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The Scattering Cross Section

Consider a target, A, consisting of a number of particles at rest, number density ρ_A particles per unit volume, and depth l_A. A beam, B, of particles with momentum q and number density ρ_B particles per unit volume is aimed at the target. In time t a beam length l_B = vt passes through the target. In real beams the number density of particles is generally larger at the centre than at the edges. The total number of scattering events per unit time, N_process, for a particular process is expected to be proportional to l_A, v, ρ_A, ρ_B, integrated over the area in which the beam intersects the target,
where σ is a constant of proportionality, called the scattering cross section».

The range of the interaction is small compared to the beam width. Number density may be taken to be slowly varying across the beam. If beam density is treated as constant, or if the above formula is used to compute an effective beam area, A, then the number of scattering events per unit time is simply the product of the cross section with the total number, N_A = Al_Aρ_A, of particles in the target and the flux», vρ_B, of particles per unit time per unit area in the beam,


Clearly the cross section has units of area. It corresponds to the cross sectional area of that part of a beam of classical particles of finite size which would collide with a single classical particle, also of finite size, in the target, on the assumption that scattering is due only to direct collision.

We are particularly interested in the dependency on the cross section on the scattering angle.

Cross Section and Probability

For definitenes I will designate the beam direction 1, and the two transverse directions 2 and 3. Particles in the beam and the target are assumed to described by wave packets with approximate momenta in the 1-direction q_B and q_A, respectively (the target may be, but is not necessarily, stationary. For example, it may be a second beam with opposite momentum). The beam may be considered as a collection of particles, described by wave packets with momentum centred on beam momentum. The initial state of a particle in the target is described by a ket,

The beam is taken to be uniform across the region containing the target particle. So, the incident wave packets may be considered identical up to the displacement, or impact parameter, b = (b², b³), transverse to beam direction. The ket describing a particle in the beam is
Without loss of generality, the initial states of beam and target particles are normalised to unity,

The number of events per unit time is the number of particles in the target multiplied by the flux, or rate of particles passing through the target, vρ_B, multiplied by the probability, P(b), of scattering, given impact parameter b, integrated over b,
Since the number density, ρ_B, of particles in the beam is taken as constant, we conclude that the cross section is

Cross Section and the Matrix Element

The detectors» for final-state particles typically measure momentum (position is not resolved at the level of the de Broglie wavelength»). It will be sufficient to treat final states as eigenstates of momentum,
The total probability for scattering is the integral of the probability density with respect to the momenta of particles in the final state,
where the probability density in momentum space is
Then the differential cross section for scattering to may be written
where it is implicit that this quantity is integrated over the resolution, dp_f = d³p₁ … d³p_n, of the detectors. There is a possibility that there will be no interaction. So, the S-matrix has the form,
Assuming we are not interested in the trivial case when no interaction takes place, we may ignore the 1 in this expression and consider only the T-matrix. T (and S) contain a factor of an energy-momentum conserving delta function. Extracting this factor gives the M-matrix, defined by
expanding and in the integrand for the differential cross section,
The integration over b gives . Together with the four delta functions in the factor
we can carry out all six integrations over . Since , only the integration in the 1-direction is non-trivial,
where the generalised scaling property of the delta function has been used, and v_A and vB are the velocities of target and beam in the laboratory frame. In the last line, we have and . In addition we have, for i = 2, 3, and , and two equations from the mass shell condition. So and . With these results, the differential cross section is
The initial states are are approximately states of pure momentum. So, we may put k_A = q_A and k_B = q_B in the M-matrix and in the delta function (since the momentum of the initial state is far more precisely determined in practice than that in the final state). Since the initial states are normalised to unity,

Two Particle Scattering

For scattering in the centre of mass frame for two particles with masses m and m', let the 4-momenta in the final state be p = (E, p) and p' = (E', p'), where the final energies are and . Integrate the differential cross section over p'. This enforces conservation of 3-momentum and sets p' = −p, giving
where E_cm is the total initial energy in the centre of mass frame. Integrate over the magnitude of final momentum to find the standard form of the differential cross section in the chosen normalisation,
From the generalised scaling property of the delta function,
where the magnitude of final momentum, |p|, and the final energies, E and E', are determined from energy momentum conservation. Simplify, using E_cm = E + E',

Rutherford Scattering

The leading order M-matrix element for scattering of two distinguishable particles with initial momenta q and q' and final momenta p and p' is
Applying Feynman rules,
For non-relativistic energies we have p ≈ (m, p), q ≈ (m, q) , so that (p − q)² ≈ −|p − q|², and u is well approximated by a two spinor,
We need consider only the first two components.
and, for a ≠ 0,
In other words spin is conserved for non-relativistic energies. The matrix element reduces to
Substituting this into the differential cross section,

In the Geiger-Marsden experiment» electrons were scattered by a stationary gold foil target, v' = 0. Velocities were non-relativistic so that, from conservation of energy, |p| = |q| ≈ mv. The mass of a gold nucleus is large compared to that of an electron, so that, E' ≈ E_cm, E ≈ m. For scattering angle θ ,
Thus, the cross section reduces to the Rutherford formula» for scattering in a Coulomb potential,
where the fine structure constant» in natural units () is
The fine structure constant is defined to be dimensionless. If e is measured in units of charge, the right hand side becomes dimensionless on division by .


Scattering ↑ Regularisation and Renormalisation →