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4.2. FORMAL DESCRIPTION OF NUCLEAR REACTIONS 101

4.2.1 Wave Function and Scattering Amplitude

The scattering is described by stationary wave functions for asymptoticstates. The incident wave is prepared as a plane wave at z → −∞, t→ −∞.The scattered wave is an outgoing spherical wave, which is asymptoticallydescribed at r → ∞, t → ∞. This situation that also corresponds to theusual experimental setup is shown in Fig. 4.9. The modification of the inci-dent wave by the scattering process is described by the scattering amplitudef(θ, φ). The total wave function

Ψ→ eikinz + f(θ, φ)eikoutr

r(4.4)

is the solution of either a Schrödinger equation with boundary conditions ora corresponding integral equation (the Lippmann-Schwinger equation, whichis normally formulated as to contain the boundary conditions automatically).These conditions are taken into account by imposing them on the wave func-tion. If there is a scattering potential , 0 the wave function in the externalregion i.e. that of the free particles is matched at a suitable point, often atthe nuclear surface r = R or the “edge” of the potential, to the wave functionin the nuclear interior i.e. that influenced by the potential. We require:

• The wave functions and their derivatives must be continuous at r = R.

• For bound states Ψ→ 0 for r →∞.

• For scattering states Ψ approaches the asymptotic solution for a freeparticle with r →∞.

• We require Ψ → 0 for r → 0 in order to avoid the singularity at theorigin.

4.2.2 Scattering Amplitude and Cross Section

We start from the general definition of the cross section

dσ =~joutd ~A∣∣∣~jin

∣∣∣

(4.5)

and use a classical continuity equation for the particle flux expressed by ~j:

~j = ρ · ~v. (4.6)

Quantum mechanically

~j =ih

2µ[Ψ∗~∇Ψ−Ψ~∇Ψ∗] and ρ = ΨΨ∗, (4.7)

which provides a connection to the wave functions that may be solutions ofthe Schrödinger equation. With

Ψin ≡ Φ = aeikinz and Ψout = af(θ, φ)eikoutr

r(4.8)

102 CHAPTER 4. CROSS SECTIONS

eikz

kout

k in

eikr

r∆Ω

θ z

Figure 4.9: Scattering quantum-mechanically.

we obtain~jin = h/2µ|a|22~kin = |a|2~vin (4.9)

~jout = h/2µ|a|2 |f(θ, φ)|2 kout

r2= |a|2~vout

|f(θ, φ)|2r2

. (4.10)

With the outoing flux ~joutd ~A = ~vout|a|2 |f(θ, φ)|2 dA/r2 through the areadA = r2dΩ we get

dσ =~joutd ~A∣∣∣~jin

∣∣∣

=vout|a|2 |f(θ, φ)|2 dΩ

vin|a|2, (4.11)

anddσ

dΩ=kout

kin|f(θ, φ)|2 =

kout

kinf · f ∗, (4.12)

anddσ

dΩ= |f(θ, φ)|2 for elastic scattering. (4.13)

Please note (for more detail see Chapter 5):

• For particles with spin the scattering amplitude has to be replaced bya matrix describing the transitions between the different spin-substatesof the entrance and exit channels (M matrix). The complex square ofthe absolute value of f ·f ∗ is replaced by taking the trace of this matrix:dσ/dΩ ∝ Tr(MM †).

• In additon to the (unpolarized) cross section there are many other(polarization) observables such as polarization, analyzing power, pola-rization-transfer coefficients, spin-correlation coefficients etc. For manymore detailed investigations in nuclear physics the study of polarizationobservables has been and still is important, in some cases indispensable.More detailed description of polarization effects in nuclear reactions andtheir measurement may be found in Refs. [HGS12, NUR13].


4.2.3 Schrödinger Equation

For the application to nuclear reactions stationary solutions are required, i.e.the time-independent Schrödinger equation is to be used. Corresponding tothe geometry of the scattering problem the use of spherical polar coordinatesis useful. For a central potential V (~r) the equation reads

− h2

2µ

[

1r2

∂

∂r

(

r2 ∂

∂r

)

+1

r2 sin θ∂

∂θ

(

sin θ∂

∂θ

)

+1

r2 sin2 θ

∂2

∂φ2

]

Ψ(~r)

+V (~r)Ψ(~r) = EΨ(~r),

(4.14)

where Ψ(~r) is an abbreviation for Ψ(+)~ki

(~r), which corresponds to the sta-tionary scattering wave function. The angle-dependent part of the Hamiltonoperator may be expressed by the angular-momentum operator

L2 = L2x + L2

y + L2z = −h2

[

1sin θ

∂

∂θ

(

sin θ∂

∂θ

)

+1

sin2 θ

∂2

∂φ2

]

. (4.15)

Eigenvalues and eigenfunctions of L2 and Lz are given by the eigenvalueequations:

L2Yℓm(θ, φ) = ℓ(ℓ+ 1)h2Yℓm(θ, φ), (4.16)

LzYℓm(θ, φ) = mhYℓm(θ, φ). (4.17)

The well-known commutation relations apply here. With these the Hamilto-nian may be written

H = − h2

2µ

[

1r2

∂

∂r

(

r2 ∂

∂r

)

− L2

h2r2

]

+ V (r), (4.18)

i.e. the substitution leads to the appearance of a term with L2, which may beadded to the potential term, providing the centrifugal potential. Because of[H,L2] and [H,Lz] = 0 one searches for the common eigenfunctions to H,L2

and Lz for a product Ansatz, which corresponds to a separation of radial andangular parts of the wave function and is at the same time a partial-waveexpansion

Ψ(~r) =∞∑

ℓ=0

ℓ∑

m=−ℓ

Cℓm(k)Rℓm(kr)Yℓm(θ, φ). (4.19)

The separated radial equation (for m = 0 we have azimuthal independenceof the scattering problem) is a Bessel differential equation

[

d2

dr2+ k2 − ℓ(ℓ+ 1)

r2− U(r)

]

uℓ(k, r) = 0, (4.20)

with the substitutions uℓ(k, r) = rRℓ(k, r) and U(r) = 2µV (r)/h2.The solutions of this equation for free particles (U = 0) are

•

jℓ(kr)→r→∞=sin(kr − ℓπ/2)

kr≡ ei[kr−ℓπ/2] − e−i[kr−ℓπ/2]

2ikr. (4.21)

These Spherical Bessel Functions are regular for r → 0


•

nℓ(kr)→r→∞cos(kr − ℓπ/2)

kr≡ ei[kr−ℓπ/2] − e−i[kr−ℓπ/2]

−2kr. (4.22)

These Spherical Neumann Functions are irregular for r → 0. Fig. 4.10shows the few lowest-order Spherical Bessel and Neumann Functions(also: Bessel Functions of the second kind or Weber Functions).

• Solutions are also the linear combinations

h(1,2)ℓ (kr) = jℓ(kr)± inℓ(kr) (4.23)

(Hankel Functions).

For bound states, because of the behavior at the origin, only the solu-tion jℓ(kr) can be used.

The scattering from a potential shifts the phase of the scattering waveby δℓ: The wave numbers in free space (kfree =

√

2µE/h2) and inside the

potential region (kpot =√

2µ(E − V )/h2) are different. After passage of awave through a potential “layer” of thickness d the phase of the transmittedwave has changed relative to that of the free wave by δ = (kpot − kfree)d. Inclassical optics terms like phase shift, refractive index and optical path lengthhave a similar origin and meaning.

In order to also expand the total wave function into partial waves a similarexpansion of the plane incident wave eikz is needed. This is given by themathematical identity (Rayleigh)

eikz =∞∑

ℓ=0

(2ℓ+ 1)iℓjℓ(kr)Pℓ(cos θ) (4.24)

where the jℓ(kr) are the Bessel Functions of the first kind and the Pℓ(cos θ)the Legendre Polynomials. The lowest-order Legendre Polynomials are de-picted in Fig. 4.11. Basically, this expansion is the angular-momentum rep-resentation of the wave function.

Thus the total wave function becomes

Ψtot →∝∞∑

ℓ=0

(2ℓ+ 1)iℓei[kr−ℓπ/2] − e−i[kr−ℓπ/2]

2ikrPℓ(cos θ) + f(θ, φ)

eikr

r(4.25)

On the other hand this total wave function must also satisfy a simple partial-wave expansion – but with a phase shift caused by the potential.

δ = (kpot − kfree) · d (4.26)

with

kfree =

√

2µEh2 and kpot =

√

2µ(E − V )h2 (4.27)

The simplified Fig. 4.12 illustrates the connection between the potential Vand the phase shift of a wave.


j (kr) n (kr)= 0 = 0

1

1

23

4

2 3 4

krkr

Figure 4.10: The behavior of the lowest-order Spherical Bessel and NeumannFunctions jℓ(kr) and nℓ(kr) as functions of x = kr.

Figure 4.11: The four lowest-order Legendre Polynomials PL(cos θ), whichdetermine the angular distributions of unpolarized cross sections, see alsoChapters 5 and 7. Only the left half of the picture is relevant for angulardistributions, θ being a polar angle.


Free Space

Potential Region

Phase Shift

d

= k dδ

Figure 4.12: Illustration (in one dimension) of the action of a potential regionon the wavelength (or wave number k) of a wave, causing a phase shift δrelative to the free wave. The sign of the phase shift δ depends on whetherthe potential is attractive (δ < 0) or repulsive (δ > 0).

Ψtot →∞∑

ℓ=0

Cℓ1

2ikr

[

ei(kr−ℓπ/2+δℓ) − e−i(kr−ℓπ/2+δℓ)]

Pℓ(cos θ). (4.28)

By comparing coefficients in the incoming and the outgoing waves in bothexpansions one obtains on the one hand a normalization:

Cℓ(k) = (2ℓ+ 1)iℓeiδℓ , (4.29)

and on the other an expression for f(θ)∑

ℓ

(2ℓ+ 1)iℓeiδℓ1

2ikrPℓ(cos θ)ei(kr−ℓπ/2+δℓ)

=∑

ℓ

(2ℓ+ 1)iℓ1

2ikrei(kr−ℓπ/2)Pℓ(cos θ) + f(θ, φ)

eikr

r, (4.30)

from which follows

f(θ) =i

2k

∞∑

ℓ=0

(2ℓ+ 1)(1− e2iδℓ)Pℓ(cos θ)

=1k

∞∑

ℓ=0

(2ℓ+ 1) sin δℓeiδℓPℓ(cos θ). (4.31)

The quantity ηℓ = exp(2iδℓ) is the scattering function and identical with thesimplest form of the general scattering matrix Sℓ(k). This function containsthe dynamics of the interaction and – via the scattering amplitude (or moregenerally: via the transfer(T) matrix or, for particles with spin, the M ma-trix) – determines the observables like dσ/dΩ and polarization components.


4.2.4 The Optical Theorem

Following from the definition of the scattering amplitude f, as derived above,and the (integrated) cross section σint,el for purely elastic scattering is aninteresting relation between both, the Optical Theorem. The imaginary partof the scattering amplitude

f(θ = 0) =1k

∞∑

ℓ=0

(2ℓ+ 1)eiδℓ sin δℓ (4.32)

=1

2ik

∞∑

ℓ=0

(2ℓ+ 1)[ηℓ − 1] (4.33)

is

Im(f(θ = 0)) =1k

∞∑

ℓ=0

(2ℓ+ 1) sin2 δℓ. (4.34)

When comparing this with

σint,el =4πk2

∞∑

ℓ=0

(2ℓ+ 1) sin2 δℓ (4.35)

σint,el =4πk· Im(f(θ = 0)) (4.36)

for elastic scattering results. In the case of contributions from non-elasticchannels (absorption, see Section 8.2) this is

σtot ≡ σint,el + σabs =4πk

Im[fα(θ = 0)] (4.37)

with fα the scattering amplitude of the elastic channel. The optical the-orem connects a global quantity, the total cross section, with the forwardscattering amplitude. It arises from the conservation of probability flux (or,equivalently, the unitarity of the S-matrix) requiring a destructive interfer-ence between the incident and the scattered waves in the forward direction,something like a “shadow” of the incident beam, cast by the target, andremoving particles from it in proportion of the total cross section, see e.g.Ref. [JOA83].


Remark and Exercise

4.1. For charged particles the solutions of the Schrödinger equation withthe Coulomb potential only, see Eq. 2.25, are the regular and irregularCoulomb Functions

Fℓ −→ sin(kr − ℓπ/2− ηS ln 2kr + σℓ), (4.38)

Gℓ −→ cos(kr − ℓπ/2− ηS ln 2kr + σℓ). (4.39)

with the Coulomb phase shifts σℓ = argΓ(ℓ + 1 + iηS). The Coulombpotential is of long range and increasingly so towards lower energies,often requiring a large number of partial waves before, at a certainlarge “screening” radius, the series can be truncated. For short-rangepotentials such as the hadronic interaction between nuclei often theseries can be truncated after a few partial waves, especially at verylow energies with s-waves acting only. If both types of potentials areacting (which is the normal case for charged particles) this different be-havior suggests treating them separately by adding the correspondingscattering amplitudes. This leads to the sum of the two cross sections(for Rutherford scattering one of them is just the closed-form Ruther-ford cross section), but in addition there is an interference term, whichneeds summing over the many partial waves of the long-range Coulombamplitude fC

dσ/dΩ = |f(θ)|2 = |fC(θ)|2 + |f(θ)|2 + 2Re[f ∗C(θ) · f(θ)]. (4.40)

However, in this case the short-range amplitude f is different from thatwithout any Coulomb potential

f(θ) =i

2k

∑

(2ℓ+ 1)e2iσℓ(1− e2iδℓ)Pℓ(cos θ), (4.41)

(compare Eq. 4.31. For details see Ref. [JOA83]).

Show that for the case of neutral projectiles like neutrons the Coulombfunctions in the asymptotic limit become the spherical Bessel functionsjℓ(kr) and nℓ(kr) (now often yℓ(kr)).

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