Abstract
We consider a one-dimensional linear lattice of particles whose mass, pair potential, and nearest neighbor separation are those of a real rare gas crystal. Numerical solution of the Hartree equation shows that the model behaves as a quantum crystal in the low mass, weak attraction case. In the basic Nosanow cluster approximation the cohesive energy of this helium-like system drops from 6.903°K/N (Hartree) to 3.64°K/N. When all except nearest neighbor correlations in the Jastrow function are taken as unity, the result is 3.69°K/N. For the case of nearest neighbor correlations only, we introduce a positive integral operator with properties akin to those of a transfer matrix and thus form a rigorous upper bound on the cohesive energy of the model system. The convergence rate of the Nosanow expansion is shown to depend on the ratio of the two highest eigenvalues of this operator.
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References
F. W. de Wette and B. R. A. Nijboer,Phys. Letters 18, 19 (1965).
L. H. Nosanow and G. L. Shaw,Phys. Rev. 128, 546 (1962).
D. Rosenwald,Phys. Rev. 154, 160 (1967).
L. H. Nosanow,Phys. Rev. 146, 120 (1966).
J. H. Hetherington, W. J. Mullin, and L. H. Nosanow,Phys. Rev. 154, 175 (1967).
K. A. Brueckner and J. Frohberg,Progr. Theoret. Phys. (Kyoto),Suppl. “Yukawa,” 383 (1965).
K. A. Brueckner and J. Frohberg, inMany Body Theory, 1965 Tokyo Summer Lectures in Theoretical Physics, R. Kubo, ed. (W. A. Benjamin, Inc., New York, 1966), p. 134.
W. E. Massey and C.-W. Woo,Phys. Rev. 169, 241 (1968).
H. A. Kramers and G. H. Wannier,Phys. Rev. 60, 252 (1941).
L. Van Hove,Physica 16, 137 (1950).
G. F. Newell and E. W. Montroll,Rev. Mod. Phys. 25, 353 (1953).
F. R. Gantmacher,The Theory of Matrices, transl. by K. A. Hirsch (Chelsea Publishing Co., New York, 1960), p. 53.
M. A. Krasnosel'skii,Positive Solutions of Operator Equations, transl. by R. E. Flaherty, L. F. Boron, ed. (P. Noordhoff Ltd., Groningen, The Netherlands, 1964), Chaps. 1 and 2.
L. V. Kantorovich and G. P. Akhilov,Functional Analysis in Normed Spaces, transl. by D. E. Brown, A. P. Robertson, ed. (Pergamon Press, New York, 1964), p. 302.
L. V. Kantorovich and G. P. Akhilov,Functional Analysis in Normed Spaces, transl. by D. E. Brown, A. P. Robertson, ed. (Pergamon Press, New York, 1964), p. 517.
Carl-Erik Froberg,Introduction to Numerical Analysis (Addison-Wesley Publishing Co., Reading, Mass., 1965), p. 103.
S. B. Trickey,Phys. Rev. 166, 177 (1968).
J. Nuttall,Bull. Am. Phys. Soc. 13, 901 (1968).
J. W. Essam,J. Math. Phys. 8, 741 (1967).
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Supported in part by the U.S. Air Force Office of Scientific Research under Grant No. AFOSR 918-67.
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Trickey, S.B., Nuttall, J. Convergence and energy bounding properties of the Nosanow cluster expansion. J Low Temp Phys 1, 109–122 (1969). https://doi.org/10.1007/BF00628266
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DOI: https://doi.org/10.1007/BF00628266