensor which inverts under time-reversal symmetry operation • Intrinsic symmetry symbol: aeV </td></tr></table></center> </div> <div id='legend_13' style="display:none"> <center><table style="width:500px" bgcolor="#CCEEFF"><tr><td><center><big>Information about the selected tensor</big></center> • 1 st rank Magnetocaloric tensor qTi • Axial tensor which inverts under time-reversal symmetry operation • Defining equation: ΔS=qTiHi • Relates Magnetic field H with Entropy variation ΔS • Intrinsic symmetry symbol: aeV </td></tr></table></center> </div> <div id='legend_14' style="display:none"> <center><table style="width:500px" bgcolor="#CCEEFF"><tr><td><center><big>Information about the selected tensor</big></center> • 1 st rank Magnetothermal effect tensor ti • Axial tensor which inverts under time-reversal symmetry operation • Defining equation: Hi=-tiΔT • Relates Temperature variation ΔT with magnetic field H • Intrinsic symmetry symbol: aeV </td></tr></table></center> </div> <div id='legend_15' style="display:none"> <center><table style="width:500px" bgcolor="#CCEEFF"><tr><td><center><big>Information about the selected tensor</big></center> • 1 st rank Pyromagnetic tensor qi (direct effect) • Axial tensor which inverts under time-reversal symmetry operation • Defining equation: Mi=qiΔT • Relates Temperature variation ΔT with Magnetization M • Intrinsic symmetry symbol: aeV </td></tr></table></center> </div> <div id='legend_16' style="display:none"> <center><table style="width:500px" bgcolor="#CCEEFF"><tr><td><center><big>Information about the selected tensor</big></center> • 2 nd rank Dielectric impermeability tensor βij • Polar tensor invariant under time-reversal symmetry operation • Defining equation: Ei=βijDj • Relates Electric displacement field D with Electric field E • Intrinsic symmetry symbol: [V2] • Symmetrized indexes due to intrinsic symmetry:      • βij = βji </td></tr></table></center> </div> <div id='legend_17' style="display:none"> <center><table style="width:500px" bgcolor="#CCEEFF"><tr><td><center><big>Information about the selected tensor</big></center> • 2 nd rank Dielectric permittivity tensor εij • Polar tensor invariant under time-reversal symmetry operation • Defining equation: Di=εijEj • Relates Electric field E with Electric displacement field D • Intrinsic symmetry symbol: [V2] • Symmetrized indexes due to intrinsic symmetry:      • εij = εji </td></tr></table></center> </div> <div id='legend_18' style="display:none"> <center><table style="width:500px" bgcolor="#CCEEFF"><tr><td><center><big>Information about the selected tensor</big></center> • 2 nd rank Dielectric susceptibility tensor χeij • Polar tensor invariant under time-reversal symmetry operation • Defining equation: Pi=χeijEj • Relates Electric field E with Polarization vector P variation • Intrinsic symmetry symbol: [V2] • Symmetrized indexes due to intrinsic symmetry:      • χeij = χeji </td></tr></table></center> </div> <div id='legend_19' style="display:none"> <center><table style="width:500px" bgcolor="#CCEEFF"><tr><td><center><big>Information about the selected tensor</big></center> • 2 nd rank Heat of deformation tensor βij • Polar tensor invariant under time-reversal symmetry operation • Defining equation: ΔS=βijεij • Relates Strain tensor εij with Entropy variation ΔS • Intrinsic symmetry symbol: [V2] • Symmetrized indexes due to intrinsic symmetry:      • βij = βji </td></tr></table></center> </div> <div id='legend_20' style="display:none"> <center><table style="width:500px" bgcolor="#CCEEFF"><tr><td><center><big>Information about the selected tensor</big></center> • 2 nd rank Magnetic permeability tensor μmij • Polar tensor invariant under time-reversal symmetry operation • Defining equation: Bi=μmijHj • Relates Magnetic field H with Magnetic field B • Intrinsic symmetry symbol: [V2] • Symmetrized indexes due to intrinsic symmetry:      • μmij = μmji </td></tr></table></center> </div> <div id='legend_21' style="display:none"> <center><table style="width:500px" bgcolor="#CCEEFF"><tr><td><center><big>Information about the selected tensor</big></center> • 2 nd rank Magnetic susceptibility tensor χmij • Polar tensor invariant under time-reversal symmetry operation • Defining equation: Mi=χmijHj • Relates Magnetic field H with Magnetization M • Intrinsic symmetry symbol: [V2] • Symmetrized indexes due to intrinsic symmetry:      • χmij = χmji </td></tr></table></center> </div> <div id='legend_22' style="display:none"> <center><table style="width:500px" bgcolor="#CCEEFF"><tr><td><center><big>Information about the selected tensor</big></center> • 2 nd rank Piezocaloric effect tensor αij • Polar tensor invariant under time-reversal symmetry operation • Defining equation: ΔS=αijσij • Relates Stress tensor σij with Entropy variation ΔS • Intrinsic symmetry symbol: [V2] • Symmetrized indexes due to intrinsic symmetry:      • αij = αji </td></tr></table></center> </div> <div id='legend_23' style="display:none"> <center><table style="width:500px" bgcolor="#CCEEFF"><tr><td><center><big>Information about the selected tensor</big></center> • 2 nd rank Strain by hydrostatic pressure sijkk • Polar tensor invariant under time-reversal symmetry operation • Defining equation: εij=-sijkkp • Relates Hydrostatic pressure p with Strain tensor εij. • This tensor of rank 2 is obtained from the contraction of the last two indices of the elastic compliance tensor sijkl. • Intrinsic symmetry symbol: [V2] • Symmetrized indexes due to intrinsic symmetry:      • sij = sji </td></tr></table></center> </div> <div id='legend_24' style="display:none"> <center><table style="width:500px" bgcolor="#CCEEFF"><tr><td><center><big>Information about the selected tensor</big></center> • 2 nd rank Strain tensor εij • Polar tensor invariant under time-reversal symmetry operation • Intrinsic symmetry symbol: [V2] • Symmetrized indexes due to intrinsic symmetry:      • εij = εji </td></tr></table></center> </div> <div id='legend_25' style="display:none"> <center><table style="width:500px" bgcolor="#CCEEFF"><tr><td><center><big>Information about the selected tensor</big></center> • 2 nd rank Susceptibility inverse tensor χij (inverse effect) • Polar tensor invariant under time-reversal symmetry operation • Defining equation: Hi=χijMj • Relates Magnetization M with Magnetic field H • Intrinsic symmetry symbol: [V2] • Symmetrized indexes due to intrinsic symmetry:      • χijij = χijji </td></tr></table></center> </div> <div id='legend_26' style="display:none"> <center><table style="width:500px" bgcolor="#CCEEFF"><tr><td><center><big>Information about the selected tensor</big></center> • 2 nd rank Thermal expansion tensor αij • Polar tensor invariant under time-reversal symmetry operation • Defining equation: εij=αijΔT • Relates Temperature variation ΔT with Strain tensor εij • Intrinsic symmetry symbol: [V2] • Symmetrized indexes due to intrinsic symmetry:      • αij = αji </td></tr></table></center> </div> <div id='legend_27' style="display:none"> <center><table style="width:500px" bgcolor="#CCEEFF"><tr><td><center><big>Information about the selected tensor</big></center> • 2 nd rank Thermoelasticity tensor βij • Polar tensor invariant under time-reversal symmetry operation • Defining equation: σij=-βijΔT • Relates Temperature variation ΔT with Stress tensor σij • Intrinsic symmetry symbol: [V2] • Symmetrized indexes due to intrinsic symmetry:      • βij = βji </td></tr></table></center> </div> <div id='legend_28' style="display:none"> <center><table style="width:500px" bgcolor="#CCEEFF"><tr><td><center><big>Information about the selected tensor</big></center> • 2 nd rank Toroidic susceptibility tensor τij • Polar tensor invariant under time-reversal symmetry operation • Defining equation: Ti=τijSj • Relates Toroidal field S with Toroidal moment T • Intrinsic symmetry symbol: [V2] • Symmetrized indexes due to intrinsic symmetry:      • τij = τji </td></tr></table></center> </div> <div id='legend_29' style="display:none"> <center><table style="width:500px" bgcolor="#CCEEFF"><tr><td><center><big>Information about the selected tensor</big></center> • 2 nd rank Magnetotoroidic tensor ζij (direct effect) • Axial tensor invariant under time-reversal symmetry operation • Defining equation: Mi=ζijSj • Relates Toroidal field S with Magnetization M • Intrinsic symmetry symbol: eV2 </td></tr></table></center> </div> <div id='legend_30' style="display:none"> <center><table style="width:500px" bgcolor="#CCEEFF"><tr><td><center><big>Information about the selected tensor</big></center> • 2 nd rank Magnetotoroidic tensor ζTij (inverse effect) • Axial tensor invariant under time-reversal symmetry operation • Defining equation: Ti=ζTijHj • Relates Magnetic field H with Toroidal moment T • Intrinsic symmetry symbol: eV2 </td></tr></table></center> </div> <div id='legend_31' style="display:none"> <center><table style="width:500px" bgcolor="#CCEEFF"><tr><td><center><big>Information about the selected tensor</big></center> • 2 nd rank Electrotoroidic tensor θij (direct effect) • Polar tensor which inverts under time-reversal symmetry operation • Defining equation: Pi=θijSj • Relates Toroidal field S with Polarization P • Intrinsic symmetry symbol: aV2 </td></tr></table></center> </div> <div id='legend_32' style="display:none"> <center><table style="width:500px" bgcolor="#CCEEFF"><tr><td><center><big>Information about the selected tensor</big></center> • 2 nd rank Electrotoroidic tensor θTij (inverse effect) • Polar tensor which inverts under time-reversal symmetry operation • Defining equation: Ti=θTijEj • Relates Electric field E with Toroidal moment T • Intrinsic symmetry symbol: aV2 </td></tr></table></center> </div> <div id='legend_33' style="display:none"> <center><table style="width:500px" bgcolor="#CCEEFF"><tr><td><center><big>Information about the selected tensor</big></center> • 2 nd rank Isothermal magnetoelectric effect tensor Aij • Axial tensor which inverts under time-reversal symmetry operation • Defining equation: Mi=AijDj • Relates Electric displacement field D with Magnetization M • Intrinsic symmetry symbol: aeV2 </td></tr></table></center> </div> <div id='legend_34' style="display:none"> <center><table style="width:500px" bgcolor="#CCEEFF"><tr><td><center><big>Information about the selected tensor</big></center> • 2 nd rank Isothermal magnetoelectric effect tensor ATij (inverse effect) • Axial tensor which inverts under time-reversal symmetry operation • Defining equation: Ei=-ATijHj • Relates Magnetic field H with Electric field E • Intrinsic symmetry symbol: aeV2 </td></tr></table></center> </div> <div id='legend_35' style="display:none"> <center><table style="width:500px" bgcolor="#CCEEFF"><tr><td><center><big>Information about the selected tensor</big></center> • 2 nd rank Magnetoelectric tensor αij (direct effect) • Axial tensor which inverts under time-reversal symmetry operation • Defining equation: Mi=αijEj • Relates Electric field E with Magnetization M • Intrinsic symmetry symbol: aeV2 </td></tr></table></center> </div> <div id='legend_36' style="display:none"> <center><table style="width:500px" bgcolor="#CCEEFF"><tr><td><center><big>Information about the selected tensor</big></center> • 2 nd rank Magnetoelectric tensor αTij (inverse effect) • Axial tensor which inverts under time-reversal symmetry operation • Defining equation: Pi=αTijHj • Relates Magnetic field H with Polarization P • Intrinsic symmetry symbol: aeV2 </td></tr></table></center> </div> <div id='legend_37' style="display:none"> <center><table style="width:500px" bgcolor="#CCEEFF"><tr><td><center><big>Information about the selected tensor</big></center> • 3 rd rank Acoustoelectricity tensor ρijk • Polar tensor invariant under time-reversal symmetry operation • Defining equation: σij=ρijkJk • Relates Alternating electric current density J with Stress tensor σij • Intrinsic symmetry symbol: [V2]V • Symmetrized indexes due to intrinsic symmetry:      • ρijk = ρjik • Abbreviated notation: ρijk → ραβ     • ij → α     • k → β     Convention used for the substitutions:     11 → 1 | 22 → 2 | 33 → 3 | 23,32 → 4 | 13,31 → 5 | 12,21 → 6</td></tr></table></center> </div> <div id='legend_38' style="display:none"> <center><table style="width:500px" bgcolor="#CCEEFF"><tr><td><center><big>Information about the selected tensor</big></center> • 3 rd rank Isothermal piezoelectric tensor eTijk (inverse effect) • Polar tensor invariant under time-reversal symmetry operation • Defining equation: σij=-eTijkEk • Relates Electric field E with Stress tensor σij • Intrinsic symmetry symbol: [V2]V • Symmetrized indexes due to intrinsic symmetry:      • eTijk = eTjik • Abbreviated notation: eTijk → eTαβ     • ij → α     • k → β     Convention used for the substitutions:     11 → 1 | 22 → 2 | 33 → 3 | 23,32 → 4 | 13,31 → 5 | 12,21 → 6</td></tr></table></center> </div> <div id='legend_39' style="display:none"> <center><table style="width:500px" bgcolor="#CCEEFF"><tr><td><center><big>Information about the selected tensor</big></center> • 3 rd rank Piezoelectric tensor dTijk (inversee ffect) dTijk • Polar tensor invariant under time-reversal symmetry operation • Defining equation: εij=dTijkEk • Relates Electric field E with Strain tensor εij • Intrinsic symmetry symbol: [V2]V • Symmetrized indexes due to intrinsic symmetry:      • dTijk = dTjik • Abbreviated notation: dTijk → dTαβ     • ij → α     • k → β     Convention used for the substitutions:     11 → 1 | 22 → 2 | 33 → 3 | 23,32 → 4 | 13,31 → 5 | 12,21 → 6</td></tr></table></center> </div> <div id='legend_40' style="display:none"> <center><table style="width:500px" bgcolor="#CCEEFF"><tr><td><center><big>Information about the selected tensor</big></center> • 3 rd rank Isothermal piezoelectric tensor eijk (direct effect) • Polar tensor invariant under time-reversal symmetry operation • Defining equation: Di=eijkεjk • Relates Strain tensor εij with Electric displacement field D • Intrinsic symmetry symbol: V[V2] • Symmetrized indexes due to intrinsic symmetry:      • eijk = eikj • Abbreviated notation: eijk → eαβ     • i → α     • jk → β     Convention used for the substitutions:     11 → 1 | 22 → 2 | 33 → 3 | 23,32 → 4 | 13,31 → 5 | 12,21 → 6</td></tr></table></center> </div> <div id='legend_41' style="display:none"> <center><table style="width:500px" bgcolor="#CCEEFF"><tr><td><center><big>Information about the selected tensor</big></center> • 3 rd rank Piezoelectric tensor dijk(directeffect) dijk • Polar tensor invariant under time-reversal symmetry operation • Defining equation: Pi=dijkσjk • Relates Stress tensor σij with Polarization vector P • Intrinsic symmetry symbol: V[V2] • Symmetrized indexes due to intrinsic symmetry:      • dijk = dikj • Abbreviated notation: dijk → dαβ     • i → α     • jk → β     Convention used for the substitutions:     11 → 1 | 22 → 2 | 33 → 3 | 23,32 → 4 | 13,31 → 5 | 12,21 → 6</td></tr></table></center> </div> <div id='legend_42' style="display:none"> <center><table style="width:500px" bgcolor="#CCEEFF"><tr><td><center><big>Information about the selected tensor</big></center> • 3 rd rank Second order magnetoelectric tensor αTijk (inverse effect) • Polar tensor invariant under time-reversal symmetry operation • Defining equation: Pi=αTijkHjHk • Relates Magnetic field H with Polarization P • Intrinsic symmetry symbol: V[V2] • Symmetrized indexes due to intrinsic symmetry:      • αTijk = αTikj • Abbreviated notation: αTijk → αTαβ     • i → α     • jk → β     Convention used for the substitutions:     11 → 1 | 22 → 2 | 33 → 3 | 23,32 → 4 | 13,31 → 5 | 12,21 → 6</td></tr></table></center> </div> <div id='legend_43' style="display:none"> <center><table style="width:500px" bgcolor="#CCEEFF"><tr><td><center><big>Information about the selected tensor</big></center> • 3 rd rank Piezotoroidic tensor γijk (direct effect) • Polar tensor which inverts under time-reversal symmetry operation • Defining equation: εij=γijkSk • Relates Toroidal field S with Strain tensor εij • Intrinsic symmetry symbol: a[V2]V • Symmetrized indexes due to intrinsic symmetry:      • γijk = γjik • Abbreviated notation: γijk → γαβ     • ij → α     • k → β     Convention used for the substitutions:     11 → 1 | 22 → 2 | 33 → 3 | 23,32 → 4 | 13,31 → 5 | 12,21 → 6</td></tr></table></center> </div> <div id='legend_44' style="display:none"> <center><table style="width:500px" bgcolor="#CCEEFF"><tr><td><center><big>Information about the selected tensor</big></center> • 3 rd rank Piezotoroidic tensor γTijk (inverse effect) • Polar tensor which inverts under time-reversal symmetry operation • Defining equation: Ti=γTijkσjk • Relates Stress tensor σij with Toroidal moment T • Intrinsic symmetry symbol: aV[V2] • Symmetrized indexes due to intrinsic symmetry:      • γTijk = γTikj • Abbreviated notation: γTijk → γTαβ     • i → α     • jk → β     Convention used for the substitutions:     11 → 1 | 22 → 2 | 33 → 3 | 23,32 → 4 | 13,31 → 5 | 12,21 → 6</td></tr></table></center> </div> <div id='legend_45' style="display:none"> <center><table style="width:500px" bgcolor="#CCEEFF"><tr><td><center><big>Information about the selected tensor</big></center> • 3 rd rank Isothermal piezomagnetic tensor emTijk (inverse effect) • Axial tensor which inverts under time-reversal symmetry operation • Defining equation: σij=-emTijkHk • Relates Magnetic field H with Stress tensor σij • Intrinsic symmetry symbol: ae[V2]V • Symmetrized indexes due to intrinsic symmetry:      • emTijk = emTjik • Abbreviated notation: emTijk → emTαβ     • ij → α     • k → β     Convention used for the substitutions:     11 → 1 | 22 → 2 | 33 → 3 | 23,32 → 4 | 13,31 → 5 | 12,21 → 6</td></tr></table></center> </div> <div id='legend_46' style="display:none"> <center><table style="width:500px" bgcolor="#CCEEFF"><tr><td><center><big>Information about the selected tensor</big></center> • 3 rd rank Piezomagnetic tensor ΛTijk (inverse effect) • Axial tensor which inverts under time-reversal symmetry operation • Defining equation: εij=ΛTijkHk • Relates Magnetic field H with Strain tensor εij • Intrinsic symmetry symbol: ae[V2]V • Symmetrized indexes due to intrinsic symmetry:      • ΛTijk = ΛTjik • Abbreviated notation: ΛTijk → ΛTαβ     • ij → α     • k → β     Convention used for the substitutions:     11 → 1 | 22 → 2 | 33 → 3 | 23,32 → 4 | 13,31 → 5 | 12,21 → 6</td></tr></table></center> </div> <div id='legend_47' style="display:none"> <center><table style="width:500px" bgcolor="#CCEEFF"><tr><td><center><big>Information about the selected tensor</big></center> • 3 rd rank Isothermal piezomagnetic tensor emijk (direct effect) • Axial tensor which inverts under time-reversal symmetry operation • Defining equation: Mi=emijkεjk • Relates Strain tensor εij with Magnetization M • Intrinsic symmetry symbol: aeV[V2] • Symmetrized indexes due to intrinsic symmetry:      • emijk = emikj • Abbreviated notation: emijk → emαβ     • i → α     • jk → β     Convention used for the substitutions:     11 → 1 | 22 → 2 | 33 → 3 | 23,32 → 4 | 13,31 → 5 | 12,21 → 6</td></tr></table></center> </div> <div id='legend_48' style="display:none"> <center><table style="width:500px" bgcolor="#CCEEFF"><tr><td><center><big>Information about the selected tensor</big></center> • 3 rd rank Piezomagnetic tensor Λijk (direct effect) • Axial tensor which inverts under time-reversal symmetry operation • Defining equation: Mi=Λijkσjk • Relates Stress tensor σij with Magnetization M • Intrinsic symmetry symbol: aeV[V2] • Symmetrized indexes due to intrinsic symmetry:      • Λijk = Λikj • Abbreviated notation: Λijk → Λαβ     • i → α     • jk → β     Convention used for the substitutions:     11 → 1 | 22 → 2 | 33 → 3 | 23,32 → 4 | 13,31 → 5 | 12,21 → 6</td></tr></table></center> </div> <div id='legend_49' style="display:none"> <center><table style="width:500px" bgcolor="#CCEEFF"><tr><td><center><big>Information about the selected tensor</big></center> • 3 rd rank Second order magnetoelectric tensor αijk (direct effect) • Axial tensor which inverts under time-reversal symmetry operation • Defining equation: Mi=αijkEjEk • Relates Electric field E with Magnetization M • Intrinsic symmetry symbol: aeV[V2] • Symmetrized indexes due to intrinsic symmetry:      • αijk = αikj • Abbreviated notation: αijk → ααβ     • i → α     • jk → β     Convention used for the substitutions:     11 → 1 | 22 → 2 | 33 → 3 | 23,32 → 4 | 13,31 → 5 | 12,21 → 6</td></tr></table></center> </div> <div id='legend_50' style="display:none"> <center><table style="width:500px" bgcolor="#CCEEFF"><tr><td><center><big>Information about the selected tensor</big></center> • 4 th rank Elastic compliance tensor Sijkl • Polar tensor invariant under time-reversal symmetry operation • Defining equation: εij=Sijklσkl • Relates Stress tensor σij with Strain tensor εij • Intrinsic symmetry symbol: [[V2][V2]] • Symmetrized indexes due to intrinsic symmetry:      • Sijkl = Sjikl      • Sijkl = Sijlk      • Sijkl = Sklij • Abbreviated notation: Sijkl → Sαβ     • ij → α     • kl → β     Convention used for the substitutions:     11 → 1 | 22 → 2 | 33 → 3 | 23,32 → 4 | 13,31 → 5 | 12,21 → 6</td></tr></table></center> </div> <div id='legend_51' style="display:none"> <center><table style="width:500px" bgcolor="#CCEEFF"><tr><td><center><big>Information about the selected tensor</big></center> • 4 th rank Elastic stiffness tensor Cijkl Cijkl • Polar tensor invariant under time-reversal symmetry operation • Defining equation: σij=Cijklεkl • Relates Strain tensor εij with Stress tensor σij • Intrinsic symmetry symbol: [[V2][V2]] • Symmetrized indexes due to intrinsic symmetry:      • Cijkl = Cjikl      • Cijkl = Cijlk      • Cijkl = Cklij • Abbreviated notation: Cijkl → Cαβ     • ij → α     • kl → β     Convention used for the substitutions:     11 → 1 | 22 → 2 | 33 → 3 | 23,32 → 4 | 13,31 → 5 | 12,21 → 6</td></tr></table></center> </div> <div id='legend_52' style="display:none"> <center><table style="width:500px" bgcolor="#CCEEFF"><tr><td><center><big>Information about the selected tensor</big></center> • 4 th rank Viscosity tensor ηijkl • Polar tensor invariant under time-reversal symmetry operation • Defining equation: σij=ηijkl∂εkl/∂t • Relates Strain tensor rate ∂εij/∂t with Stress tensor σij • Intrinsic symmetry symbol: [[V2][V2]] • Symmetrized indexes due to intrinsic symmetry:      • ηijkl = ηjikl      • ηijkl = ηijlk      • ηijkl = ηklij • Abbreviated notation: ηijkl → ηαβ     • ij → α     • kl → β     Convention used for the substitutions:     11 → 1 | 22 → 2 | 33 → 3 | 23,32 → 4 | 13,31 → 5 | 12,21 → 6</td></tr></table></center> </div> <div id='legend_53' style="display:none"> <center><table style="width:500px" bgcolor="#CCEEFF"><tr><td><center><big>Information about the selected tensor</big></center> • 4 th rank Damage effect tensor Dijkl • Polar tensor invariant under time-reversal symmetry operation • Defining equation: σij=Dijklσkl • Relates Stress tensor σij (before damage) with Effective stress tensor σij (after damage) • Intrinsic symmetry symbol: [V2][V2] • Symmetrized indexes due to intrinsic symmetry:      • Dijkl = Djikl      • Dijkl = Dijlk • Abbreviated notation: Dijkl → Dαβ     • ij → α     • kl → β     Convention used for the substitutions:     11 → 1 | 22 → 2 | 33 → 3 | 23,32 → 4 | 13,31 → 5 | 12,21 → 6</td></tr></table></center> </div> <div id='legend_54' style="display:none"> <center><table style="width:500px" bgcolor="#CCEEFF"><tr><td><center><big>Information about the selected tensor</big></center> • 4 th rank Electrostriction tensor γijkl • Polar tensor invariant under time-reversal symmetry operation • Defining equation: εij=γijklEkEl • Relates Electric field E and Electric field E with Strain tensor εij • Intrinsic symmetry symbol: [V2][V2] • Symmetrized indexes due to intrinsic symmetry:      • γijkl = γjikl      • γijkl = γijlk • Abbreviated notation: γijkl → γαβ     • ij → α     • kl → β     Convention used for the substitutions:     11 → 1 | 22 → 2 | 33 → 3 | 23,32 → 4 | 13,31 → 5 | 12,21 → 6</td></tr></table></center> </div> <div id='legend_55' style="display:none"> <center><table style="width:500px" bgcolor="#CCEEFF"><tr><td><center><big>Information about the selected tensor</big></center> • 4 th rank Magnetostriction tensor Nijkl • Polar tensor invariant under time-reversal symmetry operation • Defining equation: εij=NijklMkMl • Relates Magnetization M and Magnetization M with Strain tensor εij • Intrinsic symmetry symbol: [V2][V2] • Symmetrized indexes due to intrinsic symmetry:      • Nijkl = Njikl      • Nijkl = Nijlk • Abbreviated notation: Nijkl → Nαβ     • ij → α     • kl → β     Convention used for the substitutions:     11 → 1 | 22 → 2 | 33 → 3 | 23,32 → 4 | 13,31 → 5 | 12,21 → 6</td></tr></table></center> </div> <div id='legend_56' style="display:none"> <center><table style="width:500px" bgcolor="#CCEEFF"><tr><td><center><big>Information about the selected tensor</big></center> • 4 th rank Flexoelectric (modified) μijkl • Polar tensor invariant under time-reversal symmetry operation • Defining equation: Pi=μijkl∇jεkl • Relates strain tensor gradient ∇jεkl with polarization vector P • The intrinsic symmetry of this tensor has been claimed to be V[V3] instead of V2[V2] by Eliseev and Morozovska,Phys.Rev.B 98,094108 (2018). This allows to further reduce the number of independent coefficients, as shown here. However, this claim is not generally accepted as correct (C. Dreyer, M. Stengel, D. Vanderbilt, private communication). • Intrinsic symmetry symbol: V[V3] • Symmetrized indexes due to intrinsic symmetry:      • μijkl = μilkj      • μijkl = μikjl      • μijkl = μijlk • Abbreviated notation: μijkl → μαβγ     • i → α     • jk → β     • l → γ     Convention used for the substitutions:     11 → 1 | 22 → 2 | 33 → 3 | 23,32 → 4 | 13,31 → 5 | 12,21 → 6</td></tr></table></center> </div> <div id='legend_57' style="display:none"> <center><table style="width:500px" bgcolor="#CCEEFF"><tr><td><center><big>Information about the selected tensor</big></center> • 4 th rank Elastothermoelectric power tensor Eijkl • Polar tensor invariant under time-reversal symmetry operation • Defining equation: ΔΣij=Eijklεkl • Relates Strain tensor εij with Thermoelectric power tensor variation ΔΣij • Intrinsic symmetry symbol: V2[V2] • Symmetrized indexes due to intrinsic symmetry:      • Eijkl = Eijlk • Abbreviated notation: Eijkl → Eαβγ     • i → α     • j → β     • kl → γ     Convention used for the substitutions:     11 → 1 | 22 → 2 | 33 → 3 | 23,32 → 4 | 13,31 → 5 | 12,21 → 6</td></tr></table></center> </div> <div id='legend_58' style="display:none"> <center><table style="width:500px" bgcolor="#CCEEFF"><tr><td><center><big>Information about the selected tensor</big></center> • 4 th rank Flexoelectric tensor μijkl • Polar tensor invariant under time-reversal symmetry operation • Defining equation: Pi=μijkl∇jεkl • Relates Strain tensor gradient ∇iεjk with Polarization vector P • Intrinsic symmetry symbol: V2[V2] • Symmetrized indexes due to intrinsic symmetry:      • μijkl = μijlk • Abbreviated notation: μijkl → μαβγ     • i → α     • j → β     • kl → γ     Convention used for the substitutions:     11 → 1 | 22 → 2 | 33 → 3 | 23,32 → 4 | 13,31 → 5 | 12,21 → 6</td></tr></table></center> </div> <div id='legend_59' style="display:none"> <center><table style="width:500px" bgcolor="#CCEEFF"><tr><td><center><big>Information about the selected tensor</big></center> • 4 th rank Piezothermoelectric power tensor Πijkl • Polar tensor invariant under time-reversal symmetry operation • Defining equation: ΔΣij=Πijklσkl • Relates Stress tensor σij with Thermoelectric power tensor variation ΔΣij • Intrinsic symmetry symbol: V2[V2] • Symmetrized indexes due to intrinsic symmetry:      • Πijkl = Πijlk • Abbreviated notation: Πijkl → Παβγ     • i → α     • j → β     • kl → γ     Convention used for the substitutions:     11 → 1 | 22 → 2 | 33 → 3 | 23,32 → 4 | 13,31 → 5 | 12,21 → 6</td></tr></table></center> </div> <div id='legend_60' style="display:none"> <center><table style="width:500px" bgcolor="#CCEEFF"><tr><td><center><big>Information about the selected tensor</big></center> • 4 th rank Flexomagnetic tensor Qijkl • Axial tensor which inverts under time-reversal symmetry operation • Defining equation: Mi=Qijkl∇jσkl • Relates Stress tensor gradient ∇iσjk with Magnetization M • Intrinsic symmetry symbol: aeV2[V2] • Symmetrized indexes due to intrinsic symmetry:      • Qijkl = Qijlk • Abbreviated notation: Qijkl → Qαβγ     • i → α     • j → β     • kl → γ     Convention used for the substitutions:     11 → 1 | 22 → 2 | 33 → 3 | 23,32 → 4 | 13,31 → 5 | 12,21 → 6</td></tr></table></center> </div> <div id='legend_61' style="display:none"> <center><table style="width:500px" bgcolor="#CCEEFF"><tr><td><center><big>Information about the selected tensor</big></center> • 4 th rank Piezomagnetoelectric tensor πijkl • Axial tensor which inverts under time-reversal symmetry operation • Defining equation: Pi=πijklHjσkl • Relates Magnetization M and Stress tensor σij with Polarization vector P • Intrinsic symmetry symbol: aeV2[V2] • Symmetrized indexes due to intrinsic symmetry:      • πijkl = πijlk • Abbreviated notation: πijkl → παβγ     • i → α     • j → β     • kl → γ     Convention used for the substitutions:     11 → 1 | 22 → 2 | 33 → 3 | 23,32 → 4 | 13,31 → 5 | 12,21 → 6</td></tr></table></center> </div> <div id='legend_62' style="display:none"> <center><table style="width:500px" bgcolor="#CCEEFF"><tr><td><center><big>Information about the selected tensor</big></center> • 5 th rank Acoustic activity tensor bijklm • Polar tensor invariant under time-reversal symmetry operation • Defining equation: σij=bijklm∇mεkl • Relates Strain tensor gradient ∇lεij with Stress tensor σij • Intrinsic symmetry symbol: [[V2][V2]]V • Symmetrized indexes due to intrinsic symmetry:      • bijklm = bjiklm      • bijklm = bijlkm      • bijklm = bklijm • Abbreviated notation: bijklm → bαβγ     • ij → α     • kl → β     • m → γ     Convention used for the substitutions:     11 → 1 | 22 → 2 | 33 → 3 | 23,32 → 4 | 13,31 → 5 | 12,21 → 6</td></tr></table></center> </div> <div id='legend_63' style="display:none"> <center><table style="width:500px" bgcolor="#CCEEFF"><tr><td><center><big>Information about the selected tensor</big></center> • 5 th rank Second-order piezoelectric tensor dijklm • Polar tensor invariant under time-reversal symmetry operation • Defining equation: Pi=dijklmσjkσlm • Relates Stress tensor σij and Stress tensor σij with Polarization vector P • Intrinsic symmetry symbol: V[[V2][V2]] • Symmetrized indexes due to intrinsic symmetry:      • dijklm = dikjlm      • dijklm = dijkml      • dijklm = dilmjk • Abbreviated notation: dijklm → dαβγ     • i → α     • jk → β     • lm → γ     Convention used for the substitutions:     11 → 1 | 22 → 2 | 33 → 3 | 23,32 → 4 | 13,31 → 5 | 12,21 → 6</td></tr></table></center> </div> <div id='legend_64' style="display:none"> <center><table style="width:500px" bgcolor="#CCEEFF"><tr><td><center><big>Information about the selected tensor</big></center> • 6 th rank Third order elastic compliance tensor Sijklmn • Polar tensor invariant under time-reversal symmetry operation • Defining equation: εij=Sijklmnσklσmn • Relates Stress tensor σij and Stress tensor σij with Strain tensor εij • Intrinsic symmetry symbol: [[V2][V2][V2]] • Symmetrized indexes due to intrinsic symmetry:      • Sijklmn = Sjiklmn      • Sijklmn = Sijlkmn      • Sijklmn = Sijklnm      • Sijklmn = Sijmnkl = Sklijmn = Sklmnij = Smnijkl = Smnklij • Abbreviated notation: Sijklmn → Sαβγ     • ij → α     • kl → β     • mn → γ     Convention used for the substitutions:     11 → 1 | 22 → 2 | 33 → 3 | 23,32 → 4 | 13,31 → 5 | 12,21 → 6</td></tr></table></center> </div> <div id='legend_65' style="display:none"> <center><table style="width:500px" bgcolor="#CCEEFF"><tr><td><center><big>Information about the selected tensor</big></center> • 6 th rank Third order elastic stiffness tensor Cijklmn • Polar tensor invariant under time-reversal symmetry operation • Defining equation: σij=Cijklmnεklεmn • Relates Strain tensor εij and Strain tensor εij with Stress tensor σij • Intrinsic symmetry symbol: [[V2][V2][V2]] • Symmetrized indexes due to intrinsic symmetry:      • Cijklmn = Cjiklmn      • Cijklmn = Cijlkmn      • Cijklmn = Cijklnm      • Cijklmn = Cijmnkl = Cklijmn = Cklmnij = Cmnijkl = Cmnklij • Abbreviated notation: Cijklmn → Cαβγ     • ij → α     • kl → β     • mn → γ     Convention used for the substitutions:     11 → 1 | 22 → 2 | 33 → 3 | 23,32 → 4 | 13,31 → 5 | 12,21 → 6</td></tr></table></center> </div> <div id='legend_66' style="display:none"> <center><table style="width:500px" bgcolor="#CCEEFF"><tr><td><center><big>Information about the selected tensor</big></center> • 8 th rank Damage tensor Rijklmnpq • Polar tensor invariant under time-reversal symmetry operation • Defining equation: Cijkl=RijklmnpqCmnpq • Relates Elastic stiffness tensor Cijkl (before damage) with Elastic stiffness tensor Cijkl (after damage) • Intrinsic symmetry symbol: [[[V2][V2]][[V2][V2]]] • Symmetrized indexes due to intrinsic symmetry:      • Rijklmnpq = Rjiklmnpq      • Rijklmnpq = Rijlkmnpq      • Rijklmnpq = Rijklnmpq      • Rijklmnpq = Rijklmnqp      • Rijklmnpq = Rklijmnpq      • Rijklmnpq = Rijklpqmn      • Rijklmnpq = Rmnpqijkl • Abbreviated notation: Rijklmnpq → Rαβγδ     • ij → α     • kl → β     • mn → γ     • pq → δ     Convention used for the substitutions:     11 → 1 | 22 → 2 | 33 → 3 | 23,32 → 4 | 13,31 → 5 | 12,21 → 6</td></tr></table></center> </div> <div id='legend_67' style="display:none"> <center><table style="width:500px" bgcolor="#CCEEFF"><tr><td><center><big>Information about the selected tensor</big></center> • 8 th rank Fourth order elastic compliance tensor Sijklmnpq • Polar tensor invariant under time-reversal symmetry operation • Defining equation: εij=Sijklmnpqσklσmnσpq • Relates Stress tensor σij and Stress tensor σij and Stress tensor σij with Strain tensor εij • Intrinsic symmetry symbol: [[V2][V2][V2][V2]] • Symmetrized indexes due to intrinsic symmetry:      • Sijklmnpq = Sjiklmnpq      • Sijklmnpq = Sijlkmnpq      • Sijklmnpq = Sijklnmpq      • Sijklmnpq = Sijklmnqp      • Sijklmnpq = Sijklpqmn = Sijmnklpq = Sijmnpqkl = Sijpqklmn = Sijpqmnkl = Sklijmnpq = Sklijpqmn = Sklmnijpq = Sklmnpqij = Sklpqijmn = Sklpqmnij = Smnijklpq = Smnijpqkl = Smnklijpq = Smnklpqij = Smnpqijkl = Smnpqklij = Spqijklmn = Spqijmnkl = Spqklijmn = Spqklmnij = Spqmnijkl = Spqmnklij • Abbreviated notation: Sijklmnpq → Sαβγδ     • ij → α     • kl → β     • mn → γ     • pq → δ     Convention used for the substitutions:     11 → 1 | 22 → 2 | 33 → 3 | 23,32 → 4 | 13,31 → 5 | 12,21 → 6</td></tr></table></center> </div> <div id='legend_68' style="display:none"> <center><table style="width:500px" bgcolor="#CCEEFF"><tr><td><center><big>Information about the selected tensor</big></center> • 8 th rank Fourth order elastic stiffness tensor Cijklmnpq • Polar tensor invariant under time-reversal symmetry operation • Defining equation: σij=Cijklmnpqεklεmnεpq • Relates Strain tensor εij and Strain tensor εij and Strain tensor εij with Stress tensor σij • Intrinsic symmetry symbol: [[V2][V2][V2][V2]] • Symmetrized indexes due to intrinsic symmetry:      • Cijklmnpq = Cjiklmnpq      • Cijklmnpq = Cijlkmnpq      • Cijklmnpq = Cijklnmpq      • Cijklmnpq = Cijklmnqp      • Cijklmnpq = Cijklpqmn = Cijmnklpq = Cijmnpqkl = Cijpqklmn = Cijpqmnkl = Cklijmnpq = Cklijpqmn = Cklmnijpq = Cklmnpqij = Cklpqijmn = Cklpqmnij = Cmnijklpq = Cmnijpqkl = Cmnklijpq = Cmnklpqij = Cmnpqijkl = Cmnpqklij = Cpqijklmn = Cpqijmnkl = Cpqklijmn = Cpqklmnij = Cpqmnijkl = Cpqmnklij • Abbreviated notation: Cijklmnpq → Cαβγδ     • ij → α     • kl → β     • mn → γ     • pq → δ     Convention used for the substitutions:     11 → 1 | 22 → 2 | 33 → 3 | 23,32 → 4 | 13,31 → 5 | 12,21 → 6</td></tr></table></center> </div> <div id='legend_69' style="display:none"> <center><table style="width:500px" bgcolor="#CCEEFF"><tr><td><center><big>Information about the selected tensor</big></center> • 2 nd rank Index ellipsoid βij • Polar tensor invariant under time-reversal symmetry operation • Intrinsic symmetry symbol: [V2] • Symmetrized indexes due to intrinsic symmetry:      • βij = βji </td></tr></table></center> </div> <div id='legend_70' style="display:none"> <center><table style="width:500px" bgcolor="#CCEEFF"><tr><td><center><big>Information about the selected tensor</big></center> • 2 nd rank Second-order thermo-optical effect tensor Tij. • Polar tensor invariant under time-reversal symmetry operation • Defining equation: Δβij=Tij(ΔT)2 • Relates Temperature variation ΔT and Temperature variation ΔT with the symmetric part of the Dielectric impermeability tensor variation Δβij • Intrinsic symmetry symbol: [V2] • Symmetrized indexes due to intrinsic symmetry:      • Tij = Tji </td></tr></table></center> </div> <div id='legend_71' style="display:none"> <center><table style="width:500px" bgcolor="#CCEEFF"><tr><td><center><big>Information about the selected tensor</big></center> • 2 nd rank Thermo-optical effect tensor tij • Polar tensor invariant under time-reversal symmetry operation • Defining equation: Δβij=tijΔT • Relates Temperature variation ΔT with the symmetric part of the Dielectric impermeability tensor variation Δβij • Intrinsic symmetry symbol: [V2] • Symmetrized indexes due to intrinsic symmetry:      • tij = tji </td></tr></table></center> </div> <div id='legend_72' style="display:none"> <center><table style="width:500px" bgcolor="#CCEEFF"><tr><td><center><big>Information about the selected tensor</big></center> • 2 nd rank Verdet tensor (related to Faraday effect) Vij • Polar tensor invariant under time-reversal symmetry operation • Defining equation: Δβij=εijmVmkHk (with εijm Levi-Civita axial antisymmetric tensor) • Relates magnetic field H with the antisymmetric part of the dielectric impermeability tensor variation Δβij • Related with Faraday effect coefficients F: Fijk=εijmVmk, where εijm Levi-Civita axial antisymmtric tensor. • Intrinsic symmetry symbol: V2 </td></tr></table></center> </div> <div id='legend_73' style="display:none"> <center><table style="width:500px" bgcolor="#CCEEFF"><tr><td><center><big>Information about the selected tensor</big></center> • 2 nd rank Optical activity tensor gij • Axial tensor invariant under time-reversal symmetry operation • Defining equation: G=gijlilj • Relates Direction cosines li and Direction cosines lj with Optical activity coefficient G • Intrinsic symmetry symbol: e[V2] • Symmetrized indexes due to intrinsic symmetry:      • gij = gji </td></tr></table></center> </div> <div id='legend_74' style="display:none"> <center><table style="width:500px" bgcolor="#CCEEFF"><tr><td><center><big>Information about the selected tensor</big></center> • 2 nd rank Thermogyration tensor gij • Axial tensor invariant under time-reversal symmetry operation • Defining equation: Δgij=gijT • Relates Temperature T with Optical activity tensor variation Δgij • Intrinsic symmetry symbol: e[V2] • Symmetrized indexes due to intrinsic symmetry:      • gij = gji </td></tr></table></center> </div> <div id='legend_75' style="display:none"> <center><table style="width:500px" bgcolor="#CCEEFF"><tr><td><center><big>Information about the selected tensor</big></center> • 2 nd rank Spontaneous Faraday effect βij • Polar tensor which inverts under time-reversal symmetry operation • Intrinsic symmetry symbol: a{V2} • Symmetrized indexes due to intrinsic symmetry:      • βij = -βji </td></tr></table></center> </div> <div id='legend_76' style="display:none"> <center><table style="width:500px" bgcolor="#CCEEFF"><tr><td><center><big>Information about the selected tensor</big></center> • 3 rd rank Natural optical activity γijk • Polar tensor invariant under time-reversal symmetry operation • Defining equation: Δβij=i γijkkk • Relates light wave vector with dielectric impermeability tensor variation (antisymmetric part). • Connected with gyrotropic second-rank axial tensor glk=k0/2εijkgijl, with εijk Levi-Civita axial antisymmetric tensor and k0 modulus of light wave vector in vacuum. Gyration given by G=glkklkk/k02. • Real in non-disipative media. • Intrinsic symmetry symbol: {V2}V • Symmetrized indexes due to intrinsic symmetry:      • γijk = -γjik </td></tr></table></center> </div> <div id='legend_77' style="display:none"> <center><table style="width:500px" bgcolor="#CCEEFF"><tr><td><center><big>Information about the selected tensor</big></center> • 3 rd rank Pockels (electrooptic) effect zijk • Polar tensor invariant under time-reversal symmetry operation • Defining equation: Δβij=zijkEk • Relates Electric field E with the symmetric part of the Dielectric impermeability tensor variation Δβij • Intrinsic symmetry symbol: [V2]V • Symmetrized indexes due to intrinsic symmetry:      • zijk = zjik • Abbreviated notation: zijk → zαβ     • ij → α     • k → β     Convention used for the substitutions:     11 → 1 | 22 → 2 | 33 → 3 | 23,32 → 4 | 13,31 → 5 | 12,21 → 6</td></tr></table></center> </div> <div id='legend_78' style="display:none"> <center><table style="width:500px" bgcolor="#CCEEFF"><tr><td><center><big>Information about the selected tensor</big></center> • 3 rd rank Thermoelectro-optical effect tensor r(T)ijk • Polar tensor invariant under time-reversal symmetry operation • Defining equation: Δβij=r(T)ijkEkΔT • Relates Electric field E and Temperature variation ΔT with the symmetric part of the Dielectric impermeability tensor variation Δβij • Intrinsic symmetry symbol: [V2]V • Symmetrized indexes due to intrinsic symmetry:      • r(T)ijk = r(T)jik • Abbreviated notation: r(T)ijk → r(T)αβ     • ij → α     • k → β     Convention used for the substitutions:     11 → 1 | 22 → 2 | 33 → 3 | 23,32 → 4 | 13,31 → 5 | 12,21 → 6</td></tr></table></center> </div> <div id='legend_79' style="display:none"> <center><table style="width:500px" bgcolor="#CCEEFF"><tr><td><center><big>Information about the selected tensor</big></center> • 3 rd rank Magneto-optical tensor (Faraday effect) Fijk • Axial tensor invariant under time-reversal symmetry operation • Defining equation: Δβij=FijkHk • Relates Magnetic field H with the antisymmetric part of the Dielectric impermeability tensor variation Δβij. • Pure imaginary in non-dissipative media. • Intrinsic symmetry symbol: e{V2}V • Symmetrized indexes due to intrinsic symmetry:      • Fijk = -Fjik </td></tr></table></center> </div> <div id='legend_80' style="display:none"> <center><table style="width:500px" bgcolor="#CCEEFF"><tr><td><center><big>Information about the selected tensor</big></center> • 3 rd rank Thermomagneto-optical effect tensor f(T)ijk • Axial tensor invariant under time-reversal symmetry operation • Defining equation: Δβij=f(T)ijkHkΔT • Relates Magnetic field H and Temperature variation ΔT with the antisymmetric part of the Dielectric impermeability tensor variation Δβij • Intrinsic symmetry symbol: e{V2}V • Symmetrized indexes due to intrinsic symmetry:      • f(T)ijk = -f(T)jik </td></tr></table></center> </div> <div id='legend_81' style="display:none"> <center><table style="width:500px" bgcolor="#CCEEFF"><tr><td><center><big>Information about the selected tensor</big></center> • 3 rd rank Electrogyration effect tensor γijk • Axial tensor invariant under time-reversal symmetry operation • Defining equation: Δgij=γijkEk • Relates Electric field E with Optical activity tensor variation Δgij • Intrinsic symmetry symbol: e[V2]V • Symmetrized indexes due to intrinsic symmetry:      • γijk = γjik • Abbreviated notation: γijk → γαβ     • ij → α     • k → β     Convention used for the substitutions:     11 → 1 | 22 → 2 | 33 → 3 | 23,32 → 4 | 13,31 → 5 | 12,21 → 6</td></tr></table></center> </div> <div id='legend_82' style="display:none"> <center><table style="width:500px" bgcolor="#CCEEFF"><tr><td><center><big>Information about the selected tensor</big></center> • 3 rd rank Spontaneous Gyrotropic Birefringence γijk • Polar tensor which inverts under time-reversal symmetry operation • Defining equation: Δβij=i γijkkk • Relates light wave vector with dielectric impermeability tensor variation (symmetric part). • Pure imaginary in non-dissipative media. • Intrinsic symmetry symbol: a[V2]V • Symmetrized indexes due to intrinsic symmetry:      • γijk = γjik • Abbreviated notation: γijk → γαβ     • ij → α     • k → β     Convention used for the substitutions:     11 → 1 | 22 → 2 | 33 → 3 | 23,32 → 4 | 13,31 → 5 | 12,21 → 6</td></tr></table></center> </div> <div id='legend_83' style="display:none"> <center><table style="width:500px" bgcolor="#CCEEFF"><tr><td><center><big>Information about the selected tensor</big></center> • 3 rd rank Magneto-optic Kerr effect qijk • Axial tensor which inverts under time-reversal symmetry operation • Defining equation: Δβij=qijkHk • Relates Magnetic field H with Dielectric impermeability tensor variation Δβij (symmetric part) • Intrinsic symmetry symbol: ae[V2]V • Symmetrized indexes due to intrinsic symmetry:      • qijk = qjik • Abbreviated notation: qijk → qαβ     • ij → α     • k → β     Convention used for the substitutions:     11 → 1 | 22 → 2 | 33 → 3 | 23,32 → 4 | 13,31 → 5 | 12,21 → 6</td></tr></table></center> </div> <div id='legend_84' style="display:none"> <center><table style="width:500px" bgcolor="#CCEEFF"><tr><td><center><big>Information about the selected tensor</big></center> • 4 th rank Birefringence in Cubic Crystals γijkl • Polar tensor invariant under time-reversal symmetry operation • Defining equation: Δβij= γijlmklkm • Relates light wave vector and light wave vector with dielectric impermeability tensor variation (symmetric part). • Real in non-dissipative media. • Intrinsic symmetry symbol: [V2][V2] • Symmetrized indexes due to intrinsic symmetry:      • γijkl = γjikl      • γijkl = γijlk • Abbreviated notation: γijkl → γαβ     • ij → α     • kl → β     Convention used for the substitutions:     11 → 1 | 22 → 2 | 33 → 3 | 23,32 → 4 | 13,31 → 5 | 12,21 → 6</td></tr></table></center> </div> <div id='legend_85' style="display:none"> <center><table style="width:500px" bgcolor="#CCEEFF"><tr><td><center><big>Information about the selected tensor</big></center> • 4 th rank Elasto-optical tensor pijkl • Polar tensor invariant under time-reversal symmetry operation • Defining equation: Δβij=pijklεkl • Relates Strain tensor εij with the symmetric part of the Dielectric impermeability tensor variation Δβij • Intrinsic symmetry symbol: [V2][V2] • Symmetrized indexes due to intrinsic symmetry:      • πijkl = πjikl      • πijkl = πijlk • Abbreviated notation: πijkl → παβ     • ij → α     • kl → β     Convention used for the substitutions:     11 → 1 | 22 → 2 | 33 → 3 | 23,32 → 4 | 13,31 → 5 | 12,21 → 6</td></tr></table></center> </div> <div id='legend_86' style="display:none"> <center><table style="width:500px" bgcolor="#CCEEFF"><tr><td><center><big>Information about the selected tensor</big></center> • 4 th rank Kerr effect tensor Rijkl • Polar tensor invariant under time-reversal symmetry operation • Defining equation: Δβij=RijklEkEl • Relates Electric field E and Electric field E with the symmetric part of the Dielectric impermeability tensor variation Δβij • Intrinsic symmetry symbol: [V2][V2] • Symmetrized indexes due to intrinsic symmetry:      • Rijkl = Rjikl      • Rijkl = Rijlk • Abbreviated notation: Rijkl → Rαβ     • ij → α     • kl → β     Convention used for the substitutions:     11 → 1 | 22 → 2 | 33 → 3 | 23,32 → 4 | 13,31 → 5 | 12,21 → 6</td></tr></table></center> </div> <div id='legend_87' style="display:none"> <center><table style="width:500px" bgcolor="#CCEEFF"><tr><td><center><big>Information about the selected tensor</big></center> • 4 th rank Piezo-optical tensor πijkl • Polar tensor invariant under time-reversal symmetry operation • Defining equation: Δβij=πijklσkl • Relates Stress tensor σij with the symmetric part of the Dielectric impermeability tensor variation Δβij • Intrinsic symmetry symbol: [V2][V2] • Symmetrized indexes due to intrinsic symmetry:      • πijkl = πjikl      • πijkl = πijlk • Abbreviated notation: πijkl → παβ     • ij → α     • kl → β     Convention used for the substitutions:     11 → 1 | 22 → 2 | 33 → 3 | 23,32 → 4 | 13,31 → 5 | 12,21 → 6</td></tr></table></center> </div> <div id='legend_88' style="display:none"> <center><table style="width:500px" bgcolor="#CCEEFF"><tr><td><center><big>Information about the selected tensor</big></center> • 4 th rank Second-order magneto-optical (Cotton-Mouton effect) tensor Cijkl • Polar tensor invariant under time-reversal symmetry operation • Defining equation: Δβij=CijklHkHl • Relates Magnetic field H and Magnetic field H with the symmetric part of the Dielectric impermeability tensor variation Δβij • Intrinsic symmetry symbol: [V2][V2] • Symmetrized indexes due to intrinsic symmetry:      • Cijkl = Cjikl      • Cijkl = Cijlk • Abbreviated notation: Cijkl → Cαβ     • ij → α     • kl → β     Convention used for the substitutions:     11 → 1 | 22 → 2 | 33 → 3 | 23,32 → 4 | 13,31 → 5 | 12,21 → 6</td></tr></table></center> </div> <div id='legend_89' style="display:none"> <center><table style="width:500px" bgcolor="#CCEEFF"><tr><td><center><big>Information about the selected tensor</big></center> • 4 th rank Thermopiezo-optical effect tensor π(T)ijkl • Polar tensor invariant under time-reversal symmetry operation • Defining equation: Δβij=π(T)ijklσklΔT • Relates Stress tensor σij and Temperature variation ΔT with the symmetric part of the Dielectric impermeability tensor variation Δβij • Intrinsic symmetry symbol: [V2][V2] • Symmetrized indexes due to intrinsic symmetry:      • π(T)ijkl = π(T)jikl      • π(T)ijkl = π(T)ijlk • Abbreviated notation: π(T)ijkl → π(T)αβ     • ij → α     • kl → β     Convention used for the substitutions:     11 → 1 | 22 → 2 | 33 → 3 | 23,32 → 4 | 13,31 → 5 | 12,21 → 6</td></tr></table></center> </div> <div id='legend_90' style="display:none"> <center><table style="width:500px" bgcolor="#CCEEFF"><tr><td><center><big>Information about the selected tensor</big></center> • 4 th rank Magnetoelectro-optical effect tensor mijkl • Axial tensor invariant under time-reversal symmetry operation • Defining equation: Δβij=mijklHkEl • Relates Magnetic field H and Electric field E with the antisymmetric part of the Dielectric impermeability tensor variation Δβij • Intrinsic symmetry symbol: e{V2}V2 • Symmetrized indexes due to intrinsic symmetry:      • mijkl = -mjikl </td></tr></table></center> </div> <div id='legend_91' style="display:none"> <center><table style="width:500px" bgcolor="#CCEEFF"><tr><td><center><big>Information about the selected tensor</big></center> • 4 th rank Piezogyration tensor Cijkl • Axial tensor invariant under time-reversal symmetry operation • Defining equation: Δgij=Cijklσkl • Relates Stress tensor σij with Optical activity tensor variation Δgij • Intrinsic symmetry symbol: e[V2][V2] • Symmetrized indexes due to intrinsic symmetry:      • Cijkl = Cjikl      • Cijkl = Cijlk • Abbreviated notation: Cijkl → Cαβ     • ij → α     • kl → β     Convention used for the substitutions:     11 → 1 | 22 → 2 | 33 → 3 | 23,32 → 4 | 13,31 → 5 | 12,21 → 6</td></tr></table></center> </div> <div id='legend_92' style="display:none"> <center><table style="width:500px" bgcolor="#CCEEFF"><tr><td><center><big>Information about the selected tensor</big></center> • 4 th rank Quadratic electrogyration effect tensor βijkl • Axial tensor invariant under time-reversal symmetry operation • Defining equation: Δgij=βijklEkEl • Relates Electric field E and Electric field E with Optical activity tensor variation Δgij • Intrinsic symmetry symbol: e[V2][V2] • Symmetrized indexes due to intrinsic symmetry:      • βijkl = βjikl      • βijkl = βijlk • Abbreviated notation: βijkl → βαβ     • ij → α     • kl → β     Convention used for the substitutions:     11 → 1 | 22 → 2 | 33 → 3 | 23,32 → 4 | 13,31 → 5 | 12,21 → 6</td></tr></table></center> </div> <div id='legend_93' style="display:none"> <center><table style="width:500px" bgcolor="#CCEEFF"><tr><td><center><big>Information about the selected tensor</big></center> • 4 th rank Nonmutual Optical Activity γijkl • Polar tensor which inverts under time-reversal symmetry operation • Defining equation: Δβij= γijlmklkm • Relates light wave vector and light wave vector with dielectric impermeability tensor variation (antisymmetric part). • Pure imaginary in non-dissipative media. • Intrinsic symmetry symbol: a{V2}[V2] • Symmetrized indexes due to intrinsic symmetry:      • γijkl = -γjikl      • γijkl = γijlk • Abbreviated notation: γijkl → γαβγ     • i → α     • j → β     • kl → γ     Convention used for the substitutions:     11 → 1 | 22 → 2 | 33 → 3 | 23,32 → 4 | 13,31 → 5 | 12,21 → 6</td></tr></table></center> </div> <div id='legend_94' style="display:none"> <center><table style="width:500px" bgcolor="#CCEEFF"><tr><td><center><big>Information about the selected tensor</big></center> • 4 th rank Quadratic magneto-optic Kerr effect Bijkl • Polar tensor which inverts under time-reversal symmetry operation • Defining equation: Δβij=BijklHkHl • Relates Magnetic field H and Magnetic field H with the antisymmetric part of the Dielectric impermeability tensor variation Δβij • Intrinsic symmetry symbol: a{V2}[V2] • Symmetrized indexes due to intrinsic symmetry:      • Bijkl = -Bjikl      • Bijkl = Bijlk • Abbreviated notation: Bijkl → Bαβγ     • i → α     • j → β     • kl → γ     Convention used for the substitutions:     11 → 1 | 22 → 2 | 33 → 3 | 23,32 → 4 | 13,31 → 5 | 12,21 → 6</td></tr></table></center> </div> <div id='legend_95' style="display:none"> <center><table style="width:500px" bgcolor="#CCEEFF"><tr><td><center><big>Information about the selected tensor</big></center> • 5 th rank Piezoelectro-optical effect tensor zijklm • Polar tensor invariant under time-reversal symmetry operation • Defining equation: Δβij=zijklmσklEm • Relates Stress tensor σij and Electric field E with the symmetric part of the Dielectric impermeability tensor variation Δβij • Intrinsic symmetry symbol: [V2][V2]V • Symmetrized indexes due to intrinsic symmetry:      • zijklm = zjiklm      • zijklm = zijlkm • Abbreviated notation: zijklm → zαβγ     • ij → α     • kl → β     • m → γ     Convention used for the substitutions:     11 → 1 | 22 → 2 | 33 → 3 | 23,32 → 4 | 13,31 → 5 | 12,21 → 6</td></tr></table></center> </div> <div id='legend_96' style="display:none"> <center><table style="width:500px" bgcolor="#CCEEFF"><tr><td><center><big>Information about the selected tensor</big></center> • 5 th rank Piezomagneto-optical effect tensor ωijklm • Axial tensor invariant under time-reversal symmetry operation • Defining equation: Δβij=ωijklmHkσlm • Relates Magnetic field H and Stress tensor σij with the antisymmetric part of the Dielectric impermeability tensor variation Δβij • Intrinsic symmetry symbol: e{V2}V[V2] • Symmetrized indexes due to intrinsic symmetry:      • ωijklm = -ωjiklm      • ωijklm = ωijkml • Abbreviated notation: ωijklm → ωαβγδ     • i → α     • j → β     • k → γ     • lm → δ     Convention used for the substitutions:     11 → 1 | 22 → 2 | 33 → 3 | 23,32 → 4 | 13,31 → 5 | 12,21 → 6</td></tr></table></center> </div> <div id='legend_97' style="display:none"> <center><table style="width:500px" bgcolor="#CCEEFF"><tr><td><center><big>Information about the selected tensor</big></center> • 5 th rank Gradient piezogyration tensor βijklm • Axial tensor invariant under time-reversal symmetry operation • Defining equation: Δgij=βijklm<vec>∇</vec>mσkl • Relates Stress tensor gradient ∇kσij with Optical activity tensor variation Δgij • Intrinsic symmetry symbol: e[V2][V2]V • Symmetrized indexes due to intrinsic symmetry:      • βijklm = βjiklm      • βijklm = βijlkm • Abbreviated notation: βijklm → βαβγ     • ij → α     • kl → β     • m → γ     Convention used for the substitutions:     11 → 1 | 22 → 2 | 33 → 3 | 23,32 → 4 | 13,31 → 5 | 12,21 → 6</td></tr></table></center> </div> <div id='legend_98' style="display:none"> <center><table style="width:500px" bgcolor="#CCEEFF"><tr><td><center><big>Information about the selected tensor</big></center> • 6 th rank Second-order piezo-optical tensor Πijklmn • Polar tensor invariant under time-reversal symmetry operation • Defining equation: Δβij=Πijklmnσklσmn • Relates Stress tensor σij and Stress tensor σij with the symmetric part of Dielectric impermeability tensor variation Δβij • Intrinsic symmetry symbol: [V2][[V2][V2]] • Symmetrized indexes due to intrinsic symmetry:      • Πijklmn = Πjiklmn      • Πijklmn = Πijlkmn      • Πijklmn = Πijklnm      • Πijklmn = Πijmnkl • Abbreviated notation: Πijklmn → Παβγ     • ij → α     • kl → β     • mn → γ     Convention used for the substitutions:     11 → 1 | 22 → 2 | 33 → 3 | 23,32 → 4 | 13,31 → 5 | 12,21 → 6</td></tr></table></center> </div> <div id='legend_99' style="display:none"> <center><table style="width:500px" bgcolor="#CCEEFF"><tr><td><center><big>Information about the selected tensor</big></center> • 6 th rank Quadratic piezogyration tensor Cijklmn • Axial tensor invariant under time-reversal symmetry operation • Defining equation: Δgij=Cijklmnσklσmn • Relates Stress tensor σij and Stress tensor σij with Optical activity tensor variation Δgij • Intrinsic symmetry symbol: e[V2][[V2][V2]] • Symmetrized indexes due to intrinsic symmetry:      • Cijklmn = Cjiklmn      • Cijklmn = Cijlkmn      • Cijklmn = Cijklnm      • Cijklmn = Cijmnkl • Abbreviated notation: Cijklmn → Cαβγ     • ij → α     • kl → β     • mn → γ     Convention used for the substitutions:     11 → 1 | 22 → 2 | 33 → 3 | 23,32 → 4 | 13,31 → 5 | 12,21 → 6</td></tr></table></center> </div> <div id='legend_100' style="display:none"> <center><table style="width:500px" bgcolor="#CCEEFF"><tr><td><center><big>Information about the selected tensor</big></center> • 3 rd rank General second-order susceptibility (non dissipative media and no dispersion) χ(ω3;ω2,ω1)ijk • Polar tensor invariant under time-reversal symmetry operation • Defining equation: Pi(ω3)=χijk(ω3;ω2,ω1)Ej(ω2)Ek(ω1) • Relates Electric field of frequency ω1 and Electric field of frequency ω2 with Polarization of frequency ω3. • Tensor of real coefficients (imaginary part null). Kleinman symmetry. • Intrinsic symmetry symbol: [V3] • Symmetrized indexes due to intrinsic symmetry:      • χ(ω3;ω2,ω1)ijk = χ(ω3;ω2,ω1)jik      • χ(ω3;ω2,ω1)ijk = χ(ω3;ω2,ω1)kji      • χ(ω3;ω2,ω1)ijk = χ(ω3;ω2,ω1)ikj • Abbreviated notation: χ(ω3;ω2,ω1)ijk → χ(ω3;ω2,ω1)αβ     • ij → α     • k → β     Convention used for the substitutions:     11 → 1 | 22 → 2 | 33 → 3 | 23,32 → 4 | 13,31 → 5 | 12,21 → 6</td></tr></table></center> </div> <div id='legend_101' style="display:none"> <center><table style="width:500px" bgcolor="#CCEEFF"><tr><td><center><big>Information about the selected tensor</big></center> • 3 rd rank Optical rectification (non-dissipative media and no dispersion) χ(0;ω,-ω)ijk • Polar tensor invariant under time-reversal symmetry operation • Defining equation: Pi(0)=χijk(0;ω,-ω)Ej(ω)Ek(-ω) • Relates Electric field of frequency ω and Electric field of frequency ω with Polarization of frequency 0. • Tensor of real coefficients (imaginary part null). Kleinman symmetry. • Intrinsic symmetry symbol: [V3] • Symmetrized indexes due to intrinsic symmetry:      • χ(0;ω,-ω)ijk = χ(0;ω,-ω)jik      • χ(0;ω,-ω)ijk = χ(0;ω,-ω)kji      • χ(0;ω,-ω)ijk = χ(0;ω,-ω)ikj • Abbreviated notation: χ(0;ω,-ω)ijk → χ(0;ω,-ω)αβ     • ij → α     • k → β     Convention used for the substitutions:     11 → 1 | 22 → 2 | 33 → 3 | 23,32 → 4 | 13,31 → 5 | 12,21 → 6</td></tr></table></center> </div> <div id='legend_102' style="display:none"> <center><table style="width:500px" bgcolor="#CCEEFF"><tr><td><center><big>Information about the selected tensor</big></center> • 3 rd rank Second-harmonic generation (non-dissipative media and no dispersion) χ(2ω;ω,ω)ijk • Polar tensor invariant under time-reversal symmetry operation • Defining equation: Pi(2ω)=χijk(2ω;ω,ω)Ej(ω)Ek(ω) • Relates Electric field of frequency ω and Electric field of frequency ω with Polarization of frequency 2ω. • Tensor of real coefficients (imaginary part null). Kleinman symmetry. • Intrinsic symmetry symbol: [V3] • Symmetrized indexes due to intrinsic symmetry:      • χ(2ω;ω,ω)ijk = χ(2ω;ω,ω)jik      • χ(2ω;ω,ω)ijk = χ(2ω;ω,ω)kji      • χ(2ω;ω,ω)ijk = χ(2ω;ω,ω)ikj • Abbreviated notation: χ(2ω;ω,ω)ijk → χ(2ω;ω,ω)αβ     • ij → α     • k → β     Convention used for the substitutions:     11 → 1 | 22 → 2 | 33 → 3 | 23,32 → 4 | 13,31 → 5 | 12,21 → 6</td></tr></table></center> </div> <div id='legend_103' style="display:none"> <center><table style="width:500px" bgcolor="#CCEEFF"><tr><td><center><big>Information about the selected tensor</big></center> • 3 rd rank General optical rectification (dissipative media) χ(0;ω,-ω)ijk • Polar tensor invariant under time-reversal symmetry operation • Defining equation: Pi(0)=χijk(0;ω,-ω)Ej(ω)Ek(-ω) • Relates Electric field of frequency ω and Electric field of frequency ω with Polarization of frequency 0. • Tensor of complex coefficients. Real in non-dissipative media. • Onsager relations imply 1'χijk(0;ω,-ω)=χikj(0;ω,-ω) • Intrinsic symmetry symbol: V[V2]* </td></tr></table></center> </div> <div id='legend_104' style="display:none"> <center><table style="width:500px" bgcolor="#CCEEFF"><tr><td><center><big>Information about the selected tensor</big></center> • 3 rd rank Optical rectification (non-dissipative media)Real part χ(0;ω,-ω)ijk • Polar tensor invariant under time-reversal symmetry operation • Defining equation: Pi(0)=χijk(0;ω,-ω)Ej(ω)Ek(-ω) • Relates Electric field of frequency ω and Electric field of frequency ω with Polarization of frequency 0. • Tensor of complex coefficients. Only the real part of χijk(0;ω,-ω) is considered. • Intrinsic symmetry symbol: V[V2] • Symmetrized indexes due to intrinsic symmetry:      • χ(0;ω,-ω)ijk = χ(0;ω,-ω)ikj • Abbreviated notation: χ(0;ω,-ω)ijk → χ(0;ω,-ω)αβ     • i → α     • jk → β     Convention used for the substitutions:     11 → 1 | 22 → 2 | 33 → 3 | 23,32 → 4 | 13,31 → 5 | 12,21 → 6</td></tr></table></center> </div> <div id='legend_105' style="display:none"> <center><table style="width:500px" bgcolor="#CCEEFF"><tr><td><center><big>Information about the selected tensor</big></center> • 3 rd rank Second-harmonic generation (non-dissipative media). Real part. χ(2ω;ω,ω)ijk • Polar tensor invariant under time-reversal symmetry operation • Defining equation: Pi(2ω)=χijk(2ω;ω,ω)Ej(ω)Ek(ω) • Relates Electric field of frequency ω and Electric field of frequency ω with Polarization of frequency 2ω. • Tensor of complex coefficients. Only the real part of χijk(2ω;ω,ω) is considered. • Intrinsic symmetry symbol: V[V2] • Symmetrized indexes due to intrinsic symmetry:      • χ(2ω;ω,ω)ijk = χ(2ω;ω,ω)ikj • Abbreviated notation: χ(2ω;ω,ω)ijk → χ(2ω;ω,ω)αβ     • i → α     • jk → β     Convention used for the substitutions:     11 → 1 | 22 → 2 | 33 → 3 | 23,32 → 4 | 13,31 → 5 | 12,21 → 6</td></tr></table></center> </div> <div id='legend_106' style="display:none"> <center><table style="width:500px" bgcolor="#CCEEFF"><tr><td><center><big>Information about the selected tensor</big></center> • 3 rd rank General second-order susceptibility (non-dissipative media). Real part. χ(ω3;ω2,ω1)ijk • Polar tensor invariant under time-reversal symmetry operation • Defining equation: Pi(ω3)=χijk(ω3;ω2,ω1)Ej(ω2)Ek(ω1) • Relates Electric field of frequency ω1 and Electric field of frequency ω2 with Polarization of frequency ω3. • Tensor of complex coefficients. Only the real part of χijk(ω3;ω2,ω1) is considered. • Intrinsic symmetry symbol: V3 </td></tr></table></center> </div> <div id='legend_107' style="display:none"> <center><table style="width:500px" bgcolor="#CCEEFF"><tr><td><center><big>Information about the selected tensor</big></center> • 3 rd rank General second-order susceptibility by magnetic dipole (non-dissipative media and no dispersion) χm(ω3;ω2,ω1)ijk • Axial tensor invariant under time-reversal symmetry operation • Defining equation: Mi(ω3)=χmijk(ω3;ω2,ω1)Ej(ω2)Ek(ω1) • Relates Electric field of frequency ω1 and Electric field of frequency ω2 with Magnetization of frequency ω3. • Tensor of pure imaginary coefficients (real part null). Kleinman symmetry. • Intrinsic symmetry symbol: e[V3] • Symmetrized indexes due to intrinsic symmetry:      • χm(ω3;ω2,ω1)ijk = χm(ω3;ω2,ω1)jik      • χm(ω3;ω2,ω1)ijk = χm(ω3;ω2,ω1)kji      • χm(ω3;ω2,ω1)ijk = χm(ω3;ω2,ω1)ikj • Abbreviated notation: χm(ω3;ω2,ω1)ijk → χm(ω3;ω2,ω1)αβ     • ij → α     • k → β     Convention used for the substitutions:     11 → 1 | 22 → 2 | 33 → 3 | 23,32 → 4 | 13,31 → 5 | 12,21 → 6</td></tr></table></center> </div> <div id='legend_108' style="display:none"> <center><table style="width:500px" bgcolor="#CCEEFF"><tr><td><center><big>Information about the selected tensor</big></center> • 3 rd rank Second-harmonic generation by magnetic dipole (non-dissipative media and no dispersion) χm(2ω;ω,ω)ijk • Axial tensor invariant under time-reversal symmetry operation • Defining equation: Mi(2ω)=χmijk(2ω;ω,ω)Ej(ω)Ek(ω) • Relates Electric field of frequency ω and Electric field of frequency ω with Polarization of frequency 2ω. • Tensor of pure imaginary coefficients (real part null). Kleinman symmetry. • Intrinsic symmetry symbol: e[V3] • Symmetrized indexes due to intrinsic symmetry:      • χm(2ω;ω,ω)ijk = χm(2ω;ω,ω)jik      • χm(2ω;ω,ω)ijk = χm(2ω;ω,ω)kji      • χm(2ω;ω,ω)ijk = χm(2ω;ω,ω)ikj • Abbreviated notation: χm(2ω;ω,ω)ijk → χm(2ω;ω,ω)αβ     • ij → α     • k → β     Convention used for the substitutions:     11 → 1 | 22 → 2 | 33 → 3 | 23,32 → 4 | 13,31 → 5 | 12,21 → 6</td></tr></table></center> </div> <div id='legend_109' style="display:none"> <center><table style="width:500px" bgcolor="#CCEEFF"><tr><td><center><big>Information about the selected tensor</big></center> • 3 rd rank Second-harmonic generation by magnetic dipole (non-dissipative media). Imaginary part. χm(2ω;ω,ω)ijk • Axial tensor invariant under time-reversal symmetry operation • Defining equation: Mi(2ω)=χmijk(2ω;ω,ω)Ej(ω)Ek(ω) • Relates Electric field of frequency ω and Electric field of frequency ω with Polarization of frequency 2ω. • Tensor of complex coefficients. Only the imaginary part of χmijk(2ω;ω,ω) is considered. • Intrinsic symmetry symbol: eV[V2] • Symmetrized indexes due to intrinsic symmetry:      • χm(2ω;ω,ω)ijk = χm(2ω;ω,ω)ikj • Abbreviated notation: χm(2ω;ω,ω)ijk → χm(2ω;ω,ω)αβ     • i → α     • jk → β     Convention used for the substitutions:     11 → 1 | 22 → 2 | 33 → 3 | 23,32 → 4 | 13,31 → 5 | 12,21 → 6</td></tr></table></center> </div> <div id='legend_110' style="display:none"> <center><table style="width:500px" bgcolor="#CCEEFF"><tr><td><center><big>Information about the selected tensor</big></center> • 3 rd rank General second-order susceptibility by magnetic dipole (non-dissipative media) Imaginary part χm(ω3;ω2,ω1)ijk • Axial tensor invariant under time-reversal symmetry operation • Defining equation: Mi(ω3)=χmijk(ω3;ω2,ω1)Ej(ω2)Ek(ω1) • Relates Electric field of frequency ω1 and Electric field of frequency ω2 with Magnetization of frequency ω3. • Tensor of complex coefficients. Only the imaginary part of χmijk(ω3;ω2,ω1) is considered. • Intrinsic symmetry symbol: eV3 </td></tr></table></center> </div> <div id='legend_111' style="display:none"> <center><table style="width:500px" bgcolor="#CCEEFF"><tr><td><center><big>Information about the selected tensor</big></center> • 3 rd rank Optical rectification (non-dissipative media)Imaginary part χ(0;ω,-ω)ijk • Polar tensor which inverts under time-reversal symmetry operation • Defining equation: Pi(0)=χijk(0;ω,-ω)Ej(ω)Ek(-ω) • Relates Electric field of frequency ω and Electric field of frequency ω with Polarization of frequency 0. • Tensor of complex coefficients. Only the imaginary part of χijk(0;ω,-ω) is considered. • Intrinsic symmetry symbol: aV{V2} • Symmetrized indexes due to intrinsic symmetry:      • χ(0;ω,-ω)ijk = -χ(0;ω,-ω)ikj </td></tr></table></center> </div> <div id='legend_112' style="display:none"> <center><table style="width:500px" bgcolor="#CCEEFF"><tr><td><center><big>Information about the selected tensor</big></center> • 3 rd rank Second-harmonic generation (non-dissipative media). Imaginary part. χ(2ω;ω,ω)ijk • Polar tensor which inverts under time-reversal symmetry operation • Defining equation: Pi(2ω)=χijk(2ω;ω,ω)Ej(ω)Ek(ω) • Relates Electric field of frequency ω and Electric field of frequency ω with Polarization of frequency 2ω. • Tensor of complex coefficients. Only the imaginary part of χijk(2ω;ω,ω) is considered. • Intrinsic symmetry symbol: aV[V2] • Symmetrized indexes due to intrinsic symmetry:      • χ(2ω;ω,ω)ijk = χ(2ω;ω,ω)ikj • Abbreviated notation: χ(2ω;ω,ω)ijk → χ(2ω;ω,ω)αβ     • i → α     • jk → β     Convention used for the substitutions:     11 → 1 | 22 → 2 | 33 → 3 | 23,32 → 4 | 13,31 → 5 | 12,21 → 6</td></tr></table></center> </div> <div id='legend_113' style="display:none"> <center><table style="width:500px" bgcolor="#CCEEFF"><tr><td><center><big>Information about the selected tensor</big></center> • 3 rd rank General second-order susceptibility (non-dissipative media). Imaginary part. χ(ω3;ω2,ω1)ijk • Polar tensor which inverts under time-reversal symmetry operation • Defining equation: Pi(ω3)=χijk(ω3;ω2,ω1)Ej(ω2)Ek(ω1) • Relates Electric field of frequency ω1 and Electric field of frequency ω2 with Polarization of frequency ω3. • Tensor of complex coefficients. Only the imaginary part of χijk(ω3ω,ω) is considered. • Intrinsic symmetry symbol: aV3 </td></tr></table></center> </div> <div id='legend_114' style="display:none"> <center><table style="width:500px" bgcolor="#CCEEFF"><tr><td><center><big>Information about the selected tensor</big></center> • 3 rd rank Second-harmonic generation by magnetic dipole (non-dissipative media). Real part. χm(2ω;ω,ω)ijk • Axial tensor which inverts under time-reversal symmetry operation • Defining equation: Mi(2ω)=χmijk(2ω;ω,ω)Ej(ω)Ek(ω) • Relates Electric field of frequency ω and Electric field of frequency ω with Polarization of frequency 2ω. • Tensor of complex coefficients. Only the real part of χmijk(2ωω,ω) is considered. • Intrinsic symmetry symbol: aeV[V2] • Symmetrized indexes due to intrinsic symmetry:      • χm(2ω;ω,ω)ijk = χm(2ω;ω,ω)ikj • Abbreviated notation: χm(2ω;ω,ω)ijk → χm(2ω;ω,ω)αβ     • i → α     • jk → β     Convention used for the substitutions:     11 → 1 | 22 → 2 | 33 → 3 | 23,32 → 4 | 13,31 → 5 | 12,21 → 6</td></tr></table></center> </div> <div id='legend_115' style="display:none"> <center><table style="width:500px" bgcolor="#CCEEFF"><tr><td><center><big>Information about the selected tensor</big></center> • 3 rd rank General second-order susceptibility by magnetic dipole (non-dissipative media) Real part χm(ω3;ω2,ω1)ijk • Axial tensor which inverts under time-reversal symmetry operation • Defining equation: Mi(ω3)=χmijk(ω3;ω2,ω1)Ej(ω2)Ek(ω1) • Relates Electric field of frequency ω1 and Electric field of frequency ω2 with Magnetization of frequency ω3. • Tensor of complex coefficients. Only the real part of χmijk(ω3;ω2,ω1) is considered. • Intrinsic symmetry symbol: aeV3 </td></tr></table></center> </div> <div id='legend_116' style="display:none"> <center><table style="width:500px" bgcolor="#CCEEFF"><tr><td><center><big>Information about the selected tensor</big></center> • 3 rd rank General second-harmonic generation (dissipative media) χ(2ω;ω,ω)ijk • Polar tensor invariant under time-reversal symmetry operation • Defining equation: Pi(2ω)=χijk(2ω;ω,ω)Ej(ω)Ek(ω) • Relates Electric field of frequency ω and Electric field of frequency ω with Polarization of frequency 2ω. • Tensor of complex coefficients. • Symmetry operations including time reversal do not introduce any restriction in the tensor but they connect it with the tensor for difference frequency generation: 1'χijk(2ω;ω,ω)=χjik(ω;2ω,-ω)=χkij(ω;2ω,-ω) • Intrinsic symmetry symbol: (V[V2])* • Symmetrized indexes due to intrinsic symmetry:      • χ(2ω;ω,ω)ijk = χ(2ω;ω,ω)ikj • Abbreviated notation: χ(2ω;ω,ω)ijk → χ(2ω;ω,ω)αβ     • i → α     • jk → β     Convention used for the substitutions:     11 → 1 | 22 → 2 | 33 → 3 | 23,32 → 4 | 13,31 → 5 | 12,21 → 6</td></tr></table></center> </div> <div id='legend_117' style="display:none"> <center><table style="width:500px" bgcolor="#CCEEFF"><tr><td><center><big>Information about the selected tensor</big></center> • 3 rd rank General second-order susceptibility (dissipative media) χ(ω3;ω2,ω1)ijk • Polar tensor invariant under time-reversal symmetry operation • Defining equation: Pi(ω3)=χijk(ω3;ω2,ω1)Ej(ω2)Ek(ω1) • Relates Electric field of frequency ω1 and Electric field of frequency ω2 with Polarization of frequency ω3. • Tensor of complex coefficients. • Symmetry operations including time reversal do not introduce any restriction in the tensor but they connect it with the tensor for difference frequency generation: 1' χijk(ω3;ω2,ω1)=χjik(ω2;ω3,-ω1)=χkij(ω1;ω3,-ω2) • Intrinsic symmetry symbol: (V3)* </td></tr></table></center> </div> <div id='legend_118' style="display:none"> <center><table style="width:500px" bgcolor="#CCEEFF"><tr><td><center><big>Information about the selected tensor</big></center> • 4 th rank Degenerate four-wave mixing χ(ω;-ω,ω,ω)ijkl • Polar tensor invariant under time-reversal symmetry operation • Defining equation: Pi(ω)=χijkl(ω;-ω,ω,ω)Ej(-ω)Ek(ω)El(ω) • Relates Electric field of frequency ω and Electric field of frequency ω and Electric field of frequency ω with Polarization of frequency ω. • Tensor of complex coefficients. • Onsager relations imply: 1'χijkl(ω;-ω,ω,ω)=χklij(ω;-ω,ω,ω). • Intrinsic symmetry symbol: [[V2][V2]]* • Symmetrized indexes due to intrinsic symmetry:      • χ(ω;-ω,ω,ω)ijkl = χ(ω;-ω,ω,ω)jikl      • χ(ω;-ω,ω,ω)ijkl = χ(ω;-ω,ω,ω)ijlk • Abbreviated notation: χ(ω;-ω,ω,ω)ijkl → χ(ω;-ω,ω,ω)αβ     • ij → α     • kl → β     Convention used for the substitutions:     11 → 1 | 22 → 2 | 33 → 3 | 23,32 → 4 | 13,31 → 5 | 12,21 → 6</td></tr></table></center> </div> <div id='legend_119' style="display:none"> <center><table style="width:500px" bgcolor="#CCEEFF"><tr><td><center><big>Information about the selected tensor</big></center> • 4 th rank Electric-field induced second-harmonic generation (non-dissipative media and no dispersion) χ(2ω;0,ω,ω)ijkl • Polar tensor invariant under time-reversal symmetry operation • Defining equation: Pi(2ω)=χijkl(2ω;0,ω,ω)Ej(0)Ek(ω)El(ω) • Relates Electric field of frequency 0 and Electric field of frequency ω and Electric field of frequency ω with Polarization of frequency 2ω. • Tensor of real coefficients (imaginary part null). Kleinman symmetry. • Intrinsic symmetry symbol: [V4] • Symmetrized indexes due to intrinsic symmetry:      • χ(2ω;0,ω,ω)ijkl = χ(2ω;0,ω,ω)jikl      • χ(2ω;0,ω,ω)ijkl = χ(2ω;0,ω,ω)kjil      • χ(2ω;0,ω,ω)ijkl = χ(2ω;0,ω,ω)ljki      • χ(2ω;0,ω,ω)ijkl = χ(2ω;0,ω,ω)ikjl      • χ(2ω;0,ω,ω)ijkl = χ(2ω;0,ω,ω)ilkj      • χ(2ω;0,ω,ω)ijkl = χ(2ω;0,ω,ω)ijlk • Abbreviated notation: χ(2ω;0,ω,ω)ijkl → χ(2ω;0,ω,ω)αβ     • ij → α     • kl → β     Convention used for the substitutions:     11 → 1 | 22 → 2 | 33 → 3 | 23,32 → 4 | 13,31 → 5 | 12,21 → 6</td></tr></table></center> </div> <div id='legend_120' style="display:none"> <center><table style="width:500px" bgcolor="#CCEEFF"><tr><td><center><big>Information about the selected tensor</big></center> • 4 th rank General third-order susceptibility (non-dissipative media and no dispersion) χ(ω4;ω3,ω2,ω1)ijkl • Polar tensor invariant under time-reversal symmetry operation • Defining equation: Pi(ω4)=χijkl(ω4;ω3,ω2,ω1)Ej(ω3)Ek(ω2)El(ω1) • Relates Electric field of frequency ω1 and Electric field of frequency ω2 and Electric field of frequency ω3 with Polarization of frequency ω4. • Tensor of real coefficients (imaginary part null). Kleinman symmetry. • Intrinsic symmetry symbol: [V4] • Symmetrized indexes due to intrinsic symmetry:      • χ(ω4;ω3,ω2,ω1)ijkl = χ(ω4;ω3,ω2,ω1)jikl      • χ(ω4;ω3,ω2,ω1)ijkl = χ(ω4;ω3,ω2,ω1)kjil      • χ(ω4;ω3,ω2,ω1)ijkl = χ(ω4;ω3,ω2,ω1)ljki      • χ(ω4;ω3,ω2,ω1)ijkl = χ(ω4;ω3,ω2,ω1)ikjl      • χ(ω4;ω3,ω2,ω1)ijkl = χ(ω4;ω3,ω2,ω1)ilkj      • χ(ω4;ω3,ω2,ω1)ijkl = χ(ω4;ω3,ω2,ω1)ijlk • Abbreviated notation: χ(ω4;ω3,ω2,ω1)ijkl → χ(ω4;ω3,ω2,ω1)αβ     • ij → α     • kl → β     Convention used for the substitutions:     11 → 1 | 22 → 2 | 33 → 3 | 23,32 → 4 | 13,31 → 5 | 12,21 → 6</td></tr></table></center> </div> <div id='legend_121' style="display:none"> <center><table style="width:500px" bgcolor="#CCEEFF"><tr><td><center><big>Information about the selected tensor</big></center> • 4 th rank Third-harmonic generation (non-dissipative media and no dispersion) χ(3ω;ω,ω,ω)ijkl • Polar tensor invariant under time-reversal symmetry operation • Defining equation: Pi(3ω)=χijkl(3ω;ω,ω,ω)Ej(ω)Ek(ω)El(ω) • Relates Electric field of frequency ω and Electric field of frequency ω and Electric field of frequency ω with Polarization of frequency 3ω. • Tensor of real coefficients (imaginary part null). Kleinman symmetry. • Intrinsic symmetry symbol: [V4] • Symmetrized indexes due to intrinsic symmetry:      • χ(3ω;ω,ω,ω)ijkl = χ(3ω;ω,ω,ω)jikl      • χ(3ω;ω,ω,ω)ijkl = χ(3ω;ω,ω,ω)kjil      • χ(3ω;ω,ω,ω)ijkl = χ(3ω;ω,ω,ω)ljki      • χ(3ω;ω,ω,ω)ijkl = χ(3ω;ω,ω,ω)ikjl      • χ(3ω;ω,ω,ω)ijkl = χ(3ω;ω,ω,ω)ilkj      • χ(3ω;ω,ω,ω)ijkl = χ(3ω;ω,ω,ω)ijlk • Abbreviated notation: χ(3ω;ω,ω,ω)ijkl → χ(3ω;ω,ω,ω)αβ     • ij → α     • kl → β     Convention used for the substitutions:     11 → 1 | 22 → 2 | 33 → 3 | 23,32 → 4 | 13,31 → 5 | 12,21 → 6</td></tr></table></center> </div> <div id='legend_122' style="display:none"> <center><table style="width:500px" bgcolor="#CCEEFF"><tr><td><center><big>Information about the selected tensor</big></center> • 4 th rank Four-wave mixing χ(ω1;-ω2,ω1,ω2)ijkl • Polar tensor invariant under time-reversal symmetry operation • Defining equation: Pi(ω1)=χijkl(ω1;-ω2,ω1,ω2)Ej(-ω2)Ek(ω1)El(ω2) • Relates Electric field of frequency ω1 and Electric field of frequency ω2 and Electric field of frequency ω2 with Polarization of frequency ω1. • Tensor of complex coefficients. • Onsager relations imply: 1'χijkl(ω1;-ω2,ω1,ω2)=χklij(ω1;-ω2,ω1,ω2). • Intrinsic symmetry symbol: [V2V2]* </td></tr></table></center> </div> <div id='legend_123' style="display:none"> <center><table style="width:500px" bgcolor="#CCEEFF"><tr><td><center><big>Information about the selected tensor</big></center> • 4 th rank Third-harmonic generation (non-dissipative media). Real part. χ(3ω;ω,ω,ω)ijkl • Polar tensor invariant under time-reversal symmetry operation • Defining equation: Pi(3ω)=χijkl(3ω;ω,ω,ω)Ej(ω)Ek(ω)El(ω) • Relates Electric field of frequency ω and Electric field of frequency ω and Electric field of frequency ω with Polarization of frequency 3ω. • Tensor of complex coefficients. Only the real part of χijkl(3ω;ω,ω,ω) is considered. • Intrinsic symmetry symbol: V[V3] • Symmetrized indexes due to intrinsic symmetry:      • χ(3ω;ω,ω,ω)ijkl = χ(3ω;ω,ω,ω)ikjl      • χ(3ω;ω,ω,ω)ijkl = χ(3ω;ω,ω,ω)ilkj      • χ(3ω;ω,ω,ω)ijkl = χ(3ω;ω,ω,ω)ijlk • Abbreviated notation: χ(3ω;ω,ω,ω)ijkl → χ(3ω;ω,ω,ω)αβγ     • i → α     • jk → β     • l → γ     Convention used for the substitutions:     11 → 1 | 22 → 2 | 33 → 3 | 23,32 → 4 | 13,31 → 5 | 12,21 → 6</td></tr></table></center> </div> <div id='legend_124' style="display:none"> <center><table style="width:500px" bgcolor="#CCEEFF"><tr><td><center><big>Information about the selected tensor</big></center> • 4 th rank General third-order susceptibility (non-dissipative media). Real part. χ(ω4;ω3,ω2,ω1)ijkl • Polar tensor invariant under time-reversal symmetry operation • Defining equation: Pi(ω4)=χijkl(ω4;ω3,ω2,ω1)Ej(ω3)Ek(ω2)El(ω1) • Relates Electric field of frequency ω1 and Electric field of frequency ω2 and Electric field of frequency ω3 with Polarization of frequency ω4. • Tensor of complex coefficients. • Only the real part of χijkl(ω4;ω3,ω2,ω1) is considered. • Intrinsic symmetry symbol: V4 </td></tr></table></center> </div> <div id='legend_125' style="display:none"> <center><table style="width:500px" bgcolor="#CCEEFF"><tr><td><center><big>Information about the selected tensor</big></center> • 4 th rank Electric-field induced second-harmonic generation (non-dissipative media). Real part. χ(2ω;0,ω,ω)ijkl • Polar tensor invariant under time-reversal symmetry operation • Defining equation: Pi(2ω)=χijkl(2ω;0,ω,ω)Ej(0)Ek(ω)El(ω) • Relates Electric field of frequency 0 and Electric field of frequency ω and Electric field of frequency ω with Polarization of frequency 2ω. • Tensor of complex coefficients. Only the real part of χijkl(2ω0,ω,ω) is considered. • Intrinsic symmetry symbol: V2[V2] • Symmetrized indexes due to intrinsic symmetry:      • χ(2ω;0,ω,ω)ijkl = χ(2ω;0,ω,ω)ijlk • Abbreviated notation: χ(2ω;0,ω,ω)ijkl → χ(2ω;0,ω,ω)αβγ     • i → α     • j → β     • kl → γ     Convention used for the substitutions:     11 → 1 | 22 → 2 | 33 → 3 | 23,32 → 4 | 13,31 → 5 | 12,21 → 6</td></tr></table></center> </div> <div id='legend_126' style="display:none"> <center><table style="width:500px" bgcolor="#CCEEFF"><tr><td><center><big>Information about the selected tensor</big></center> • 4 th rank Third-harmonic generation (non-dissipative media). Imaginary part. χ(3ω;ω,ω,ω)ijkl • Polar tensor which inverts under time-reversal symmetry operation • Defining equation: Pi(3ω)=χijkl(3ω;ω,ω,ω)Ej(ω)Ek(ω)El(ω) • Relates Electric field of frequency ω and Electric field of frequency ω and Electric field of frequency ω with Polarization of frequency 3ω. • Tensor of complex coefficients. Only the imaginary part of χijkl(3ω;ω,ω,ω) is considered. • Intrinsic symmetry symbol: aV[V3] • Symmetrized indexes due to intrinsic symmetry:      • χ(3ω;ω,ω,ω)ijkl = χ(3ω;ω,ω,ω)ikjl      • χ(3ω;ω,ω,ω)ijkl = χ(3ω;ω,ω,ω)ilkj      • χ(3ω;ω,ω,ω)ijkl = χ(3ω;ω,ω,ω)ijlk • Abbreviated notation: χ(3ω;ω,ω,ω)ijkl → χ(3ω;ω,ω,ω)αβγ     • i → α     • jk → β     • l → γ     Convention used for the substitutions:     11 → 1 | 22 → 2 | 33 → 3 | 23,32 → 4 | 13,31 → 5 | 12,21 → 6</td></tr></table></center> </div> <div id='legend_127' style="display:none"> <center><table style="width:500px" bgcolor="#CCEEFF"><tr><td><center><big>Information about the selected tensor</big></center> • 4 th rank General third-order susceptibility (non-dissipative media). Imaginary part. χ(ω4;ω3,ω2,ω1)ijkl • Polar tensor which inverts under time-reversal symmetry operation • Defining equation: Pi(ω4)=χijkl(ω4;ω3,ω2,ω1)Ej(ω3)Ek(ω2)El(ω1) • Relates Electric field of frequency ω1 and Electric field of frequency ω2 and Electric field of frequency ω3 with Polarization of frequency ω4. • Tensor of complex coefficients. • Only the imaginary part of χijkl(ω4;ω3,ω2,ω1)is considered. • Intrinsic symmetry symbol: aV4 </td></tr></table></center> </div> <div id='legend_128' style="display:none"> <center><table style="width:500px" bgcolor="#CCEEFF"><tr><td><center><big>Information about the selected tensor</big></center> • 4 th rank Electric-field induced second-harmonic generation (non-dissipative media). Imaginary part. χ(2ω0,ω,ω)ijkl • Polar tensor which inverts under time-reversal symmetry operation • Defining equation: Pi(2ω)=χijkl(2ω;0,ω,ω)Ej(0)Ek(ω)El(ω) • Relates Electric field of frequency 0 and Electric field of frequency ω and Electric field of frequency ω with Polarization of frequency 2ω. • Tensor of complex coefficients.Only the imaginary part of χijkl(2ω;0,ω,ω) is considered. • Intrinsic symmetry symbol: aV2[V2] • Symmetrized indexes due to intrinsic symmetry:      • χ(2ω;0,ω,ω)ijkl = χ(2ω;0,ω,ω)ijlk • Abbreviated notation: χ(2ω;0,ω,ω)ijkl → χ(2ω;0,ω,ω)αβγ     • i → α     • j → β     • kl → γ     Convention used for the substitutions:     11 → 1 | 22 → 2 | 33 → 3 | 23,32 → 4 | 13,31 → 5 | 12,21 → 6</td></tr></table></center> </div> <div id='legend_129' style="display:none"> <center><table style="width:500px" bgcolor="#CCEEFF"><tr><td><center><big>Information about the selected tensor</big></center> • 4 th rank General third-harmonic generation (dissipative media) χ(3ω;ω,ω,ω)ijkl • Polar tensor invariant under time-reversal symmetry operation • Defining equation: Pi(3ω)=χijkl(3ω;ω,ω,ω)Ej(ω)Ek(ω)El(ω) • Relates Electric field of frequency ω and Electric field of frequency ω and Electric field of frequency ω with Polarization of frequency 3ω. • Tensor of complex coefficients. • Symmetry operations including time reversal do not introduce any restriction in the tensor but they connect it with a tensor for a different third order process: 1'χijkl(3ω;ω,ω,ω)=χjikl(ω;3ω,-ω,-ω)=χkijl(ω;3ω,-ω,-ω)=χlijk(ω;3ω,-ω,-ω). • Intrinsic symmetry symbol: (V[V3])* • Symmetrized indexes due to intrinsic symmetry:      • χ(3ω;ω,ω,ω)ijkl = χ(3ω;ω,ω,ω)ikjl      • χ(3ω;ω,ω,ω)ijkl = χ(3ω;ω,ω,ω)ilkj      • χ(3ω;ω,ω,ω)ijkl = χ(3ω;ω,ω,ω)ijlk • Abbreviated notation: χ(3ω;ω,ω,ω)ijkl → χ(3ω;ω,ω,ω)αβγ     • i → α     • jk → β     • l → γ     Convention used for the substitutions:     11 → 1 | 22 → 2 | 33 → 3 | 23,32 → 4 | 13,31 → 5 | 12,21 → 6</td></tr></table></center> </div> <div id='legend_130' style="display:none"> <center><table style="width:500px" bgcolor="#CCEEFF"><tr><td><center><big>Information about the selected tensor</big></center> • 4 th rank General third-order susceptibility (dissipative media) χ(ω4;ω3,ω2,ω1)ijkl • Polar tensor invariant under time-reversal symmetry operation • Defining equation: Pi(ω4)=χijkl(ω4;ω3,ω2,ω1)Ej(ω3)Ek(ω2)El(ω1) • Relates Electric field of frequency ω1 and Electric field of frequency ω2 and Electric field of frequency ω3 with Polarization of frequency ω4. • Tensor of complex coefficients. • Symmetry operations including time reversal do not introduce any restriction in the tensor but they connect it with a tensor for other third order processes: 1'χijkl(ω4;ω3,ω2,ω1)=χjikl(ω3;ω4,-ω2,-ω1)=χkijl(ω2;ω4,-ω3,-ω1)=χlijk(ω1;ω4,-ω3,-ω2). • Intrinsic symmetry symbol: (V4)* </td></tr></table></center> </div> <div id='legend_131' style="display:none"> <center><table style="width:500px" bgcolor="#CCEEFF"><tr><td><center><big>Information about the selected tensor</big></center> • 2 nd rank Diffusion tensor Dij • Polar tensor invariant under time-reversal symmetry operation • Defining equation: Ji=Dij<vec>∇</vec>jC • Relates Concentration gradient ∇C with Diffusive flux J • Intrinsic symmetry symbol: [V2]* </td></tr></table></center> </div> <div id='legend_132' style="display:none"> <center><table style="width:500px" bgcolor="#CCEEFF"><tr><td><center><big>Information about the selected tensor</big></center> • 2 nd rank Dufour effect (reversal thermodiffusion) tensor βij • Polar tensor invariant under time-reversal symmetry operation • Defining equation: qi=βij<vec>∇</vec>jC • Relates Concentration gradient ∇C with Heat flux q • Intrinsic symmetry symbol: [V2]* </td></tr></table></center> </div> <div id='legend_133' style="display:none"> <center><table style="width:500px" bgcolor="#CCEEFF"><tr><td><center><big>Information about the selected tensor</big></center> • 2 nd rank Electric conductivity tensor σij • Polar tensor invariant under time-reversal symmetry operation • Defining equation: Ji=σijEj • Relates Electric field E with Electric current density J • Intrinsic symmetry symbol: [V2]* </td></tr></table></center> </div> <div id='legend_134' style="display:none"> <center><table style="width:500px" bgcolor="#CCEEFF"><tr><td><center><big>Information about the selected tensor</big></center> • 2 nd rank Electric resistivity tensor ρij • Polar tensor invariant under time-reversal symmetry operation • Defining equation: Ei=ρijJj • Relates Electric current density J with Electric field E • Intrinsic symmetry symbol: [V2]* </td></tr></table></center> </div> <div id='legend_135' style="display:none"> <center><table style="width:500px" bgcolor="#CCEEFF"><tr><td><center><big>Information about the selected tensor</big></center> • 2 nd rank Electrodiffusion tensor γij (direct effect) • Polar tensor invariant under time-reversal symmetry operation • Defining equation: Ji=γijEjT • Relates Electric field E with Diffusive flux J • Intrinsic symmetry symbol: [V2]* </td></tr></table></center> </div> <div id='legend_136' style="display:none"> <center><table style="width:500px" bgcolor="#CCEEFF"><tr><td><center><big>Information about the selected tensor</big></center> • 2 nd rank Electrodiffusion tensor γTij (inverse effect) • Polar tensor invariant under time-reversal symmetry operation • Defining equation: Ji=γTij<vec>∇</vec>jC • Relates Concentration gradient ∇C with Electric current density J • Intrinsic symmetry symbol: [V2]* </td></tr></table></center> </div> <div id='legend_137' style="display:none"> <center><table style="width:500px" bgcolor="#CCEEFF"><tr><td><center><big>Information about the selected tensor</big></center> • 2 nd rank Soret effect (thermodiffusion) tensor βij • Polar tensor invariant under time-reversal symmetry operation • Defining equation: Ji=βij<vec>∇</vec>jT • Relates Temperature gradient ∇T with Diffusive flux J • Intrinsic symmetry symbol: [V2]* </td></tr></table></center> </div> <div id='legend_138' style="display:none"> <center><table style="width:500px" bgcolor="#CCEEFF"><tr><td><center><big>Information about the selected tensor</big></center> • 2 nd rank Thermal conductivity tensor κij • Polar tensor invariant under time-reversal symmetry operation • Defining equation: qi=κij<vec>∇</vec>jT • Relates Temperature gradient ∇T with Heat flux q • Intrinsic symmetry symbol: [V2]* </td></tr></table></center> </div> <div id='legend_139' style="display:none"> <center><table style="width:500px" bgcolor="#CCEEFF"><tr><td><center><big>Information about the selected tensor</big></center> • 2 nd rank Thermal diffusivity tensor αij • Polar tensor invariant under time-reversal symmetry operation • Defining equation: ∂T/∂t=αij<vec>∇</vec>iT<vec>∇</vec>jT • Relates Temperature gradient ∇T and Temperature gradient ∇T with Time derivative of the temperature ∂T/∂t • Intrinsic symmetry symbol: [V2]* </td></tr></table></center> </div> <div id='legend_140' style="display:none"> <center><table style="width:500px" bgcolor="#CCEEFF"><tr><td><center><big>Information about the selected tensor</big></center> • 2 nd rank Thermal resistivity tensor rij • Polar tensor invariant under time-reversal symmetry operation • Defining equation: <vec>∇</vec>iT=rijqj • Relates Heat flux q with Temperature gradient ∇T • Intrinsic symmetry symbol: [V2]* </td></tr></table></center> </div> <div id='legend_141' style="display:none"> <center><table style="width:500px" bgcolor="#CCEEFF"><tr><td><center><big>Information about the selected tensor</big></center> • 2 nd rank Ordinary Resistivity ρ(s)ij • Polar tensor invariant under time-reversal symmetry operation • Defining equation: Ei=ρijJj • Relates Electric current density J with Electric Field E. • Symmetric part of Electric resistivity tensor ρ(s)ij=(ρij+ρji)/2. • Intrinsic symmetry symbol: [V2] • Symmetrized indexes due to intrinsic symmetry:      • ρ(s)ij = ρ(s)ji </td></tr></table></center> </div> <div id='legend_142' style="display:none"> <center><table style="width:500px" bgcolor="#CCEEFF"><tr><td><center><big>Information about the selected tensor</big></center> • 2 nd rank Ordinary Thermal Conductivity κ(s)ij • Polar tensor invariant under time-reversal symmetry operation • Defining equation: qi=κij∇jT • Relates Temperature gradient ∇T with Heat flux q. • Symmetric part of Thermal Conductivity tensor κ(s)ij=(κij+κji)/2. • Intrinsic symmetry symbol: [V2] • Symmetrized indexes due to intrinsic symmetry:      • κ(s)ij = κ(s)ji </td></tr></table></center> </div> <div id='legend_143' style="display:none"> <center><table style="width:500px" bgcolor="#CCEEFF"><tr><td><center><big>Information about the selected tensor</big></center> • 2 nd rank Ordinary Peltier Effect Pij • Polar tensor invariant under time-reversal symmetry operation • Defining equation: qi=πijJj, Ei=βij∇jT, Pij=(πij+βji)/2 • π relates Electric current density J with Heat flux q and β relates Temperature gradient ∇T with Electric field E. • Intrinsic symmetry symbol: V2 </td></tr></table></center> </div> <div id='legend_144' style="display:none"> <center><table style="width:500px" bgcolor="#CCEEFF"><tr><td><center><big>Information about the selected tensor</big></center> • 2 nd rank Ordinary Seebeck Effect Sij • Polar tensor invariant under time-reversal symmetry operation • Defining equation: Ei=βij∇jT, qi=πijJj, Sij=(βij+πji)/2 • β relates Temperature gradient ∇T with Electric field E and π relates Electric current density J with Heat flux q. • Intrinsic symmetry symbol: V2 </td></tr></table></center> </div> <div id='legend_145' style="display:none"> <center><table style="width:500px" bgcolor="#CCEEFF"><tr><td><center><big>Information about the selected tensor</big></center> • 2 nd rank Spontaneous Hall Effect ρ(a)ij • Polar tensor which inverts under time-reversal symmetry operation • Defining equation: Ei=ρijJj • Relates Electric current density J with Electric Field E. • Antisymmetric part of Electric Resistivity tensor ρ(a)ij=(ρij-ρji)/2. • Intrinsic symmetry symbol: a{V2} • Symmetrized indexes due to intrinsic symmetry:      • ρ(a)ij = -ρ(a)ji </td></tr></table></center> </div> <div id='legend_146' style="display:none"> <center><table style="width:500px" bgcolor="#CCEEFF"><tr><td><center><big>Information about the selected tensor</big></center> • 2 nd rank Spontaneous Righi-Leduc Effect κ(a)ij • Polar tensor which inverts under time-reversal symmetry operation • Defining equation: qi=κij∇jT • Relates Temperature gradient ∇T with Heat flux q. • Antisymmetric part of Thermal Conductivity tensor κ(a)ij=(κij-κji)/2. • Intrinsic symmetry symbol: a{V2} • Symmetrized indexes due to intrinsic symmetry:      • κ(a)ij = -κ(a)ji </td></tr></table></center> </div> <div id='legend_147' style="display:none"> <center><table style="width:500px" bgcolor="#CCEEFF"><tr><td><center><big>Information about the selected tensor</big></center> • 2 nd rank Spontaneous Ettingshausen Effect SEij • Polar tensor which inverts under time-reversal symmetry operation • Defining equation: qi=πijJj, Ei=βij∇jT, SEij=(πij-βji)/2 • π relates Electric current density J with Heat flux q and β relates Temperature gradient ∇T with Electric field E. • Intrinsic symmetry symbol: aV2 </td></tr></table></center> </div> <div id='legend_148' style="display:none"> <center><table style="width:500px" bgcolor="#CCEEFF"><tr><td><center><big>Information about the selected tensor</big></center> • 2 nd rank Spontaneous Nernst Effect SNij • Polar tensor which inverts under time-reversal symmetry operation • Defining equation: Ei=βij∇jT, qi=πijJj, SNij=(βij-πji)/2 • β relates Temperature gradient ∇T with Electric field E and π relates Electric current density J with Heat flux q. • Intrinsic symmetry symbol: aV2 </td></tr></table></center> </div> <div id='legend_149' style="display:none"> <center><table style="width:500px" bgcolor="#CCEEFF"><tr><td><center><big>Information about the selected tensor</big></center> • 2 nd rank Peltier effect tensor πij • Polar tensor invariant under time-reversal symmetry operation • Defining equation: qi=πijJj • Relates Electric current density J with Heat flux q. • Symmetry operations including time reversal do not introduce any restriction in the tensor but they connect it with the Thermoelectric power (Seebeck) effect tensor β. •1'πijkl=βjikl. • Intrinsic symmetry symbol: (V2)* </td></tr></table></center> </div> <div id='legend_150' style="display:none"> <center><table style="width:500px" bgcolor="#CCEEFF"><tr><td><center><big>Information about the selected tensor</big></center> • 2 nd rank Thermoelectric power (Seebeck effect) tensor βij • Polar tensor invariant under time-reversal symmetry operation • Defining equation: Ei=βij<vec>∇</vec>jT • Relates Electric field E with Temperature gradient ∇T. • Symmetry operations including time reversal do not introduce any restriction in the tensor but they connect it with the Peltier effect tensor π. •1'βijkl=πjikl • Intrinsic symmetry symbol: (V2)* </td></tr></table></center> </div> <div id='legend_151' style="display:none"> <center><table style="width:500px" bgcolor="#CCEEFF"><tr><td><center><big>Information about the selected tensor</big></center> • 2 nd rank Thomson heat tensor τij • Polar tensor invariant under time-reversal symmetry operation • Defining equation: ∂q/∂t=τij<vec>∇</vec>iTJj • Relates Temperature gradient ∇T and Electric current density J with Heat production rate ∂q/∂t. • Symmetry operations including time reversal do not introduce any restriction in the tensor but they connect it with another tensor property. • Intrinsic symmetry symbol: (V2)* </td></tr></table></center> </div> <div id='legend_152' style="display:none"> <center><table style="width:500px" bgcolor="#CCEEFF"><tr><td><center><big>Information about the selected tensor</big></center> • 3 rd rank Ordinary Hall Effect R(a)ijk • Axial tensor invariant under time-reversal symmetry operation • Defining equation: Ei=RijkJjHk • Relates Electric current density J and Magnetic field H with Electric field E. • Antisymmetric part of Hall effect tensor R(a)ijk=(Rijk-Rjik)/2. • Intrinsic symmetry symbol: e{V2}V • Symmetrized indexes due to intrinsic symmetry:      • R(a)ijk = -R(a)jik </td></tr></table></center> </div> <div id='legend_153' style="display:none"> <center><table style="width:500px" bgcolor="#CCEEFF"><tr><td><center><big>Information about the selected tensor</big></center> • 3 rd rank Ordinary Righi-Leduc Effect Q(a)ijk • Axial tensor invariant under time-reversal symmetry operation • Defining equation: qi=Qijk∇jTHk • Relates Temperature gradient ∇T and Magnetic field H with Heat flux q. • Antisymmetric part of Righi-Leduc Effect tensor Q(a)ijk=(Qijk-Qjik)/2. • Intrinsic symmetry symbol: e{V2}V • Symmetrized indexes due to intrinsic symmetry:      • Q(a)ijk = -Q(a)jik </td></tr></table></center> </div> <div id='legend_154' style="display:none"> <center><table style="width:500px" bgcolor="#CCEEFF"><tr><td><center><big>Information about the selected tensor</big></center> • 3 rd rank Hall effect (magnetorresistance) tensor Rijk • Axial tensor invariant under time-reversal symmetry operation • Defining equation: Ei=RijkJjHk • Relates Electric current density J and Magnetic field H with Electric field E • Intrinsic symmetry symbol: e{V2}*V </td></tr></table></center> </div> <div id='legend_155' style="display:none"> <center><table style="width:500px" bgcolor="#CCEEFF"><tr><td><center><big>Information about the selected tensor</big></center> • 3 rd rank Righi-Leduc, Maggi-Righi-Leduc and magnetothermal effects tensor Qijk • Axial tensor invariant under time-reversal symmetry operation • Defining equation: qi=Qijk<vec>∇</vec>jTHk • Relates Temperature gradient ∇T and Magnetic field H with Heat flux q • Intrinsic symmetry symbol: e{V2}*V </td></tr></table></center> </div> <div id='legend_156' style="display:none"> <center><table style="width:500px" bgcolor="#CCEEFF"><tr><td><center><big>Information about the selected tensor</big></center> • 3 rd rank Ordinary Ettingshausen Effect OEij • Axial tensor invariant under time-reversal symmetry operation • Defining equation: qi=MijkJjHk, Ei=Nijk∇jTHk, OEijk=(Mijk-Njik)/2 • M relates Electric current density J and Magnetic field H with Heat flux q and N relates Temperature gradient ∇T and Magnetic field H with Electric field E. • Intrinsic symmetry symbol: eV3 </td></tr></table></center> </div> <div id='legend_157' style="display:none"> <center><table style="width:500px" bgcolor="#CCEEFF"><tr><td><center><big>Information about the selected tensor</big></center> • 3 rd rank Ordinary Nernst Effect ONijk • Axial tensor invariant under time-reversal symmetry operation • Defining equation: Ei=Nijk∇jTHk, qi=MijkJjHk, ONijk=(Nijk-Mjik)/2 • N relates Temperature gradient ∇T and Magnetic field H with Electric field E and M relates Electric current density J and Magnetic field H with Heat flux q. • Intrinsic symmetry symbol: eV3 </td></tr></table></center> </div> <div id='legend_158' style="display:none"> <center><table style="width:500px" bgcolor="#CCEEFF"><tr><td><center><big>Information about the selected tensor</big></center> • 3 rd rank Linear Magneto-heat Conductivity Effect Q(s)ijk • Axial tensor which inverts under time-reversal symmetry operation • Defining equation: qi=Qijk∇jTHk • Relates Temperature gradient ∇T and Magnetic field H with Heat flux q. • Symmetric part of Righi-Leduc Effect tensor Q(s)ijk=(Qijk+Qjik)/2. • Intrinsic symmetry symbol: ae[V2]V • Symmetrized indexes due to intrinsic symmetry:      • Q(s)ijk = Q(s)jik • Abbreviated notation: Q(s)ijk → Q(s)αβ     • ij → α     • k → β     Convention used for the substitutions:     11 → 1 | 22 → 2 | 33 → 3 | 23,32 → 4 | 13,31 → 5 | 12,21 → 6</td></tr></table></center> </div> <div id='legend_159' style="display:none"> <center><table style="width:500px" bgcolor="#CCEEFF"><tr><td><center><big>Information about the selected tensor</big></center> • 3 rd rank Linear Magnetorresistance R(s)ijk • Axial tensor which inverts under time-reversal symmetry operation • Defining equation: Ei=RijkJjHk • Relates Electric current density J and Magnetic field H with Electric Field E. • Symmetric part of Hall effect tensor R(s)ijk=(Rijk+Rjik)/2. • Intrinsic symmetry symbol: ae[V2]V • Symmetrized indexes due to intrinsic symmetry:      • R(s)ijk = R(s)jik • Abbreviated notation: R(s)ijk → R(s)αβ     • ij → α     • k → β     Convention used for the substitutions:     11 → 1 | 22 → 2 | 33 → 3 | 23,32 → 4 | 13,31 → 5 | 12,21 → 6</td></tr></table></center> </div> <div id='legend_160' style="display:none"> <center><table style="width:500px" bgcolor="#CCEEFF"><tr><td><center><big>Information about the selected tensor</big></center> • 3 rd rank Linear Magneto Peltier Effect MPijk • Axial tensor which inverts under time-reversal symmetry operation • Defining equation: qi=MijkJjHk, Ei=Nijk∇jTHk, MPijk=(Mijk+Njik)/2 • M relates Electric current density J and Magnetic field H with Heat flux q and N relates Temperature gradient ∇T and Magnetic field H with Electric field E. • Intrinsic symmetry symbol: aeV3 </td></tr></table></center> </div> <div id='legend_161' style="display:none"> <center><table style="width:500px" bgcolor="#CCEEFF"><tr><td><center><big>Information about the selected tensor</big></center> • 3 rd rank Linear Magneto Seebeck Effect MSijk • Axial tensor which inverts under time-reversal symmetry operation • Defining equation: Ei=Nijk∇jTHk, qi=MijkJjHk, MSijk=(Nijk+Mjik)/2 • N relates Temperature gradient ∇T and Magnetic field H with Electric field E and M relates Electric current density J and Magnetic field H with Heat flux q. • Intrinsic symmetry symbol: aeV3 </td></tr></table></center> </div> <div id='legend_162' style="display:none"> <center><table style="width:500px" bgcolor="#CCEEFF"><tr><td><center><big>Information about the selected tensor</big></center> • 3 rd rank Ettingshausen effect tensor Mijk • Axial tensor invariant under time-reversal symmetry operation • Defining equation: qi=MijkJjHk • Relates Electric current density J and Magnetic field H with Heat flux q. • Symmetry operations including time reversal do not introduce any restriction in the tensor but they connect it with the Nernst effect tensor N. •1'Mijkl=Njikl. • Intrinsic symmetry symbol: (eV3)* </td></tr></table></center> </div> <div id='legend_163' style="display:none"> <center><table style="width:500px" bgcolor="#CCEEFF"><tr><td><center><big>Information about the selected tensor</big></center> • 3 rd rank Nernst effect tensor Nijk • Axial tensor invariant under time-reversal symmetry operation • Defining equation: Ei=Nijk<vec>∇</vec>TjHk • Relates Temperature gradient ∇T and Magnetic field H with Electric field E. • Symmetry operations including time reversal do not introduce any restriction in the tensor but they connect it with the Ettinghausen effect tensor M. •1'Nijkl=Mjikl. • Intrinsic symmetry symbol: (eV3)* </td></tr></table></center> </div> <div id='legend_164' style="display:none"> <center><table style="width:500px" bgcolor="#CCEEFF"><tr><td><center><big>Information about the selected tensor</big></center> • 4 th rank Quadratic Magneto-heat-conductivity S(s)ijkl • Polar tensor invariant under time-reversal symmetry operation • Defining equation: qi=Sijkl∇jTHkHl • Relates Temperature gradient ∇T, Magnetic field H and Magnetic field H with Heat flux q. • Symmetric part of magneto-heat conductivity tensor S(s)ijkl=(Sijkl+Sjikl)/2. • Intrinsic symmetry symbol: [V2][V2] • Symmetrized indexes due to intrinsic symmetry:      • S(s)ijkl = S(s)jikl      • S(s)ijkl = S(s)ijlk • Abbreviated notation: S(s)ijkl → S(s)αβ     • ij → α     • kl → β     Convention used for the substitutions:     11 → 1 | 22 → 2 | 33 → 3 | 23,32 → 4 | 13,31 → 5 | 12,21 → 6</td></tr></table></center> </div> <div id='legend_165' style="display:none"> <center><table style="width:500px" bgcolor="#CCEEFF"><tr><td><center><big>Information about the selected tensor</big></center> • 4 th rank Magnetic resistance tensor Tijkl • Polar tensor invariant under time-reversal symmetry operation • Defining equation: Ei=TijklJjHkHl • Relates Electric current density J and Magnetic field H and Magnetic field H with Electric field E • Intrinsic symmetry symbol: [V2]*[V2] • Symmetrized indexes due to intrinsic symmetry:      • Tijkl = Tijlk • Abbreviated notation: Tijkl → Tαβγ     • i → α     • j → β     • kl → γ     Convention used for the substitutions:     11 → 1 | 22 → 2 | 33 → 3 | 23,32 → 4 | 13,31 → 5 | 12,21 → 6</td></tr></table></center> </div> <div id='legend_166' style="display:none"> <center><table style="width:500px" bgcolor="#CCEEFF"><tr><td><center><big>Information about the selected tensor</big></center> • 4 th rank Magneto-heat-conductivity tensor Sijkl Sijkl • Polar tensor invariant under time-reversal symmetry operation • Defining equation: qi=Sijkl∇jTHkHl • Relates Temperature gradient ∇T, Magnetic field H and Magnetic field H with Heat flux q. • Onsager relations imply 1'Sij=Sji • Intrinsic symmetry symbol: [V2]*[V2] • Symmetrized indexes due to intrinsic symmetry:      • Sijkl = Sijlk • Abbreviated notation: Sijkl → Sαβγ     • i → α     • j → β     • kl → γ     Convention used for the substitutions:     11 → 1 | 22 → 2 | 33 → 3 | 23,32 → 4 | 13,31 → 5 | 12,21 → 6</td></tr></table></center> </div> <div id='legend_167' style="display:none"> <center><table style="width:500px" bgcolor="#CCEEFF"><tr><td><center><big>Information about the selected tensor</big></center> • 4 th rank Piezoresistivity (Strain Gauge effect) tensor πijkl • Polar tensor invariant under time-reversal symmetry operation • Defining equation: Δρij=πijklσkl • Relates Stress tensor σij with Electric resistivity tensor variation Δρij • Intrinsic symmetry symbol: [V2]*[V2] • Symmetrized indexes due to intrinsic symmetry:      • πijkl = πijlk • Abbreviated notation: πijkl → παβγ     • i → α     • j → β     • kl → γ     Convention used for the substitutions:     11 → 1 | 22 → 2 | 33 → 3 | 23,32 → 4 | 13,31 → 5 | 12,21 → 6</td></tr></table></center> </div> <div id='legend_168' style="display:none"> <center><table style="width:500px" bgcolor="#CCEEFF"><tr><td><center><big>Information about the selected tensor</big></center> • 4 th rank Quadratic Anomalous Righi-Leduc Effect S(a)ijkl • Polar tensor which inverts under time-reversal symmetry operation • Defining equation: qi=Sijkl∇jTHkHl • Relates Temperature gradient ∇T, Magnetic field H and Magnetic field H with Heat flux q. • Antisymmetric part of magneto-heat conductivity tensor S(a)ijkl=(Sijkl-Sjikl)/2. • Intrinsic symmetry symbol: a{V2}[V2] • Symmetrized indexes due to intrinsic symmetry:      • S(a)ijkl = -S(a)jikl      • S(a)ijkl = S(a)ijlk • Abbreviated notation: S(a)ijkl → S(a)αβγ     • i → α     • j → β     • kl → γ     Convention used for the substitutions:     11 → 1 | 22 → 2 | 33 → 3 | 23,32 → 4 | 13,31 → 5 | 12,21 → 6</td></tr></table></center> </div> <div id='legend_169' style="display:none"> <center><table style="width:500px" bgcolor="#CCEEFF"><tr><td><center><big>Information about the selected tensor</big></center> • 4 th rank Magneto Peltier effect Pijkl • Polar tensor invariant under time-reversal symmetry operation • Defining equation: qi=PijklHkHlJj • Relates current density and magnetic field and magnetic field with heat flux. • Symmetry operations including time reversal do not introduce any restriction in the tensor but they connect it with the magneto-Seebeck effect tensor α. •1'Pijkl=αjikl • Intrinsic symmetry symbol: (V2[V2])* • Symmetrized indexes due to intrinsic symmetry:      • Pijkl = Pijlk • Abbreviated notation: Pijkl → Pαβγ     • i → α     • j → β     • kl → γ     Convention used for the substitutions:     11 → 1 | 22 → 2 | 33 → 3 | 23,32 → 4 | 13,31 → 5 | 12,21 → 6</td></tr></table></center> </div> <div id='legend_170' style="display:none"> <center><table style="width:500px" bgcolor="#CCEEFF"><tr><td><center><big>Information about the selected tensor</big></center> • 4 th rank Magneto Seebeck effect αijkl • Polar tensor invariant under time-reversal symmetry operation • Defining equation: Ei=αijkl∇jTHkHl • Relates temperature gradient ∇T and magnetic field and magnetic field with electric field. • Symmetry operations including time reversal do not introduce any restriction in the tensor but they connect it with the magneto-Peltier effect tensor P. •1'αijkl=Pjikl • Intrinsic symmetry symbol: (V2[V2])* • Symmetrized indexes due to intrinsic symmetry:      • αijkl = αijlk • Abbreviated notation: αijkl → ααβγ     • i → α     • j → β     • kl → γ     Convention used for the substitutions:     11 → 1 | 22 → 2 | 33 → 3 | 23,32 → 4 | 13,31 → 5 | 12,21 → 6</td></tr></table></center> </div> </td></tr></table> </ul> </td> </tr> </table> </td> <td width="65%"> <form method="post" action="/cgi-bin/cryst/programs/nph-mtensor" name="form1"> <input type="hidden" name="magnum" value="">  <ul>   Please, enter a magnetic point group or a magnetic space group: </ul>   <table width="100%" class="grey"> <tr> <td align=left style="white-space:nowrap;"> Magnetic Point or Space Group number:   <input type="button" name="choose" value="choose it" onClick="location.href='/cryst/mtensor.php?from=mtensor&magtr=3'"> </td> </tr> </table>  <div id='nonconvDiv' style="display:none"> <table class="grey" width=100%> <tr> <td colspan=6 align=center><a href="nph-doc-trmat">Transformation to the standard setting</a></td> </tr> <tr> <td colspan=6 align=center><a href="nph-doc-trmat#abcxyz">abc</a> format: <input type="text" name="trm" value="a,b,c"></td> </tr> <tr><td colspan=6 align=center> - OR -</td></tr> <tr> <td> </td> <td align="center" colspan="3"> <TABLE border="0" width="100%" cellpadding="10"> <TR> <TD ALIGN="center" colspan="1"> Transformation matrix <TABLE ALIGN="center"> <TR> <TD ALIGN="right"><INPUT TYPE="text" NAME="x1" VALUE="1" SIZE="6"></TD> <TD ALIGN="right"><INPUT TYPE="text" NAME="x2" VALUE="0" SIZE="6"></TD> <TD ALIGN="right"><INPUT TYPE="text" NAME="x3" VALUE="0" SIZE="6"></TD> </TR> <TR> <TD ALIGN="right"><INPUT TYPE="text" NAME="y1" VALUE="0" SIZE="6"></TD> <TD ALIGN="right"><INPUT TYPE="text" NAME="y2" VALUE="1" SIZE="6"></TD> <TD ALIGN="right"><INPUT TYPE="text" NAME="y3" VALUE="0" SIZE="6"></TD> </TR> <TR> <TD ALIGN="right"><INPUT TYPE="text" NAME="z1" VALUE="0" SIZE="6"></TD> <TD ALIGN="right"><INPUT TYPE="text" NAME="z2" VALUE="0" SIZE="6"></TD> <TD ALIGN="right"><INPUT TYPE="text" NAME="z3" VALUE="1" SIZE="6"></TD> </TR> </TABLE> </TD>  </TR> </TABLE> </td> </tr> <tr><td colspan=6 align=center>[If the matrix format is different than identity, it will be preferred over the abc format]</td></tr> </table> <INPUT TYPE="hidden" NAME="x4" VALUE="0"> <INPUT TYPE="hidden" NAME="y4" VALUE="0"> <INPUT TYPE="hidden" NAME="z4" VALUE="0">  </div>  </div> <ul> Please, choose a tensor by one of these ways:</ul> <ul><input type="radio" id="tensor_type" name="tensor_type" value="standard" checked> Choose a tensor from the lists</ul> <center><input type="checkbox" name="All" value="all">Show symmetry-adapted tensors for all the magnetic point groups in standard setting (this overrides previous choices) </center> <center><a href="javascript:;" onClick="javascript:showHideElem('eq_block');">EQUILIBRIUM TENSORS</a></center><div id="eq_block" style="display:none"><center> <table width="100%" border="5" align="center" cellspacing="10"><tr><th>Rank</th><th><a class="blue" href="/html/cryst/mtensor_help.html#intrinsic">Intrinsic symmetry</a></th><th>Tensor description</th><th>Defining equation</th><th>Select</th></tr><tr><td align="center" rowspan=15>1</td><td align="center" rowspan=6>V</td><td align="center">Electric polarization vector Pi</td><td align="center">-</td><td align="center"><input type="radio" name="choose_tensor" value="polarization" checked="checked" onChange="active_legend(1);"></td></tr> <tr><td align="center">Electrocaloric effect tensor pi</td><td align="center">ΔS=piEi</td><td align="center"><input type="radio" name="choose_tensor" value="electrocaloric" onChange="active_legend(2);"></td></tr> <tr><td align="center">Electrothermal effect tensor ti</td><td align="center">Ei=-tiΔT</td><td align="center"><input type="radio" name="choose_tensor" value="electrothermal" onChange="active_legend(3);"></td></tr> <tr><td align="center">Heat of polarization tensor ti</td><td align="center">ΔS=tiΔPi</td><td align="center"><input type="radio" name="choose_tensor" value="polarization_heat" onChange="active_legend(4);"></td></tr> <tr><td align="center">Piezoelectric polarization tensor under hydrostatic pressure dijj</td><td align="center">Pi=-dijjp</td><td align="center"><input type="radio" name="choose_tensor" value="hydr_piezo_polarization" onChange="active_legend(5);"></td></tr> <tr><td align="center">Pyroelectric tensor pi</td><td align="center">ΔPi=piΔT</td><td align="center"><input type="radio" name="choose_tensor" value="pyroelectric" onChange="active_legend(6);"></td></tr> <tr><td align="center" rowspan=1>eV</td><td align="center">Axial toroidal moment Ai</td><td align="center">-</td><td align="center"><input type="radio" name="choose_tensor" value="axialtoroidic" onChange="active_legend(7);"></td></tr> <tr><td align="center" rowspan=3>aV</td><td align="center">Polar Toroidal moment Ti</td><td align="center">-</td><td align="center"><input type="radio" name="choose_tensor" value="toroidal" onChange="active_legend(8);"></td></tr> <tr><td align="center">Pyrotoroidic tensor ri</td><td align="center">Ti=riΔT</td><td align="center"><input type="radio" name="choose_tensor" value="pyrotoroidic" onChange="active_legend(9);"></td></tr> <tr><td align="center">Toroidalcaloric tensor rTi</td><td align="center">ΔS=rTiSi</td><td align="center"><input type="radio" name="choose_tensor" value="toroidalcaloric" onChange="active_legend(10);"></td></tr> <tr><td align="center" rowspan=5>aeV</td><td align="center">Heat of Magnetization tensor ti</td><td align="center">ΔS=tiMi</td><td align="center"><input type="radio" name="choose_tensor" value="magnetization_heat" onChange="active_legend(11);"></td></tr> <tr><td align="center">Magnetization vector Mi</td><td align="center">-</td><td align="center"><input type="radio" name="choose_tensor" value="magnetization" onChange="active_legend(12);"></td></tr> <tr><td align="center">Magnetocaloric tensor qTi</td><td align="center">ΔS=qTiHi</td><td align="center"><input type="radio" name="choose_tensor" value="magnetocaloric" onChange="active_legend(13);"></td></tr> <tr><td align="center">Magnetothermal effect tensor ti</td><td align="center">Hi=-tiΔT</td><td align="center"><input type="radio" name="choose_tensor" value="magnetothermal" onChange="active_legend(14);"></td></tr> <tr><td align="center">Pyromagnetic tensor qi (direct effect)</td><td align="center">Mi=qiΔT</td><td align="center"><input type="radio" name="choose_tensor" value="pyromagnetic" onChange="active_legend(15);"></td></tr> <tr><td align="center" rowspan=21>2</td><td align="center" rowspan=13>[V2]</td><td align="center">Dielectric impermeability tensor βij</td><td align="center">Ei=βijDj</td><td align="center"><input type="radio" name="choose_tensor" value="impermeability" onChange="active_legend(16);"></td></tr> <tr><td align="center">Dielectric permittivity tensor εij</td><td align="center">Di=εijEj</td><td align="center"><input type="radio" name="choose_tensor" value="permittivity" onChange="active_legend(17);"></td></tr> <tr><td align="center">Dielectric susceptibility tensor χeij</td><td align="center">Pi=χeijEj</td><td align="center"><input type="radio" name="choose_tensor" value="susceptibility" onChange="active_legend(18);"></td></tr> <tr><td align="center">Heat of deformation tensor βij</td><td align="center">ΔS=βijεij</td><td align="center"><input type="radio" name="choose_tensor" value="deformation_heat" onChange="active_legend(19);"></td></tr> <tr><td align="center">Magnetic permeability tensor μmij</td><td align="center">Bi=μmijHj</td><td align="center"><input type="radio" name="choose_tensor" value="permeability" onChange="active_legend(20);"></td></tr> <tr><td align="center">Magnetic susceptibility tensor χmij</td><td align="center">Mi=χmijHj</td><td align="center"><input type="radio" name="choose_tensor" value="susceptibility_mag" onChange="active_legend(21);"></td></tr> <tr><td align="center">Piezocaloric effect tensor αij</td><td align="center">ΔS=αijσij</td><td align="center"><input type="radio" name="choose_tensor" value="piezocaloric" onChange="active_legend(22);"></td></tr> <tr><td align="center">Strain by hydrostatic pressure sijkk</td><td align="center">εij=-sijkkp</td><td align="center"><input type="radio" name="choose_tensor" value="compressibility" onChange="active_legend(23);"></td></tr> <tr><td align="center">Strain tensor εij</td><td align="center">-</td><td align="center"><input type="radio" name="choose_tensor" value="strain" onChange="active_legend(24);"></td></tr> <tr><td align="center">Susceptibility inverse tensor χij (inverse effect)</td><td align="center">Hi=χijMj</td><td align="center"><input type="radio" name="choose_tensor" value="susceptibility_inv" onChange="active_legend(25);"></td></tr> <tr><td align="center">Thermal expansion tensor αij</td><td align="center">εij=αijΔT</td><td align="center"><input type="radio" name="choose_tensor" value="t_expansion" onChange="active_legend(26);"></td></tr> <tr><td align="center">Thermoelasticity tensor βij</td><td align="center">σij=-βijΔT</td><td align="center"><input type="radio" name="choose_tensor" value="thermoelasticity" onChange="active_legend(27);"></td></tr> <tr><td align="center">Toroidic susceptibility tensor τij</td><td align="center">Ti=τijSj</td><td align="center"><input type="radio" name="choose_tensor" value="susceptibility_tor" onChange="active_legend(28);"></td></tr> <tr><td align="center" rowspan=2>eV2</td><td align="center">Magnetotoroidic tensor ζij (direct effect)</td><td align="center">Mi=ζijSj</td><td align="center"><input type="radio" name="choose_tensor" value="magnetotoroidic" onChange="active_legend(29);"></td></tr> <tr><td align="center">Magnetotoroidic tensor ζTij (inverse effect)</td><td align="center">Ti=ζTijHj</td><td align="center"><input type="radio" name="choose_tensor" value="magnetotoroidic_inv" onChange="active_legend(30);"></td></tr> <tr><td align="center" rowspan=2>aV2</td><td align="center">Electrotoroidic tensor θij (direct effect)</td><td align="center">Pi=θijSj</td><td align="center"><input type="radio" name="choose_tensor" value="electrotoroidic" onChange="active_legend(31);"></td></tr> <tr><td align="center">Electrotoroidic tensor θTij (inverse effect)</td><td align="center">Ti=θTijEj</td><td align="center"><input type="radio" name="choose_tensor" value="electrotoroidic_inv" onChange="active_legend(32);"></td></tr> <tr><td align="center" rowspan=4>aeV2</td><td align="center">Isothermal magnetoelectric effect tensor Aij</td><td align="center">Mi=AijDj</td><td align="center"><input type="radio" name="choose_tensor" value="isothermal_magnetoelectric" onChange="active_legend(33);"></td></tr> <tr><td align="center">Isothermal magnetoelectric effect tensor ATij (inverse effect)</td><td align="center">Ei=-ATijHj</td><td align="center"><input type="radio" name="choose_tensor" value="isothermal_magnetoelectric_inv" onChange="active_legend(34);"></td></tr> <tr><td align="center">Magnetoelectric tensor αij (direct effect)</td><td align="center">Mi=αijEj</td><td align="center"><input type="radio" name="choose_tensor" value="magnetoelectric" onChange="active_legend(35);"></td></tr> <tr><td align="center">Magnetoelectric tensor αTij (inverse effect)</td><td align="center">Pi=αTijHj</td><td align="center"><input type="radio" name="choose_tensor" value="magnetoelectric_inv" onChange="active_legend(36);"></td></tr> <tr><td align="center" rowspan=13>3</td><td align="center" rowspan=3>[V2]V</td><td align="center">Acoustoelectricity tensor ρijk</td><td align="center">σij=ρijkJk</td><td align="center"><input type="radio" name="choose_tensor" value="acoustoelectricity" onChange="active_legend(37);"></td></tr> <tr><td align="center">Isothermal piezoelectric tensor eTijk (inverse effect)</td><td align="center">σij=-eTijkEk</td><td align="center"><input type="radio" name="choose_tensor" value="isothermal_piezoelectric_inv" onChange="active_legend(38);"></td></tr> <tr><td align="center">Piezoelectric tensor dTijk (inversee ffect) dTijk</td><td align="center">εij=dTijkEk</td><td align="center"><input type="radio" name="choose_tensor" value="piezoelectric_inv" onChange="active_legend(39);"></td></tr> <tr><td align="center" rowspan=3>V[V2]</td><td align="center">Isothermal piezoelectric tensor eijk (direct effect)</td><td align="center">Di=eijkεjk</td><td align="center"><input type="radio" name="choose_tensor" value="isothermal_piezoelectric" onChange="active_legend(40);"></td></tr> <tr><td align="center">Piezoelectric tensor dijk(directeffect) dijk</td><td align="center">Pi=dijkσjk</td><td align="center"><input type="radio" name="choose_tensor" value="piezoelectric" onChange="active_legend(41);"></td></tr> <tr><td align="center">Second order magnetoelectric tensor αTijk (inverse effect)</td><td align="center">Pi=αTijkHjHk</td><td align="center"><input type="radio" name="choose_tensor" value="magnetoelectric_inv_second" onChange="active_legend(42);"></td></tr> <tr><td align="center" rowspan=1>a[V2]V</td><td align="center">Piezotoroidic tensor γijk (direct effect)</td><td align="center">εij=γijkSk</td><td align="center"><input type="radio" name="choose_tensor" value="piezotoroidic" onChange="active_legend(43);"></td></tr> <tr><td align="center" rowspan=1>aV[V2]</td><td align="center">Piezotoroidic tensor γTijk (inverse effect)</td><td align="center">Ti=γTijkσjk</td><td align="center"><input type="radio" name="choose_tensor" value="piezotoroidic_inv" onChange="active_legend(44);"></td></tr> <tr><td align="center" rowspan=2>ae[V2]V</td><td align="center">Isothermal piezomagnetic tensor emTijk (inverse effect)</td><td align="center">σij=-emTijkHk</td><td align="center"><input type="radio" name="choose_tensor" value="isothermal_piezomagnetic_inv" onChange="active_legend(45);"></td></tr> <tr><td align="center">Piezomagnetic tensor ΛTijk (inverse effect)</td><td align="center">εij=ΛTijkHk</td><td align="center"><input type="radio" name="choose_tensor" value="piezomagnetic_inv" onChange="active_legend(46);"></td></tr> <tr><td align="center" rowspan=3>aeV[V2]</td><td align="center">Isothermal piezomagnetic tensor emijk (direct effect)</td><td align="center">Mi=emijkεjk</td><td align="center"><input type="radio" name="choose_tensor" value="isothermal_piezomagnetic" onChange="active_legend(47);"></td></tr> <tr><td align="center">Piezomagnetic tensor Λijk (direct effect)</td><td align="center">Mi=Λijkσjk</td><td align="center"><input type="radio" name="choose_tensor" value="piezomagnetic" onChange="active_legend(48);"></td></tr> <tr><td align="center">Second order magnetoelectric tensor αijk (direct effect)</td><td align="center">Mi=αijkEjEk</td><td align="center"><input type="radio" name="choose_tensor" value="magnetoelectric_second" onChange="active_legend(49);"></td></tr> <tr><td align="center" rowspan=12>4</td><td align="center" rowspan=3>[[V2][V2]]</td><td align="center">Elastic compliance tensor Sijkl</td><td align="center">εij=Sijklσkl</td><td align="center"><input type="radio" name="choose_tensor" value="elasticity" onChange="active_legend(50);"></td></tr> <tr><td align="center">Elastic stiffness tensor Cijkl Cijkl</td><td align="center">σij=Cijklεkl</td><td align="center"><input type="radio" name="choose_tensor" value="elasticity_inv" onChange="active_legend(51);"></td></tr> <tr><td align="center">Viscosity tensor ηijkl</td><td align="center">σij=ηijkl∂εkl/∂t</td><td align="center"><input type="radio" name="choose_tensor" value="viscosity" onChange="active_legend(52);"></td></tr> <tr><td align="center" rowspan=3>[V2][V2]</td><td align="center">Damage effect tensor Dijkl</td><td align="center">σij=Dijklσkl</td><td align="center"><input type="radio" name="choose_tensor" value="damage_effect" onChange="active_legend(53);"></td></tr> <tr><td align="center">Electrostriction tensor γijkl</td><td align="center">εij=γijklEkEl</td><td align="center"><input type="radio" name="choose_tensor" value="electrostriction" onChange="active_legend(54);"></td></tr> <tr><td align="center">Magnetostriction tensor Nijkl</td><td align="center">εij=NijklMkMl</td><td align="center"><input type="radio" name="choose_tensor" value="magnetostriction" onChange="active_legend(55);"></td></tr> <tr><td align="center" rowspan=1>V[V3]</td><td align="center">Flexoelectric (modified) μijkl</td><td align="center">Pi=μijkl∇jεkl</td><td align="center"><input type="radio" name="choose_tensor" value="flexoelectricNew" onChange="active_legend(56);"></td></tr> <tr><td align="center" rowspan=3>V2[V2]</td><td align="center">Elastothermoelectric power tensor Eijkl</td><td align="center">ΔΣij=Eijklεkl</td><td align="center"><input type="radio" name="choose_tensor" value="elastothermoelectric" onChange="active_legend(57);"></td></tr> <tr><td align="center">Flexoelectric tensor μijkl</td><td align="center">Pi=μijkl∇jεkl</td><td align="center"><input type="radio" name="choose_tensor" value="flexoelectric" onChange="active_legend(58);"></td></tr> <tr><td align="center">Piezothermoelectric power tensor Πijkl</td><td align="center">ΔΣij=Πijklσkl</td><td align="center"><input type="radio" name="choose_tensor" value="piezothermoelectric" onChange="active_legend(59);"></td></tr> <tr><td align="center" rowspan=2>aeV2[V2]</td><td align="center">Flexomagnetic tensor Qijkl</td><td align="center">Mi=Qijkl∇jσkl</td><td align="center"><input type="radio" name="choose_tensor" value="flexomagnetic" onChange="active_legend(60);"></td></tr> <tr><td align="center">Piezomagnetoelectric tensor πijkl</td><td align="center">Pi=πijklHjσkl</td><td align="center"><input type="radio" name="choose_tensor" value="piezomagnetoelectric" onChange="active_legend(61);"></td></tr> <tr><td align="center" rowspan=2>5</td><td align="center" rowspan=1>[[V2][V2]]V</td><td align="center">Acoustic activity tensor bijklm</td><td align="center">σij=bijklm∇mεkl</td><td align="center"><input type="radio" name="choose_tensor" value="acoustic_activity" onChange="active_legend(62);"></td></tr> <tr><td align="center" rowspan=1>V[[V2][V2]]</td><td align="center">Second-order piezoelectric tensor dijklm</td><td align="center">Pi=dijklmσjkσlm</td><td align="center"><input type="radio" name="choose_tensor" value="piezoelectric_quad" onChange="active_legend(63);"></td></tr> <tr><td align="center" rowspan=2>6</td><td align="center" rowspan=2>[[V2][V2][V2]]</td><td align="center">Third order elastic compliance tensor Sijklmn</td><td align="center">εij=Sijklmnσklσmn</td><td align="center"><input type="radio" name="choose_tensor" value="third_order_elastic" onChange="active_legend(64);"></td></tr> <tr><td align="center">Third order elastic stiffness tensor Cijklmn</td><td align="center">σij=Cijklmnεklεmn</td><td align="center"><input type="radio" name="choose_tensor" value="third_order_elastic_inv" onChange="active_legend(65);"></td></tr> <tr><td align="center" rowspan=3>8</td><td align="center" rowspan=1>[[[V2][V2]][[V2][V2]]]</td><td align="center">Damage tensor Rijklmnpq</td><td align="center">Cijkl=RijklmnpqCmnpq</td><td align="center"><input type="radio" name="choose_tensor" value="damage" onChange="active_legend(66);"></td></tr> <tr><td align="center" rowspan=2>[[V2][V2][V2][V2]]</td><td align="center">Fourth order elastic compliance tensor Sijklmnpq</td><td align="center">εij=Sijklmnpqσklσmnσpq</td><td align="center"><input type="radio" name="choose_tensor" value="fourth_order_elastic" onChange="active_legend(67);"></td></tr> <tr><td align="center">Fourth order elastic stiffness tensor Cijklmnpq</td><td align="center">σij=Cijklmnpqεklεmnεpq</td><td align="center"><input type="radio" name="choose_tensor" value="fourth_order_elastic_inv" onChange="active_legend(68);"></td></tr> </table></div> </center><center><a href="javascript:;" onClick="javascript:showHideElem('opt_block');">OPTICAL TENSORS</a> <div id="opt_block" style="display:none"> <table border="5" width="100%" align="center"><tr><th>Rank</th><th><a class="blue" href="/html/cryst/mtensor_help.html#intrinsic">Intrinsic symmetry</a></th><th>Tensor description</th><th>Defining equation</th><th>Select</th></tr><tr><td align="center" rowspan=7>2</td><td align="center" rowspan=3>[V2]</td><td align="center">Index ellipsoid βij</td><td align="center">-</td><td align="center"><input type="radio" name="choose_tensor" value="Indexellipsoid" onChange="active_legend(69);"></td></tr> <tr><td align="center">Second-order thermo-optical effect tensor Tij.</td><td align="center">Δβij=Tij(ΔT)2</td><td align="center"><input type="radio" name="choose_tensor" value="thermooptical_quad" onChange="active_legend(70);"></td></tr> <tr><td align="center">Thermo-optical effect tensor tij</td><td align="center">Δβij=tijΔT</td><td align="center"><input type="radio" name="choose_tensor" value="thermooptical" onChange="active_legend(71);"></td></tr> <tr><td align="center" rowspan=1>V2</td><td align="center">Verdet tensor (related to Faraday effect) Vij</td><td align="center">Δβij=εijmVmkHk </td><td align="center"><input type="radio" name="choose_tensor" value="verdet" onChange="active_legend(72);"></td></tr> <tr><td align="center" rowspan=2>e[V2]</td><td align="center">Optical activity tensor gij</td><td align="center">G=gijlilj</td><td align="center"><input type="radio" name="choose_tensor" value="gyration" onChange="active_legend(73);"></td></tr> <tr><td align="center">Thermogyration tensor gij</td><td align="center">Δgij=gijT</td><td align="center"><input type="radio" name="choose_tensor" value="thermogyration" onChange="active_legend(74);"></td></tr> <tr><td align="center" rowspan=1>a{V2}</td><td align="center">Spontaneous Faraday effect βij</td><td align="center">-</td><td align="center"><input type="radio" name="choose_tensor" value="spontaneousFaraday" onChange="active_legend(75);"></td></tr> <tr><td align="center" rowspan=8>3</td><td align="center" rowspan=1>{V2}V</td><td align="center">Natural optical activity γijk</td><td align="center">Δβij=i γijkkk</td><td align="center"><input type="radio" name="choose_tensor" value="naturaloptic" onChange="active_legend(76);"></td></tr> <tr><td align="center" rowspan=2>[V2]V</td><td align="center">Pockels (electrooptic) effect zijk</td><td align="center">Δβij=zijkEk</td><td align="center"><input type="radio" name="choose_tensor" value="pockels" onChange="active_legend(77);"></td></tr> <tr><td align="center">Thermoelectro-optical effect tensor r(T)ijk</td><td align="center">Δβij=r(T)ijkEkΔT</td><td align="center"><input type="radio" name="choose_tensor" value="thermoelectrooptical" onChange="active_legend(78);"></td></tr> <tr><td align="center" rowspan=2>e{V2}V</td><td align="center">Magneto-optical tensor (Faraday effect) Fijk</td><td align="center">Δβij=FijkHk</td><td align="center"><input type="radio" name="choose_tensor" value="faraday" onChange="active_legend(79);"></td></tr> <tr><td align="center">Thermomagneto-optical effect tensor f(T)ijk</td><td align="center">Δβij=f(T)ijkHkΔT</td><td align="center"><input type="radio" name="choose_tensor" value="thermomagnetooptical" onChange="active_legend(80);"></td></tr> <tr><td align="center" rowspan=1>e[V2]V</td><td align="center">Electrogyration effect tensor γijk</td><td align="center">Δgij=γijkEk</td><td align="center"><input type="radio" name="choose_tensor" value="electrogyrational" onChange="active_legend(81);"></td></tr> <tr><td align="center" rowspan=1>a[V2]V</td><td align="center">Spontaneous Gyrotropic Birefringence γijk</td><td align="center">Δβij=i γijkkk</td><td align="center"><input type="radio" name="choose_tensor" value="SpontGyrotropicBirefringence" onChange="active_legend(82);"></td></tr> <tr><td align="center" rowspan=1>ae[V2]V</td><td align="center">Magneto-optic Kerr effect qijk</td><td align="center">Δβij=qijkHk</td><td align="center"><input type="radio" name="choose_tensor" value="kerr_mag" onChange="active_legend(83);"></td></tr> <tr><td align="center" rowspan=11>4</td><td align="center" rowspan=6>[V2][V2]</td><td align="center">Birefringence in Cubic Crystals γijkl</td><td align="center">Δβij= γijlmklkm</td><td align="center"><input type="radio" name="choose_tensor" value="BirefringenceCubic" onChange="active_legend(84);"></td></tr> <tr><td align="center">Elasto-optical tensor pijkl</td><td align="center">Δβij=pijklεkl</td><td align="center"><input type="radio" name="choose_tensor" value="elastooptical" onChange="active_legend(85);"></td></tr> <tr><td align="center">Kerr effect tensor Rijkl</td><td align="center">Δβij=RijklEkEl</td><td align="center"><input type="radio" name="choose_tensor" value="kerr" onChange="active_legend(86);"></td></tr> <tr><td align="center">Piezo-optical tensor πijkl</td><td align="center">Δβij=πijklσkl</td><td align="center"><input type="radio" name="choose_tensor" value="piezooptical" onChange="active_legend(87);"></td></tr> <tr><td align="center">Second-order magneto-optical (Cotton-Mouton effect) tensor Cijkl</td><td align="center">Δβij=CijklHkHl</td><td align="center"><input type="radio" name="choose_tensor" value="cotton_mouton" onChange="active_legend(88);"></td></tr> <tr><td align="center">Thermopiezo-optical effect tensor π(T)ijkl</td><td align="center">Δβij=π(T)ijklσklΔT</td><td align="center"><input type="radio" name="choose_tensor" value="thermopiezooptical" onChange="active_legend(89);"></td></tr> <tr><td align="center" rowspan=1>e{V2}V2</td><td align="center">Magnetoelectro-optical effect tensor mijkl</td><td align="center">Δβij=mijklHkEl</td><td align="center"><input type="radio" name="choose_tensor" value="magnetoelectrooptical" onChange="active_legend(90);"></td></tr> <tr><td align="center" rowspan=2>e[V2][V2]</td><td align="center">Piezogyration tensor Cijkl</td><td align="center">Δgij=Cijklσkl</td><td align="center"><input type="radio" name="choose_tensor" value="piezogyration" onChange="active_legend(91);"></td></tr> <tr><td align="center">Quadratic electrogyration effect tensor βijkl</td><td align="center">Δgij=βijklEkEl</td><td align="center"><input type="radio" name="choose_tensor" value="electrogyrational_quad" onChange="active_legend(92);"></td></tr> <tr><td align="center" rowspan=2>a{V2}[V2]</td><td align="center">Nonmutual Optical Activity γijkl</td><td align="center">Δβij= γijlmklkm</td><td align="center"><input type="radio" name="choose_tensor" value="NonmutualOA" onChange="active_legend(93);"></td></tr> <tr><td align="center">Quadratic magneto-optic Kerr effect Bijkl</td><td align="center">Δβij=BijklHkHl</td><td align="center"><input type="radio" name="choose_tensor" value="kerr_mag_quad" onChange="active_legend(94);"></td></tr> <tr><td align="center" rowspan=3>5</td><td align="center" rowspan=1>[V2][V2]V</td><td align="center">Piezoelectro-optical effect tensor zijklm</td><td align="center">Δβij=zijklmσklEm</td><td align="center"><input type="radio" name="choose_tensor" value="piezoelectrooptical" onChange="active_legend(95);"></td></tr> <tr><td align="center" rowspan=1>e{V2}V[V2]</td><td align="center">Piezomagneto-optical effect tensor ωijklm</td><td align="center">Δβij=ωijklmHkσlm</td><td align="center"><input type="radio" name="choose_tensor" value="piezomagnetooptical" onChange="active_legend(96);"></td></tr> <tr><td align="center" rowspan=1>e[V2][V2]V</td><td align="center">Gradient piezogyration tensor βijklm</td><td align="center">Δgij=βijklm<vec>∇</vec>mσkl</td><td align="center"><input type="radio" name="choose_tensor" value="gradient_piezogyration" onChange="active_legend(97);"></td></tr> <tr><td align="center" rowspan=2>6</td><td align="center" rowspan=1>[V2][[V2][V2]]</td><td align="center">Second-order piezo-optical tensor Πijklmn</td><td align="center">Δβij=Πijklmnσklσmn</td><td align="center"><input type="radio" name="choose_tensor" value="piezooptical_quad" onChange="active_legend(98);"></td></tr> <tr><td align="center" rowspan=1>e[V2][[V2][V2]]</td><td align="center">Quadratic piezogyration tensor Cijklmn</td><td align="center">Δgij=Cijklmnσklσmn</td><td align="center"><input type="radio" name="choose_tensor" value="piezogyration_quad" onChange="active_legend(99);"></td></tr> </table></div> </center><center><a href="javascript:;" onClick="javascript:showHideElem('nonopt_block');">NONLINEAR OPTICAL SUSCEPTIBILITY TENSORS</a> <div id="nonopt_block" style="display:none"> <table border="5" width="100%" align="center"><tr><th>Rank</th><th><a class="blue" href="/html/cryst/mtensor_help.html#intrinsic">Intrinsic symmetry</a></th><th>Tensor description</th><th>Defining equation</th><th>Select</th></tr><tr><td align="center" rowspan=18>3</td><td align="center" rowspan=3>[V3]</td><td align="center">General second-order susceptibility (non dissipative media and no dispersion) χ(ω3;ω2,ω1)ijk</td><td align="center">Pi(ω3)=χijk(ω3;ω2,ω1)Ej(ω2)Ek(ω1)</td><td align="center"><input type="radio" name="choose_tensor" value="general_susceptibility_second_Kleinman" onChange="active_legend(100);"></td></tr> <tr><td align="center">Optical rectification (non-dissipative media and no dispersion) χ(0;ω,-ω)ijk</td><td align="center">Pi(0)=χijk(0;ω,-ω)Ej(ω)Ek(-ω)</td><td align="center"><input type="radio" name="choose_tensor" value="OptRectification_nondissp_Kleinman" onChange="active_legend(101);"></td></tr> <tr><td align="center">Second-harmonic generation (non-dissipative media and no dispersion) χ(2ω;ω,ω)ijk</td><td align="center">Pi(2ω)=χijk(2ω;ω,ω)Ej(ω)Ek(ω)</td><td align="center"><input type="radio" name="choose_tensor" value="SHG_nondissip_Kleinman" onChange="active_legend(102);"></td></tr> <tr><td align="center" rowspan=1>V[V2]*</td><td align="center">General optical rectification (dissipative media) χ(0;ω,-ω)ijk</td><td align="center">Pi(0)=χijk(0;ω,-ω)Ej(ω)Ek(-ω)</td><td align="center"><input type="radio" name="choose_tensor" value="general_OptRectification" onChange="active_legend(103);"></td></tr> <tr><td align="center" rowspan=2>V[V2]</td><td align="center">Optical rectification (non-dissipative media)Real part χ(0;ω,-ω)ijk</td><td align="center">Pi(0)=χijk(0;ω,-ω)Ej(ω)Ek(-ω)</td><td align="center"><input type="radio" name="choose_tensor" value="OptRectification_nondissp_real" onChange="active_legend(104);"></td></tr> <tr><td align="center">Second-harmonic generation (non-dissipative media). Real part. χ(2ω;ω,ω)ijk</td><td align="center">Pi(2ω)=χijk(2ω;ω,ω)Ej(ω)Ek(ω)</td><td align="center"><input type="radio" name="choose_tensor" value="SHG_nondissip_real" onChange="active_legend(105);"></td></tr> <tr><td align="center" rowspan=1>V3</td><td align="center">General second-order susceptibility (non-dissipative media). Real part. χ(ω3;ω2,ω1)ijk</td><td align="center">Pi(ω3)=χijk(ω3;ω2,ω1)Ej(ω2)Ek(ω1)</td><td align="center"><input type="radio" name="choose_tensor" value="susceptibility_second_real" onChange="active_legend(106);"></td></tr> <tr><td align="center" rowspan=2>e[V3]</td><td align="center">General second-order susceptibility by magnetic dipole (non-dissipative media and no dispersion) χm(ω3;ω2,ω1)ijk</td><td align="center">Mi(ω3)=χmijk(ω3;ω2,ω1)Ej(ω2)Ek(ω1)</td><td align="center"><input type="radio" name="choose_tensor" value="susceptibility_second_magnetic_Kleinman" onChange="active_legend(107);"></td></tr> <tr><td align="center">Second-harmonic generation by magnetic dipole (non-dissipative media and no dispersion) χm(2ω;ω,ω)ijk</td><td align="center">Mi(2ω)=χmijk(2ω;ω,ω)Ej(ω)Ek(ω)</td><td align="center"><input type="radio" name="choose_tensor" value="SHG_nondissip_magnetic_Kleinman" onChange="active_legend(108);"></td></tr> <tr><td align="center" rowspan=1>eV[V2]</td><td align="center">Second-harmonic generation by magnetic dipole (non-dissipative media). Imaginary part. χm(2ω;ω,ω)ijk</td><td align="center">Mi(2ω)=χmijk(2ω;ω,ω)Ej(ω)Ek(ω)</td><td align="center"><input type="radio" name="choose_tensor" value="SHG_nondissip_magnetic_imag" onChange="active_legend(109);"></td></tr> <tr><td align="center" rowspan=1>eV3</td><td align="center">General second-order susceptibility by magnetic dipole (non-dissipative media) Imaginary part χm(ω3;ω2,ω1)ijk</td><td align="center">Mi(ω3)=χmijk(ω3;ω2,ω1)Ej(ω2)Ek(ω1)</td><td align="center"><input type="radio" name="choose_tensor" value="susceptibility_second_magnetic_imag" onChange="active_legend(110);"></td></tr> <tr><td align="center" rowspan=1>aV{V2}</td><td align="center">Optical rectification (non-dissipative media)Imaginary part χ(0;ω,-ω)ijk</td><td align="center">Pi(0)=χijk(0;ω,-ω)Ej(ω)Ek(-ω)</td><td align="center"><input type="radio" name="choose_tensor" value="OptRectification_nondissp_imag" onChange="active_legend(111);"></td></tr> <tr><td align="center" rowspan=1>aV[V2]</td><td align="center">Second-harmonic generation (non-dissipative media). Imaginary part. χ(2ω;ω,ω)ijk</td><td align="center">Pi(2ω)=χijk(2ω;ω,ω)Ej(ω)Ek(ω)</td><td align="center"><input type="radio" name="choose_tensor" value="SHG_nondissip_imag" onChange="active_legend(112);"></td></tr> <tr><td align="center" rowspan=1>aV3</td><td align="center">General second-order susceptibility (non-dissipative media). Imaginary part. χ(ω3;ω2,ω1)ijk</td><td align="center">Pi(ω3)=χijk(ω3;ω2,ω1)Ej(ω2)Ek(ω1)</td><td align="center"><input type="radio" name="choose_tensor" value="susceptibility_second_imag" onChange="active_legend(113);"></td></tr> <tr><td align="center" rowspan=1>aeV[V2]</td><td align="center">Second-harmonic generation by magnetic dipole (non-dissipative media). Real part. χm(2ω;ω,ω)ijk</td><td align="center">Mi(2ω)=χmijk(2ω;ω,ω)Ej(ω)Ek(ω)</td><td align="center"><input type="radio" name="choose_tensor" value="SHG_nondissip_magnetic_real" onChange="active_legend(114);"></td></tr> <tr><td align="center" rowspan=1>aeV3</td><td align="center">General second-order susceptibility by magnetic dipole (non-dissipative media) Real part χm(ω3;ω2,ω1)ijk</td><td align="center">Mi(ω3)=χmijk(ω3;ω2,ω1)Ej(ω2)Ek(ω1)</td><td align="center"><input type="radio" name="choose_tensor" value="susceptibility_second_magnetic_real" onChange="active_legend(115);"></td></tr> <tr><td align="center" rowspan=1>(V[V2])*</td><td align="center">General second-harmonic generation (dissipative media) χ(2ω;ω,ω)ijk</td><td align="center">Pi(2ω)=χijk(2ω;ω,ω)Ej(ω)Ek(ω)</td><td align="center"><input type="radio" name="choose_tensor" value="general_SHG" onChange="active_legend(116);"></td></tr> <tr><td align="center" rowspan=1>(V3)*</td><td align="center">General second-order susceptibility (dissipative media) χ(ω3;ω2,ω1)ijk</td><td align="center">Pi(ω3)=χijk(ω3;ω2,ω1)Ej(ω2)Ek(ω1)</td><td align="center"><input type="radio" name="choose_tensor" value="general_susceptibility_second" onChange="active_legend(117);"></td></tr> <tr><td align="center" rowspan=13>4</td><td align="center" rowspan=1>[[V2][V2]]*</td><td align="center">Degenerate four-wave mixing χ(ω;-ω,ω,ω)ijkl</td><td align="center">Pi(ω)=χijkl(ω;-ω,ω,ω)Ej(-ω)Ek(ω)El(ω)</td><td align="center"><input type="radio" name="choose_tensor" value="DFWM" onChange="active_legend(118);"></td></tr> <tr><td align="center" rowspan=3>[V4]</td><td align="center">Electric-field induced second-harmonic generation (non-dissipative media and no dispersion) χ(2ω;0,ω,ω)ijkl</td><td align="center">Pi(2ω)=χijkl(2ω;0,ω,ω)Ej(0)Ek(ω)El(ω)</td><td align="center"><input type="radio" name="choose_tensor" value="EFISH_nondissip_Kleinman" onChange="active_legend(119);"></td></tr> <tr><td align="center">General third-order susceptibility (non-dissipative media and no dispersion) χ(ω4;ω3,ω2,ω1)ijkl</td><td align="center">Pi(ω4)=χijkl(ω4;ω3,ω2,ω1)Ej(ω3)Ek(ω2)El(ω1)</td><td align="center"><input type="radio" name="choose_tensor" value="susceptibility_third_Kleinman" onChange="active_legend(120);"></td></tr> <tr><td align="center">Third-harmonic generation (non-dissipative media and no dispersion) χ(3ω;ω,ω,ω)ijkl</td><td align="center">Pi(3ω)=χijkl(3ω;ω,ω,ω)Ej(ω)Ek(ω)El(ω)</td><td align="center"><input type="radio" name="choose_tensor" value="THG_nondissip_Kleinman" onChange="active_legend(121);"></td></tr> <tr><td align="center" rowspan=1>[V2V2]*</td><td align="center">Four-wave mixing χ(ω1;-ω2,ω1,ω2)ijkl</td><td align="center">Pi(ω1)=χijkl(ω1;-ω2,ω1,ω2)Ej(-ω2)Ek(ω1)El(ω2)</td><td align="center"><input type="radio" name="choose_tensor" value="FWM" onChange="active_legend(122);"></td></tr> <tr><td align="center" rowspan=1>V[V3]</td><td align="center">Third-harmonic generation (non-dissipative media). Real part. χ(3ω;ω,ω,ω)ijkl</td><td align="center">Pi(3ω)=χijkl(3ω;ω,ω,ω)Ej(ω)Ek(ω)El(ω)</td><td align="center"><input type="radio" name="choose_tensor" value="THG_nondissip_real" onChange="active_legend(123);"></td></tr> <tr><td align="center" rowspan=1>V4</td><td align="center">General third-order susceptibility (non-dissipative media). Real part. χ(ω4;ω3,ω2,ω1)ijkl</td><td align="center">Pi(ω4)=χijkl(ω4;ω3,ω2,ω1)Ej(ω3)Ek(ω2)El(ω1)</td><td align="center"><input type="radio" name="choose_tensor" value="susceptibility_third_real" onChange="active_legend(124);"></td></tr> <tr><td align="center" rowspan=1>V2[V2]</td><td align="center">Electric-field induced second-harmonic generation (non-dissipative media). Real part. χ(2ω;0,ω,ω)ijkl</td><td align="center">Pi(2ω)=χijkl(2ω;0,ω,ω)Ej(0)Ek(ω)El(ω)</td><td align="center"><input type="radio" name="choose_tensor" value="EFISH_nondissip_real" onChange="active_legend(125);"></td></tr> <tr><td align="center" rowspan=1>aV[V3]</td><td align="center">Third-harmonic generation (non-dissipative media). Imaginary part. χ(3ω;ω,ω,ω)ijkl</td><td align="center">Pi(3ω)=χijkl(3ω;ω,ω,ω)Ej(ω)Ek(ω)El(ω)</td><td align="center"><input type="radio" name="choose_tensor" value="THG_nondissip_imag" onChange="active_legend(126);"></td></tr> <tr><td align="center" rowspan=1>aV4</td><td align="center">General third-order susceptibility (non-dissipative media). Imaginary part. χ(ω4;ω3,ω2,ω1)ijkl</td><td align="center">Pi(ω4)=χijkl(ω4;ω3,ω2,ω1)Ej(ω3)Ek(ω2)El(ω1)</td><td align="center"><input type="radio" name="choose_tensor" value="susceptibility_third_imag" onChange="active_legend(127);"></td></tr> <tr><td align="center" rowspan=1>aV2[V2] </td><td align="center">Electric-field induced second-harmonic generation (non-dissipative media). Imaginary part. χ(2ω0,ω,ω)ijkl</td><td align="center">Pi(2ω)=χijkl(2ω;0,ω,ω)Ej(0)Ek(ω)El(ω)</td><td align="center"><input type="radio" name="choose_tensor" value="EFISH_nondissip_imag" onChange="active_legend(128);"></td></tr> <tr><td align="center" rowspan=1>(V[V3])*</td><td align="center">General third-harmonic generation (dissipative media) χ(3ω;ω,ω,ω)ijkl</td><td align="center">Pi(3ω)=χijkl(3ω;ω,ω,ω)Ej(ω)Ek(ω)El(ω)</td><td align="center"><input type="radio" name="choose_tensor" value="general_THG" onChange="active_legend(129);"></td></tr> <tr><td align="center" rowspan=1>(V4)*</td><td align="center">General third-order susceptibility (dissipative media) χ(ω4;ω3,ω2,ω1)ijkl</td><td align="center">Pi(ω4)=χijkl(ω4;ω3,ω2,ω1)Ej(ω3)Ek(ω2)El(ω1)</td><td align="center"><input type="radio" name="choose_tensor" value="general_susceptibility_third" onChange="active_legend(130);"></td></tr> </table></div> </center><center><a href="javascript:;" onClick="javascript:showHideElem('tr_block');">TRANSPORT TENSORS</a> <div id=tr_block style="display:none"> <table border="5" width="100%" align="center"><tr><th>Rank</th><th><a class="blue" href="/html/cryst/mtensor_help.html#intrinsic">Intrinsic symmetry</a></th><th>Tensor description</th><th>Defining equation</th><th>Select</th></tr><tr><td align="center" rowspan=21>2</td><td align="center" rowspan=10>[V2]*</td><td align="center">Diffusion tensor Dij</td><td align="center">Ji=Dij<vec>∇</vec>jC</td><td align="center"><input type="radio" name="choose_tensor" value="diffusion" onChange="active_legend(131);"></td></tr> <tr><td align="center">Dufour effect (reversal thermodiffusion) tensor βij</td><td align="center">qi=βij<vec>∇</vec>jC</td><td align="center"><input type="radio" name="choose_tensor" value="dufour" onChange="active_legend(132);"></td></tr> <tr><td align="center">Electric conductivity tensor σij</td><td align="center">Ji=σijEj</td><td align="center"><input type="radio" name="choose_tensor" value="conductivity_mag" onChange="active_legend(133);"></td></tr> <tr><td align="center">Electric resistivity tensor ρij</td><td align="center">Ei=ρijJj</td><td align="center"><input type="radio" name="choose_tensor" value="resistivity_mag" onChange="active_legend(134);"></td></tr> <tr><td align="center">Electrodiffusion tensor γij (direct effect)</td><td align="center">Ji=γijEjT</td><td align="center"><input type="radio" name="choose_tensor" value="electrodiffusion" onChange="active_legend(135);"></td></tr> <tr><td align="center">Electrodiffusion tensor γTij (inverse effect)</td><td align="center">Ji=γTij<vec>∇</vec>jC</td><td align="center"><input type="radio" name="choose_tensor" value="electrodiffusion_inv" onChange="active_legend(136);"></td></tr> <tr><td align="center">Soret effect (thermodiffusion) tensor βij</td><td align="center">Ji=βij<vec>∇</vec>jT</td><td align="center"><input type="radio" name="choose_tensor" value="soret" onChange="active_legend(137);"></td></tr> <tr><td align="center">Thermal conductivity tensor κij</td><td align="center">qi=κij<vec>∇</vec>jT</td><td align="center"><input type="radio" name="choose_tensor" value="t_conductivity_mag" onChange="active_legend(138);"></td></tr> <tr><td align="center">Thermal diffusivity tensor αij</td><td align="center">∂T/∂t=αij<vec>∇</vec>iT<vec>∇</vec>jT</td><td align="center"><input type="radio" name="choose_tensor" value="t_diffusivity" onChange="active_legend(139);"></td></tr> <tr><td align="center">Thermal resistivity tensor rij</td><td align="center"><vec>∇</vec>iT=rijqj</td><td align="center"><input type="radio" name="choose_tensor" value="t_resistivity_mag" onChange="active_legend(140);"></td></tr> <tr><td align="center" rowspan=2>[V2]</td><td align="center">Ordinary Resistivity ρ(s)ij</td><td align="center">Ei=ρijJj</td><td align="center"><input type="radio" name="choose_tensor" value="resistivity" onChange="active_legend(141);"></td></tr> <tr><td align="center">Ordinary Thermal Conductivity κ(s)ij</td><td align="center">qi=κij∇jT</td><td align="center"><input type="radio" name="choose_tensor" value="thermconduct" onChange="active_legend(142);"></td></tr> <tr><td align="center" rowspan=2>V2</td><td align="center">Ordinary Peltier Effect Pij</td><td align="center">qi=πijJj, Ei=βij∇jT, Pij=(πij+βji)/2</td><td align="center"><input type="radio" name="choose_tensor" value="ordinpeltier" onChange="active_legend(143);"></td></tr> <tr><td align="center">Ordinary Seebeck Effect Sij</td><td align="center">Ei=βij∇jT, qi=πijJj, Sij=(βij+πji)/2</td><td align="center"><input type="radio" name="choose_tensor" value="ordinseebeck" onChange="active_legend(144);"></td></tr> <tr><td align="center" rowspan=2>a{V2}</td><td align="center">Spontaneous Hall Effect ρ(a)ij</td><td align="center">Ei=ρijJj</td><td align="center"><input type="radio" name="choose_tensor" value="she" onChange="active_legend(145);"></td></tr> <tr><td align="center">Spontaneous Righi-Leduc Effect κ(a)ij</td><td align="center">qi=κij∇jT</td><td align="center"><input type="radio" name="choose_tensor" value="srle" onChange="active_legend(146);"></td></tr> <tr><td align="center" rowspan=2>aV2</td><td align="center">Spontaneous Ettingshausen Effect SEij</td><td align="center">qi=πijJj, Ei=βij∇jT, SEij=(πij-βji)/2</td><td align="center"><input type="radio" name="choose_tensor" value="sponettingshausen" onChange="active_legend(147);"></td></tr> <tr><td align="center">Spontaneous Nernst Effect SNij</td><td align="center">Ei=βij∇jT, qi=πijJj, SNij=(βij-πji)/2</td><td align="center"><input type="radio" name="choose_tensor" value="sponnernst" onChange="active_legend(148);"></td></tr> <tr><td align="center" rowspan=3>(V2)*</td><td align="center">Peltier effect tensor πij</td><td align="center">qi=πijJj</td><td align="center"><input type="radio" name="choose_tensor" value="peltier" onChange="active_legend(149);"></td></tr> <tr><td align="center">Thermoelectric power (Seebeck effect) tensor βij</td><td align="center">Ei=βij<vec>∇</vec>jT</td><td align="center"><input type="radio" name="choose_tensor" value="seebeck" onChange="active_legend(150);"></td></tr> <tr><td align="center">Thomson heat tensor τij</td><td align="center">∂q/∂t=τij<vec>∇</vec>iTJj</td><td align="center"><input type="radio" name="choose_tensor" value="thomson" onChange="active_legend(151);"></td></tr> <tr><td align="center" rowspan=12>3</td><td align="center" rowspan=2>e{V2}V</td><td align="center">Ordinary Hall Effect R(a)ijk</td><td align="center">Ei=RijkJjHk</td><td align="center"><input type="radio" name="choose_tensor" value="ordinaryhall" onChange="active_legend(152);"></td></tr> <tr><td align="center">Ordinary Righi-Leduc Effect Q(a)ijk</td><td align="center">qi=Qijk∇jTHk</td><td align="center"><input type="radio" name="choose_tensor" value="ordinRL" onChange="active_legend(153);"></td></tr> <tr><td align="center" rowspan=2>e{V2}*V</td><td align="center">Hall effect (magnetorresistance) tensor Rijk</td><td align="center">Ei=RijkJjHk</td><td align="center"><input type="radio" name="choose_tensor" value="hall" onChange="active_legend(154);"></td></tr> <tr><td align="center">Righi-Leduc, Maggi-Righi-Leduc and magnetothermal effects tensor Qijk</td><td align="center">qi=Qijk<vec>∇</vec>jTHk</td><td align="center"><input type="radio" name="choose_tensor" value="righi_leduc" onChange="active_legend(155);"></td></tr> <tr><td align="center" rowspan=2>eV3</td><td align="center">Ordinary Ettingshausen Effect OEij</td><td align="center">qi=MijkJjHk, Ei=Nijk∇jTHk, OEijk=(Mijk-Njik)/2</td><td align="center"><input type="radio" name="choose_tensor" value="ordinettingshausen" onChange="active_legend(156);"></td></tr> <tr><td align="center">Ordinary Nernst Effect ONijk</td><td align="center">Ei=Nijk∇jTHk, qi=MijkJjHk, ONijk=(Nijk-Mjik)/2</td><td align="center"><input type="radio" name="choose_tensor" value="ordnern" onChange="active_legend(157);"></td></tr> <tr><td align="center" rowspan=2>ae[V2]V</td><td align="center">Linear Magneto-heat Conductivity Effect Q(s)ijk</td><td align="center">qi=Qijk∇jTHk</td><td align="center"><input type="radio" name="choose_tensor" value="linmagheat" onChange="active_legend(158);"></td></tr> <tr><td align="center">Linear Magnetorresistance R(s)ijk</td><td align="center">Ei=RijkJjHk</td><td align="center"><input type="radio" name="choose_tensor" value="linmagnetorresist" onChange="active_legend(159);"></td></tr> <tr><td align="center" rowspan=2>aeV3</td><td align="center">Linear Magneto Peltier Effect MPijk</td><td align="center">qi=MijkJjHk, Ei=Nijk∇jTHk, MPijk=(Mijk+Njik)/2</td><td align="center"><input type="radio" name="choose_tensor" value="linmagpel" onChange="active_legend(160);"></td></tr> <tr><td align="center">Linear Magneto Seebeck Effect MSijk</td><td align="center">Ei=Nijk∇jTHk, qi=MijkJjHk, MSijk=(Nijk+Mjik)/2</td><td align="center"><input type="radio" name="choose_tensor" value="linmagseebeck" onChange="active_legend(161);"></td></tr> <tr><td align="center" rowspan=2>(eV3)*</td><td align="center">Ettingshausen effect tensor Mijk</td><td align="center">qi=MijkJjHk</td><td align="center"><input type="radio" name="choose_tensor" value="ettinghausen" onChange="active_legend(162);"></td></tr> <tr><td align="center">Nernst effect tensor Nijk</td><td align="center">Ei=Nijk<vec>∇</vec>TjHk</td><td align="center"><input type="radio" name="choose_tensor" value="nernst" onChange="active_legend(163);"></td></tr> <tr><td align="center" rowspan=7>4</td><td align="center" rowspan=1>[V2][V2]</td><td align="center">Quadratic Magneto-heat-conductivity S(s)ijkl</td><td align="center">qi=Sijkl∇jTHkHl</td><td align="center"><input type="radio" name="choose_tensor" value="quadmagheat" onChange="active_legend(164);"></td></tr> <tr><td align="center" rowspan=3>[V2]*[V2]</td><td align="center">Magnetic resistance tensor Tijkl</td><td align="center">Ei=TijklJjHkHl</td><td align="center"><input type="radio" name="choose_tensor" value="magnetic_resistance" onChange="active_legend(165);"></td></tr> <tr><td align="center">Magneto-heat-conductivity tensor Sijkl Sijkl</td><td align="center">qi=Sijkl∇jTHkHl</td><td align="center"><input type="radio" name="choose_tensor" value="magnetoheatconductivity" onChange="active_legend(166);"></td></tr> <tr><td align="center">Piezoresistivity (Strain Gauge effect) tensor πijkl</td><td align="center">Δρij=πijklσkl</td><td align="center"><input type="radio" name="choose_tensor" value="piezoresistivity" onChange="active_legend(167);"></td></tr> <tr><td align="center" rowspan=1>a{V2}[V2]</td><td align="center">Quadratic Anomalous Righi-Leduc Effect S(a)ijkl</td><td align="center">qi=Sijkl∇jTHkHl</td><td align="center"><input type="radio" name="choose_tensor" value="quadanomRL" onChange="active_legend(168);"></td></tr> <tr><td align="center" rowspan=2>(V2[V2])*</td><td align="center">Magneto Peltier effect Pijkl</td><td align="center">qi=PijklHkHlJj</td><td align="center"><input type="radio" name="choose_tensor" value="MagnetoPeltier" onChange="active_legend(169);"></td></tr> <tr><td align="center">Magneto Seebeck effect αijkl</td><td align="center">Ei=αijkl∇jTHkHl</td><td align="center"><input type="radio" name="choose_tensor" value="MagnetoSeebeck" onChange="active_legend(170);"></td></tr> </table></center> <ul><input type="radio" id="tensor_type" name="tensor_type" value="by_hand"> Build your own tensor</ul><ul>- Introduce Jahn's symbol without superscripts. Examples: (1) [[V2][V2]], (2) a{V2}*, (3) (V2[V2])* </center> <center><input type="text" name="jahnsymbol" value=""></ul><left><center> <input type='hidden' name="database_type" value = "mtensor"> <center> <input type="submit" name="enter" value="Get results"> </center> </form> </td> </tr> </table> <center> [<a class="blue" href="/">Bilbao Crystallographic Server Main Menu</a>] <table class="signature" width="100%"> <tr> <td align="left"> <table> <tr> <td><a href="http://www.cryst.ehu.es">Bilbao Crystallographic Server</a> http://www.cryst.ehu.es</td> <td> <a rel="license" href="http://creativecommons.org/licenses/by-nc-sa/4.0/"><img alt="Licencia de Creative Commons" style="border-width:0" src="https://i.creativecommons.org/l/by-nc-sa/4.0/88x31.png" /></a> </td> </tr> </table> </td> <td align="right" rowspan="2"> For comments, please mail to <a class="blue" href= "mailto:administrador.bcs@ehu.eus">administrador.bcs@ehu.eus</a> </td> </tr> </table> </center> </body> </html>

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