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Current issue   Ukr. J. Phys. 2017, Vol. 62, N 3, p.217-229
https://doi.org/10.15407/ujpe62.03.0217    Paper

Lev B.I., Tymchyshyn V.B., Zagorodny A.G.

Bogolyubov Institute for Theoretical Physics, Nat. Acad. of Sci. of Ukraine
(14b, Metrolohichna Str., Kyiv 03680, Ukraine; e-mail: blev@bitp.kiev.ua, yu.binkukoku@gmail.com, azagorodny@bitp.kiev.ua)

Potential Energy Analysis for a System of Interacting Particles Arranged in a Bravais Lattice

Section: Plasmas and Gases
Original Author's Text: Ukrainian/English

Abstract: We propose a method to calculate the type of a lattice formed by grains in dusty plasma and estimate its potential energy. Basically, this task is complicated by the interparticle potential that appertains to “catastrophic potentials”. This kind of potentials needs special approaches to avoid divergences during potential energy calculations. In the current contribution, we will develop all the necessary modifications to appropriate methods. It will be shown that the obtained potential energy expression can be used to determine lattice parameters and these parameters comply to known experimental data.

Key words:  Coulomb potential, dusty plasma, potential energy, Bravais lattice.


  1. V.E. Fortov, A.V. Ivlev, S.A. Khrapak, A.G. Khrapak, G.E. Morfill. Complex (dusty) plasma: Current status, open issues, perspectives. Phys. Rep. 421, 1 (2005).
  2. H. L¨owen. Melting, freezing and colloidal suspensions. Phys. Rep. 237, 249 (1994).
  3. G.E. Morfill, H.M. Thomas, U. Konopka, M. Zuzic. The plasma condensation: Liquid and crystalline plasmas. Phys. Plasmas 6, 1769 (1999).
  4. Y. Bilotsky. On calculation of lattice energy in spatially confined domains. Advances in Materials Science and Applications 2, (4), 127 (2013).
  5. L.N. Kantorovich, I.I. Tupitsyn. Coulomb potential inside a large finite crystal. J. Phys.: Condens. Matter 11, 6159 (1999).
  6. L.P. Buhler, R.E. Crandal. On the convergence problem for lattice sums. J. Phys. A: Math. Gen. 23, 2523 (1990).
  7. D. Borwein, J.M. Borwein, R. Shail, I.J. Zucker. Energy of static electron lattices. J. Phys. A: Math. Gen. 20, 1519 (1988).
  8. T.R.S. Prasanna. Physical meaning of the Ewald sum method. Phil. Magaz. Lett. 92, No. 1, 29 (2012).
  9. Y. Bilotsky, M. Gasik. A new approach for modelling lattice energy in finite crystal domains. Phys.: Conf. Ser. 633, 012014 (2015).
  10. D.M. Heyes, A.C. Bra’nka. Lattice summations for spread out particles: Applications to neutral and charged systems. J. Chem. Phys. 138, 034504 (2013).
  11. B.I. Lev, V.P. Ostroukh, V.B. Tymchyshyn, A.G. Zagorodny. Statistical description of the system electrons on the liquid helium surface. Eur. Phys. J. B 87:253 (2014).
  12. D. Ruelle. Statistical Mechanics: Rigorous Results (Benjamin, 1969).
  13. A. Isihara. Statistical Mechanics (Academic Press, 1971).
  14. K. Huang. Statistical Mechanics (Wiley, 1963).
  15. R. Baxter. Exactly Solved Models in Statistical Mechanics (Academic Press, 1982).
  16. B. Klumov, G. Joyce, C. R¨ath, P. Huber, H. Thomas, G.E. Morfill, V. Molotkov, V. Fortov. Structural properties of 3D complex plasmas under microgravity conditions. Europhys. Lett. 92 (1), 15003 (2010).
  17. B.A. Klumov, G.E. Morfill. Structural properties of complex (dusty) plasma upon crystallization and melting. JETP Lett. 90 (6), 444 (2009).
  18. B.I. Lev, A.G. Zagorodny. Structure formation in system of Brownian particle in dusty plasma. Phys. Lett. A 373, 1101 (2009).
  19. B.I. Lev, V.B. Tymchyshyn, A.G. Zagorodny. Brownian particle in non-equilibrium plasma. Condens. Phys. 12, 593 (2009).
  20. H. Thomas, G.E. Morfill, V. Demmel, J. Goree, B. Feuerbacher, D. Mohlmann. Plasma crystal: Coulomb crystallization in a dusty plasma. Phys. Rev. Lett. 73, 652 (1994).
  21. P. Ewal . Die Berechnung optischer und elektrostatischer Gitterpotentiale. Ann. Phys. 369 (3), 253 (1921).
  22. J.H. Chu, Lin I. Direct observation of Coulomb crystals and liquids in strongly coupled rf dusty plasmas. Phys. Rev. Lett. 72, 4009 (1994).
  23. A. Melzer, T. Trottenberg, A. Piel. Experimental determination of the charge on dust particles forming Coulomb lattices. Phys. Lett. A 191, 301 (1994).
  24. S.V. Vladimirov, S.A. Khrapak, M. Chaudhuri, G.E. Morfill. Superfluidlike motion of an absorbing body in a collisional plasma. Phys. Rev. Lett. 100, 055002 (2008).
  25. H. Ikezi. Coulomb solid of small particles in plasmas. Phys. Fluids 29, 1764 (1986).
  26. A. Melzer, A. Homann, A. Piel. Experimental investigation of the melting transition of the plasma crystal. Phys. Rev. E 53, 2757 (1996).
  27. A.G. Sitenko, A.G. Zagrodny, V.N. Tsytovich. Fluctuation phenomena in dusty plasmas. AIP Conf. Proc. 345, 311 (1995).
  28. S.A. Brazovsky. Phase transition of an isotropic system to a nonuniform state. Sov. Phys. JETP 41 (1), 85 (1975).
  29. B.I. Lev, H. Yokoyama. Selection of states and fluctuation under the first order phase transitions. Int. J. Mod. Phys. B 17, 4913 (2003) .
  30. H. Totsuji, T. Kishimot o, C. Totsuji. Structure of confined Yukawa system (dusty plasma). Phys. Rev. Lett. 78, 3113 (1997).
  31. I.S. Gradshteyn, I.M. Ryzhik. Tables of Integrals, Series, and Products (Academic Press, 2007).
  32. A. Erd’elyi, W. Magnus, F. Oberhettinger, F. G. Tricomi. Higher Transcendental Functions (Krieger, 1981), Vol. 2.
  33. R. Bellman. Introduction to Matrix Analysis (Society for Industrial and Applied Mathematics, 1970).
  34. R. Bellman. A Brief Introduction to Theta Functions (Holt, Rinehart and Winston, 1961).
  35. M. Chiani, D. Dardari, M.K. Simon. New exponential bounds and approximations for the computation of error probability in fading channels. IEEE Transactions on Wireless Communications 4 (2), 840 (2003).
  36. C. Meyer. Matrix Analysis and Applied Linear Algebra (SIAM, 2000).
  37. M. Abramowitz, I. Stegun. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Dover, 1974).
  38. B.R. Brooks, R.E. Bruccoleri, B.D. Olafson, D.J. States, S. Swaminathan, M. Karplus. HARMM: A program for macromolecular energy, minimization, and dynamics calculations. J. Comput. Chem. 4, 187 (1983).
  39. T. Darden, D. York, L. Pederson. Particle mesh Ewald: An · log() method for Ewald sums in large systems. J. Chem. Phys. 98, 10089 (1993) .
  40. B. Nijboer, E. De Wette. On the calculation of lattice sums. Physica 23, 309 (1957).
  41. J. Kolafa, J. Perram. Cutoff errors in the Ewald summation formulae for point charge systems. Mol. Simulation 9, 351 (1992).
  42. J. Perram, H. Petersen, S. De Leeuw. An algorithm for the simulation of condensed matter which grows as the 3/2 power of the number of particles. Mol. Phys. 65, 875 (1988).
  43. Z. Rycerz, P. Jacobs. Ewald summation in the molecular dynamics simulation of large ionic systems. Mol. Simulation 8, 197 (1992).
  44. W. Smith. FORTRAN code for the Ewald summation method. Information Quarterly for Computer Simulation of Condensed Phases 21, 37 (1986).
  45. D. York, W. Yang. The fast Fourier Poisson method for calculating Ewald sums. J. Chem. Phys. 101, 3298 (1994).