Lecture 19

Recall from last lecture:

Scattering Cross Sections

Hard-sphere scattering

Consider the elastic collision between two spheres of radius R, one fixed rigidly in place. From the figure we see that b=R sinα, and also that 2α + θ = π. Use the second expression to eliminate sinα from the first. We have sinα = sin(π/2 -θ/2) = cos(θ/2), so that

b = Rcos(θ/2).

Solving for θ we find

θ(b) = 2cos^-1(b/R).

If the impact parameter lies in the ring between b and b+db, then it will have a scattering angle between θ and θ+dθ. The initial ring the particle passes through has area dσ = 2πb db, and it scatters into a solid angle dΩ = 2πsinθ d&theta = -2πd(cosθ). (θ lies between 0 and π such that dΩ is always positive.) We call the ratio dσ/dΩ the differential cross section, and in this example it is

dσ/dΩ = (b/sinθ)(db/dθ)

Differentiating the expression for b(θ) we have

dσ/dΩ = Rb sin(θ/2)/2 sinθ = R² cos(θ/2)sin(θ/2)/2 sinθ = R²/4 .

The total cross section is obtained by integrating over the full solid angle

σ = Int{(dσ/dΩ)dΩ} = Int {R²/4 dΩ} = πR².

This is exactly the answer we expect for this example.

Rutherford scattering

The classic(al) scattering experiment is Rutherford's scattering experiment where α particles (⁴He nuclei) emitted by radioactive decay are collimated into a narrow beam impinging on a thin gold foil. The α particles scatter from the gold nuclei and are detected. In previous courses you have probably worked out the 1/sin⁴(θ/2) distribution of scattered particles. Let's review how this is done to make connection with quantities we will use.

The relation between impact parameter and scattering angle for a charge q₁ with initial kinetic energy T scattering in the electrostatic potential from a fixed charge q₂ is

b = (q₁q₂/2T)cot(θ/2).

The differential cross section is

dσ/dΩ = {q₁q₂/(4Tsin²(θ/2))}².

The total cross section is

σ = 2π (q₁q₂/4T)² Int{ (sinθ dθ)/sin⁴(θ/2)} .

which is infinite. This shouldn't cause concern since Rutherford's actual experiment used gold atoms, not gold nuclei, and gold atoms have orbiting electrons which cancel the field of the nucleus beyond some radius. If we integrate out to a finite impact parameter (translated to scattering angle θ) then the cross section is finite.

In scattering experiments we characterize the beam by its luminosity, L, defined as the number of beam particles per unit area per second passing through an imaginary plane. The number of interactions per second is N = Lσ. The number of interactions scattering into solid angle dΩ per second is dN = L(dσ/dΩ)dΩ. We can also write dσ/dΩ = (1/L)dN/dΩ.

Fermi's Golden Rule for Scattering

In the general case of particles 1 and 2 colliding and producing particles 3, 4, ... , N

1 2 --> 3 4 ... N

the differential cross section is given the formula

dσ = |Mif|² hbar²S/4Sqrt{(p₁·p₂)² - (m₁m₂c²)²} {Π_j=1^n_f [(c d³p_j)/((2π)³ 2E_j)]} (2π)⁴ δ⁴(p_i - Σ_j p_j)

Example: Two-body scattering in the CM frame

Consider a two-body to two-body scattering process

1 2 --> 3 4

in the CM frame. Assume the matrix element of this interaction is M, and determine the differential cross section.

In the CM, p₁ = -p₂, therefore p_1μp₂^μ = E₁E₂ + p₁². After some massaging,

Sqrt{(p₁·p₂)² - (m₁m₂c²)²} = (E₁ + E₂)|p₁|/c .

Inserting this into the expression for dσ yields:

dσ = (hbarc/8π)² {S|M|²c/(E₁ + E₂)|p₁|} (d³p₃/E₃) (d³p₄/E₄) δ⁴(p₁ + p₂ - p₃ - p₄)

The delta function becomes

δ⁴(p₁ + p₂ - p₃ - p₄) = δ(E₁ + E₂ - E₃ - E₄) δ³(-p₃ - p₄)

Relation to Perkins notation

Consider a two-body to two-body scattering process

A B --> C D

in which a parallel beam of particles A, moving with speed v_i, impinges on a target containing particles of type B. For instance, A could be a beam of protons, pions, electrons, or neutrinos, and B is some material containing nuclei with protons and neutrons. (When the typical interaction energies exceed the binding energy of protons and neutrons in nuclei (about 10MeV) then we can consider the nucleus as a collection of quasi-free protons and neutrons.) The outgoing particles are called C and D. If the scattering is elastic then C and D are the same as A and B. If the scattering is inelastic then C and D are different from A and B (different types of particles; particles of the same type are indistinguishable).

Consider the target to be of thickness dx, and to have n_B B particles per unit volume. The flux of incoming particles (particles/cm²/sec) is Φ = n_Av_i where n_A is the density of A particles in the beam and v_i is their velocity. For a single nucleus interaction cross section, σ, the fraction of the target area covered by the cross section of B particles is σ n_B dx. The scattering rate is the rate that A particles will hit this area

f s n_B dx

The reaction rate per target particle is