- Source: Scattering amplitude
In quantum physics, the scattering amplitude is the probability amplitude of the outgoing spherical wave relative to the incoming plane wave in a stationary-state scattering process. At large distances from the centrally symmetric scattering center, the plane wave is described by the wavefunction
ψ
(
r
)
=
e
i
k
z
+
f
(
θ
)
e
i
k
r
r
,
{\displaystyle \psi (\mathbf {r} )=e^{ikz}+f(\theta ){\frac {e^{ikr}}{r}}\;,}
where
r
≡
(
x
,
y
,
z
)
{\displaystyle \mathbf {r} \equiv (x,y,z)}
is the position vector;
r
≡
|
r
|
{\displaystyle r\equiv |\mathbf {r} |}
;
e
i
k
z
{\displaystyle e^{ikz}}
is the incoming plane wave with the wavenumber k along the z axis;
e
i
k
r
/
r
{\displaystyle e^{ikr}/r}
is the outgoing spherical wave; θ is the scattering angle (angle between the incident and scattered direction); and
f
(
θ
)
{\displaystyle f(\theta )}
is the scattering amplitude. The dimension of the scattering amplitude is length. The scattering amplitude is a probability amplitude; the differential cross-section as a function of scattering angle is given as its modulus squared,
d
σ
=
|
f
(
θ
)
|
2
d
Ω
.
{\displaystyle d\sigma =|f(\theta )|^{2}\;d\Omega .}
The asymptotic form of the wave function in arbitrary external field takes the form
ψ
=
e
i
k
r
n
⋅
n
′
+
f
(
n
,
n
′
)
e
i
k
r
r
{\displaystyle \psi =e^{ikr\mathbf {n} \cdot \mathbf {n} '}+f(\mathbf {n} ,\mathbf {n} '){\frac {e^{ikr}}{r}}}
where
n
{\displaystyle \mathbf {n} }
is the direction of incidient particles and
n
′
{\displaystyle \mathbf {n} '}
is the direction of scattered particles.
Unitary condition
When conservation of number of particles holds true during scattering, it leads to a unitary condition for the scattering amplitude. In the general case, we have
f
(
n
,
n
′
)
−
f
∗
(
n
′
,
n
)
=
i
k
2
π
∫
f
(
n
,
n
″
)
f
∗
(
n
,
n
″
)
d
Ω
″
{\displaystyle f(\mathbf {n} ,\mathbf {n} ')-f^{*}(\mathbf {n} ',\mathbf {n} )={\frac {ik}{2\pi }}\int f(\mathbf {n} ,\mathbf {n} '')f^{*}(\mathbf {n} ,\mathbf {n} '')\,d\Omega ''}
Optical theorem follows from here by setting
n
=
n
′
.
{\displaystyle \mathbf {n} =\mathbf {n} '.}
In the centrally symmetric field, the unitary condition becomes
I
m
f
(
θ
)
=
k
4
π
∫
f
(
γ
)
f
(
γ
′
)
d
Ω
″
{\displaystyle \mathrm {Im} f(\theta )={\frac {k}{4\pi }}\int f(\gamma )f(\gamma ')\,d\Omega ''}
where
γ
{\displaystyle \gamma }
and
γ
′
{\displaystyle \gamma '}
are the angles between
n
{\displaystyle \mathbf {n} }
and
n
′
{\displaystyle \mathbf {n} '}
and some direction
n
″
{\displaystyle \mathbf {n} ''}
. This condition puts a constraint on the allowed form for
f
(
θ
)
{\displaystyle f(\theta )}
, i.e., the real and imaginary part of the scattering amplitude are not independent in this case. For example, if
|
f
(
θ
)
|
{\displaystyle |f(\theta )|}
in
f
=
|
f
|
e
2
i
α
{\displaystyle f=|f|e^{2i\alpha }}
is known (say, from the measurement of the cross section), then
α
(
θ
)
{\displaystyle \alpha (\theta )}
can be determined such that
f
(
θ
)
{\displaystyle f(\theta )}
is uniquely determined within the alternative
f
(
θ
)
→
−
f
∗
(
θ
)
{\displaystyle f(\theta )\rightarrow -f^{*}(\theta )}
.
Partial wave expansion
In the partial wave expansion the scattering amplitude is represented as a sum over the partial waves,
f
=
∑
ℓ
=
0
∞
(
2
ℓ
+
1
)
f
ℓ
P
ℓ
(
cos
θ
)
{\displaystyle f=\sum _{\ell =0}^{\infty }(2\ell +1)f_{\ell }P_{\ell }(\cos \theta )}
,
where fℓ is the partial scattering amplitude and Pℓ are the Legendre polynomials. The partial amplitude can be expressed via the partial wave S-matrix element Sℓ (
=
e
2
i
δ
ℓ
{\displaystyle =e^{2i\delta _{\ell }}}
) and the scattering phase shift δℓ as
f
ℓ
=
S
ℓ
−
1
2
i
k
=
e
2
i
δ
ℓ
−
1
2
i
k
=
e
i
δ
ℓ
sin
δ
ℓ
k
=
1
k
cot
δ
ℓ
−
i
k
.
{\displaystyle f_{\ell }={\frac {S_{\ell }-1}{2ik}}={\frac {e^{2i\delta _{\ell }}-1}{2ik}}={\frac {e^{i\delta _{\ell }}\sin \delta _{\ell }}{k}}={\frac {1}{k\cot \delta _{\ell }-ik}}\;.}
Then the total cross section
σ
=
∫
|
f
(
θ
)
|
2
d
Ω
{\displaystyle \sigma =\int |f(\theta )|^{2}d\Omega }
,
can be expanded as
σ
=
∑
l
=
0
∞
σ
l
,
where
σ
l
=
4
π
(
2
l
+
1
)
|
f
l
|
2
=
4
π
k
2
(
2
l
+
1
)
sin
2
δ
l
{\displaystyle \sigma =\sum _{l=0}^{\infty }\sigma _{l},\quad {\text{where}}\quad \sigma _{l}=4\pi (2l+1)|f_{l}|^{2}={\frac {4\pi }{k^{2}}}(2l+1)\sin ^{2}\delta _{l}}
is the partial cross section. The total cross section is also equal to
σ
=
(
4
π
/
k
)
I
m
f
(
0
)
{\displaystyle \sigma =(4\pi /k)\,\mathrm {Im} f(0)}
due to optical theorem.
For
θ
≠
0
{\displaystyle \theta \neq 0}
, we can write
f
=
1
2
i
k
∑
ℓ
=
0
∞
(
2
ℓ
+
1
)
e
2
i
δ
l
P
ℓ
(
cos
θ
)
.
{\displaystyle f={\frac {1}{2ik}}\sum _{\ell =0}^{\infty }(2\ell +1)e^{2i\delta _{l}}P_{\ell }(\cos \theta ).}
X-rays
The scattering length for X-rays is the Thomson scattering length or classical electron radius, r0.
Neutrons
The nuclear neutron scattering process involves the coherent neutron scattering length, often described by b.
Quantum mechanical formalism
A quantum mechanical approach is given by the S matrix formalism.
Measurement
The scattering amplitude can be determined by the scattering length in the low-energy regime.
See also
Veneziano amplitude
Plane wave expansion
References
Kata Kunci Pencarian:
- Scattering amplitude
- Atomic form factor
- Unitarity (physics)
- Regge theory
- Amplituhedron
- Veneziano amplitude
- Scattering
- High resolution electron energy loss spectroscopy
- X-ray diffraction
- Associahedron
