In Taylor, read sections 14.5 to 14.6 for today, 15 to 15. for Monday.
The master formula for the number os particles scattered into an area of solid angle
To calculate the differential cross section, we need to know the part of the cross section particles must hit
to scatter into the direction of interest
The differential cross section is then the ratio of these two quantities (more precisely, we find Δσ, and ΔΩ, and we calculate the limit of the ratio as ΔΩ goes to zero, but dividing differentials works equally well here)
where the absolute value sign is inserted because normally the scattering angle θ decreases as the impact parameter b increases. The problem now is to determine the relation between impact parameter and scattering angle and evaluate the derivative db/dθ.
Find the differential cross section for scattering of a point projectile off a fixed rigid sphere of radius R. Integrate the differential cross section to find the total cross section.
Use the knowledge that for hard sphere scattering, the angle of reflection equals the angle of incidence. The angle of incidence is α = sin-1(b/R) where b is the impact parameter. The scattering angle is θ = π - 2α = π - 2sin-1(b/R). Invert this to find b as a function of θ. This yields b = R sin(π/2 - θ/2) = R cos(θ/2). We need the derivative db/dθ = -(R/2)sin(θ/2). Put the pieces together to find (for 0 < θ < π, check sign for -π < θ < 0)
The differential cross section is isotropic, meaning there is equal probability to scatter into any element of solid angle. The total cross section is
That is, the total cross section is the circular cross section of the sphere, as expected.
For scattering of particles by a conservative force, we can begin by choosing a coordinate system so that the orbit lies in the x-y plane, and writing down the Lagrangian in polar coordinates
where U(r) is the potential for the conservative force. The Lagrange equation for φ yields m r² φdot = constant = L, where L is the total angular momentum. The total angular momentum is related to the impact parameter by L = m v∞ b where m v∞ is the momentum of the projectile when far from the target. When far from the target, the projectile's energy is purely kinetic, so E = ½m v∞², so we can also write L = b √(2mE). Instead of proceeding to the r equation, it is easier to start from the knowledge that the total energy is constant.
Use the angular momentum relation to eliminate φdot, separate variables (r and t), and integrate
Find rmin by setting rdot to zero and solving for r.
Find rmin by setting rdot to zero and solving for r. We'll do this after specifying the potential, just note that there is a minimum radius, also known as a distance of closest approach.
Since we are interested in solving for the scattering angle, not the orbit. The angular change of the orbit can be found by again using the angular momentum expression to replace dt by (mr²/L)dφ and separating differentials to integrate
And finally, the scattering angle is θ = π - 2α. For Rutherford scattering, we are interested in the case of an electrostatic potential.
Now we'll find the differential cross section for the case that U = c/r. (The constant c = kqQ = qQ/4πε0 where q and Q are the charges of the projectile and target, respectively.) The potential is attractive if c is negative (opposite sign charges between projectile and target) and repulsive if c is positive (same sign charges). We will see that the scattering is the same in either case.
We solved this problem in chapter 8, and found
where E = (c² m/2L²)(ε² - 1). (This differs from 8.49 by a minus sign. This arises because the derivation in chapter 8 contains an explicit minus sign on the force. If you propagate a change in sign through the derivation, you will arrive at the result above.)
The energy is greater than zero for scattering, meaning that ε > 1. The distance of closest approach occurs when φ = 0 yielding rmin = L²/[c m (ε - 1)]. The radius of the orbit goes to infinity for φ = ±φ0 where φ0 = cos-1(1/ε). The scattering angle is θ = π - 2φ0. Since 1/ε = cosφ0, ε² - 1 = cot²φ0, and using L² = 2m E b² we find that ε² - 1 = (2Eb/c)². Therefore cotφ0 = 2Eb/c, or
The derivative is
Use this to find the differential cross section
Use the relation sinθ = 2sin(θ/2)cos(θ/2) and simplify to get
This is the Rutherford Scattering formula. It predicts a strong probability for scattering at small angles in the forward direction, but yet does allow for a measurable amount of scattering into the backward hemisphere. Were the atom constructed with its charges more or less mixed, like in the so-called "plum pudding" model, then the force law would be considerably different, much more like a 1/r³ force.
The important part is that, while the scattering is largest in the forward direction, near θ = 0, there is a measureable amount predicted at large angles, all the way to θ = π. Models of the atom that have charge and mass more distributed throughout the volume predict much smaller amounts of scattering at large angles.
Optional, won't be covered in lecture.
Optional, won't be covered in lecture.