22.101 Applied Nuclear Physics (Fall 2004)
Lecture 7 (9/27/04)
Overview of Cross Section Calculation
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References – Appendix A: Concepts of Cross Sections Appendix B: Cross Section Calculation: Method of Phase Shifts ________________________________________________________________________
The method of phase shifts has been discussed in Appendix B. Here we will summarize
the key steps in this method, going from the introduction of the scattering amplitude
f (θ ) to the expression for the angular differential cross section σ (θ ) .
Expressing σ (θ ) in terms of the Scattering Amplitude f (θ )
We consider a scattering scenario sketched in Fig.B.1.
Fig.B.1. Scattering of an incoming plane wave by a potential field V(r), resulting in
spherical outgoing wave. The scattered current crossing an element of surface area dΩ
about the direction Ω is used to define the angular differential cross section
dσ / d Ω ≡ σ (θ ) , where the scattering angle θ is the angle between the direction of
incidence and direction of scattering.
We write the incident plane wave as
Ψ in = bei ( k
r −ω t )
(B.1)
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where the wavenumber k is set by the energy of the incoming effective particle E, and the
scattered spherical outgoing wave as
Ψ sc = f (θ )b
ei ( kr −ωt )
r
(B.2)
where f (θ ) is the scattering amplitude. The angular differential cross section for
scattering through d Ω about Ω is
J sc ⋅ Ω
2
= f (θ )
J in
(B.5)
⎡⎣ Ψ * (∇Ψ ) − Ψ (∇Ψ * ) ⎤⎦
2µ i
(B.3)
σ (θ ) =
where we have used the expression
J=
h
Calculating f (θ ) from the Schrödinger wave equation
The Schrödinger equation to be solved is of the form
⎛ h2 2
⎞
⎜ − 2 µ ∇ + V (r) ⎟ψ (r ) = Eψ (r )
⎝
⎠
(B.6)
where µ = m1m2 /( m1 + m2 ) is the reduced mass, and E = µ v 2 / 2 , with v being the relative
speed, is positive. To obtain a solution to our particular scattering set-up, we impose
the boundary condition
ψ k ( r ) → r >>r eikz + f (θ )
o
eikr
r
(B.7)
2
where ro is the range of force, V(r) = 0 for r > ro. In the region beyond the force range the
wave equation describes a free particle, so the free-particle solution to is what we want to
match up with the RHS of (B.7). The most convenient form of the free-particle is an
expansion in terms of partial waves,
∞
ψ ( r ,θ ) = ∑ Rl (r ) Pl (cosθ )
(B.8)
l=0
where
Pl (cos θ )
is the Legendre polynomial of order l . Inserting (B.8) into (B.6), and
setting ul ( r ) = rRl ( r ) , we obtain
⎛ d2
2µ
l(l +1) ⎞
2
⎜ dr 2 + k − h 2 V (r ) − r 2 ⎟ ul (r ) = 0 ,
⎝
⎠
(B.10)
Eq.(B.10) describes the wave function everywhere. Its solution clearly depends on the
form of V(r). Outside of the interaction region, r > ro, Eq.(B.10) reduces to the radial
wave equation for a free particle,
⎛ d2
l(l +1) ⎞
2
⎜ dr 2 + k − r 2 ⎟ ul (r ) = 0
⎝
⎠
(B.11)
with general solution
ul ( r) = Bl rjl ( kr ) + Cl rnl ( kr )
(B.12)
where Bl and Cl are integration constants, and jl and nl are spherical Bessel and
Neumann functions respectively (see Appendix B for their properties).
Introduction of the Phase Shift δ l
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We rewrite the general solution (B.12) as
ul (r) →kr >>1 (Bl / k )sin(kr − lπ / 2) − (Cl / k ) cos(kr − lπ / 2)
= ( al / k )sin[kr − (lπ / 2) + δ l ]
(B.14)
where we have replaced B and C by two other constants, a and δ , the latter is seen to be
a phase shift. Combining (B.14) with (B.8) the partial-wave expansion of the freeparticle wave function in the asymptotic region becomes
ψ ( r ,θ ) →kr >>1 ∑ al
sin[kr − (lπ / 2) + δ l ]
kr
l
Pl (cosθ )
(B.15)
This is the LHS of (B.7). Now we prepare the RHS of (B.7) to have the same form of
partial wave expansion by writing
f (θ ) = ∑ f l Pl (cosθ )
(B.16)
l
and
eikr cosθ = ∑ i l (2l + 1) jl ( kr ) Pl (cosθ )
l
→ kr >>1 ∑ i l (2l + 1)
sin( kr − lπ / 2)
l
kr
Pl (cosθ )
(B.17)
Inserting both (B.16) and (B.17) into the RHS of (B.7), we match the coefficients of
exp(ikr) and exp(-ikr) to obtain
fl =
1
2ik
iδ l
( −i ) l [al e − i l (2l + 1)]
(B.18)
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al = i l (2l + 1)eiδ
(B.19)
l
Combing (B.18) and (B.16) we obtain
∞
f (θ ) = (1/ k )∑ (2l + 1)eiδ sin δ l Pl (cosθ )
l
(B.20)
l =0
Final Expressions for σ (θ ) and σ
In view of (B.20) (B.5), becomes
σ (θ ) = D
∞
2
∑ (2l +1)e
2
iδ l
l =0
sin δ l Pl (cosθ )
(B.21)
where D = 1/ k is the reduced wavelength. Correspondingly,
∞
σ = ∫ dΩσ (θ ) = 4π D 2 ∑ (2l +1)sin 2 δ l
(B.22)
l=0
S-wave scattering
We have seen that if kro is appreciably less than unity, then only the l = 0 term
contributes in (B.21) and (B.22). The differential and total cross sections for s-wave
scattering are therefore
σ (θ ) = D 2 sin 2 δ o (k )
(B.23)
σ = 4π D 2 sin 2 δ o (k )
(B.24)
Notice that s-wave scattering is spherically symmetric, or σ (θ ) is independent of the
scattering angle. This is true in CMCS, but not in LCS. From (B.18) we see
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iδ o
f o = (e sin δ o ) / k .
Since the cross section must be finite at low energies, as k → 0 fo has
to remain finite, or δ o ( k ) → 0 . We can set
lim k→0 [e
iδ o ( k )
sin δ o (k )] = δ o (k ) = −ak
(B.25)
where the constant a is called the scattering length. Thus for low-energy scattering, the
differential and total cross sections depend only on knowing the scattering length of the
target nucleus,
σ (θ ) = a 2
(B.26)
σ = 4π a 2
(B.27)
Physical significance of sign of scattering length
Fig. B.2 shows two sine waves, one is the reference wave sin kr which has not had
Fig. B.2. Comparison of unscattered and scattered waves showing a phase shift δ o in the
asymptotic region as a result of the scattering. any interaction (unscattered) and the other one is the wave sin( kr + δ o ) which has suffered a phase shift by virtue of the scattering. The entire effect of the scattering is seen to be
represented by the phase shift δ o , or equivalently the scattering length through (B.25).
In the vicinity of the potential, we take kro to be small (this is again the condition of low6
energy scattering), so that uo ~ k ( r − a ) , in which case a becomes the distance at which
the wave function extrapolates to zero from its value and slope at r = ro. There are two
ways in which this extrapolation can take place, depending on the value of kro. As shown
in Fig. B.3, when kro >
π /2,
the wave function has reached more than a quarter of its
wavelength at r = ro. So its slope is downward and the extrapolation gives a distance a
which is positive. If on the other hand, kro <
π /2,
then the extrapolation gives a distance
a which is negative. The significance is that a > 0 means the potential is such that it can
have a bound state, whereas a < 0 means that the potential can only give rise to a virtual
state.
Fig. B.3. Geometric interpretation of positive and negative scattering lengths as the
distance of extrapolation of the wave function at the interface between interior and
exterior solutions, for potentials which can have a bound state and which can only virtual
state respectively.
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