Vegard's law

<h2 id="relationship-to-band-gaps-in-semiconductors">Relationship to band gaps in semiconductors</h2>
<p>In many binary semiconducting systems, the <a href="/facts/Band_gap/IjIeh95l">band gap</a> in semiconductors is approximately a linear function of the lattice parameter. Therefore, if the lattice parameter of a semiconducting system follows Vegard's law, one can also write a linear relationship between the band gap and composition. Using InP<i>x</i>As(1-<i>x</i>) as before, the band gap energy, 
  
    
      
        
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    {\displaystyle E_{g}}
  
,  can be written as:
</p>

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    {\displaystyle E_{g,\mathrm {InPAs} }=x\ E_{g,\mathrm {InP} }+(1-x)\ E_{g,\mathrm {InAs} }}

<p>Sometimes, the linear interpolation between the band gap energies is not accurate enough, and a second term to account for the curvature of the band gap energies as a function of composition is added. This curvature correction is characterized by the bowing parameter, <i>b</i>:
</p>

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    {\displaystyle E_{g,\mathrm {InPAs} }=x\ E_{g,\mathrm {InP} }+(1-x)\ E_{g,\mathrm {InAs} }-bx\ (1-x)}

<h2 id="mineralogy">Mineralogy</h2>
<p>The following excerpt from Takashi Fujii (1960)<a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a> summarises well the limits of Vegard’s law in the context of <a href="/facts/Mineralogy/MeIK9lx9">mineralogy</a> and also makes the link with the <a href="/facts/Gladstone%25E2%2580%2593Dale_relation/tR2ushbl">Gladstone–Dale equation</a>:
</p>
<blockquote><p>In mineralogy, the tacit assumption for the <a href="/facts/Linear_correlation/egkluAEm">linear correlation</a> of the density and the chemical composition of a solid solution is twofold: one is an ideal solid solution and the other identical or nearly identical molar volumes of the components. …
Coefficients of <a href="/facts/Thermal_expansion/BP2vjX0N">thermal expansion</a> and <a href="/facts/Compressibility/eePznleN">compressibilities</a> of the ideal solid solution can be discussed in the same manner. But when the solid solution is ideal, the linear correlation of molar heat capacities and chemical composition is possible. The linear correlation of <a href="/facts/Refractive_index/yjkzvgeE">refractive index</a> and chemical composition of an <a href="/facts/Isotropic/mc3QEY2x">isotropic</a> solid solution can be derived from the <a href="/facts/Gladstone%25E2%2580%2593Dale_relation/tR2ushbl">Gladstone–Dale equation</a>, but it is required that the system must be ideal and the <a href="/facts/Molar_volume/eIkRv7Mj">molar volumes</a> of the components are equal or nearly equal. If the concept of the <a href="/facts/Volume_fraction/nj2akBNM">volume fraction</a> is introduced, density, <a href="/facts/Thermal_expansion/BP2vjX0N">coefficient of thermal expansion</a>, compressibility and refractive index can be correlated linearly with the volume fraction in an ideal system.“<a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a></p></blockquote>
<h2 id="see-also">See also</h2>
<p>When considering the empirical correlation of some physical properties and the chemical composition of solid compounds, other relationships, rules, or laws, also closely resembles Vegard's law, and in fact the more general rule of mixtures: 
</p>
<ul><li><a href="/facts/Amagat%2527s_law/v9G5S3wN">Amagat's law</a></li>
<li><a href="/facts/Gladstone%25E2%2580%2593Dale_relation/tR2ushbl">Gladstone–Dale equation</a></li>
<li><a href="/facts/Kopp%2527s_law/WvjA18GX">Kopp's law</a></li>
<li><a href="/facts/Kopp%2527s_law/WvjA18GX">Kopp–Neumann law</a></li>
<li><a href="/facts/Rule_of_mixtures/PoNlqesu">Rule of mixtures</a></li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1"><p>Vegard, L. (1921). "Die Konstitution der Mischkristalle und die Raumfüllung der Atome". Zeitschrift für Physik. 5 (1): 17–26. Bibcode:1921ZPhy....5...17V. doi:10.1007/BF01349680. S2CID 120699637. <a href="/wiki/Zeitschrift_f%C3%BCr_Physik" target="_blank">/wiki/Zeitschrift_f%C3%BCr_Physik</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>Denton, A.R.; Ashcroft, N.W. (1991). "Vegard's law". Phys. Rev. A. 43 (6): 3161–3164. Bibcode:1991PhRvA..43.3161D. doi:10.1103/PhysRevA.43.3161. PMID 9905387. <a href="/wiki/Physical_Review_A" target="_blank">/wiki/Physical_Review_A</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
<li id="fn:3"><p>King, H.W. (1966). "Quantitative size-factors for metallic solid solutions". Journal of Materials Science. 1 (1): 79–90. Bibcode:1966JMatS...1...79K. doi:10.1007/BF00549722. ISSN 0022-2461. S2CID 97859635. <a href="/wiki/Bibcode_(identifier)" target="_blank">/wiki/Bibcode_(identifier)</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li>
<li id="fn:4"><p>Cordero, Zachary C.; Schuh, Christopher A. (2015). "Phase strength effects on chemical mixing in extensively deformed alloys". Acta Materialia. 82 (1): 123–136. Bibcode:2015AcMat..82..123C. doi:10.1016/j.actamat.2014.09.009. <a href="/wiki/Bibcode_(identifier)" target="_blank">/wiki/Bibcode_(identifier)</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li>
<li id="fn:5"><p>Fujii, Takashi (1960). Correlation of some physical properties and chemical composition of solid solution. The American Mineralogist, 45 (3-4), 370-382. http://www.minsocam.org/ammin/AM45/AM45_370.pdf <a href="http://www.minsocam.org/ammin/AM45/AM45_370.pdf" target="_blank">http://www.minsocam.org/ammin/AM45/AM45_370.pdf</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li>
<li id="fn:6"><p>Zen, E.-AN (1956). Validity of Vegard’s law. American Mineralogist (1956) 41 (5-6), 523-524. https://pubs.geoscienceworld.org/msa/ammin/article-abstract/41/5-6/523/539644 <a href="https://pubs.geoscienceworld.org/msa/ammin/article-abstract/41/5-6/523/539644" target="_blank">https://pubs.geoscienceworld.org/msa/ammin/article-abstract/41/5-6/523/539644</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li>
</ol>

Vegard's law open-in-new

Vegard's law