A New Approach for Calculating Phonon Dispersions and Elastic Constants of Simple Metals
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A NEW APPROACH FOR CALCULATING PHONON DISPERSIONS AND ELASTIC CONSTANTS OF SIMPLE METALS M. LI, S.J. LIU AND N.X. CHEN Institute of Applied Physics Beijing University of Sci. and Tech. Beijing 100083, China. ABSTRACT A new method to calculate the phonon dispersions for simple metals is first introduced, based on the Mobius transform. The cohesive energy can be obtained from experiment or theory, then the pair and three-body potentials between atoms in metals can be obtained by use of the modified Mobius theorem. Based on only the pair potentials the phonon dispersions for Cu, Al and Ni have been obtained without any adjustable parameters. The calculation of elastic constants has involved both the 2-and 3-body interactions, and the restriction of the Cauchy relation has been erased. The above result represents a significant improvement over the Carlsson - Gelatt - Ehrenreich method. INTRODUCTION, During the past few years the Mobius inverse theorem in number theory"1 has been successfully applied to some inverse problems for Boson or Fermion systems1 2-5] Recently, it has been neatly used to solve some lattice problems1 6 .71. Therefore, the pair potential between atoms can be derived from the cohesive energy of a crystal for simple metals. If a phenomenological expression of the 3-body potential is given, the 2- and 3body potentials can be obtained simultaneously by the above inversion formalism. Based on only the pair potential, the atomic force constants for fcc metals Cu, Al, Ni have been evaluated, and the dispersion relations are obtained. The results are in good agreement with experiments. When we evaluated the deformed energies, both the 2- and 3- body potentials are considered in the calculation. Finally, the calculated elastic con11 stants are in good agreement with experiments I] and the restriction of the Cauchy relation has been erased. CALCULATION METHOD 1. Modified Mobius transform The 21modified Mobius transform in the real function field can be expressed as follows:[ if then
F(x) = E .1(nx), Aix) = E p(n)F(nx),
where A(n) is the Mobius function defined asIll if n = 1, if n includes s distinct primes, i(n) 0, if n includes repeat factors
and F(x) and f(x) are any real functions.
Mat. Res. Soc. Symp. Proc. Vol. 291. 01993 Materials Research Society
(1) (2)
(3)
498
2. Calculating pair potential from cohesive energy 6
For fee simple metals the cohesive energy E(r) is assumed to be written as [67
E(r)=(1/2) E
JA• 00
I)=3 1 q(nr)+ 6 1: f(o([p 2 +q 2 ] " 2 r)
q'(I R
mI
.4-1I
+gp([(p-l/ +3go([(p-
)
6 2[ko(nr/2
1/2)2 +u
)2 +(q-
2
] /2r)} 2
+ (1/2)q(nr)] + 6 V [9o([p + q
n-1
2
pa-I
2
p([(p-
+2
1/2)
(q-
2 1/22
r)] +
r)
2
M
1/2
1/2)
+u 2]
, [go([p2 +q
]/r)+4
+(q-l/2)
1/2
2
12
2
2
4
1/2
]r) 2
[p+[p
+q 2
2 1/2
U2/2 r)
PA., - I
(4)
)2+(q_-/2)2+u21/2 r);,
+3g([(p--/2
Where V' denotes the sum without p =q terms, q0(I RI) is the pair. potential and r the lattice constant. Operators T, and T are introduced as T, (p(r) = 6 1 [.p(nr / 2
/2)
(5)
+ (1 / 2)ip(n
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