The Born approximation

Problem:

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 U₀(r) = -Ze²/r.  This potential is screened by the atomic electron cloud.  As a result, the total potential energy of the incident electron is U(r) = (-Ze²/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 U₀(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') = [μ²/(4п²ħ⁴)]|∫d³r' exp(-iq∙r')U(r')|²,
  where q = k' - k, k = μv₀/h, k' = μv₀/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) = (-Ze²/r)exp(-r/a) we have
  ∫d³r' exp(-iq∙r')U(r') = -Ze²∫∫∫r'dr'sinθ'dθ'dφ'exp(-r'/a)exp(-iqr'cosθ')
  = -Ze²2π∫₀^∞r'dr'exp(-r'/a)∫_-1¹exp(-iqr'cosθ')dcosθ'
  = -(Ze²2π/q)∫₀^∞dr'exp(-r'/a)∫_-qr'^qr'exp(-ix)dx
  = -(Ze²2π/q)∫₀^∞dr'exp(-r'/a)[-i(e^iqr' - e^-iqr')]
  = -(Ze²4π/q)∫₀^∞dr'exp(-r'/a)sin(qr') = -(Ze²4π/q²)q²a²/(1 + q²a²).
  Therefore σ_k^B(θ,φ) = [4μ²Z²e⁴/ħ⁴][1/((1/a²) + q²)]². 
  With  q = 2ksin(θ/2) we have σ_k^B(θ,φ) = [4μ²Z²e⁴/ħ⁴][1/((1/a²) + 4k²sin²(θ/2))]².
  (b)  Let a --> ∞, then  U(r) = (-Ze²/r)exp(-r/a) -->  U₀(r) = -Ze²/r.
  Then  σ_k^B(θ,φ) = [4μ²Z²e⁴/ħ⁴]/(16k⁴sin⁴(θ/2)) = [Z²e⁴]/(16E²sin⁴(θ/2)).
  This is the Rutherford cross section.
  (c)  Ratio: σ_k^B(θ,φ)_{part a}/σ_k^B(θ,φ)_{part b} = (q²a²/(1 + q²a²))² = (x²/(1 + x²))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.

Problem:

Consider an electron of energy E₀ and velocity v₀ 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') = [μ²/(4п²ħ⁴)]|∫d³r' exp(-iq∙r')U(r')|²,
  where q = k' - k, k = μv₀/h, k' = μv₀/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) = -2e²/r + e²∫d³r' ρ(r')/|r - r'| = -e²∫d³r' ρ_t(r')/|r - r'|,
  where ρ_t(r') = 2 δ(r') - ρ(r').

  ∫d³r exp(-iq∙r)U(r) = -e²∫d³r exp(-iq∙r)∫d³r' ρ_t(r')/|r - r'|
  = -e²∫d³r' ρ_t(r')exp(-iq∙r')∫d³r exp(-iq∙(r - r')/|r - r'|
  = -(4πe²/q²)∫d³r' ρ_t(r')exp(-iq∙r') = -(4πe²/q²)[2 - F(q)],
  where F(q) = ∫d³r' ρ(r')exp(-iq∙r') is the atomic scattering form factor.

  [To evaluate ∫d³r exp(-iq∙(r - r')/|r - r'| we have used
  ∫d³r e^-λr exp(iq∙r)/r = 2π∫₀^∞dr r e^-λr∫_-1¹d(cosθ)e^iqrcosθ
  = 2π∫₀^∞dr e^-λr(e^iqr - e^-iqr)/(iq) = (4π/q)Im(∫₀∞dre^-(λ-iq)r)
  = (4π/q)Im(1/(λ - iq)) = 4π/(λ² + q²).
  Take the limit as λ --> 0.]

  σ_k^B(θ,φ)  = [4m_e²e⁴/(q⁴ħ⁴)][2 - F(q)]².
  q² = 4k²sin²(θ/2).
  σ_k^B(θ,φ)  = [4m²e⁴/(16ħ⁴k⁴sin⁴(θ/2))][2 - F(q)]²
  = (e⁴/(16E²sin⁴(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.

Problem:

Evaluate the differential scattering cross section for a particle of mass m in a repulsive potential U(r) = A/r² 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') = [μ²/(4π²ħ⁴)]|∫d³r' exp(-iq∙r')U(r')|²,
  where q = k' - k, k = μv₀/ħ, k' = μv₀/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/r² we have
  ∫d³r' exp(-iq∙r')U(r') = A∫∫∫dr'sinθ'dθ'dφ'exp(-iqr'cosθ')
  = 2πA∫₀^∞dr'∫_-1¹exp(-iqr'cosθ')dcosθ'
  = (4πA/q)∫₀^∞dr'sin(qr')/r' = 2π²A/q.
  σ_k^B(θ,φ) = [μ²/(4π²ħ⁴)][4π⁴A²/q²] = μ²π²A²/(ħ⁴q²).
  With  q = 2ksin(θ/2) we have σ_k^B(θ,φ) = μ²π²A²/(4ħ⁴k²sin²(θ/2)).

Problem:

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') = [μ²/(4п²ħ⁴)]|∫d³r' exp(-iq∙r')U(r')|².
  
  
  σ_k^B(θ,φ) = [Bμ²/(4π²ħ⁴)]|∫d³r' exp(-iq∙r')δ(r')|² = Bμ²/(4π²ħ⁴).
  The differential scattering cross section is independent of scattering angle and velocity.
  σ_k^B = Bμ²/(πħ⁴) is the total scattering cross section.

Problem:

Consider two distinguishable spin-½ particles interacting via the interaction
H_int = gS₁·S₂δ³(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 = r₁ - r₂ 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') = [μ²/(4π²ħ⁴)]|∫d³r' exp(-iq∙r')U(r')|²,
  where q = k' - k, k = μv₀/ħ, k' = μv₀/ħ (k'/k'), and μ is the reduced mass.
  Here ħq 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 = H₀ + 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|²δ_E-Ei,ħω, where W_Ei = <Φ_E|W|Φ_i>.
  Here we consider an initial state with momentum ħk and a final state with momentum ħk'.  The box normalized wave functions are given as
  Φ_i(r) = L^-3/2exp(ik∙r),   Φ_f(r) = L^-3/2exp(ik'∙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∫d³r' exp(iq∙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^-3exp(i2π(n_xx + n_yy + n_zz)/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 = k²dΩdk/(2π/L)³  = L³k²dΩdk/(2π)³.
  dN/dk = L³k²dΩ/(2π)³.
  The density of states is ρ(E,dΩ) = dN/dE = (dN/dk)(dk/dE).
  With E = ħ²k²/(2μ) we have dN/dE = μħkdΩL³/(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/(4π²ħ⁴L³)]|∫d³r' exp(-iq∙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)|²v = L^-3v = L^-3ħk/μ.  Therefore we have
  σ_k^B(k,k')dΩ =  [μ²/(4π²ħ⁴)]|∫d³r' exp(-iq∙r')U(r')|²dΩ.
  This derivation makes it clear how to incorporate a spin-dependent potential into the Born approximation.

• Details of the calculation:
  (a)  Assume  U = gS₁·S₂ d³(r) and the two particles are in an eigenstate of S² and S_z. 
  Then σ_k^B(k,k') =  [g²m²/(4π²ħ⁴)]|∫d³r' exp(-iq∙r')δ³(r')<S,M_s|S₁·S₂|S,M_s>|².
  S₁∙S₂= ½(S² - S₁² - S₂²).
  S₁∙S₂|S,M_s> = (ħ²/2)(S(S+1) - ½(3/2) - ½(3/2))|S,M_s> = (ħ²/2)(S(S+1) - (3/2))|S,M_s>.
  For the singlet state we have <0,0|S₁·S₂|0,0> = -3ħ²/4, and for the triplet state we have <1,M_s|S₁·S₂|1,M_s> = ħ²/4. 
  Therefore σ_k^B(k,k') = <S,M_s|S₁·S₂|S,M_s>²[g²m²/(4π²ħ⁴)]|∫d³r' exp(-iq∙r')δ³(r')|².
  [g²m²/(4π²ħ⁴)]|∫d³r' exp(-iq∙r')δ³(r')|² = g²m²/(4π²ħ⁴).
  For particles in the singlet state we have σ_k^B(k,k')_singlet = 9g²μ²/(64π²).
  For particles in the triplet state we have σ_k^B(k,k')_triplet = g²μ²/(64π²).
  (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 = 3g²μ²/(64π²).

Problem:

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) = [μ²/(4π²ħ⁴)]|∫d³r' exp(-iq∙r')U(r')|²,
  where q = k_f - k_fi, k_i = μv₀/ħ, k_f = μv₀/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(ik_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.)

  ∫d³r' exp(-iq∙r')U(r') = ∑_-∞^+∞C(k_m)∫d³r' 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.