Editing Melting point (section)

==Predicting the melting point of substances (Lindemann's criterion)==
An attempt to predict the bulk melting point of crystalline materials was first made in 1910 by [[Frederick Lindemann]].<ref name="Lindemann1910">{{cite journal |author=Lindemann FA |title=The calculation of molecular vibration frequencies |journal=Phys. Z. |volume=11 |year=1910 |pages=609–612 |author-link=Frederick Lindemann}}</ref> The idea behind the theory was the observation that the average amplitude of thermal vibrations increases with increasing temperature. Melting initiates when the amplitude of vibration becomes large enough for adjacent atoms to partly occupy the same space. The '''Lindemann criterion''' states that melting is expected when the vibration [[root mean square amplitude]] exceeds a threshold value.

Assuming that all atoms in a crystal vibrate with the same frequency ''ν'', the average thermal energy can be estimated using the [[equipartition theorem]] as<ref name=Sorkin>Sorkin, S., (2003), [http://phycomp.technion.ac.il/~phsorkin/thesis/thesis.html Point defects, lattice structure, and melting] {{Webarchive|url=https://web.archive.org/web/20161005221157/http://phycomp.technion.ac.il/~phsorkin/thesis/thesis.html |date=5 October 2016 }}, Thesis, Technion, Israel.</ref>
:<math>
 E = 4\pi^2 m \nu^2~u^2 = k_{\rm B} T
 </math>
where ''m'' is the [[atomic mass]], ''ν'' is the [[frequency]], ''u'' is the average vibration amplitude, ''k''<sub>B</sub> is the [[Boltzmann constant]], and ''T'' is the [[Thermodynamic temperature|absolute temperature]]. If the threshold value of ''u<sup>2</sup>'' is ''c<sup>2</sup>a<sup>2</sup>'' where ''c'' is the [[Lindemann index|Lindemann constant]] and ''a'' is the [[atomic spacing]], then the melting point is estimated as
:<math>
 T_{\rm m} = \cfrac{4\pi^2 m \nu^2 c^2 a^2}{k_{\rm B}} .
 </math>
Several other expressions for the estimated melting temperature can be obtained depending on the estimate of the average thermal energy. Another commonly used expression for the Lindemann criterion is<ref name=Hofmann>{{cite book|author=Philip Hofmann|title=Solid state physics: an introduction|url=https://books.google.com/books?id=XfIpxi4kcM4C&pg=PA67|access-date=13 March 2011|year=2008|publisher=Wiley-VCH|isbn=978-3-527-40861-0|page=67}}{{Dead link|date=August 2023 |bot=InternetArchiveBot |fix-attempted=yes }}</ref>
:<math>
 T_{\rm m} = \cfrac{4\pi^2 m \nu^2 c^2 a^2}{2k_{\rm B}} .
 </math>
From the expression for the [[Debye frequency]] for ''ν'',
:<math>
 T_{\rm m} = \cfrac{2\pi^2 m c^2 a^2 \theta_{\rm D}^2 k_{\rm B}}{h^2}
 </math>
where ''θ''<sub>D</sub> is the [[Debye temperature]] and ''h'' is the [[Planck constant]]. Values of ''c'' range from 0.15 to 0.3 for most materials.<ref name=Nelson>Nelson, D. R., (2002), [https://books.google.com/books?id=YtYFAqswRzUC Defects and geometry in condensed matter physics], Cambridge University Press, {{ISBN|0-521-00400-4}}</ref>

=== Databases and automated prediction ===
In February 2011, [[Alfa Aesar]] released over 10,000 melting points of compounds from their catalog as [[open data]]<ref name="LangGuha">{{cite journal |last1=Bradley |first1=Jean-Claude |last2=Lang |first2=Andrew |last3=Williams |first3=Antony |last4=Curtin |first4=Evan |date=11 August 2011 |title=ONS Open Melting Point Collection |journal=Nature Precedings |doi=10.1038/npre.2011.6229.1 |doi-access=free}}  [http://www.qsardb.org/repository/predictor/10967/104?model=rf Model] published on [[QsarDB]] retrieved 13 September 2013.</ref> and similar data has been [[data mining|mined]] from [[patent]]s.<ref name=Tetko/>  The Alfa Aesar and patent data have been summarized in (respectively) [[random forest]]<ref name=LangGuha/> and [[support vector machine]]s.<ref name="Tetko">{{cite journal |last1=Tetko |first1=Igor V |last2=m. Lowe |first2=Daniel |last3=Williams |first3=Antony J |year=2016 |title=The development of models to predict melting and pyrolysis point data associated with several hundred thousand compounds mined from PATENTS |journal=Journal of Cheminformatics |volume=8 |pages=2 |doi=10.1186/s13321-016-0113-y |pmc=4724158 |pmid=26807157 |doi-access=free}}  [http://ochem.eu/article/99826 Model]{{Dead link|date=March 2024|bot=InternetArchiveBot|fix-attempted=yes}} published on [[OCHEM]] retrieved 18 June 2016.</ref>