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Evaluation of a new lattice sum
Abstract
The lattice sum Sigma r-n cos(ar) exp(i alpha .r) has not yet been tackled for an arbitrary value of n. Here the authors obtain quite easily a rapidly convergent expression for this sum, by using a Fourier transform relationship. This result is applied to evaluate the Fourier transform of the Ruderman-Kittel interaction. It is shown that from a special case of the Fourier transform relationship, one may derive the Ewald transformation.
... In particular, one can try to directly evaluate the world-sheet integrals in closed-string genusone amplitudes thus obtaining lattice-sum representations of MGFs [1][2][3][4]18]. Although it is possible to extract the asymptotic expansion at the cusp τ → i∞ from some of these lattice-sum representations [39,48], this is nonetheless a hard task suggesting that a different approach might in general be necessary. ...
... 48) and then collect the appropriate powers of t in the integrand, we arrive at φ . . . .All of the results here discussed can be checked for comparison with the examples given in section 3.3 and are consistent with the Laplace equation (2.15). ...
A bstract
Modular graph functions arise in the calculation of the low-energy expansion of closed-string scattering amplitudes. For toroidal world-sheets, they are SL(2 , ℤ)-invariant functions of the torus complex structure that have to be integrated over the moduli space of inequivalent tori. We use methods from resurgent analysis to construct the non-perturbative corrections arising for two-loop modular graph functions when the argument of the function approaches the cusp on this moduli space. SL(2 , ℤ)-invariance will in turn strongly constrain the behaviour of the non-perturbative sector when expanded at the origin of the moduli space.
... In particular, one can try to directly evaluate the world-sheet integrals in closed-string genus-one amplitudes thus obtaining lattice-sum representations of MGFs [1][2][3][4]18]. Although it is possible to extract the asymptotic expansion at the cusp τ → i∞ from some of these lattice-sum representations [48,39], this is nonetheless a hard task suggesting that a different approach might in general be necessary. ...
Modular graph functions arise in the calculation of the low-energy expansion of closed-string scattering amplitudes. For toroidal world-sheets, they are -invariant functions of the torus complex structure that have to be integrated over the moduli space of inequivalent tori. We use methods from resurgent analysis to construct the non-perturbative corrections arising when the argument of the modular graph function approaches the cusp on this moduli space. -invariance will in turn strongly constrain the behaviour of the non-perturbative sector when expanded at the origin of the moduli space.
In order to calculate the electrostatic contributions to the elastic constants of piezoelectric crystals, Fuller and Naimon [Phys. Rev. B. 6, 3609 (1972)] used the conventional method of homogeneous deformation by evaluating the lattice sums using the Ewald transformation and omitting the zero-wave-vector term. This procedure, however, lacks justification, since it is well known that the conventional homogeneous deformation theory breaks down for piezoelectric crystals. We develop here the full theory for this case. (The use of a new technique for performing the theta transformation makes the treatment much simpler.) The theory verifies that the procedure of Fuller and Naimon will give the correct contribution to the elastic constants. We also explain why the zero-wave-vector term remained absent in their treatment. Moreover, we derive for an arbitrary crystal structure some relationships between the electrostatic contributions to the different second- and third-order elastic constants. In view of these relationships, one has to calculate a lesser number of electrostatic contributions for a given crystal structure, and some of the evaluations of Fuller and Naimon become redundant.
In previous papers by Misra(1) and by Born and Misra(2) lattice sums of the type required in discussing the stability of a cubic crystal of the Bravais type in which the forces are central have been calculated. In the investigation of the thermodynamic properties of crystals a more general type of lattice sum occurs, which involves the phases of the waves. In the present paper a method of calculating these sums is developed and tables are computed.(Received November 14 1942)