^{a)}, L. C. Botten

^{2}, N. A. Nicorovici

^{1,2}and I. J. Zucker

^{3}

### Abstract

We discuss general properties of doubly periodic sums over the square lattice, linking phased, Bloch-type sums in the direct lattice with displaced sums in the reciprocal lattice using the Poisson summation formula. We discuss cardinal points, where the sums reduce to a single product of Dirichlet functions, and exhibit all cardinal points for the square lattice. We introduce a new method for evaluating lattice sums and illustrate this by solving low order systems of sums of integer order 2,3,4,5. For the last case, the analytic expressions for the sums involve complementary functions, or alternatively functions with complex characters.

This work was supported by the Australian Research Council. We acknowledge helpful discussions with M. L. Glasser and M. Watkins.

I. INTRODUCTION

II. THE LATTICE SUMS GREEN’S FUNCTION

III. MACDONALD FUNCTION SERIES

IV. GENERAL PROPERTIES OF THE PHASED ARRAY SUMS

V. CARDINAL POINTS IN THE BRILLOUIN ZONE

VI. THE REFLECTION FORMULAS FOR GENERAL

VII. EXAMPLES OF EVEN/ODD DECOMPOSITIONS

A.

B.

C.

D.

VIII. SOLVED USING COMPLEX DIRICHLET FUNCTIONS

IX. CONCLUSION

### Key Topics

- Real functions
- 18.0
- Brillouin scattering
- 13.0
- Cardinal points
- 12.0
- Eigenvalues
- 12.0
- Functional equations
- 6.0

## Figures

(Color online) The top plot shows a surface plot of , while on the bottom is a contour plot of the same function.

(Color online) The top plot shows a surface plot of , while on the bottom is a contour plot of the same function.

The modulus of is shown for 80 equally spaced points with angles ranging from to , on the contour , and with corresponding to the first zero of on the critical line.

The modulus of is shown for 80 equally spaced points with angles ranging from to , on the contour , and with corresponding to the first zero of on the critical line.

## Tables

Definition of the main symbols in this paper.

Definition of the main symbols in this paper.

The value of the Green’s function at particular points in the Brillouin zone.

The value of the Green’s function at particular points in the Brillouin zone.

The first zero of each , the values at one and the values of from Eq. (41) for primitive Dirichlet functions with .

The first zero of each , the values at one and the values of from Eq. (41) for primitive Dirichlet functions with .

Sums which may be derived from the results of Sec. VII A.

Sums which may be derived from the results of Sec. VII A.

Sums which may be derived from the results of Sec. VII B.

Sums which may be derived from the results of Sec. VII B.

Sums which may be derived from the results of Sec. VII C.

Sums which may be derived from the results of Sec. VII C.

Sums which may be derived from the results of Sec. VII D. For each sum in the first column, the coefficients are given of the functions in the heads of the other columns.

Sums which may be derived from the results of Sec. VII D. For each sum in the first column, the coefficients are given of the functions in the heads of the other columns.

Sums which may be derived from the results of Sec. VIII, involving functions with complex characters. For each sum in the first column, the coefficients are given of the functions in the heads of the other columns.

Sums which may be derived from the results of Sec. VIII, involving functions with complex characters. For each sum in the first column, the coefficients are given of the functions in the heads of the other columns.

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