共查询到20条相似文献,搜索用时 635 毫秒
1.
S. Dvořák 《Czechoslovak Journal of Physics》1968,18(7):840-846
The identity $$\sum\limits_{v = 0} {\left( {\begin{array}{*{20}c} {n + 1} \\ v \\ \end{array} } \right)\left[ {\left( {\begin{array}{*{20}c} {n - v} \\ v \\ \end{array} } \right) - \left( {\begin{array}{*{20}c} {n - v} \\ {v - 1} \\ \end{array} } \right)} \right] = ( - 1)^n } $$ is proved and, by means of it, the coefficients of the decomposition ofD 1 n into irreducible representations are found. It holds: ifD 1 n \(\mathop {\sum ^n }\limits_{m = 0} A_{nm} D_m \) , then $$A_{nm} = \mathop \sum \limits_{\lambda = 0} \left( {\begin{array}{*{20}c} n \\ \lambda \\ \end{array} } \right)\left[ {\left( {\begin{array}{*{20}c} \lambda \\ {n - m - \lambda } \\ \end{array} } \right) - \left( {\begin{array}{*{20}c} \lambda \\ {n - m - \lambda - 1} \\ \end{array} } \right)} \right].$$ 相似文献
2.
Tetz Yoshimura 《International Journal of Theoretical Physics》1976,15(8):605-615
Scalar fields with interaction
${*{20}c} {\lambda \sum\limits_{i \ne j} {\phi i^2 \phi j^2 } } & {or} & {\lambda '\sum\limits_{i,j,k,l pairwise disticnt} {\phi i\phi j\phi k\phi l} } \\ $\begin{array}{*{20}c} {\lambda \sum\limits_{i \ne j} {\phi i^2 \phi j^2 } } & {or} & {\lambda '\sum\limits_{i,j,k,l pairwise disticnt} {\phi i\phi j\phi k\phi l} } \\ \end{array} 相似文献
3.
A. Yu. Gufan O. V. Kukin Yu. M. Gufan I. A. Osipenko 《Bulletin of the Russian Academy of Sciences: Physics》2012,76(3):328-338
The third-order elastic modulus of α-Fe were calculated based on the computation of lattice sums. The lattice sums were determined
using an integer rational basis of invariants composed by vectors connecting equilibrium atomic positions in the crystal lattice.
Irreducible interactions within clusters consisting of atomic pairs and triplets were taken into account in performing the
calculations. Comparison with experimental data showed that the potential can be written in the form of e9 = - ?i,k A19 rik - 6 + ?i,k A29 rik - 12 + ?i,k,l Q9 I9 - 1\varepsilon _9 = - \sum\nolimits_{i,k} {A_{19} r_{ik}^{ - 6} } + \sum\nolimits_{i,k} {A_{29} r_{ik}^{ - 12} + \sum\nolimits_{i,k,l} {Q_9 I_9^{ - 1} } }, where I9 = [(r)\vec]ik2 [ ( [(r)\vec]ik [(r)\vec]kl )2 + ( [(r)\vec]li [(r)\vec]ik )2 ] + [(r)\vec]kl2 [ ( [(r)\vec]ik [(r)\vec]kl )2 + ( [(r)\vec]kl [(r)\vec]li )2 ] + [(r)\vec]li2 [ ( [(r)\vec]li [(r)\vec]ik )2 + ( [(r)\vec]kl [(r)\vec]li )2 ]I_9 = \vec r_{ik}^2 \left[ {\left( {\vec r_{ik} \vec r_{kl} } \right)^2 + \left( {\vec r_{li} \vec r_{ik} } \right)^2 } \right] + \vec r_{kl}^2 \left[ {\left( {\vec r_{ik} \vec r_{kl} } \right)^2 + \left( {\vec r_{kl} \vec r_{li} } \right)^2 } \right] + \vec r_{li}^2 \left[ {\left( {\vec r_{li} \vec r_{ik} } \right)^2 + \left( {\vec r_{kl} \vec r_{li} } \right)^2 } \right]. If the values of [(r)\vec]ik\vec r_{ik} are scaled in half-lattice constant units, then A19 = 1.22
ë t9
û GPa, A29 = 5.07 ×102
ë t15
û GPa, Q9 = 5.31
ë t9
û GPaA_{19} = 1.22\left\lfloor {\tau ^9 } \right\rfloor GPa, A_{29} = 5.07 \times 10^2 \left\lfloor {\tau ^{15} } \right\rfloor GPa, Q_9 = 5.31\left\lfloor {\tau ^9 } \right\rfloor GPa, and τ = 1.26 ?. It is shown that the condition of thermodynamic stability of a crystal requires that we allow for irreducible
interactions in atom triplets in at least four coordination spheres. The analytical expressions for the lattice sums determining
the contributions from irreducible interactions in the atom triplets to the second- and third-order elastic moduli of cubic
crystals in the case of interactions determined by I
9 are presented in the appendix. 相似文献
4.
Let S
2 be the 2-dimensional unit sphere and let J
α
denote the nonlinear functional on the Sobolev space H
1(S
2) defined by
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