One-body potential theory, which includes the effect of exchange and correlation forces, is used to calculate the change in the electron density due to small displacements of the ions. The final result contains a Dirac density matrix for the perfect crystal, the diagonal element being the exact ground state density
ρ
0
(
r
). The basic quantity
R
(
r
) determining the electronic contribution to the dynamical matrix is such that the gradient of
ρ
0
(
r
) is obtained by superposition of
R
(
r
-
l
) on each lattice site
l
. An integral equation is obtained which gives
R
(
r
) uniquely once the exchange and correlation energy is known. The Fourier transform
R
k
of
R
(
r
) is given in term s of the Fourier components
ρ
K
n
of the charge density, which are known from X-ray scattering, by
R
K
n
= i
ρKn
K
n
the reciprocal lattice vectors
K
n
. This is the same result as the rigid-ion model at the
K
n
's, which makes the assumption that this is true for all
k
. Deviations from rigid ions can be evaluated quantitatively from the integral equation obtained here. Such deviations reflect the role of many-body forces in lattice dynamics and the present theory provides a systematic basis for their calculation.
High-resolution diffuse X-ray scattering study of defect structure in silicon single crystals by exploring reciprocal space around different lattice points with a four-crystal X-ray diffractometer