LectureLecture 55 ScatteringScattering theory,theory, BornBorn ApproximationApproximation
SS2011: ‚Introduction to Nuclear and Particle Physics, Part 2‘ SS2011: ‚Introduction to Nuclear and Particle Physics, Part 2‘ 1 ScatteringScattering amplitudeamplitude
We are going to show here that we can obtain the differential cross section in the CM frame from an asymptotic form of the solution of the Schrödinger equation:
(1.1)
Let us first focus on the determination of the scattering amplitude f (θ, φ), it can be obtained from the solutions of (1.1), which in turn can be rewritten as
2 2μE (1.2) where k = 2 h The general solution of the equation (1.2) consists of a sum of two components: 1) a general solution to the homogeneous equation: (1.3)
In (1.3) is the incident plane wave
2) and a particular solution of (1.2) with the interaction potential
2 GeneralGeneral solutionsolution ofof SchrSchröödingerdinger eq.eq. inin termsterms ofof GreenGreen‘‘ss functionfunction
The general solution of (1.2) can be expressed in terms of Green’s function.
(1.4)
is the Green‘s function corresponding to the operator on the left side of eq.(1.3)
The Green‘s function is obtained by solving the point source equation:
(1.5)
(1.6)
(1.7)
3 GreenGreen‘‘ss functionfunction
A substitution of (1.6) and (1.7) into (1.5) leads to (1.8)
The expression for can be obtained by inserting (1.8) into (1.6)
(1.9)
(1.10)
To integrate over angle in (1.10) we need to make the variable change x=cosθ
(1.11)
4 MethodMethod ofof residuesresidues
Thus, (1.9) becomes
(1.12)
(1.13)
The integral in (1.13) can be evaluated by the method of residues by closing the contour in the upper half of the q-plane: The integral is equal to 2π i times the residue of the integrand at the poles.
5 GreenGreen‘‘ss functionsfunctions
Since there are two poles, q =+k, the integral has two possible values: the value corresponding to the pole at q =k, which lies inside the contour of integration in Figure 1a, is given by
(1.14)
the value corresponding to the pole at q =-k, Figure 1b, is
(1.15)
Green’s function represents an outgoing spherical wave emitted from r‘ and the function corresponds to an incoming wave that converges onto r . Since the scattered waves are outgoing waves, only is of interest to us.
6 BornBorn seriesseries
Inserting (1.14) into (1.4) we obtain for the total scattered wave function:
(1.16)
This is an integral equation. All we have done is to rewrite the Schrödinger (differential) equation (1.1) in an integral form (1.16), which is more suitable for scattering theory. ¾Note that (1.16) can be solved approximately by means of a series of successive or iterative approximations, known as the Born series. the zero-order solution is given by the first-order solution is obtained by inserting into the integral of (1.16):
(1.17)
7 BornBorn seriesseries
the second order solution is obtained by inserting into (1.16):
(1.18)
continuing in this way, we can obtain to any desired order; the nth order approximation for the wave function is a series which can be obtained by analogy to (1.18) .
8 AsymptoticAsymptotic limitlimit ofof thethe wavewave functionfunction
In a scattering experiment, since the detector is located at distances (away from the target) that are much larger than the size of the target, we have r>>r‘, where r represents the distance from the target to the detector and r’ the size of the target. If r >> r‘ we may approximate:
(1.19)
9 AsymptoticAsymptotic limitlimit ofof thethe wavewave functionfunction
Substitute (1.19) to (1.16): φ μ (1.16) π r eikr− er ikr for r>>r‘ = (rr ) − V (rr ′ψ) (rr ′)d 3 r′ inc ∫ 2 h r From the previous two approximations (1.19), we may write the asymptotic form of (1.16) as follows: (1.20)
(1.21)
where ia a plane wave and k is the wave vector of scattered wave; the integration variable r‘ extends over the spacial degrees of freedom of the target. The differential cross section is given by
(1.22)
10 TheThe firstfirst BornBorn approximationapproximation
If the potential V(r) is weak enough, it will distort only slightly the incident plane wave.
The first Born approximation consists then of approximating the scattered wave function Ψ(r ) by a plane wave.
This approximation corresponds to the first iteration in the Born series of (1.16):
(1.16)
that is, Ψ(r ) is given by (1.17):
(1.23)
Thus, using (1.21) we can write the scattering amplitude in the first Born approximation as follows:
(1.24)
11 TheThe firstfirst BornBorn approximationapproximation
Using (1.23), we can write the differential cross section in the first Born approximation as follows:
(1.25)
where is the momentum transfer; are the linear momenta of the incident and scattered particles, respectively.
In elastic scattering, the magnitudes of are equal (Figure 2); hence
(1.26)
Figure 2: Momentum transfer for elastic scattering:
12 TheThe firstfirst BornBorn approximationapproximation
If the potential is spherically symmetric, and choosing the z-axis along q (Figure 1), then and therefore
(1.27)
Inserting (1.27) into (1.24) and (1.25) we obtain
(1.28)
(1.29)
In summary, we have shown that by solving the Schrödinger equation (1.1) in first-order Born approximation (where the potential V(r)is weak enough that the scattered wave function is only slightly different from the incident plane wave), the differential cross section is given by equation (1.29) for a spherically symmetric potential. 13 ValidityValidity ofof thethe firtsfirts BornBorn approximationapproximation
The first Born approximation is valid whenever the wave function Ψ(r)is only slightly different from the incident plane wave; that is, whenever the second term in (1.23) is very small compared to the first:
(1.23)
(1.30)
Since we have
(1.31) ValidityValidity ofof thethe firtsfirts BornBorn approximationapproximation
In elastic scattering k0= k and assuming that the scattering potential is largest near r=0, we have
(1.32)
(1.33)
Since the energy of the incident particle is proportional to k (it is purely kinetic, ) we infer from (1.33) that the Born approximation is valid for large incident energies and weak scattering potentials. That is, when the average interaction energy between the incident particle and the scattering potential is much smaller than the particle’s incident kinetic energy, the scattered wave can be considered to be a plane wave. BornBorn approximationapproximation forfor CoulombCoulomb potentialpotential
Let‘s calculate the differential cross section in the first Born approximation for a Coulomb potential (1.34) where Z1e and Z1e are the charges of the projectile and target particles, respectively. In a case of Coulomb potential, eq.(1.29):
(1.35)
becomes (1.36)
(1.37) BornBorn approximationapproximation forfor CoulombCoulomb potentialpotential
Now, since an insertion of (1.37) into (1.36) leads to
(1.38)
where is the kinetic energy of the incident particle.
Eq. (1.38) is Rutherford formula
Note: (1.38) is identical to the purely classical case! – cf. Lecture 4, eq.(2.16)