Electromagnetic Waves - Introduction to Imaging from Scattered Fields - page 22

3

=− �

1
2

-
‚

e
KK

− iK ′ ,( rr K − ′ ) d ′

∫

(2.47)

Grr (, ) ′

π

2

− () ′

2

− ∫

ψϕ () r = () r GrrUr (, )( )( ) ′ ′ ψ r ′ d r ′ (2.48)

where ψ( r ) is the total wave, φ( r ) is the incident or unscattered wave and the
integral represents the total scattered wave. If the scattering is weak then
ψ ≃ φ, which is the first Born approximation. Examples of when the first Born
approximation might be valid include the case when an electron incident on
an atom has extremely high energy or when an electron approaches a nucleus
and needs to be very close for a significant interaction to take place. In elec-
tromagnetic interactions, x-ray illumination of materials is typically a weak
interaction.
Now we consider

1
2

e K
KK

iK ′ , r

d ′

∫

Gr (, ) 0 =−

() π

3 2

− () ′

2

(2.49)

∞ π 2 π

e ()

()

K ′ sindd d θθ ϕ K ′

1
2

iK r ′ cos θ 2

∫∫ ∫

=−

() π

3

KK

2

− ′

2

K ′ = 00 0 , θ == , ϕ

Here the expression is written in spherical polar coordinates with
d

2 θθ ϕ
If there is no φ dependence (i.e., only the scattering angle important and not
direction) then

KK ′ = () ′ sindd d K ′ .

π π

1

∫ ∫ ∫

e

iK r ′ cos θ

sind θθ = ( e

iKr ′

− e

− iK r ′

) if ex e

iKx ′

d =−

iK r ′ cos dcos θ

( θ ) (2.50)

iK r ′

0 0

∞

− 2 π e

iK r ′

e KK
KK

−

− iKr ′

′ d ′

∫

Gr (, ) 0 =

() 2 π

3 2

− () ′

2

0

(2.51)

∞

1 1 [ e

iK r iK e KK

KK

−

−

]
()

′

′ d ′

∫

=

4 π

2

i r

2

− ′

2

0

This function has poles at K ′ = ± K . The limit can now be taken as follows:

∞

1
4

1 ( e

iK r ′

−
+−

e

− iKr ′

) KK ′ d ′

∫

Gr (, ) 0 =− lim (2.52)

π ir

( Ki

2

ε ) () K ′

2

ε → 0

0

There are three different approaches to taking this limit, giving three dif-
ferent answers. We could write ( K 2 − i ε) − ( K ′) 2 which switches from outgoing
to ingoing wave, that is, this switches the time origin.