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<div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Quenched disorder</span> </div> </a> <ul id="toc-Quenched_disorder-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Annealed_disorder" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Annealed_disorder"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Annealed disorder</span> </div> </a> <ul id="toc-Annealed_disorder-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Further_reading" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" 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Unsourced material may be challenged and removed.<br /><small><span class="plainlinks"><i>Find sources:</i> <a rel="nofollow" class="external text" href="https://www.google.com/search?as_eq=wikipedia&q=%22Order+and+disorder%22">"Order and disorder"</a> – <a rel="nofollow" class="external text" href="https://www.google.com/search?tbm=nws&q=%22Order+and+disorder%22+-wikipedia&tbs=ar:1">news</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.google.com/search?&q=%22Order+and+disorder%22&tbs=bkt:s&tbm=bks">newspapers</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.google.com/search?tbs=bks:1&q=%22Order+and+disorder%22+-wikipedia">books</a> <b>·</b> <a rel="nofollow" class="external text" href="https://scholar.google.com/scholar?q=%22Order+and+disorder%22">scholar</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.jstor.org/action/doBasicSearch?Query=%22Order+and+disorder%22&acc=on&wc=on">JSTOR</a></span></small></span> <span class="date-container"><i>(<span class="date">February 2024</span>)</i></span><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table> <p>In <a href="/wiki/Physics" title="Physics">physics</a>, the terms <b>order</b> and <b>disorder</b> designate the presence or absence of some <a href="/wiki/Symmetry" title="Symmetry">symmetry</a> or <a href="/wiki/Correlation" title="Correlation">correlation</a> in a many-particle system.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (February 2024)">citation needed</span></a></i>]</sup> </p><p>In <a href="/wiki/Condensed_matter_physics" title="Condensed matter physics">condensed matter physics</a>, systems typically are ordered at low <a href="/wiki/Temperature" title="Temperature">temperatures</a>; upon heating, they undergo one or several <a href="/wiki/Phase_transition" title="Phase transition">phase transitions</a> into less ordered states. Examples for such an <b>order-disorder transition</b> are: </p> <ul><li>the <a href="/wiki/Melting" title="Melting">melting</a> of <a href="/wiki/Ice" title="Ice">ice</a>: solid–liquid transition, loss of <a href="/wiki/Crystallinity" title="Crystallinity">crystalline order</a>;</li> <li>the <a href="/wiki/Demagnetization" class="mw-redirect" title="Demagnetization">demagnetization</a> of <a href="/wiki/Iron" title="Iron">iron</a> by heating above the <a href="/wiki/Curie_temperature" title="Curie temperature">Curie temperature</a>: <a href="/wiki/Ferromagnetic" class="mw-redirect" title="Ferromagnetic">ferromagnetic</a>–<a href="/wiki/Paramagnetic" class="mw-redirect" title="Paramagnetic">paramagnetic</a> transition, loss of magnetic order.</li></ul> <p>The degree of freedom that is ordered or disordered can be translational (<a href="/wiki/Crystal" title="Crystal">crystalline</a> ordering), rotational (<a href="/wiki/Ferroelectricity" title="Ferroelectricity">ferroelectric</a> ordering), or a spin state (<a href="/wiki/Magnetism" title="Magnetism">magnetic</a> ordering). </p><p>The order can consist either in a full crystalline <a href="/wiki/Space_group" title="Space group">space group</a> symmetry, or in a correlation. Depending on how the correlations decay with distance, one speaks of <a href="/wiki/Long_range_order" class="mw-redirect" title="Long range order">long range order</a> or <a href="/wiki/Short_range_order" title="Short range order">short range order</a>. </p><p>If a disordered state is not in <a href="/wiki/Thermodynamic_equilibrium" title="Thermodynamic equilibrium">thermodynamic equilibrium</a>, one speaks of <b>quenched disorder</b>. For instance, a <a href="/wiki/Glass" title="Glass">glass</a> is obtained by quenching (<a href="/wiki/Supercooling" title="Supercooling">supercooling</a>) a liquid. By extension, other quenched states are called <a href="/wiki/Spin_glass" title="Spin glass">spin glass</a>, <a href="/wiki/Orientational_glass" title="Orientational glass">orientational glass</a>. In some contexts, the opposite of quenched disorder is <b>annealed disorder</b>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Characterizing_order">Characterizing order</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Order_and_disorder&action=edit&section=1" title="Edit section: Characterizing order"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Lattice_periodicity_and_X-ray_crystallinity">Lattice periodicity and X-ray crystallinity</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Order_and_disorder&action=edit&section=2" title="Edit section: Lattice periodicity and X-ray crystallinity"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The strictest form of order in a solid is <b><a href="/wiki/Lattice_(order)" title="Lattice (order)">lattice</a> periodicity</b>: a certain pattern (the arrangement of atoms in a <a href="/wiki/Unit_cell" title="Unit cell">unit cell</a>) is repeated again and again to form a translationally invariant <a href="/wiki/Tessellation" title="Tessellation">tiling</a> of space. This is the defining property of a <a href="/wiki/Crystal" title="Crystal">crystal</a>. Possible symmetries have been classified in 14 <a href="/wiki/Bravais_lattice" title="Bravais lattice">Bravais lattices</a> and 230 <a href="/wiki/Space_group" title="Space group">space groups</a>. </p><p>Lattice periodicity implies <b>long-range order</b>:<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> if only one unit cell is known, then by virtue of the translational symmetry it is possible to accurately predict all atomic positions at arbitrary distances. During much of the 20th century, the converse was also taken for granted – until the discovery of <a href="/wiki/Quasicrystal" title="Quasicrystal">quasicrystals</a> in 1982 showed that there are perfectly deterministic tilings that do not possess lattice periodicity. </p><p>Besides structural order, one may consider <a href="/wiki/Charge_ordering" title="Charge ordering">charge ordering</a>, <a href="/wiki/Spin_(physics)" title="Spin (physics)">spin</a> ordering, <a href="/wiki/Magnetic_ordering" class="mw-redirect" title="Magnetic ordering">magnetic ordering</a>, and compositional ordering. Magnetic ordering is observable in <a href="/wiki/Neutron_diffraction" title="Neutron diffraction">neutron diffraction</a>. </p><p>It is a <a href="/wiki/Thermodynamic" class="mw-redirect" title="Thermodynamic">thermodynamic</a> <a href="/wiki/Entropy_(order_and_disorder)" title="Entropy (order and disorder)">entropy</a> concept often displayed by a second-order <a href="/wiki/Phase_transition" title="Phase transition">phase transition</a>. Generally speaking, high thermal energy is associated with disorder and low thermal energy with ordering, although there have been violations of this. Ordering peaks become apparent in diffraction experiments at low energy. </p> <div class="mw-heading mw-heading3"><h3 id="Long-range_order">Long-range order</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Order_and_disorder&action=edit&section=3" title="Edit section: Long-range order"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><b>Long-range order</b> characterizes physical <a href="/wiki/System" title="System">systems</a> in which remote portions of the same sample exhibit <a href="/wiki/Correlation" title="Correlation">correlated</a> behavior. </p><p>This can be expressed as a <a href="/wiki/Correlation_function" title="Correlation function">correlation function</a>, namely the <a href="/wiki/Spin_(physics)" title="Spin (physics)">spin-spin correlation function</a>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G(x,x')=\langle s(x),s(x')\rangle .\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <msup> <mi>x</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> <mo>=</mo> <mo fence="false" stretchy="false">⟨</mo> <mi>s</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mi>s</mi> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">⟩</mo> <mo>.</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G(x,x')=\langle s(x),s(x')\rangle .\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fa46980117822c1c41aaf348e0cd06910b614faf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.133ex; height:3.009ex;" alt="{\displaystyle G(x,x')=\langle s(x),s(x')\rangle .\,}"></span></dd></dl> <p>where <i>s</i> is the spin quantum number and <i>x</i> is the distance function within the particular system. </p><p>This function is equal to unity when <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=x'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <msup> <mi>x</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=x'}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/97c3f46e04d75950815b03645addc5b644e2ba29" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.443ex; height:2.509ex;" alt="{\displaystyle x=x'}"></span> and decreases as the distance <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |x-x'|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>x</mi> <mo>−</mo> <msup> <mi>x</mi> <mo>′</mo> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |x-x'|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee2387353866aacde6bccfb2fc0586c1105b2b37" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.478ex; height:3.009ex;" alt="{\displaystyle |x-x'|}"></span> increases. Typically, it <a href="/wiki/Exponential_decay" title="Exponential decay">decays exponentially</a> to zero at large distances, and the system is considered to be disordered. But if the correlation function decays to a constant value at large <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |x-x'|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>x</mi> <mo>−</mo> <msup> <mi>x</mi> <mo>′</mo> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |x-x'|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee2387353866aacde6bccfb2fc0586c1105b2b37" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.478ex; height:3.009ex;" alt="{\displaystyle |x-x'|}"></span> then the system is said to possess long-range order. If it decays to zero as a power of the distance then it is called quasi-long-range order (for details see Chapter 11 in the textbook cited below. See also <a href="/wiki/Kosterlitz-Thouless_transition" class="mw-redirect" title="Kosterlitz-Thouless transition">Berezinskii–Kosterlitz–Thouless transition</a>). Note that what constitutes a large value of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |x-x'|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>x</mi> <mo>−</mo> <msup> <mi>x</mi> <mo>′</mo> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |x-x'|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee2387353866aacde6bccfb2fc0586c1105b2b37" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.478ex; height:3.009ex;" alt="{\displaystyle |x-x'|}"></span> is understood in the sense of <a href="/wiki/Asymptotic_analysis" title="Asymptotic analysis">asymptotics</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Quenched_disorder">Quenched disorder</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Order_and_disorder&action=edit&section=4" title="Edit section: Quenched disorder"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In <a href="/wiki/Statistical_physics" class="mw-redirect" title="Statistical physics">statistical physics</a>, a system is said to present <b>quenched disorder</b> when some parameters defining its behavior are <a href="/wiki/Random_variable" title="Random variable">random variables</a> which do not evolve with time. These parameters are said to be quenched or frozen. <a href="/wiki/Spin_glass" title="Spin glass">Spin glasses</a> are a typical example. Quenched disorder is contrasted with <a href="/wiki/Annealed_disorder" class="mw-redirect" title="Annealed disorder">annealed disorder</a> in which the parameters are allowed to evolve themselves. </p><p>Mathematically, quenched disorder is more difficult to analyze than its annealed counterpart as averages over thermal noise and quenched disorder play distinct roles. Few techniques to approach each are known, most of which rely on approximations. Common techniques used to analyzed systems with quenched disorder include the <a href="/wiki/Replica_trick" title="Replica trick">replica trick</a>, based on <a href="/wiki/Analytic_continuation" title="Analytic continuation">analytic continuation</a>, and the <a href="/wiki/Cavity_method" title="Cavity method">cavity method</a>, where a system's response to the perturbation due to an added constituent is analyzed. While these methods yield results agreeing with experiments in many systems, the procedures have not been formally mathematically justified. Recently, rigorous methods have shown that in the <a href="/wiki/Sherrington-Kirkpatrick_model" class="mw-redirect" title="Sherrington-Kirkpatrick model">Sherrington-Kirkpatrick model</a>, an archetypal spin glass model, the replica-based solution is exact. The <a href="/wiki/Moment-generating_function" title="Moment-generating function">generating functional formalism</a>, which relies on the computation of <a href="/wiki/Functional_integration" title="Functional integration">path integrals</a>, is a fully exact method but is more difficult to apply than the replica or cavity procedures in practice. </p> <figure class="mw-halign-center" typeof="mw:File/Thumb"><a href="/wiki/File:Ordering.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/5/57/Ordering.png" decoding="async" width="506" height="191" class="mw-file-element" data-file-width="506" data-file-height="191" /></a><figcaption>Transition from disordered (left) to ordered (right) states</figcaption></figure> <div class="mw-heading mw-heading2"><h2 id="Annealed_disorder">Annealed disorder</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Order_and_disorder&action=edit&section=5" title="Edit section: Annealed disorder"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A system is said to present <b>annealed disorder</b> when some parameters entering its definition are <a href="/wiki/Random_variable" title="Random variable">random variables</a>, but whose evolution is related to that of the <a href="/wiki/Degrees_of_freedom_(physics_and_chemistry)" title="Degrees of freedom (physics and chemistry)">degrees of freedom</a> defining the system. It is defined in opposition to quenched disorder, where the random variables may not change their values. </p><p>Systems with annealed disorder are usually considered to be easier to deal with mathematically, since the average on the disorder and the <a href="/w/index.php?title=Thermal_average&action=edit&redlink=1" class="new" title="Thermal average (page does not exist)">thermal average</a> may be treated on the same footing. </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Order_and_disorder&action=edit&section=6" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>In <a href="/wiki/High_energy_physics" class="mw-redirect" title="High energy physics">high energy physics</a>, the formation of the <a href="/wiki/Chiral_condensate" class="mw-redirect" title="Chiral condensate">chiral condensate</a> in <a href="/wiki/Quantum_chromodynamics" title="Quantum chromodynamics">quantum chromodynamics</a> is an ordering transition; it is discussed in terms of <a href="/wiki/Superselection" title="Superselection">superselection</a>.</li> <li><a href="/wiki/Entropy" title="Entropy">Entropy</a></li> <li><a href="/wiki/Topological_order" title="Topological order">Topological order</a></li> <li><a href="/wiki/Impurity" class="mw-redirect" title="Impurity">Impurity</a></li> <li><a href="/wiki/Superstructure_(physics)" class="mw-redirect" title="Superstructure (physics)">superstructure (physics)</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Order_and_disorder&action=edit&section=7" title="Edit section: Further reading"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>H Kleinert: <a rel="nofollow" class="external text" href="http://www.physik.fu-berlin.de/~kleinert/kleiner_reb1"><i>Gauge Fields in Condensed Matter</i></a> (<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><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9971-5-0210-0" title="Special:BookSources/9971-5-0210-0">9971-5-0210-0</a>, 2 volumes) Singapore: World Scientific (1989).</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBürgi2000" class="citation journal cs1">Bürgi, H. B. (2000). "Motion and Disorder in Crystal Structure Analysis: Measuring and Distinguishing them". <i>Annual Review of Physical Chemistry</i>. <b>51</b>: 275–296. <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/2000ARPC...51..275B">2000ARPC...51..275B</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1146%2Fannurev.physchem.51.1.275">10.1146/annurev.physchem.51.1.275</a>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/11031283">11031283</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Annual+Review+of+Physical+Chemistry&rft.atitle=Motion+and+Disorder+in+Crystal+Structure+Analysis%3A+Measuring+and+Distinguishing+them&rft.volume=51&rft.pages=275-296&rft.date=2000&rft_id=info%3Apmid%2F11031283&rft_id=info%3Adoi%2F10.1146%2Fannurev.physchem.51.1.275&rft_id=info%3Abibcode%2F2000ARPC...51..275B&rft.aulast=B%C3%BCrgi&rft.aufirst=H.+B.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOrder+and+disorder" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMüller2009" class="citation web cs1">Müller, Peter (2009). <a rel="nofollow" class="external text" href="http://ocw.mit.edu/courses/chemistry/5-067-crystal-structure-refinement-fall-2009/lecture-notes/MIT5_067F09_lec4.pdf">"5.067 Crystal Structure Refinement"</a> <span class="cs1-format">(PDF)</span>. Cambridge: MIT OpenCourseWare<span class="reference-accessdate">. Retrieved <span class="nowrap">13 October</span> 2013</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=5.067+Crystal+Structure+Refinement&rft.place=Cambridge&rft.pub=MIT+OpenCourseWare&rft.date=2009&rft.aulast=M%C3%BCller&rft.aufirst=Peter&rft_id=http%3A%2F%2Focw.mit.edu%2Fcourses%2Fchemistry%2F5-067-crystal-structure-refinement-fall-2009%2Flecture-notes%2FMIT5_067F09_lec4.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOrder+and+disorder" class="Z3988"></span></li></ul> <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=Order_and_disorder&action=edit&section=8" 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-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.britannica.com/science/long-range-order">"Long-range order | chemistry | Britannica"</a>. <i>www.britannica.com</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2024-02-09</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=www.britannica.com&rft.atitle=Long-range+order+%7C+chemistry+%7C+Britannica&rft_id=https%3A%2F%2Fwww.britannica.com%2Fscience%2Flong-range-order&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOrder+and+disorder" class="Z3988"></span></span> </li> </ol></div></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Order_and_disorder&oldid=1210934079">https://en.wikipedia.org/w/index.php?title=Order_and_disorder&oldid=1210934079</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Categories</a>: <ul><li><a href="/wiki/Category:Statistical_mechanics" title="Category:Statistical mechanics">Statistical mechanics</a></li><li><a href="/wiki/Category:Crystallography" title="Category:Crystallography">Crystallography</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Hidden categories: <ul><li><a href="/wiki/Category:Articles_with_short_description" title="Category:Articles with short description">Articles with short description</a></li><li><a href="/wiki/Category:Short_description_is_different_from_Wikidata" title="Category:Short description is different from Wikidata">Short description is different from Wikidata</a></li><li><a href="/wiki/Category:Articles_needing_additional_references_from_February_2024" title="Category:Articles needing additional references from February 2024">Articles needing additional references from February 2024</a></li><li><a href="/wiki/Category:All_articles_needing_additional_references" title="Category:All articles needing additional references">All articles needing additional references</a></li><li><a href="/wiki/Category:All_articles_with_unsourced_statements" title="Category:All articles with unsourced statements">All articles with unsourced statements</a></li><li><a href="/wiki/Category:Articles_with_unsourced_statements_from_February_2024" title="Category:Articles with unsourced statements from February 2024">Articles with unsourced statements from February 2024</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> This page was last edited on 29 February 2024, at 01:16<span class="anonymous-show"> (UTC)</span>.</li> <li id="footer-info-copyright">Text is available under the <a href="/wiki/Wikipedia:Text_of_the_Creative_Commons_Attribution-ShareAlike_4.0_International_License" title="Wikipedia:Text of the Creative Commons Attribution-ShareAlike 4.0 International License">Creative Commons Attribution-ShareAlike 4.0 License</a>; additional terms may apply. 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