An electron of incident momentum **k**_{i} is scattered
elastically by the electric field of an atom of atomic number Z. The
potential due to the nucleus is of the form U_{0}(r) = -Ze^{2}/r.
This potential is screened by the atomic electron cloud. As a result, the **
total** potential energy of the incident electron is U(r) = (-Ze^{2}/r)exp(-r/a),
where a is the radius of the atom. Let **k**_{f} be the final
momentum of the electron and **q **=** k**_{f }-** k**_{i}
be the momentum transfer.

(a) Calculate the differential cross section, dσ/dΩ,
in the Born approximation for the scattering of the electron by the atom (using the screened potential
U(r)).

(b) Now calculate the differential cross section, dσ/dΩ, in the
Born approximation for the scattering of the electron by the nucleus only (using
the unscreened potential U_{0}(r)).

(c) Plot the ratio of the two cross sections as a function of x = a|q| and
briefly discuss the limits x --> 0 and x --> ∞.

Solution:

- Concepts:

The Born approximation - Reasoning:

We are asked to evaluate the scattering cross section in the Born approximation. - Details of the calculation:

(a) In the Born approximation we have

σ_{k}^{B}(θ,φ) = σ_{k}^{B}(**k**,**k**') = [μ^{2}/(4п^{2}ħ^{4})]|∫d^{3}r' exp(-i**q∙r**')U(**r**')|^{2},

where**q**=**k**' -**k**,**k**= μ**v**_{0}/h,**k**' = μv_{0}/ħ (**k**'/k'), and μ is the reduced mass.

Here μ = m_{e}.

**k**is the incident wave vector, and (**k**'/k') is a unit vector pointing in the direction (θ,φ).

With U(**r**) = U(r) = (-Ze^{2}/r)exp(-r/a) we have

∫d^{3}r' exp(-i**q∙r**')U(**r**') = -Ze^{2}∫∫∫r'dr'sinθ'dθ'dφ'exp(-r'/a)exp(-iqr'cosθ')

= -Ze^{2}2π∫_{0}^{∞}r'dr'exp(-r'/a)∫_{-1}^{1}exp(-iqr'cosθ')dcosθ'

= -(Ze^{2}2π/q)∫_{0}^{∞}dr'exp(-r'/a)∫_{-qr'}^{qr'}exp(-ix)dx

= -(Ze^{2}2π/q)∫_{0}^{∞}dr'exp(-r'/a)[-i(e^{iqr'}- e^{-iqr'})]

= -(Ze^{2}4π/q)∫_{0}^{∞}dr'exp(-r'/a)sin(qr') = -(Ze^{2}4π/q^{2})q^{2}a^{2}/(1 + q^{2}a^{2}).

Therefore σ_{k}^{B}(θ,φ) = [4μ^{2}Z^{2}e^{4}/ħ^{4}][1/((1/a^{2}) + q^{2})]^{2}.

With q = 2ksin(θ/2) we have σ_{k}^{B}(θ,φ) = [4μ^{2}Z^{2}e^{4}/ħ^{4}][1/((1/a^{2}) + 4k^{2}sin^{2}(θ/2))]^{2}.

(b) Let a --> ∞, then U(r) = (-Ze^{2}/r)exp(-r/a) --> U_{0}(r) = -Ze^{2}/r.

Then σ_{k}^{B}(θ,φ) = [4μ^{2}Z^{2}e^{4}/ħ^{4}]/(16k^{4}sin^{4}(θ/2)) = [Z^{2}e^{4}]/(16E^{2}sin^{4}(θ/2)).

This is the Rutherford cross section.

(c) Ratio: σ_{k}^{B}(θ,φ)_{part a}/σ_{k}^{B}(θ,φ)_{part b}= (q^{2}a^{2}/(1 + q^{2}a^{2}))^{2}= (x^{2}/(1 + x^{2}))^{2}with x = |qa|.

As x --> 0, the ratio approaches 0.

As a --> 0, x --> 0, the nuclear charge is completely screened, we have a neutral object.

As x --> ∞, the ratio approaches 1.

As a --> ∞, x --> ∞, the screening vanishes, we have a bare nucleus.

Consider an electron of energy E_{0} and velocity **v**_{0}
in the z-direction incident on an ionized Helium atom He^{+}, with just
one electron in its ground state. Compute the differential cross section dσ/dΩ
for the incident electron to scatter into the solid angle dΩ
about the spherical angles (θ,φ). Explain your
assumptions and approximations.

Solution:

- Concepts:

The Born approximation - Reasoning:

For fast electrons we can use the Born approximation to calculate the scattering cross section. - Details of the calculation:

In the Born approximation we have

σ_{k}^{B}(θ,φ) = σ_{k}^{B}(**k**,**k**') = [μ^{2}/(4п^{2}ħ^{4})]|∫d^{3}r' exp(-i**q∙r**')U(**r**')|^{2},

where**q**=**k**' -**k**,**k**= μ**v**_{0}/h,**k**' = μv_{0}/ħ (**k**'/k'), and μ is the reduced mass.

Here μ = m_{e},**k**is the incident wave vector, and (**k**'/k') is a unit vector pointing in the direction (θ,φ).

U(**r**) = -2e^{2}/r + e^{2}∫d^{3}r' ρ(**r**')/|**r**-**r**'| = -e^{2}∫d^{3}r' ρ_{t}(**r**')/|**r**-**r**'|,

where ρ_{t}(**r**') = 2 δ(**r**') - ρ(**r**').∫d

^{3}r exp(-i**q∙r**)U(**r**= -e^{2}∫d^{3}r exp(-i**q∙r**)∫d^{3}r' ρ_{t}(**r**')/|**r**-**r**'|

= -e^{2}∫d^{3}r' ρ_{t}(**r**')exp(-i**q∙r**')∫d^{3}r exp(-i**q∙**(**r**-**r**')/|**r**-**r**'|

= -(4πe^{2}/q^{2})∫d^{3}r' ρ_{t}(**r**')exp(-i**q∙r**') = -(4πe^{2}/q^{2})[2 - F(q)],

where F(q) = ∫d^{3}r' ρ(**r**')exp(-i**q∙r**') is the**atomic scattering form factor**.[To evaluate ∫d

^{3}r exp(-i**q∙**(**r**-**r**')/|**r**-**r**'| we have used

∫d^{3}r e^{-λr }exp(i**k∙r**)/r = 2π∫_{0}^{∞}dr r e^{-λr}∫_{-1}^{1}d(cosθ)e^{ikrcosθ}

= 2π∫_{0}^{∞}dr e^{-λr}(e^{ikr}- e^{-ikr})/(ik) = (4π/k)Im(∫_{0}∞dre^{-(λ-ik)r})^{ }= (4π/k)Im(1/(λ - ik)) = 4π/(λ^{2}+ k^{2}).

Take the limit as λ --> 0.]σ

_{k}^{B}(θ,φ) = [4m_{e}^{2}e^{4}/(q^{4}ħ^{4})][2 - F(q)]^{2}.

q^{2}= 4k^{2}sin^{2}(θ/2).

σ_{k}^{B}(θ,φ) = [4m^{2}e^{4}/(16ħ^{4}k^{4}sin^{4}(θ/2))][2 - F(q)]^{2 }= (e^{4}/(16E^{2}sin^{4}(θ/2))[2 - F(q)]^{2}.

If we assume that F(q) = 1, i.e. that the incoming electron sees a screened nucleus of charge 1, we obtain the Rutherford scattering cross section.

Evaluate the differential
scattering cross section for a particle of mass m in a repulsive
potential U(r) = A/r^{2} in the Born approximation.

Solution:

- Concepts:

The Born approximation - Reasoning:

We are asked to evaluate the scattering cross section in the Born approximation. -
Details of the calculation:

In the Born approximation we have

σ_{k}^{B}(θ,φ) = σ_{k}^{B}(**k**,**k**') = [μ^{2}/(4π^{2}ħ^{4})]|∫d^{3}r' exp(-i**q∙r**')U(**r**')|^{2},

where**q**=**k**' -**k**,**k**= μ**v**_{0}/ħ,**k**' = μv_{0}/ħ (**k**'/k'), and μ is the reduced mass.

Here μ = m,**k**is the incident wave vector, and (**k**'/k') is a unit vector pointing in the direction (θ,φ).

With U(r) = A/r^{2}we have

∫d^{3}r' exp(-i**q∙r**')U(**r**') = A∫∫∫dr'sinθ'dθ'dφ'exp(-iqr'cosθ')

= 2πA∫_{0}^{∞}dr'∫_{-1}^{1}exp(-iqr'cosθ')dcosθ'

= (4πA/q)∫_{0}^{∞}dr'sin(qr')/r' = 2π^{2}A/q.

σ_{k}^{B}(θ,φ) = [μ^{2}/(4π^{2}ħ^{4})][4π^{4}A^{2}/q^{2}] = μ^{2}π^{2}A^{2}/(ħ^{4}q^{2}).

With q = 2ksin(θ/2) we have σ_{k}^{B}(θ,φ) = μ^{2}π^{2}A^{2}/(4ħ^{4}k^{2}sin^{2}(θ/2)).

Evaluate, in the Born Approximation, the differential cross section for the
scattering of a particle of mass m by a delta-function potential U(**r**) = Bδ(**r**). Comment on the angular and velocity dependence.
Find the total cross section.

Solution:

- Concepts:

The Born Approximation - Reasoning:

We are asked to evaluate the scattering cross section in the Born approximation. - Details of the calculation:

In the Born approximation

σ_{k}^{B}(θ,φ) = σ_{k}^{B}(**k**,**k**') = [μ^{2}/(4п^{2}ħ^{4})]|∫d^{3}r' exp(-i**q∙r**')U(**r**')|^{2}.

σ_{k}^{B}(θ,φ) = [Bμ^{2}/(4π^{2}ħ^{4})]|∫d^{3}r' exp(-i**q∙r**')δ(**r**')|^{2}= Bμ^{2}/(4π^{2}ħ^{4}).

The differential scattering cross section is independent of scattering angle and velocity.

σ_{k}^{B}= Bμ^{2}/(πħ^{4}) is the total scattering cross section.

Consider two distinguishable spin-½ particles interacting
via the interaction

H_{int} = g**S**_{1}·**S**_{2}δ^{3}(**r**),

where g is a constant, **S**_{i} are the spin operators of the
two particles, and **r** is the separation of the two particles.

(a) Using the Born approximation, calculate the differential cross section in
the center of mass frame for particles in the spin-singlet and spin-triplet
states, assuming the particles are non-identical.

(b) What is the total cross section if the particles are unpolarized?

Solution:

- Concepts:

Born approximation, two spin ½ particles - Reasoning:

In the CM frame we are only interested in the relative motion of the particles. They are treated as a fictitious particle with wave function Φ(**r**)⊗Χ(S,M_{s}), where**r**=**r**_{1}-**r**_{2}is the relative coordinate and Χ(S,M_{s}) characterizes the spin of the system.

For this fictitious particle In the Born approximation, neglecting spin, we have

dσ_{k}^{B}/dΩ = σ_{k}^{B}(θ,φ) = σ_{k}^{B}(**k**,**k**') = [μ^{2}/(4π^{2}ħ^{4})]|∫d^{3}r' exp(-i**q∙r**')U(**r**')|^{2},

where**q**=**k**' -**k**,**k**= μ**v**_{0}/ħ,**k**' = μv_{0}/ħ (**k**'/k'), and μ is the reduced mass.

Here ħ**q**is the momentum transfer and q = 2ksin(θ/2).

The differential scattering cross section is proportional to the Fourier transform of the potential.

Note:

We can derive the above expression for dσ_{k}^{B}/dΩ using time-dependent perturbation theory and Fermi’s golden rule. If H = H_{0 }+ Wexp(±iωt), then the transition probability per unit time to a group of states nearly equal in energy E = E_{i}+ ħω is given by

w(i,βE) = (2π/ħ)ρ(β,E)|W_{Ei}|^{2}δ_{E-Ei,ħω}, where W_{Ei}= <Φ_{E}|W|Φ_{i}>.

Here we consider an initial state with momentum ħ**k**and a final state with momentum ħ**k**'. The box normalized wave functions are given as

Φ_{i}(**r**) = L^{-3/2}exp(i**k∙r**), Φ_{f}(**r**) = L^{-3/2}exp(i**k**'**∙r**),

where L is the dimension of the box.

We let W = U(**r**), ω = 0, and β = dΩ. Then

W_{Ei}= <Φ_{E}|U|Φ_{i}> = L^{-3}∫d^{3}r' exp(i**q∙r**')U(**r**').

The energy eigenstates of a particle confined to a cubical box with periodic boundary conditions are

Φ_{nx,ny,nz}(x,y,z) = L^{-3}exp(i2π(n_{x}x + n_{y}y + n_{z}z)/L),

with n_{x}, n_{y}, n_{z}= 0, ±1, ±2, ... .

We have k_{x}= 2πn_{x}/L, k_{y}= 2πn_{y}/L, k_{z}= 2πn_{z}/L.

If k is large, then the number of states with wave vectors whose magnitudes lie between k and k + dk within an element of solid angle dΩ is

dN = k^{2}dΩdk/(2π/L)^{3}= L^{3}k^{2}dΩdk/(2π)^{3}.

dN/dk = L^{3}k^{2}dΩ/(2π)^{3}.

The density of states is ρ(E,dΩ) = dN/dE = (dN/dk)(dk/dE).

With E = ħ^{2}k^{2}/(2μ) we have dN/dE = μħkdΩL^{3}/(2πħ)^{3}.

We now can insert W_{Ei}and ρ(E,dΩ) into the expression for the transition probability per unit time.

w(i,dΩE) = [μħkdΩ/(4π^{2}ħ^{4}L^{3})]|∫d^{3}r' exp(-i**q∙r**')U(**r**')|^{2}.

Here w(i,dΩE) is the probability per unit time that a particle scatters into the solid angle dΩ. The cross section is the transition rate per incident flux. The incident flux is the incident probability density times the velocity, |Φ_{i}(**r**)|^{2}v = L^{-3}v = L^{-3}ħk/μ. Therefore we have

σ_{k}^{B}(**k**,**k**')dΩ = [μ^{2}/(4π^{2}ħ^{4})]|∫d^{3}r' exp(-i**q∙r**')U(**r**')|^{2}dΩ.

This derivation makes it clear how to incorporate a spin-dependent potential into the Born approximation. - Details of the calculation:

(a) Assume U = g**S**_{1}·**S**_{2}d^{3}(**r**) and the two particles are in an eigenstate of S^{2}and S_{z}.

Then σ_{k}^{B}(**k**,**k**') = [g^{2}m^{2}/(4π^{2}ħ^{4})]|∫d^{3}r' exp(-i**q∙r**')δ^{3}(**r**')<S,M_{s}|**S**_{1}·**S**_{2}|S,M_{s}>|^{2}.^{ }**S**_{1}∙**S**_{2}= ½(S^{2}- S_{1}^{2}- S_{2}^{2}).

**S**_{1}∙**S**_{2}|S,M_{s}> = (ħ^{2}/2)(S(S+1) - ½(3/2) - ½(3/2))|S,M_{s}> = (ħ^{2}/2)(S(S+1) - (3/2))|S,M_{s}>.

For the singlet state we have <0,0|**S**_{1}·**S**_{2}|0,0> = -3ħ^{2}/4, and for the triplet state we have <1,M_{s}|**S**_{1}·**S**_{2}|1,M_{s}> = ħ^{2}/4.

Therefore σ_{k}^{B}(**k**,**k**') = <S,M_{s}|**S**_{1}·**S**_{2}|S,M_{s}>^{2}[g^{2}m^{2}/(4π^{2}ħ^{4})]|∫d^{3}r' exp(-i**q∙r**')δ^{3}(**r**')|^{2}.

[g^{2}m^{2}/(4π^{2}ħ^{4})]|∫d^{3}r' exp(-i**q∙r**')δ^{3}(**r**')|^{2}= g^{2}m^{2}/(4π^{2}ħ^{4}).

For particles in the singlet state we have σ_{k}^{B}(**k**,**k**')_{singlet}= 9g^{2}μ^{2}/(64π^{2}).

For particles in the triplet state we have σ_{k}^{B}(**k**,**k**')_{triplet}= g^{2}μ^{2}/(64π^{2}).

(b) For unpolarized particles we have

σ_{k}^{B}(**k**,**k**')_{random}= (σ_{k}^{B}(**k**,**k**')_{singlet}+ 3σ_{k}^{B}(**k**,**k**')_{triplet})/4 = 3g^{2}μ^{2}/(64π^{2}).

Consider a periodic scattering potential with translational
invariance U(**r** + **R**) = U(**r**), where **R** is a constant
vector. Show that in the Born approximation scattering occurs only in the
directions defined by (**k**_{i} – **k**_{f})**·R** =
2πn where **k**_{i} is the initial wave vector, **k**_{f} is the final wave vector and n is an integer.

Solution:

- Concepts:

The Born approximation - Reasoning:

We are asked to use the Born approximation - Details of the calculation:

The elastic scattering cross section in the Born Approximation is

σ_{k}^{B}(θ,φ) = σ_{k}^{B}(**k**_{i},**k**_{f}) = [μ^{2}/(4π^{2}ħ^{4})]|∫d^{3}r' exp(-i**q∙r**')U(**r**')|^{2},

where**q**=**k**_{f}–**k**_{fi},**k**_{i}= μ**v**_{0}/ħ,**k**_{f}= μv_{0}/ħ (**k**_{f}/k_{f}), and μ is the reduced mass.

U is a periodic potential, we can write it in terms of a Fourier series.U(

**r**) = ∑_{-∞}^{+∞}C(**k**_{m})exp(i**k**_{m}**∙r**).Here

**k**_{n }= n2π/R(**R**/R), Δk = k_{n+1 }- k_{n}= 2π/R, n = integer.

(Note:**R**/R denotes the unit vector in the direction of**R**.)∫d

^{3}r' exp(-i**q∙r**')U(**r**') = ∑_{-∞}^{+∞}C(**k**_{m})∫d^{3}r' exp(-i(**q**-**k**_{m})**∙r**') = 2π ∑_{-∞}^{+∞}C(**k**_{m})δ(**k**_{m}-**q**)

The integral is zero unless (**k**_{m}-**q**) = 0, (**k**_{f}–**k**_{i})**k**_{m}, (**k**_{f}–**k**_{i})**·R**= m2π.

Scattering occurs only in the directions defined by (**k**_{f}–**k**_{i})**·R**= 2πm.