Bogoliubov Excitations of Inhomogeneous Bose-Einstein ...

k
+ −
w (1)
k ′ k
(a)
2.4. Single scattering event
Figure 2.6.: First-order scattering envelope functions. (a) Envelope function w (1)
k ′ k
first-order scattering element (2.45c), plotted in the k ′ -plane. The circle of elastic scattering
is shown as a solid line. There is a node in the scattering amplitude (dashed
line) that separates forward scattering from backscattering with opposite signs. (b) Polar
plot [A(kξ, θ)] 2 of the angular envelope function (2.59) of elastic scattering for
kξ = 0.2 (red), 0.5, 1, 2, 5 (violet). The envelope is close to a dipole-radiation (p-wave)
pattern for sound waves kξ ≪ 1, and tends to an isotropic (s-wave) pattern for singleparticle
excitations kξ ≫ 1. In the intermediate regime, backscattering is favored over
forward scattering.
0.5
(b)
0.5
of the
simplifies significantly. As function of the momentum k and the scattering
angle θ = ∡(k, k ′ ), the elastic scattering amplitude writes
with a remarkably simple angular envelope
W (1)
k ′ �
�
k�
=
k ′ =k ɛ0 k A(kξ, θ)V ′
k −k, (2.58)
ɛk
A(kξ, θ) = k2 ξ 2 (1 − cos θ) − cos θ
k 2 ξ 2 (1 − cos θ) + 1
, (2.59)
see figure 2.6(b). The node of vanishing scattering amplitude of this envelope
is found to be at
cos θ0 = k2ξ2 1 + k2 . (2.60)
ξ2 In the deep sound-wave regime kξ → 0, the envelope A(θ) = − cos(θ)
presents the dipole radiation pattern. In the opposite limit kξ → ∞, the
nodes move to the forward direction. Finally, when the healing length ξ
becomes larger than the system size L, the node angle θ0 ≈ √ 2/kξ becomes
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