Tricritical point

<h2 id="solid-state">Solid state</h2>
It seems more convenient experimentally<a class="footnote-ref" id="fnref:6" href="#fn:6">6</a> to consider mixtures with four components for which one thermodynamic variable (usually the pressure or the volume) is kept fixed. The situation then reduces to the one described for mixtures of three components.
Historically, it was for a long time unclear whether a <a href="/facts/Superconduction/pqQAlSzJ">superconductor</a> undergoes a first- or a second-order phase transition. The question was finally settled in 1982.<a class="footnote-ref" id="fnref:7" href="#fn:7">7</a> If the Ginzburg–Landau parameter 
 
 
 
 κ
 
 
 {\displaystyle \kappa }
 
 that distinguishes <a href="/facts/Type_I_superconductor/xsosaCVK">type-I</a> and <a href="/facts/Type_II_superconductor/8V4UTGjY">type-II</a> superconductors (see also <a href="/facts/Ginzburg%E2%80%93Landau_theory/VAHPgSzY">here</a>) is large enough, vortex fluctuations become important which drive the transition to second order.<a class="footnote-ref" id="fnref:8" href="#fn:8">8</a>
The tricritical point lies at roughly 
 
 
 
 κ
 =
 0.76
 
 /
 
 
 
 2
 
 
 
 
 {\displaystyle \kappa =0.76/{\sqrt {2}}}
 
, slightly below the value 
 
 
 
 κ
 =
 1
 
 /
 
 
 
 2
 
 
 
 
 {\displaystyle \kappa =1/{\sqrt {2}}}
 
 where type-I goes over into type-II superconductor. The prediction was confirmed in 2002 by Monte Carlo <a href="/facts/Monte_Carlo_method/AkHjY7jc">computer simulations</a>.<a class="footnote-ref" id="fnref:9" href="#fn:9">9</a>

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

<ol>
<li id="fn:1">Landau, L. D. (1937). On the theory of phase transitions. I. Zh. Eksp. Teor. Fiz., 11, 19. <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Landau, L. D., & Lifshitz, E. M. (2013). Statistical Physics: Volume 5 (Vol. 5). Elsevier. <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">Griffiths, R. B. (1970). Thermodynamics near the two-fluid critical mixing point in He 3-He 4. Physical Review Letters, 24(13), 715. <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">B. Widom, Theory of Phase Equilibrium, J. Phys. Chem. 1996, 100, 13190-13199 <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">ibid. <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn:6">A. S. Freitas & Douglas F. de Albuquerque (2015). "Existence of a tricritical point in the antiferromagnet KFe3(OH)6(SO4)2 on a kagome lattice". Phys. Rev. E. 91 (1): 012117. Bibcode:2015PhRvE..91a2117F. doi:10.1103/PhysRevE.91.012117. PMID 25679580. <a href="/wiki/Kagome_lattice" target="_blank">/wiki/Kagome_lattice</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
<li id="fn:7">H. Kleinert (1982). "Disorder Version of the Abelian Higgs Model and the Order of the Superconductive Phase Transition" (PDF). Lettere al Nuovo Cimento. 35 (13): 405–412. doi:10.1007/BF02754760. S2CID 121012850. <a href="/wiki/Hagen_Kleinert" target="_blank">/wiki/Hagen_Kleinert</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></li>
<li id="fn:8">H. Kleinert (2006). "Vortex Origin of Tricritical Point in Ginzburg-Landau Theory" (PDF). Europhys. Lett. 74 (5): 889–895. arXiv:cond-mat/0509430. Bibcode:2006EL.....74..889K. doi:10.1209/epl/i2006-10029-5. S2CID 55633766. <a href="/wiki/Hagen_Kleinert" target="_blank">/wiki/Hagen_Kleinert</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></li>
<li id="fn:9">J. Hove; S. Mo; A. Sudbo (2002). "Vortex interactions and thermally induced crossover from type-I to type-II superconductivity" (PDF). Phys. Rev. B 66 (6): 064524. arXiv:cond-mat/0202215. Bibcode:2002PhRvB..66f4524H. doi:10.1103/PhysRevB.66.064524. S2CID 13672575. <a href="http://users.physik.fu-berlin.de/~kleinert/papers/sudbotre064524.pdf" target="_blank">http://users.physik.fu-berlin.de/~kleinert/papers/sudbotre064524.pdf</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></li>
</ol>

Tricritical point open-in-new

Tricritical point