Skip to main content
SpringerLink
Log in

Phonon density of states from the experimental heat capacity: an improved distribution function for solid aluminium using an inverse framework

  • Original Paper
  • Published:
Journal of Molecular Modeling Aims and scope Submit manuscript

Abstract

In this study it is reported the retrieval of the phonon density of states for solid aluminium from the temperature dependent heat capacity, the inverse heat capacity problem. The singularity in this ill posed problem was removed by the Tikhonov approach with the regularization parameter calculated as the L curve maximum curvature. A sensitivity analysis was also coupled to the numerical inversion. For temperatures ranging from 15 K to 300 K the heat capacity results, calculated from the inverted phonon density of states, yields an average error of about 0.3 % , within the experimental errors that ranged from 2 % to 3 %. The predicted entropy, enthalpy and Gibbs free energy are also within experimental errors.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5

Similar content being viewed by others

References

  1. Kittel C (2004) Introduction to solid state physics. Wiley, New York

    Google Scholar 

  2. Einstein A (1907) Die plancksche theorie der strahlung und die theorie der spezifischen wärmen. Ann Phys (Berlin,Ger) 327(1):180–190

    Article  Google Scholar 

  3. Debye P (1912) Zur theorie der spezifischen wärmen. Ann Phys (Berlin,Ger) 344(14):789–839

    Article  Google Scholar 

  4. Hill TL (1986) An Introduction to Statistical Thermodynamics. Dover, New York

    Google Scholar 

  5. Walker CB (1956) X-ray study of lattice vibrations in aluminum. Phys Rev 103:547–557

    Article  CAS  Google Scholar 

  6. Dawidowski J, Cuello G, Koza M, Blostein J, Aurelio G, Guillermet AF, Donato P (2002) Analysis of multiple scattering and multiphonon contributions in inelastic neutron scattering experiments. Nucl Instrum Methodys Phys Res Sect B 195(34):389–399

    Article  CAS  Google Scholar 

  7. Ferreira AR, Martins MJ, Konstantinova E, Capaz RB, Souza WF, Chiaro SSX, Leitão AA (2011) Direct comparison between two structural models by dft calculations. J Solid State Chem 184(5):1105–1111

    Article  CAS  Google Scholar 

  8. Wing GM, Zahrt JA (1991) A primer on integral equations of the first kind. SIAM, Philadelphia

    Book  Google Scholar 

  9. Montroll EW (1942) Frequency spectrum of crystalline solids. J Chem Phys 10(4):218–229

    CAS  Google Scholar 

  10. Chen N-X (2010) Mobius inversion problem. World Scientific, New York

    Google Scholar 

  11. Chen N-X (1990) Modified möbius inverse formula and its applications in physics. Phys Rev Lett 64:1193–1195

    Article  Google Scholar 

  12. Chen N-X, Chen Y, Li G-Y (1990) Theoretical investigation on inversion for the phonon density of states. Phys Lett A 149(78):357–364

    Article  Google Scholar 

  13. Chen N-X, Rong E-G (1998) Unified solution of the inverse capacity problem. Phys Rev E 57(2):1302–1307

    Article  Google Scholar 

  14. Lemes NHT, Braga JP, Belchior JC (1998) Spherical potential energy function from second virial coefficient using tikhonov regularization and truncated singular value decomposition. Chem Phys Lett 296(34):233–238

    CAS  Google Scholar 

  15. Lemes NHT, Sebastião RCO, Braga JP (2006) Potential energy function from second virial data using sensitivity analysis. Inverse Probl Sci Eng 14(6):581–587

    Article  CAS  Google Scholar 

  16. Tikhonov AN, (1987) Ill-posed problems in the natural sciences. Mir, Moscow

    Google Scholar 

  17. Braga JP (2001) Numerical comparison between Tikhonov regularization and singular value decomposition methods using the L curve criterion. J Math Chem 29:151–161

    Article  CAS  Google Scholar 

  18. Lemes NHT, Borges E, Souza RV, Braga JP (2008) Potential energy function from differential cross-section data: An inverse quantum scattering theory approach. Int J Quantum Chem 108:2623–2627

    Article  CAS  Google Scholar 

  19. Sun X, Jaggard DL (1987) The inverse blackbody radiation problem: A regularization solution. J Appl Phys 62(11):4382–4386

    Article  Google Scholar 

  20. Hauge JP (2005) Determining the phonon density of states specific hear measurements via maximum entropy methods. J Phys: Condens Matter 17:2397–2405

    Article  Google Scholar 

  21. Makovetskii GI, Smolik ChK, Severin GM (2001) Calculation of the phonon density states function of aluminum from temperature dependence of specific heat. In: Proceedings of the Natl Academy of Sciences of Belarus: Phys Math Sci, vol 2, pp 102–105

  22. Hadamard J (1923) Lectures on cauchy’s problems in linear partial differential equations. Yale University Press, New Haven

    Google Scholar 

  23. Riele HJJ (1985) A program for solving first kind fredholm integral equations by means of regularization. Comput Phys Commun 36:423–432

    Article  Google Scholar 

  24. Giauque WF, Meads PF (1941) The Heat Capacities and Entropies of Aluminum and Copper from 15 to 300 o K. J Am Chem Soc 63:1897–1901

    Article  CAS  Google Scholar 

  25. Leon SJ (2009) Linear algebra with applications. Pearson, New Jersey

    Google Scholar 

  26. Hansen PC (1997) Rank-deficient and discrete Ill-posed problems. Numer Aspects of Linear Inversion. SIAM, Philadelphia

    Google Scholar 

  27. Park HW, Shin S, Lee HS (2001) Determination of an optimal regularization factor in system identification with Tikhonov regularization for linear elastic continua. Int J Numer Meth Engng 151:1211–1230

    Article  Google Scholar 

Download references

Acknowledgments

This work was supported by CNPq and Fapemig, Brazil.

Author information

Authors and Affiliations

  1. Instituto de Química, Universidade Federal de Alfenas, Rua Gabriel Monteiro da Silva, 700 - Centro, Alfenas/MG, 37130-000, Brazil

    Éderson D’M. Costa & Nelson H. T. Lemes

  2. Departamento de Química, Universidade Federal de Minas Gerais, Av. Antônio Carlos, 6627 - Pampulha, Belo Horizonte/MG, 31270-010, Brazil

    Márcio O. Alves & João P. Braga

Authors
  1. Éderson D’M. Costa
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Nelson H. T. Lemes
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Márcio O. Alves
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. João P. Braga
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Márcio O. Alves.

Additional information

This paper belongs to Topical Collection Brazilian Symposium of Theoretical Chemistry (SBQT2013)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Costa, É.D., Lemes, N.H.T., O. Alves, M. et al. Phonon density of states from the experimental heat capacity: an improved distribution function for solid aluminium using an inverse framework. J Mol Model 20, 2360 (2014). https://doi.org/10.1007/s00894-014-2360-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00894-014-2360-z

Keywords

Search

Navigation