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searchaux" style="display:none">Solid material whose physical properties are independent of orientation</div> <p>In <a href="/wiki/Condensed_matter_physics" title="Condensed matter physics">condensed matter physics</a> and <a href="/wiki/Continuum_mechanics" title="Continuum mechanics">continuum mechanics</a>, an <b>isotropic solid</b> refers to a <a href="/wiki/Solid" title="Solid">solid</a> material for which <a href="/wiki/Physical_property" title="Physical property">physical properties</a> are independent of the orientation of the system. While the finite sizes of <a href="/wiki/Atom" title="Atom">atoms</a> and <a href="/wiki/Chemical_bond" title="Chemical bond">bonding</a> considerations ensure that true <a href="/wiki/Isotropy" title="Isotropy">isotropy</a> of atomic position will not exist in the solid state, it is possible for measurements of a given property to yield isotropic results, either due to the <a href="/wiki/Symmetry_(physics)" title="Symmetry (physics)">symmetries</a> present within a <a href="/wiki/Crystal_system" title="Crystal system">crystal system</a>, or due to the effects of orientational averaging over a sample (e.g. in an <a href="/wiki/Amorphous_solid" title="Amorphous solid">amorphous solid</a> or a <a href="/wiki/Polycrystalline" class="mw-redirect" title="Polycrystalline">polycrystalline</a> metal). Isotropic solids tend to be of interest when developing models for physical behavior of materials, as they tend to allow for dramatic simplifications of theory; for example, <a href="/wiki/Electrical_resistivity_and_conductivity" title="Electrical resistivity and conductivity">conductivity</a> in metals of the <a href="/wiki/Cubic_crystal_system" title="Cubic crystal system">cubic crystal system</a> can be described with single <a href="/wiki/Scalar_(physics)" title="Scalar (physics)">scalar</a> value, rather than a <a href="/wiki/Tensor" title="Tensor">tensor</a>.<sup id="cite_ref-AshcroftMermin1976_1-0" class="reference"><a href="#cite_note-AshcroftMermin1976-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> Additionally, cubic crystals are isotropic with respect to <a href="/wiki/Thermal_expansion" title="Thermal expansion">thermal expansion</a><sup id="cite_ref-NewnhamCh9_2-0" class="reference"><a href="#cite_note-NewnhamCh9-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> and will expand equally in all directions when heated.<sup id="cite_ref-Vail2003_3-0" class="reference"><a href="#cite_note-Vail2003-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p><p>Isotropy should not be confused with <a href="/wiki/Homogeneity_(physics)" title="Homogeneity (physics)">homogeneity</a>, which characterizes a system’s properties as being independent of position, rather than orientation. Additionally, all crystal structures, including the cubic crystal system, are anisotropic with respect to certain properties, and isotropic to others (such as <a href="/wiki/Density" title="Density">density</a>). <sup id="cite_ref-Nye1985_4-0" class="reference"><a href="#cite_note-Nye1985-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> The anisotropy of a crystal’s properties depends on the rank of the tensor used to describe the property, as well as the symmetries present within the crystal. The rotational symmetries within cubic crystals, for example, ensure that the <a href="/wiki/Dielectric_constant" class="mw-redirect" title="Dielectric constant">dielectric constant</a> (a 2nd rank tensor property) will be equal in all directions, whereas the symmetries in hexagonal systems dictate that the measurement will vary depending on whether the measurement is made within the <a href="/wiki/Basal_plane" class="mw-redirect" title="Basal plane">basal plane</a>.<sup id="cite_ref-NewnhamCh11_5-0" class="reference"><a href="#cite_note-NewnhamCh11-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> Due to the relationship between the dielectric constant and the optical index of refraction, it would be expected for cubic crystals to be optically isotropic, and hexagonal crystals to be optically anisotropic; Measurements of the optical properties of cubic and hexagonal CdSe confirm this understanding.<sup id="cite_ref-Ninomiya&Adachi1995_6-0" class="reference"><a href="#cite_note-Ninomiya&Adachi1995-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p><p>Nearly all single crystal systems are anisotropic with respect to mechanical properties, with <a href="/wiki/Tungsten" title="Tungsten">Tungsten</a> being a very notable exception, as it is a cubic metal with stiffness tensor coefficients that exist in the proper ratio to allow for mechanical isotropy. In general, however, cubic crystals are not mechanically isotropic. However, many materials, such as <a href="/wiki/Structural_steel" title="Structural steel">structural steel</a>, tend to be encountered and utilized in a polycrystalline state. Due to random orientation of the grains within the material, measured mechanical properties tend to be averages of the values associated with different crystallographic directions, with the net effect of apparent isotropy. As a result, it is typical for parameters such as the <a href="/wiki/Young%27s_Modulus" class="mw-redirect" title="Young's Modulus">Young's Modulus</a> to be reported independent of crystallographic direction.<sup id="cite_ref-Courtney2005_7-0" class="reference"><a href="#cite_note-Courtney2005-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> Treating solids as mechanically isotropic greatly simplifies analysis of deformation and fracture (as well as of the elastic fields produced by dislocations <sup id="cite_ref-Cai&Nix_8-0" class="reference"><a href="#cite_note-Cai&Nix-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup>). However, preferential orientation of grains (called texture) can occur as a result of certain types of deformation and recrystallization processes, which will create anisotropy in mechanical properties of the solid.<sup id="cite_ref-Courtney2005_7-1" class="reference"><a href="#cite_note-Courtney2005-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Isotropic_solid&action=edit&section=1" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-AshcroftMermin1976-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-AshcroftMermin1976_1-0">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFAshcroftMermin1976" class="citation book cs1">Ashcroft, Neil W.; Mermin, N.David (1976-01-02). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/solidstatephysic00ashc/page/250"><i>Solid State Physics</i></a></span>. Cengage Learning. pp. <a rel="nofollow" class="external text" href="https://archive.org/details/solidstatephysic00ashc/page/250">250</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780030839931" title="Special:BookSources/9780030839931"><bdi>9780030839931</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Solid+State+Physics&rft.pages=250&rft.pub=Cengage+Learning&rft.date=1976-01-02&rft.isbn=9780030839931&rft.aulast=Ashcroft&rft.aufirst=Neil+W.&rft.au=Mermin%2C+N.David&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fsolidstatephysic00ashc%2Fpage%2F250&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIsotropic+solid" class="Z3988"></span></span> </li> <li id="cite_note-NewnhamCh9-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-NewnhamCh9_2-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFNewnham2005" class="citation book cs1">Newnham, Robert. E. (2005-01-27). <i>Properties of Materials: Anisotropy, Symmetry, Structure</i>. Oxford University Press. pp. 60–64. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780198520764" title="Special:BookSources/9780198520764"><bdi>9780198520764</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Properties+of+Materials%3A+Anisotropy%2C+Symmetry%2C+Structure&rft.pages=60-64&rft.pub=Oxford+University+Press&rft.date=2005-01-27&rft.isbn=9780198520764&rft.aulast=Newnham&rft.aufirst=Robert.+E.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIsotropic+solid" class="Z3988"></span></span> </li> <li id="cite_note-Vail2003-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-Vail2003_3-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFVail2003" class="citation book cs1">Vail, J.M. (2003-04-24). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=WcKSjs1_CTMC&pg=PA37"><i>Topics in the Theory of Solid Materials</i></a>. CRC Press. pp. 34–47. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780750307291" title="Special:BookSources/9780750307291"><bdi>9780750307291</bdi></a><span class="reference-accessdate">. Retrieved <span class="nowrap">31 January</span> 2014</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Topics+in+the+Theory+of+Solid+Materials&rft.pages=34-47&rft.pub=CRC+Press&rft.date=2003-04-24&rft.isbn=9780750307291&rft.aulast=Vail&rft.aufirst=J.M.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DWcKSjs1_CTMC%26pg%3DPA37&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIsotropic+solid" class="Z3988"></span></span> </li> <li id="cite_note-Nye1985-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-Nye1985_4-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFNye1985" class="citation book cs1">Nye, J.F. (1985-07-11). <i>Physical Properties of Crystals: Their Representation by Tensors and Matrices</i>. Oxford University Press. pp. xv–xvi. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780198511656" title="Special:BookSources/9780198511656"><bdi>9780198511656</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Physical+Properties+of+Crystals%3A+Their+Representation+by+Tensors+and+Matrices&rft.pages=xv-xvi&rft.pub=Oxford+University+Press&rft.date=1985-07-11&rft.isbn=9780198511656&rft.aulast=Nye&rft.aufirst=J.F.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIsotropic+solid" class="Z3988"></span></span> </li> <li id="cite_note-NewnhamCh11-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-NewnhamCh11_5-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFNewnham2005" class="citation book cs1">Newnham, Robert. E. (2005-01-27). <i>Properties of Materials: Anisotropy, Symmetry, Structure</i>. Oxford University Press. pp. 79–85. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780198520764" title="Special:BookSources/9780198520764"><bdi>9780198520764</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Properties+of+Materials%3A+Anisotropy%2C+Symmetry%2C+Structure&rft.pages=79-85&rft.pub=Oxford+University+Press&rft.date=2005-01-27&rft.isbn=9780198520764&rft.aulast=Newnham&rft.aufirst=Robert.+E.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIsotropic+solid" class="Z3988"></span></span> </li> <li id="cite_note-Ninomiya&Adachi1995-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-Ninomiya&Adachi1995_6-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFNinomiyaAdachi1995" class="citation journal cs1">Ninomiya, Susumu; Adachi, Sadao (1995-06-19). "Optical properties of cubic and hexagonal CdSe". <i>Journal of Applied Physics</i>. <b>78</b> (7): 4681–4689. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1995JAP....78.4681N">1995JAP....78.4681N</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1063%2F1.359815">10.1063/1.359815</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Applied+Physics&rft.atitle=Optical+properties+of+cubic+and+hexagonal+CdSe&rft.volume=78&rft.issue=7&rft.pages=4681-4689&rft.date=1995-06-19&rft_id=info%3Adoi%2F10.1063%2F1.359815&rft_id=info%3Abibcode%2F1995JAP....78.4681N&rft.aulast=Ninomiya&rft.aufirst=Susumu&rft.au=Adachi%2C+Sadao&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIsotropic+solid" class="Z3988"></span></span> </li> <li id="cite_note-Courtney2005-7"><span class="mw-cite-backlink">^ <a href="#cite_ref-Courtney2005_7-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Courtney2005_7-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCourtney2005" class="citation book cs1">Courtney, Thomas H. (2005-12-01). <i>Mechanical Behavior of Materials</i>. Waveland Press Inc. pp. 47–61. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9781577664253" title="Special:BookSources/9781577664253"><bdi>9781577664253</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mechanical+Behavior+of+Materials&rft.pages=47-61&rft.pub=Waveland+Press+Inc.&rft.date=2005-12-01&rft.isbn=9781577664253&rft.aulast=Courtney&rft.aufirst=Thomas+H.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIsotropic+solid" class="Z3988"></span></span> </li> <li id="cite_note-Cai&Nix-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-Cai&Nix_8-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCaiNix2016" class="citation book cs1">Cai, Wei; Nix, William D. (2016-08-21). <i>Imperfections in Crystalline Solids</i>. Cambridge University Press. pp. 369–417. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9781107123137" title="Special:BookSources/9781107123137"><bdi>9781107123137</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Imperfections+in+Crystalline+Solids&rft.pages=369-417&rft.pub=Cambridge+University+Press&rft.date=2016-08-21&rft.isbn=9781107123137&rft.aulast=Cai&rft.aufirst=Wei&rft.au=Nix%2C+William+D.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIsotropic+solid" class="Z3988"></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Isotropic_solid&action=edit&section=2" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLiu2002" class="citation book cs1">Liu, I-Shih (2002-05-28). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=-gWqM4uMV6wC&pg=PA87"><i>Continuum Mechanics</i></a>. Springer. pp. 86–88. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9783540430193" title="Special:BookSources/9783540430193"><bdi>9783540430193</bdi></a><span class="reference-accessdate">. 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