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Calculation of lattice sums of general type. (English) Zbl 1469.82034

Summary: An efficient calculation of lattice sums is given. The proposed algorithm is based on Ewald decomposition and gives a new insight into the calculation of lattice sums with an arbitrary degree of \(\vert\mathbf{R}-\mathbf{r}\vert\), that arise in a solution to many problems of the crystalline state. Its implementation to the calculation of the Madelung constant is presented for PC, BCC and FCC lattices.

MSC:

82D25 Statistical mechanics of crystals
33F05 Numerical approximation and evaluation of special functions
40A25 Approximation to limiting values (summation of series, etc.)
40A30 Convergence and divergence of series and sequences of functions
40B05 Multiple sequences and series
40C15 Function-theoretic methods (including power series methods and semicontinuous methods) for summability
40F05 Absolute and strong summability
40H05 Functional analytic methods in summability
65D20 Computation of special functions and constants, construction of tables
82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics
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