Scattering Theory of Molecules, Atoms and Nuclei || Spin and Identical Particles

September 21, 2012 15:54 8012 - Scattering Theory of Molecules, Atoms and Nuclei canto-hussein

Chapter 6

Spin and Identical Particles

In the preceding chapters we dealt with spherically symmetric potentials

and distinguishable spinless particles. In such cases, the only particular

orientation in space is that of the incident beam, and the exchange symme-

try of the wave function can be ignored. In the present chapter we discuss

how the collisions are influenced by the spin of the collision partners and

the modifications required when the projectile and the target are identical

particles.

6.1 Collisions of particles with spin

In collisions of discernible spinless particles and rotationally invariant in-

teractions, the scattering amplitude has axial symmetry with respect to

the orientation of the collision plane. In this way, taking the z-axis along

the incident beam, one eliminates m 6= 0 components in partial-wave ex-

pansions (see section 2.3). The situation is more complicated when the

projectile or/and the target has/have internal degrees of freedom and the

intrinsic state does not behave as a scalar with respect to rotations. This

occurs when the system has intrinsic spin or any other kind of angular mo-

mentum, e.g. that associated with the rotation of a deformed target1. For

simplicity2, we consider collisions of a spinless partner with another with

spin s. We assume that the incident beam is polarized so that the spin

component along the z-axis is equal to ν, and that the experimental setup

is capable of distinguishing among the possible final spin projections, ν′.

Other situations will be considered at the end of this section.

1Here, we loosely use the term spin for intrinsic angular momentum of any kind.2For a general treatment of the scattering of particles with spin, we refer to [Satchler

(1983)] and [Newton (2002)].

247

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248 Scattering Theory of Molecules, Atoms and Nuclei

The spin states, |sν〉, have the properties

S2 |sν〉 = 2s(s+ 1) |sν〉 (6.1)

Sz |sν〉 = ν |sν〉 . (6.2)

In the time-independent approach, the incident wave is

φ(νk; r) =1

(2π)3/2

eik·r |sν〉 (6.3)

and the scattering state satisfies the Schrodinger equation[E −H

]|ψ(+)(νk)〉 = 0. (6.4)

It satisfies also the Lippmann-Schwinger equation,

|ψ(+)(νk)〉 = |φ(νk)〉+G(+)

0 V |ψ(+)(νk)〉 . (6.5)

Above, H = K + V is the system’s Hamiltonian, with K representing the

kinetic energy operator and V the potential, which may act on the collision

and spin coordinates, and G(+)

0 = [E −K]−1

is the free particle’s Green

operator.

Usually, the scattering wave function is written as

ψ(+)(νk; r) = φ(νk; r) + ψsc(νk; r) . (6.6)

In Eq. (6.6),

ψsc(νk; r) =+s∑

ν′ =−sψscν′(νk; r) |sν′〉 (6.7)

is the scattered wave, which has the asymptotic behavior

ψsc(νk; r→∞) =1

(2π)3/2

+s∑ν′ =−s

|sν′〉[fν′,ν(Ω)

eikr

r

]. (6.8)

In this way, ψ(+)(νk; r) has the asymptotic form

ψ(+)(νk; r)→ 1

(2π)3/2

+s∑ν′ =−s

|sν′〉[δνν′ e

ik·r + fν′,ν(Ω)eikr

r

](6.9)

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Spin and Identical Particles 249

The distribution of the scattering amplitude over the spin components,

ν′, is determined by the spin-dependence of the projectile-target interaction.

Clearly, when the potential V is spin-indedependent, we have

fν′,ν(Ω) = fν,ν(Ω) δνν′ .

with Ω standing for the angular coordinates, θ, ϕ. Assuming that the po-

tential is spherically symmetric3 and independent of spin, the scattering

amplitude is independent of ϕ and conserves the z-component of the spin.

That is,

fν,ν′(θ, ϕ) ≡ f(θ) δνν′ .

If the potential contains a Coulomb term, modifications along the lines

of chapter 3 are necessary. Is this case, the asymptotic form of Eq. (6.9)

becomes

ψ(+)(νk; r)→ φC (k; r) |sν) +1

(2π)3/2

+s∑ν ′ =−s

fν′,ν(θ)eiΘ(r)

r|sν′) , (6.10)

where φC (k; r) is the Coulomb scattering wave function of section 3.2,

fν′,ν(θ) is the partial scattering amplitude arising from the short-range

part of the interaction, and Θ(r) is the asymptotic phase distorted by the

Coulomb field,

Θ(r) = kr − η ln 2kr. (6.11)

The scattering amplitude is then given by

fν′,ν(θ) = fC(θ) δνν′ + fν′,ν(θ), (6.12)

where fC(θ) is the Coulomb amplitude of Eq. (3.29), and the elastic cross

section for a final state with spin component ν′ is

dσν′,ν(θ)

dΩ= |fν′,ν(θ)|2 . (6.13)

For unpolarized beams and inclusive experiments, where the final com-

ponent of the spin along the z-axis is not determined, the cross section

should be averaged over the initial spin orientations and summed over the

final spin states. That is

dσ(θ)

dΩ=

1

(2s+ 1)

∑ν′ν

|fν′,ν(θ)|2 . (6.14)

3We make this assumption throughout this chapter.

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6.1.1 Partial-wave expansions

The aim of the partial-wave expansion method is to reduce the three-

dimensional Schrodinger equation (Eq. (6.4)) to an angular momentum

dependent equation on a single variable, the radial coordinate r. The re-

duced equation is then solved for each value of the angular momentum and

its contribution to the scattering amplitude is evaluated.

To generalize the partial-wave expansion method to systems with spin,

we re-introduce it using a more general approach. The first step is to single

out the radial coordinate and look for an appropriate basis for the vector

space associated with the remaining degrees of freedom, which we formally

represent by ζ. In the case of spinless particles, discussed in chapter 2, ζ

stands for the orientation variables r ≡θ, ϕ . We call Yn (ζ) the states

of this basis, with the label n standing for the set of quantum numbers

α(1)n , α

(2)n , α

(3)n , · · · , which specify the state. The wave funtion is then

expanded as

ψ(+) (νk;r, ζ) =∑n′

Yn′ (ζ) Xn′ (r) . (6.15)

Using this expansion in the Schrodinger equation and taking scalar product4

with (Yn| , one gets ∑n′

[E δnn′ −Hnn′ ] Xn′ (r) = 0 , (6.16)

with

Hn,n′ = (Yn| H |Yn′) . (6.17)

Eq. (6.16) is a matrix equation in the vector space spanned by |Yn), cou-

pling the r-dependent components of the vector ψ(+), Xn′ (r) , and their

derivatives. The essential point of the method is to take advantage of

the symmetries of the system, looking for a basis in which the Hamilto-

nian is diagonal. This aim is achieved if each quantum number of the set

α(1)n , α

(2)n , α

(3)n , · · · is a constant of motion5. The equations for Xn (r) are

then completely decoupled and can be solved as described in section 2.2.

In the case of spinless particles, the rotational invariance of H leads to4We use round brackets to represent scalar producs in the space of the coordinates ζ.5In some situations, e.g. multi-channel scattering (see chapter 9), one cannot find a

basis which diagonalizes the Hamiltonian. One should then look for a basis in which the

setα

(1)n , α

(2)n , α

(3)n , · · ·

contains as many constants of motion as possible. The Hamil-

tonian matrix then breaks into idependent blocks, leading to sets of coupled differential

equations of smaller dimensions.

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orbital angular momentum conservation, so that the set of spherical har-

monics is the appropriate expansion basis. In the Ylm representation, the

Hamiltonian matrix is diagonal and independent of m. That is,

Hn,n′ ≡ Hlm,l′m′ = δll′ δmm′

− ~2

2µ

[1

r2

∂

∂r

(r2 ∂

∂r

)− l(l + 1)

r2

]+ V (r)

.

(6.18)

Inserting these matrix-elements into Eq. (6.16), with ψ(+) (νk; r) expanded

as in Eq. (2.8), and taking scalar product with (Ylm|, we have obtained the

radial equation for spinless systems (Eqs. (2.6) and (2.9)).

The partial-wave expansion method for particles with spin is more com-

plicated. If the potential is spin-dependent, the rotational invariance of H

may not guarantee conservation of the orbital angular momentum and the

spherical harmonics may no longer be a convenient expansion basis. We

clarify this point with the example of a potential which contains a spin-orbit

term, as in the expression

V (r) = V0(r) + Vso(r) L · s

≡ V0(r) +1

2Vso(r)

[J2 − L2 − s2

], (6.19)

where J = L + s is the total angular momentum operator. In this case, the

method requires a basis for the vector space associated with the orientation

and spin coordinates (i.e., ζ ≡ r, ν) which diagonalizes the Hamiltonian.

Since [Lz, H] 6= 0, the quantum number m is not conserved. Therefore,

a trivial generalization like |Ylmν) = |sν) |Ylm) would not be satisfactory.

However, rotational invariance guarantees conservation of any component

of the total angular momentum, J. In this case, the appropriate basis is

the set of spin-angle wave functions6 defined as7

Y lJM (ζ) = il∑m′ν′

〈lm′sν′ |JM〉 Ylm′(r) |sν′〉 , (6.20)

6We write the quantum numbers of the constants of motion of the spin-angle basis assuperscripts. This choice will prove convenient for the extension of the procedures ofthis chapter to coupled-channel problems (see chapter 9). For simplicity of notation, wedo not include the quantum number s, because it has a fixed value.

7It is convenient to define the spin-angle wave functions with the arbitrary phase il.With this phase, the wave function has convenient time-reversal properties [Huby (1954);

Lane (1962); Thomas (1958); Satchler (1983)].

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where 〈lmsν |JM〉 are the usual Clebsch-Gordan coefficients [Edmonds

(1974)]. Since the above Clebsch-Gordan coefficients vanish if m′+ν′ 6= M ,

we can write

Y lJM (ζ) = il∑ν′

〈l (M − ν′) sν′ |JM〉 Yl(M−ν′)(r) |sν′〉 . (6.21)

Using properties of the Clebsch-Gordan coefficients and the orthogonality

of angular momentum eigestates (in the coordinate and in the spin spaces),

it can easily be checked that(Y lJM

∣∣∣ Y l′J′M ′) = δll′ δJJ ′ δMM ′ . (6.22)

The rotational invariance of the Hamiltonian guarantees that the H

is diagonal in the total angular momentum representation. On the other

hand, the commutators[L2, H

]and

[s2, H

]depend on the nature of the

potential. In most cases they vanish8, so that H is diagonal in the basis of

spin-angle wave functions. One then obtains a set of fully decoupled radial

equations. Otherwise, the partial-wave method leads to a set of differential

equations for each invariant subspace, in which radial wave functions with

the same J and different l are coupled. This situation is analogous to the

coupled-channel equations in many-body scattering, which will be discussed

in detail in chapter 9. We consider below the simpler situation where the

orbital angular momentum is conserved. In this case, the set of spin-angle

wave functions diagonalizes the Hamiltonian and we can write

(Y lJM |H |Y l′J′M ′) =

− ~2

2µ

[1

r2

∂

∂r

(r2 ∂

∂r

)− l(l + 1)

r2

]+ V Jl (r)

× δll′ δJJ ′ δMM ′ . (6.23)

The incident wave can be expanded in the spin-angle basis as9

φ(νk; r) =1

(2π)3/2

∑lJM

CJM (νlk)∣∣Y lJM) l(kr)

kr. (6.24)

To determine the coefficients CJM (νlk), we first use Bauer’s expansion

(Eq. (2.43a)) in Eq. (6.3). We get

φ(νk; r) =1

(2π)3/2

∑lm′

4π Y ∗lm′(k)[il Ylm′(r) |sν〉

] l(kr)

kr. (6.25)

8There are, however, important exceptions, like the tensor force in the nucleon-nucleoninteractions.

9This equation corresponds to Eq. (6.15), with the replacement Xn(r) ≡ XlJM (r) →(2π)−3/2 ClJM l(kr)/kr.

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Using the inverse of Eq. (6.21),

il Ylm′(r) |sν〉 =∑JM

〈JM |lm′sν〉 Y lJM (ζ) , (6.26)

Eq. (6.25) takes the form

φ(νk; r) =1

(2π)3/2

∑lm′JM

4π Y ∗lm′(k) 〈JM |lm′sν〉 Y lJM (ζ)l(kr)

kr.

Since the non-vanishing Clebsch-Gordan coefficients have M = m′ + ν, we

can replace m′ →M − ν and the above equation becomes

φ(νk; r) =1

(2π)3/2

∑lJM

[4π 〈JM |l (M − ν) sν〉

×Y ∗l (M−ν)(k)]Y lJM (ζ)

l(kr)

kr. (6.27)

Comparing Eqs. (6.24) and (6.27), we get

CJM (νlk) = 4π 〈JM |l(M − ν) sν〉 Y ∗l (M−ν)(k). (6.28)

We now evaluate Eq. (6.25) in the r →∞ limit and express it in terms of

incoming and outgoing waves, as we did in section 2.3. Using in Eq. (6.25)

the asymptotic form of the Ricatti-Bessel function (Eq. (2.22a)),

l(kr)

kr→ sin (kr − lπ/2)

kr=e−ikr

r

(−il

2ik

)+eikr

r

(i−l

2ik

), (6.29)

we get

φ(νk; r)→ 1

(2π)3/2

∑ν′

|sν′〉[e−ikr

rZν′

− +eikr

rZν′

+

], (6.30)

with

Zν′

− = − δνν′1

2ik

∑lm

4π Y ∗lm(k)Ylm(r) i2l (6.31)

and

Zν′

+ = δνν′1

2ik

∑lm

4π Y ∗lm(k)Ylm(r) . (6.32)

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If the potential contains a Coulomb term, changes along the lines of

chapter 3 are required. In first place, Eq. (6.3) becomes

φ(νk; r) = φC(k; r) |sν〉 . (6.33)

Using the partial-wave expansion of φC(k; r) (Eq. (3.102a)) instead of

Bauer’s expansion, Eq. (6.24) takes the form

φ(νk; r) =1

(2π)3/2

∑CJM (νlk)

∣∣Y lJM) Fl(η, r)kr

, (6.34)

with the coefficients CJM (νlk) given by

CJM (νlk) = 4π 〈JM |l(M − ν) sν〉 Y ∗l (M−ν)(k) eiσl (6.35)

Above, Fl(η, r) is the regular Coulomb function and σl is the Coulomb

phase shift at the lth partial-wave.

Evaluating Eq. (6.34) in the r →∞ limit (see Eq. (3.103)) we obtain

φ(νk; r)→ 1

(2π)3/2

∑ν′

|sν′〉[e−iΘ(r)

rZν′

− +eiΘ(r)

rZν′

+

], (6.36)

with Zν′

− and Θ given respectively by Eqs. (6.31) and (6.11), and

Zν′

+ = δνν′1

2ik

∑lm

4π Y ∗lm(k)Ylm(r) e2iσl . (6.37)

We now consider the wave function in the case of scattering from a short

range potential. First, we expand it in the spin-angle basis as

ψ(+)(νk; r) =1

(2π)3/2

∑lJM

CJM (νlk)∣∣Y lJM) uJl (k, r)

kr. (6.38)

Inserting the expansion into the Schrodinger equation, taking scalar prod-

uct with(Y lJM

∣∣ and using the orthogonality of the spin-angle wave func-

tions, we obtain the radial equation[− 2

2µ

(d2

dr2− l(l + 1)

r2

)+ V Jl (r)

]uJl (k, r) = E uJl (k, r), (6.39)

where10

V Jl (r) =(Y lJM

∣∣ V ∣∣Y lJM) . (6.40)

10For simplicity, we assume that the interaction is local.

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For the potential of Eq. (6.19), for example, these matrix-elements are

V Jl (r) = V0(r) +1

2Vso(r) [J(J + 1)− l(l + 1)− s(s+ 1)] .

Beyond the range of the interaction, R, the radial wave functions can be

written (Eq. (2.35) with γ = i/2)

uJl (k, r > R) =i

2

[h(−)

l (kr)− SJl h(+)

l (kr)]

(6.41)

and in the r →∞ limit they become

uJl (k, r →∞) = e−ikr(− i

l

2i

)+ eikr

(i−l

2iSJl

). (6.42)

The coefficients CJM (νlk) in the partial-wave expansion of ψ(+)

(Eq. (6.38)) are still given by Eq. (6.28). That is, they are the same as

those in the expansion of the incident wave. This statement is justified

as follows. First, we remark that the difference between ψ(+)(νk; r) and

φ(νk; r) is the scattered wave ψsc(νk; r), which has the asymptotic behav-

ior of an outgoing wave (Eq. (6.8)). In this way, ψ(+)(νk; r) and φ(νk; r)

have identical incoming contents. The same happens for the radial wave

functions uJl and l. This becomes clear if the Ricatti-Bessel function is

written as

l(kr) =h(+)

l (kr)− h(−)

l (kr)

2i=i

2

[h(−)

l (kr)− h(+)

l (kr)]. (6.43)

Comparing Eqs. (6.41) and (6.43), we conclude that these radial wave func-

tions have the same incoming part (the term in h(−)

l ) and their outgoing

parts (the term in h(+)

l ) differ only by the factor SJl . The equality of the

incoming parts,

[ψ(+)(νk; r→∞)]in = [φ(νk; r→∞)]in[uJl (k, r →∞)

]in

= [l(kr →∞)]in ,

implies that the partial-wave expansions of ψ(+) and φ have the same coef-

ficients.

The scattering wave function can then be written

ψ(+)(νk; r) =1

(2π)3/2

∑lJM

4π 〈JM |l(M − ν) sν〉

×Y ∗l (M−ν)(k) Y lJM (ζ)uJl (k, r)

kr. (6.44)

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Using Eq. (6.21) and carrying out some algebra, the above equation can be

put in the form

ψ(+)(νk; r) =1

(2π)3/2

∑ν′

|sν′〉

[ ∑Jlm

4π Y ∗lm(k)Yl(m+ν−ν′)(r)

× il 〈l (m+ ν − ν′)sν′ |J(m+ ν)〉

× 〈J(m+ ν) |lmsν〉 uJl (k, r)

kr

]. (6.45)

Eq. (6.45) can be considerably simplified if one chooses the z-axis along

the incident beam. In this case we can replace Y ∗lm(k) → Y ∗lm(z) =

δm0

√(2l + 1) /4π and obtain

ψ(+)(νk; r) =1

(2π)3/2

∑ν′

|sν′〉

[ ∑Jl

√4π (2l + 1)Yl(ν−ν′)(r) il

×〈l (ν − ν′)sν′ |Jν〉〈Jν |l0sν〉 uJl (k, r)

kr

]. (6.46)

If the potential contains a Coulomb term, the coefficients of the partial-

wave expansion of ψ(+) are given by Eq. (6.35). Using the same arguments

as above, we conclude that they are the same as those in the expansion of

φC(k; r) |sν〉 . Therefore, we can write

ψ(+)(νk; r) =1

(2π)3/2

∑ν′

|sν′〉

[ ∑Jl

√4π (2l + 1)Yl(ν−ν′)(r) il

×〈l (ν − ν′)sν′ |Jν〉〈Jν |l0sν〉 eiσl uJl (k, r)

kr

]. (6.47)

Now, however, the asymptotic radial wave functions are expressed in terms

of Coulomb functions (see chapter 3)

uJl (k, r > R) =i

2

[H(−)

l (kr)− SJl H(+)

l (kr)]

(6.48)

and in the r →∞ limit it becomes

uJl (k, r →∞) = e−iΘ(r)

(− i

l

2i

)+ eiΘ(r)

(i−l

2iSJl

). (6.49)

with Θ(r) given by Eq. (6.11).

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6.1.1.1 Partial-wave projected scattering amplitudes and cross

sections

To find the scattering amplitude, we compare the asymptotic form of

Eq. (6.45) with the scattering boundary condition of Eq. (6.9). Using

Eq. (6.42) in Eq. (6.45), we obtain

ψ(+)(νk; r)→ 1

(2π)3/2

∑ν′

|sν′〉[e−ikr

rZν′

− +eikr

rZν′

+

], (6.50)

with the amplitude Zν′

− given by Eq. (6.31) and Zν′

+ given by

Zν′

+ =1

2ik

∑Jlm

4π Y ∗lm(k)Yl(m+ν−ν′)(r) 〈J(m+ ν) |lmsν〉

×SJl 〈l (m+ ν − ν′) sν′ |J(m+ ν)〉 . (6.51)

The partial-wave expansion of the Eq. (6.9) can be easily obtained if we

consider Eqs. (6.6) and (6.8). According to these equations, the incoming

part of ψ(+) is identical to that of φ. On the other hand, the outgoing part

of ψ(+) is the sum of the outgoing part of φ with the contribution of ψsc,

given in Eq. (6.8). We get

ψ(+)(νk; r)→ 1

(2π)3/2

∑ν′

|sν′〉[e−ikr

rZν′

− +eikr

rZν′

+

], (6.52)

with Zν′

− still given by Eq. (6.31) and the outgoing amplitude

Zν′

+ = Zν′

+ + fν′,ν(Ω) . (6.53)

Setting Zν′

+ = Zν′

+ , and defining k′ as the vector parallel to r with modulus

k, we obtain

fν′,ν(k′, k) =1

2ik

∑Jlm

4π Y ∗lm(k) Yl(m+ν−ν′)(k′)[〈J(m+ ν) |lmsν〉

×SJl 〈l (m+ ν − ν′) sν′ |J(m+ ν)〉 − δν′ν]. (6.54)

A simpler expression for the scattering amplitude is obtained choos-

ing the z-axis along the incident beam. Substituting Y ∗lm(k) →δm0

√(2l + 1) /4π, the above equation becomes

fν′,ν(k′, k) =1

2ik

∑lJ

√4π (2l + 1) Yl(ν−ν′)(k

′)

×[〈l (ν − ν′) sν′ |Jν〉 SJl 〈Jν |l 0 sν〉 − δν′ν

]. (6.55)

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It is convenient to use the orthogonality property of the Clebsch-Gordan

coefficients [Edmonds (1974)],∑JM

〈j1m1 j2m2|JM〉〈JM |j1m′1 j2m′2〉 = δm′1m1δm′2m2

(6.56)

to write

δν′ν =∑JM

〈l (ν − ν′) s ν′ |J M〉〈JM |l 0 sν〉

=∑J

〈l (ν − ν′) sν′ |Jν〉〈Jν |l 0 sν〉 .

Note that the sum of the first line of the above equation reduces to the sum

of the second line because the Clebsch-Gordan coefficients with M 6= ν

vanish. Using this result, Eq. (6.56) becomes

fν′,ν(k′, k) =1

2ik

∑lJ

√4π (2l + 1)Yl(ν−ν′)(k

′)

×〈l (ν − ν′) sν′ |Jν〉〈Jν |l 0 sν〉[SJl − 1

]. (6.57)

For long range potentials, one takes the r →∞ limit in Eq. (6.47) and

in the partial-wave expansion of Eq. (6.10), and the comparison leads to

the partial amplitude fν′,ν(k′, k). Following the same procedures as above,

we obtain

fν′,ν(k′, k) =1

2ik

∑lJ

√4π (2l + 1)Yl(ν−ν′)(r)

×〈l (ν − ν′) sν′ |Jν〉〈Jν |l 0 sν〉 e2iσl[SJl − 1

]. (6.58)

This expression is analogous to Eq. (6.57), except for the inclusion of the

Coulomb phase factor, exp (2iσl) . Above, we use the notation SJl (instead

of SJl ) to stress that it is associated with the partial amplitude fν′,ν . The

bar indicates association with the short-range part of the interaction. The

full scattering amplitude is then obtained adding the Coulomb amplitude,

according to Eq. (6.12).

Eq. (6.54) is the analog of Eq. (2.53a) when the system’s spin is different

from zero. This can be seen more clearly if we introduce the S-matrix

elements averaged over the total angular momentum J , for the transition

ν → ν′,⟨Sν,ν

′

ls

⟩=∑J

〈J(m+ν) |lmsν〉 SJls 〈l (m+ ν − ν′) sν′ |J(m+ ν)〉 . (6.59)

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Using the properties of Clebsch-Gordan coefficients, it is straightforward to

show that Eq. (6.54) can be written as

fν,ν′(k, k′) =

1

2ik

∑lm

4π Y ∗lm(k) Yl(m−∆ν)(k′)[⟨Sν,ν

′

ls

⟩− 1], (6.60)

where ∆ν = ν′ − ν. In the case of spin zero (s = 0, ν = ν′ = 0) the above

equation reduces to Eq. (2.53a).

In practical situations, we must solve numerically the radial equation

for l = 0, 1, ..., lmax. For the partial-wave expansion to converge, lmax should

satisfy the usual condition lmax kR, where R is the range of the potential.

However, for each partial wave the radial equation must be solved several

times, once for each of the possible J-values. That is, 2s + 1 times for

partial-waves where l ≥ s or 2l + 1 times for partial-waves where l < s. In

this way, the procedure to obtain the scattering amplitude is very similar

to the one described in chapter 2, except that, for the same potential range,

the radial equation must be solved a larger number of times.

In a more general situation, both the projectile and the target may

have spin and the algebra involved becomes more complicated. However,

a generalization of the present section to this case is straightforward. For

problems of this kind, we refer to [Satchler (1983)].

6.1.1.2 Partial-wave projected T-matrix, S-matrix and

Lippmann-Schwinger equations

First, we derive Lippmann-Schwinger equations for the radial wave func-

tions of systems with spin. They are similar to the ones for spinless systems.

We start from Eq. (6.5) in the coordinate representation11,

ψ(+)(νk; r) = φ(νk; r) +

∫d3r′ G(+)

0 (E; r, r′) V (r′) ψ(+)(νk; r′). (6.61)

Using the partial-wave expansions for φ(νk; r) (Eq. (6.24)) and ψ(+)(νk; r)

(Eq. (6.38)) in Eq. (6.61), we obtain12

uJl (k, r) = l(kr) +

∫dr′ g(+)

0,l (E; r, r′) V Jl (r′) uJl (k, r′). (6.62)

11For simplicity, we consider the case of a rotationally invariant local potential.12We are assuming that the potential is diagonal in the representation of orbital angularmomentum. The general situation where this condition is not satisfied is included in the

discussion of multi-channel scattering (chapter 9).

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Above, V Jl (r′) is given by Eq. (6.40) and the angular momentum projected

components of the free Green’s function are

g(+)

0,l (E; r, r′) =

∫dζ dζ ′

(Y lJM (ζ)

)∗G(+)

0 (E; r, r′) Y lJM (ζ ′). (6.63)

When the potential contains a Coulomb part, the Born series does not

converge and the above expressions are not useful. However the results of

section 4.5 may be applied. The potential can be split into two parts,

V = V1 + V2, (6.64)

with the long-range Coulomb interaction included in V1. The Lippman-

Schwinger equation can then be written in terms of the exact solutions for

the potential V1, χ(+)(νk; r), which satisfies Eq. (4.141). Proceeding as in

the case of spinless particles (see chapter 4), we get

ψ(+)(νk; r) = χ(+)(νk; r) +

∫d3r′ G(+)

1 (E; r, r′) V2(r′) ψ(+)(νk; r′), (6.65)

where

G(+)

1 (E; r, r′) =

⟨r

∣∣∣∣ 1

E −K − V1 + iε

∣∣∣∣ r′⟩ . (6.66)

It is straightforward to carry out angular momentum expansions in

Eq. (6.65) and derive the projected Lippmann-Schwinger equation involving

distorted waves,

uJl (k, r) = wJl (k, r) +

∫dr′ g(+)

1,l (E; r, r′) V J2,l(r′) uJl (k, r′). (6.67)

Above, g(+)

1,l (E; r, r′) and V J2,l(r′) are given by Eqs. (6.63) and (6.40), re-

placing respectively G(+)

0 (E; r, r′)→ G(+)1 (E; r, r′) and V (r′)→ V2(r

′), and

wJl (k, r) is the radial wave function distorted by the potential V1 (solution of

the radial equation setting V2(r) = 0). In the particular case where V1 = VC,

the distorted wave is equal to the Coulomb wave function, Fl(η, kr).

Now we derive the angular momentum projection of the T- and the S-

matrices. For this purpose we introduce the normalized energy and angular

momentum eigenfunctions,

φ (ElJM ; ζ) =

(2µk

π~2

)1/2

Y lJM (ζ)(kr)

kr(6.68)

ψ (ElJM ; ζ) =

(2µk

π~2

)1/2

Y lJM (ζ)uJl (k, r)

kr. (6.69)

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One can easily prove that they satisfy the orthonormality relations (see

section 4.6.1)⟨φ (ElJM)

∣∣φ (E′l′J ′M ′)⟩

= δll′ δJJ ′ δMM ′ δ(E − E′), (6.70)⟨φ (ElJM)

∣∣φ (E′l′J ′M ′)⟩

= δll′ δJJ ′ δMM ′ δ(E − E′). (6.71)

The partial wave projected T-matrix is then defined as

T Jl (E) ≡⟨φ(ElJM)

∣∣T ∣∣φ(ElJM)⟩

=⟨φ(ElJM)

∣∣V ∣∣ψ(ElJM)⟩.

The above matrix-elements can be evaluated in the coordinate representa-

tion, using the explicit forms of φ (ElJM ; ζ) and ψ (ElJM ; ζ) (Eqs. (6.68)

and (6.69)). We get

T Jl (E) =k

πE

∫dr l(kr)V

Jl (r)uJl (k, r). (6.72)

To express the T-matrix in momentum space,

Tν′k′;νk =⟨φ(ν′k′)

∣∣V ∣∣ψ(+)(νk)⟩, (6.73)

in terms of T Jl (E), we re-write the angular momentum expansions of

φ(ν′k′, ζ) and ψ(+)(νk; ζ) in terms of YlJM (ζ). Using Eqs. (6.68) and (6.69)

in Eqs. (6.27) and (6.44), we get the expansion

Tν′k′;νk =~2

µk

∑Jlm

Yl(m+ν−ν′)(k′) Y ∗lm(k) 〈lmsν| J(m+ ν)〉

× 〈J(m+ ν)|l(m+ ν − ν′)sν〉 T Jl (E). (6.74)

This expression may be simplified if one takes the z-axis along the incident

beam. One can then replace: Ylm(k) → Ylm(z) = δm0

√(2l + 1)/4π, and

get

Tν′k′;νk =~2

µk

∑Jl

√2l + 1

4πYl(ν−ν′)(k

′) 〈l0sν| Jν〉

× 〈Jν|l(ν − ν′)sν〉 T Jl (E). (6.75)

A relation between the angular momentum projected T- and S-matrices

can easily be derived. This can be achieved multiplying the above equa-

tion by the factor −4π2µ/~2 and comparing the result with the scattering

amplitude of Eq. (6.57). According to the general relation between the T-

matrix and the scattering amplitude (Eq. (4.64)), they should be identical.

In this way, we get

SJl (E) = 1− 2πi T Jl (E). (6.76)

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6.2 The scattering of a spin 1/2 particle

Nucleons and electrons are spin- 1/2 particles and their scattering and spin

polarization are quite important. The formalism developed above becomes

quite simple in the case of a spin- 1/2 particle colliding with a particle with

spin-zero. In this case, it is convenient to write the wave function and the

scattering amplitude as vectors and matrices in the two-dimensional space

of spin. This procedure becomes particularly useful when the initial state

is not polarized along the beam axis and the incident state is described in

terms of the density matrix.

In the collision of one spin- 1/2 particle with one spin-zero particle, the

spin quantum numbers take the values, s = 1/2 and ν = ±1/2. We then

use the short hand notations |+〉 and |−〉 to represent the spin-up and spin-

down states, respectively. The scattering amplitude is a 2x2 matrix in the

spinor space, which can be written as

f(θ, ϕ) =∑

ν,ν′=±|ν′〉 fν′,ν(θ, ϕ) 〈ν| . (6.77)

The matrix-elements fν′,ν are given by Eq. (6.57), for s = 1/2 and ν, ν′ =

±1/2. The amplitudes f++ and f+− are

f++(k′) =1

2ik

∑l

√4π (2l + 1) Yl0(k′)

×

∣∣⟨l 0 1/21/2 |(l + 1/2) 1/2

⟩∣∣2 [S+l − 1

]+∣∣⟨l 0 1/2

1/2 |(l − 1/2) 1/2

⟩∣∣2 [S−l − 1]

(6.78)

and

f−+(k′) =1

2ik

∑l

√4π (2l + 1) Yl1(k′)

×

⟨l 1 1/2

(−1/2

)|(l + 1/2) 1/2

⟩ ⟨(l + 1/2) 1/2|l 0 1/2

1/2

⟩ [S+l − 1

]+⟨l 1 1/2

(−1/2

)|(l − 1/2) 1/2

⟩ ⟨(l − 1/2) 1/2|l 0 1/2

1/2

⟩ [S−l − 1

].

(6.79)

Above, we used the short-hand notation: S l±1/2

l = S±l . These expressions

are greatly simplified if one uses the explicit values of the Clebsch-Gordan

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coefficients ([Edmonds (1974)], table 5.2, p.89). The non-spin-flip ampli-

tudes become

f++(k′) =∑l

[(l + 1)

[S+l − 1

]+ l

[S−l − 1

]2ik

]Pl(cos θ) (6.80)

f−−(k′) = f++(k′) (6.81)

and the spin-flip amplitudes can be put in the form

f−+(k′) =∑l

−[S+l − S

−l

2ik

]dPl(cos θ)

d cos θsin θ eiϕ (6.82)

f+−(k′) =∑l

[S+l − S

−l

2ik

]dPl(cos θ)

d cos θsin θ e−iϕ. (6.83)

To get the above results we have used the relations [Jackson (1975)]

Yl1(θ, φ) = −

√2l + 1

4π l(l + 1)

[dPl(cos θ)

d cos θ

]sin θ eiϕ (6.84)

and

Yl(−1)(θ, ϕ) = −Y ∗l1(θ, ϕ). (6.85)

It is convenient to express the matrix elements of the scattering ampli-

tude in terms of two scalar functions of θ:

g(θ) =∑l

[(l + 1)

[S+l − 1

]+ l

[S−l − 1

]2ik

]Pl(cos θ) (6.86)

and

h(θ) =∑l

[S+l − S

−l

2ik

]sin θ

dPl(cos θ)

d cos θ. (6.87)

In this way, we get

f++(θ) = f−−(θ) = g(θ) (6.88)

f+−(θ) = h(θ) e−i ϕ(θ) (6.89)

f−+(θ) = −h(θ) ei ϕ(θ) (6.90)

or, in matrix form

f2x2(θ) ≡(f++(θ) f+−(θ)

f−+(θ) f−−(θ)

)=

(g(θ) h(θ) e− iϕ

−h(θ) ei ϕ g(θ)

). (6.91)

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The scattering amplitude matrix can be expressed as a linear combina-

tion of the Pauli matrices and the identity,

I2x2 =

(1 0

0 1

), σx =

(0 1

1 0

), σy =

(0 −ii 0

)and σz =

(1 0

0 −1

).

These four matrices form a complete set for the space of complex 2x2 ma-

trices. It can be easily checked that Eq. (6.91) can be written as (see

problem 4)

f2x2(θ) = g(θ) I2x2 + i h(θ) n · σ, (6.92)

where σ ≡ σx x + σy y + σz z and n = k × k′/|k × k′| is the unit vector

perpendicular to the scattering plane. Taking the incident beam along the

z-axis, k = z and we can write

n =z× k′

|z× k′|=− sin θ sinϕ x + sin θ cosϕ y

sin θ= − sinϕ x+cosϕ y. (6.93)

The spin of the scattered wave, |χsc〉, is obtained applying the 2x2

scattering amplitude matrix on the spin of the incident wave,∣∣χin

⟩. That is

|χsc〉 = f(k′)∣∣χin

⟩. (6.94)

The cross section is then given by

dσ(k′)

dΩ= 〈χsc|χsc〉 . (6.95)

For incident beams polarized with spin-up and spin-down the corresponding

cross sections are respectively

dσ+(k′)

dΩ= |f++(k′)|2 + |f−+(k′)|2 (6.96)

dσ−(k′)

dΩ= |f−−(k′)|2 + |f+−(k′)|2 . (6.97)

For the arbitrary polarization,∣∣χin⟩

=

(α

β

), (6.98)

the cross section is

dσ(k′)

dΩ=∣∣∣αf++(k′) + βf−+(k′)

∣∣∣2 +∣∣∣αf+−(k′) + βf−−(k′)

∣∣∣2+ 2 Re

(α∗f∗++(k′) + β∗f∗−+(k′)

)(αf+−(k′) + βf−−(k′))

.

(6.99)

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The above discussion assumes that the incident beam is in a pure state

in the spin space. However in most experiments the incident beam is in a

mixed state. The only available information is the spin polarization of the

incident beam. For a pure state |χ〉, the components of the polarization

P = Px x + Py y + Pz z are the expectation values of the Pauli matrices,

Pi = 〈χ|σi |χ〉 , (6.100)

with i = x, y, z. On the other hand, mixed states are represented by den-

sity matrices, ρ, in the spin space. In this case, the components of the

polarization are given by

Pi = tr σi ρ . (6.101)

It can be easily checked that an incident beam in a mixed state with

polarization Pin is described by the density matrix13

ρin =1

2

[I2x2 + σ ·Pin

]=

1

2

I2x2 +∑

i=x,y,z

P ini σi

. (6.102)

We have seen that in the scattering process a pure incident spin state,

χin, is transformed into an scattered spin state. The transformation is

performed through the action of the matrix f . That is,∣∣χin⟩−→ |χsc〉 = f

∣∣χin⟩. (6.103)

Accordingly, an operator O acting on the spin space is transformed as O →fOf†. Thus, the density matrix describing the incident wave transforms

into the scattered density matrix

ρin −→ ρsc = fρinf† (6.104)

and the cross section is given by

dσ(k′)

dΩ= tr ρsc = tr

fρinf†

. (6.105)

In experiments with polarized beams described by an initial density matrix,

ρin, the cross section should be expressed in terms of the known polarization

of the incident beam, Pin. For this purpose, we first use Eq. (6.102) and

(6.92) in Eq. (6.104), to get

ρsc =1

2

[g(θ) I2x2 + ih(θ) n · σ

] [I2x2 + σ ·Pin

] [g∗(θ) I2x2 − i h∗(θ) n · σ

].

(6.106)

13It is an immediate consequence of the facts that tr I2x2 = 2 and the Pauli matrices are

traceless.

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The above expression can be considerably simplified using the property of

Pauli matrices,

(σ ·A) (σ ·B) = (A ·B) I2x2 + iσ · (A×B) , (6.107)

where A and B are vectors and I2x2 is the identity matrix in 2 dimensions.

Thus, one gets

ρsc =1

2

[|g(θ)|2 + |h(θ)|2

+ 2 Im h∗(θ) g(θ n ·Pin]I2x2 +

1

2σ ·∆. (6.108)

Above, we have introduced the angle-dependent vector, ∆,

∆ = 2 Im g(θ)h∗(θ) n +(|g(θ)|2 − |h(θ)|2

)Pin

+ 2 |h(θ)|2(n ·Pin

)n− 2 Re g(θ)h∗(θ) n×Pin. (6.109)

Since the traces of the Pauli matrices are zero, the cross section for a initial

polarization Pin is given by,[dσ(k′)

dΩ

]Pin

= tr ρsc = |g(θ)|2 + |h(θ)|2 + 2 Im g(θ)h∗(θ) n ·Pin,

(6.110)

or, [dσ(k′)

dΩ

]Pin

=

(dσ(k′)

dΩ

) [1 + P(θ) ·Pin

]. (6.111)

Above, we have introduced the polarization vector,

P(θ) =

[2 Im g(θ)h∗(θ)|g(θ)|2 + |h(θ)|2

]n (6.112)

and the unpolarized cross section,

dσ(k′)

dΩ= |g(θ)|2 + |h(θ)|2 . (6.113)

Note that the polarization vector is independent of the polarization of the

incident beam.

The final spin polarization,

Psc(θ) =tr σρsc(θ)tr ρsc(θ)

, (6.114)

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then becomes

Psc(θ) =P(θ) + Pin −Q(θ)×Pin

1 + P(θ) ·Pin

+2 |g(θ)|2

(n ·Pin

)n− |h(θ)|2 Pin

(1 + P(θ) ·Pin)(|g(θ)|2 + |h(θ)|2

) , (6.115)

where Q(θ) is called the spin rotation vector,

Q(θ) =

[2 Re g(θ)h∗(θ)|g(θ)|2 + |h(θ)|2

]n. (6.116)

The important feature of Eq. (6.115) is that even if the incident beam is

unpolarized (Pin = 0), the final polarization is not. Thus one can produce

polarized particles by merely scattering them. The interaction responsible

for this final polarization is the spin-orbit term. Thus by measuring the

final polarization of unpolarized spin- 1/2 beam of particles one can learn

about the nature of the spin-orbit interaction.

To summarize, for unpolarized spin- 1/2 particles, the cross section and

final polarization of the scattered beam are given by,

dσ(k′)

dΩ= |g(θ)|2 + |h(θ)|2 .

and

Psc(θ) = P(θ) =

[2 Im g(θ)h∗(θ)|g(θ)|2 + |h(θ)|2

]n.

The final polarization is perpendicular to the scattering plane. Any compo-

nent of the polarization which lies on the scattering plane is an indication

of violation of parity. Such component would be present if the interaction

were to contain a term of the type σ · p, where p is the linear momentum

vector.

How to measure the polarization? If the initial polarization is not zero,

then the cross section is given by Eq. (6.110). Taking Pin to be pointing

at an azimuthal angle of ϕ from n, namely, Pin · n = P in cosϕ, then by

flipping the direction of n we can measure the cross section for scattering to

the right and to the left. The difference is directly related to the magnitude

of initial polarization. We take Pin to be parallel to n, namely ϕ = 0. This

defines, say, the scattering to the right. Then the scattering to the left is

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defined by ϕ = π. The right-left asymmetry, called the analyzing power,

A(θ), is then given by

A(θ) =[dσ/dΩ]Pin

R− [dσ/dΩ]Pin

L

[dσ/dΩ]PinR

+ [dσ/dΩ]PinL

=2 |Im g(θ)h∗(θ)||g(θ)|2 + |h(θ)|2

P in, (6.117)

with L and R standing for scattering to the left and to the right, respec-

tively. The above type of measurement requires a rather complicated ex-

perimental set up with magnetic field used to fix the value and direction of

Pin. An easier way to measure the strength of the spin-orbit interaction is

through a double scattering experiment. The initial polarization is taken

to be zero. After the first scattering, the cross section is given by,

dσ(k′)

dΩ= |g(θ)|2 + |h(θ)|2 . (6.118)

The scattered beam is polarized according to Eq. (6.2). This first target is

thus a polarizer. The beam, now with an ‘initial’ polarization vector P(θ)

along n, is then scattered from another target in the same scattering plane,

both to the right and to the left. This second target acts as an analyzer.

According to Eq. (6.117), we get, for the analyzing power, or right-left

asymmetry,

A2(θ) =

[2 |Im g(θ)h∗(θ)||g(θ)|2 + |h(θ)|2

]2

(6.119)

From the first scattering, the unpolarized cross section dσ/dΩ = |g(θ)|2 +

|h(θ)|2 is determined. For all practical purposes the term |g|2 dominates

over |h|2, and thus one can safely say that the first scattering determines

|g|2. After the second scattering, the asymmtery is found proportional

to[Im g(θ)h∗(θ) /|g|2

]2. In this way, the absolute value of the term

Im g(θ)h∗(θ) is determined. The above double scattering experiment

with a spin unpolarized beam determines only the absolute value of g.

The determination of the phase requires a more complicated set up. The

spin rotation vector Q(θ) of Eq. (6.116) is a quantity which can furnish this

information. The determination of the spin rotation is more involved as it

requires the triple experiment set up.

It is instructive to calculate the amplitudes h and g at very low energies.

In this case, we have,

g =1

2ik

[S

1/20 − 1

](6.120)

h =1

2ik

[S

3/21 − S

1/21

], (6.121)

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and the spin polarization and rotation projected along the normal n,

n ·P(θ) = 2 sin θ

Im(S

1/20 − 1

) (S

3/21 − S

1/21

)∗|S

1/20 − 1|2

(6.122)

n ·Q(θ) = 2 sin θ

Re(S

1/20 − 1

) (S

3/21 − S

1/21

)∗|S

1/20 − 1|2

(6.123)

6.2.1 The eikonal approximation in collisions of a

spin- 1/2 particle

At high energies, the eikonal approximation is applicable (see section 5.2.1).

Here, we use the eikonal approximation in collisions of a spin- 1/2 projectile

with a spin-zero target. We consider the case of a spherically symmetric

potential containing a spin-orbit term, with the form

V (r) = V0(r) + V1(r)σ · l ≡ V0(r) + V1(r)σ · (r× k) . (6.124)

In high energy collisions at forward angles, k ' k z. One can then approx-

imate r× k ' b× k and the eikonal phase of Eq. (5.118) becomes

χ(b) =k

2E

∫ ∞−∞

dz[V0

(√z2 + b2

)+ σ · (b× k) V1

(√z2 + b2

)],

(6.125)

or

χ(b) = ∆0(b) + ∆1(b) σ · (b× k) (6.126)

with

∆i(b) = − k

2E

∫ ∞−∞

dz Vi

(√z2 + b2

). (6.127)

The S-matrix within the eikonal approximation then can be written

S(b) = ei∆0(b)+i∆1(b)σ·(b×k) = ei∆0(b) ei∆1(b)σ·(b×k). (6.128)

The functions ∆0(b) and ∆1(b) can easily be calculated using numerical

methods. Thus, the evaluation of the first factor on the RHS of the above

equation is straightforward. Let us then discuss the second factor. It has

the general form exp[iaσ · A

], where a is a scalar function and A is a

vector. In our case they are given by

A =b× k

|b× k|and a = |b× k| ∆1(b). (6.129)

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Since at forward angles k ' k z, k is nearly perpendicular to b and thus

|b× k| ' kb. In this way, we can approximate

a = kb∆1(b). (6.130)

To evaluate exp[iaσ · A

], we use the series expansion of the exponential,

writing the even and the odd terms separately. That is

eiaσ·A =∞∑n=0

(ia)n

n!

(σ · A

)n=∞∑m=0

(ia)2m

(2m)!

(σ · A

)2m

+∞∑m=0

(ia)2m+1

(2m+ 1)!

(σ · A

)2m+1

, (6.131)

or

eiaσ·A =∞∑m=0

(−)ma2m

(2m)!

(σ · A

)2m

+ iσ · A∞∑m=0

(−)ma2m+1

(2m+ 1)!

(σ · A

)2m

. (6.132)

According to Eq. (6.107), we can replace(σ · A

)2m

= I2x2. (6.133)

Using this result in Eq. (6.132), identifying the cosinus (even terms) and

sinus (odd terms) series, and replacing a by its explicit form (Eq. (6.130)),

we get

eiaσ·A = cos(kb∆1(b)

)I2x2 + iσ · A sin

(kb∆1(b)

). (6.134)

Using Eq. (6.134), the eikonal S-matrix becomes

S(b) = eiΞ(b) = ei∆0(b)[cos(kb∆1(b)

)I2x2 + iσ · A sin

(kb∆1(b)

)].

(6.135)

The scattering amplitude is obtained inserting the above S-matrix into

Eq. (5.120). We get

fe(k′) ≡ fe(θ, ϕ) =ik

2π

∫db eiq·b

[ [ei∆0(b) cos (kb∆1(b))− 1

]I2x2

+ i ei∆0(b) sin (kb∆1(b)) σ · A

]. (6.136)

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The above expression can be written as a sum of two terms,

fe(θ, ϕ) = fnsf(θ) + fsf(θ, ϕ), (6.137)

corresponding respectively to the non-spin-flip and spin-flip, amplitudes.

The first one is given by

fnsf(θ) = I2x2

[ik

∫b db

[ei∆0(b) cos

(kb∆1(b)

)− 1]

×(

1

2π

∫ 2π

0

eikb cosφ dϕb

)]. (6.138)

Above, ϕb is the azimuthal angle associated with the variable b, whereas φ

is the angle between the vectors q and b. With the neglect of longitudinal

momentum transfer, both q and b are vectors parallel to the x-y plane.

Thus, the angle between them is simply the difference φ = ϕb − ϕ, with ϕ

being the azimuthal angle describing q. Note that ϕ is also the azimuthal

angle determining the scattering plane. This plane contains the z-axis and

form an angle ϕ with the plane x-z. To evaluate the integral within round

brackets, we change variable ϕb → φ = ϕb−ϕ. This transformation leaves

the differential unchanged and the integration limits become −ϕ, 2π−ϕ.However, this change does not affect the result of the integration14. In

this way, the integral over φ reduces to the integral representation of the

Bessel function of order zero (Eq. (5.124)) and the non-spin-flip scattering

amplitude becomes

fnsp(θ) = I2x2[ik

∫b db

[ei∆0(b) cos

(kb∆1(b)

)− 1]J0(qb)

], (6.140)

where the momentum transfer can be expressed in terms of the scattering

angle through the relation q = 2k sin(θ/2).

We now turn to the spin-flip amplitude,

fsf(θ, ϕ) = − k

2πσ ·

[∫b db ei∆0(b) sin

(kb∆1(b))

∫A eiqb cosφ dϕb

].

(6.141)

14Since cos (−φ) = cos (2π − φ),

1

2π

∫ 2π−ϕ

−ϕeikb cosφ dφ =

1

2π

∫ 2π

0eikb cosφ dφ. (6.139)

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To evaluate the integral over ϕb, we must determine A. First, we point

out that the vectors b and k are orthogonal and thus we can write

A ≡ b× k

|b× k|= b× k. (6.142)

The unit vectors k and b are: k = z (incident beam parallel to the z-axis)

and b = cosϕb x + sinϕb y. The unit vector A then is

A ≡ b× k = (cosϕb x + sinϕb y)× z = − cosϕb y + sinϕb x. (6.143)

Using this result, the integral over ϕb of Eq. (6.141) becomes∫ 2π

0

A eiqb cosφ dϕb = −y

∫ 2π

0

eiqb cosφ cosϕb dϕb

+ x

∫ 2π

0

eiqb cosφ sinϕb dϕb. (6.144)

Changing the integration variable to φ = ϕb − ϕ and replacing

cosϕb = cos (φ+ ϕ) = cosφ cosϕ− sinφ sinϕ

sinϕb = sin (φ+ ϕ) = sinφ cosϕ+ cosφ sinϕ,

Eq. (6.144) takes the form∫ 2π

0

A eiqb cosφ dϕb = (− cosϕ y + sinϕ x)

∫ 2π

0

eiqb cosφ cosφ dφ

+ (sinϕ y + cosϕ x)

∫ 2π

0

eiqb cosφ sinφ dφ.

(6.145)

Again, we are keeping the original integration limits because the modi-

fications induced by the variable change do not change the integral (see

footnote on the previous page). It can easily be checked that∫ 2π

0

eiqb cosφ sinφ dφ = 0. (6.146)

and ∫ 2π

0

eiqb cosφ cosφ dφ = −i d

d(qb)

(∫ 2π

0

eiqb cosφ dφ

)= −2πi

d

d(qb)J0(qb) = 2πi J1(qb). (6.147)

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The above result was obtained with the help of the property of Bessel

functions: dJ0(x)/dx = −J1(x). Inserting the integrals of Eqs. (6.146) and

(6.147) in Eq. (6.145) we get the result of the integration over ϕb:∫ 2π

0

A eiqb cosφ dϕb = 2πi [sinϕ x− cosϕ y] J1(qb). (6.148)

The vector within square brackets in Eq. (6.148) is orthogonal to the scat-

tering plane. Comparing it with Eq. (6.93), we conclude that

sinϕ x− cosϕ y = −n. (6.149)

Thus, we can write∫ 2π

0

A eiqb cosφ dϕb = −2πi J1(qb) n. (6.150)

Inserting this result in Eq. (6.141) we get the spin-flip amplitude

fsf(θ, ϕ) = iσ · n[k

∫b db ei∆0(b) sin

(kb∆1(b)

)J1(qb)

]. (6.151)

Adding Eqs. (6.140) and (6.151), the scattering amplitude takes the general

form of Eq. (6.92), with

g(θ) = ik

∫b db

[ei∆0(b) cos

(kb∆1(b)

)− 1]J0(qb) (6.152)

and

h(θ, ϕ) = k

∫b db ei∆0(b) sin

(kb∆1(b)

)J1(qb). (6.153)

6.3 Identical particles

If the projectile and the target are identical, Quantum Mechanics imposes a

few changes in the results of the previous chapters. When a particle reaches

a detector, the experiment cannot tell if this particle is the projectile or

the target. On the other hand, momentum conservation guarantees that

whenever one collision partner emerges at the orientation r ≡ θ, ϕ (in the

center of mass frame), the other emerges at the opposite orientation, i.e.,

− r ≡ π − θ, ϕ+ π . Therefore, the cross sections for these orientations

will be mixed in some way. From the point of view of classical mechanics,

the cross section, which we denote σinc, is an incoherent sum of the cross

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sections for the corresponding scattering of discernible particles at these

orientations15. That is16

σinc (θ, ϕ) = σ (θ, ϕ) + σ (π − θ, ϕ+ π) . (6.154)

Note that the total elastic scattering cross section obtained by angular in-

tegration of the above equation is twice that for the collision of discernible

particle under the same conditions. This result is not surprising since σinc

contains equal contributions from the projectile and from the target parti-

cles. The Quantum Mechanical description of such collisions is more com-

plicated, as discussed below.

In Quantum mechanics, the total wave function for pairs of identical

particles with integer spins, i.e., two bosons, must be symmetric with re-

spect to the exchange of these particles. On the other hand, in the case of

particles with half-integer spin, i.e., two fermions, the wave function must

be anti-symmetric with respect to exchange. The wave function of a parti-

cle with spin can be expressed in terms of the set of coordinates q ≡ r, µ,where r is the position vector and µ is the spin component along the z-axis.

In collisions of identical particles, the wave function must be an eigenstate

of the exchange operator PPT (subscripts p and t stand for projectile and

target). This operator can be written

PPT = P rPT P

sPT ,

where P rPT exchanges rP rT and P sPT exchanges µP µT. That is

PPT ψλ(qP, qT) = ψλ(qT, qP) = λψλ(qP, qT),

with λ = 1 for boson-boson collisions and λ = −1 for fermion-fermion col-

lisions. Symmetric (ψ+(qP, qT)) or anti-symmetric (ψ−(qP, qT)) wave func-

tions can be generated with the help of the projector Aλ, given by

Aλ =1

2[1 + λPPT] . (6.155)

That is,

ψ+(qP, qT) = N A+ ψ(qP, qT) =1√2

[ψ(qP, qT) + ψ(qT, qP)] (6.156)

ψ−(qP, qT) = N A− ψ(qP, qT) =1√2

[ψ(qP, qT)− ψ(qT, qP)] , (6.157)

where N =√

2 is an appropriate normalization factor.15In our discussion, the cross-section dσinc (θ, ϕ) /dΩ is only classical in the sense that

the contributions from the target and the projectile to the cross section are summedincoherently. The cross sections dσ (θ, ϕ) /dΩ and dσ (π − θ, π + ϕ) /dΩ are evaluated

by Quantum Mechanics.16Henceforth, we adopt the compact notation:

dσ(θ)dΩ

→ σ(θ).

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6.3.1 Collisions of identical spinless particles and particles

with aligned spins

We consider here the simple situation were the wave function in the spin

space is symmetric with respect to projectile-target exchange. This condi-

tion is trivially satisfied when the identical particles have spin s = 0. A

general discussion of the s 6= 0 case, which is more complicated, will be

postponed to the next section. However, a simple situation is also encoun-

tered when both the projectile and the target are polarized with their spins

aligned. The total spin is then S = 2s, and the spin wave function is sym-

metric with respect to exchange (see section 6.3.2). In this case, the spin

degree of freedom can be ignored and the eigenvalue λ is given by the ex-

change symmetry in the coordinate space. The wave function should then

be written

ψ±(rP, rT) =1√2

[ψ(rP, rT) ± ψ(rT, rP)] , (6.158)

with ‘ + ’ for bosons and ‘ − ’ for fermions.

The usual treatment of the scattering problem replaces the position

vectors rP and rT by center of mass (Rc.m.) and relative (r) coordinates,

which are given by the transformations (see section 1.3.1),

rP; rT → Rc.m. =rP + rT

2; r = rP − rT, (6.159)

Therefore, we should discuss the effects of exchange on these variables.

Inspection of Eq. (6.159) indicates that they transform as

P †PT Rc.m. PPT = Rc.m. (6.160)

P †PT rPPT = − r . (6.161)

In this way, we have

PPT = Π .

Above, Π is the parity operator, which carries out space inversion of the

vector r. Eq. (6.158) then yields

ψ±(r) =1√2

[ψ(r)± ψ(−r)] . (6.162)

Therefore, the exchange symmetry requires that the wave function have

even parity, π = +, in the case of bosons, or odd parity, π = −, in the case

of fermions.

Eq. (6.162) leads to a modification of the scattering boundary condition

of Eq. (2.3). Writing

ψ(+)

± (r) = φ±(r) + ψsc± (r) ,

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the asymptotic forms of the incident plane wave and the scattered wave are

respectively17

φ±(r)→ 1

(2π)3/2

[eik·r ± e−ik·r√

2

](6.163)

and

ψsc± (r) → 1

(2π)3/2

[f(θ)± f(π − θ)√

2

]eikr

r. (6.164)

Above, we have assumed that the potential is spherically symmetric, so that

the wave function has axial symmetry and we can write f(r) ≡ f(θ, ϕ) =

f(θ) and f(−r) = f(π − θ). We now consider the implications of these

modifications on the cross section. Inspecting Eq. (6.163), we see that

there are two incident currents. One approaching from the left, arising from

exp (ik · r) , and one approaching from the right, arising from exp (−ik · r) .

The incident flux appearing in Eq. (1.68) should have contributions from

both. Since the opposite signs of the currents are compensated by the

opposite orientation of the normals, the two contributions are identical. In

this way one gets a factor 2 that cancels the factor 1/2 arising from the

1/√

2 normalization of φ±(r). Therefore, the incident flux is unchanged

with respect to the value obtained in chapter 1. We now consider the flux

scattered onto the detector. Comparing to section 1.5, one should make

the replacement

|f(θ)|2 →∣∣∣∣ f(θ)± f(π − θ)√

2

∣∣∣∣2 =1

2

∣∣f(θ)± f(π − θ)∣∣2,

where f is the scattering amplitude for discernible particles under the same

conditions. Since the factor 1/2 is cancelled by the factor 2 arising from

the indiscernibility of projectile and target, discussed in the beginning of

the present section, we get the cross section

σ±(θ) = |f±(θ)|2 = |f(θ)± f(π − θ)|2 . (6.165)

Above, we have introduced the notation

f±(θ) = f(θ)± f(π − θ). (6.166)

Evaluating Eq. (6.165) and comparing with Eq. (6.154), we can write

σ±(θ) = σinc(θ) + σint(θ), (6.167)

where σint(θ) is the interference term

σint(θ) = ± 2 Ref∗(θ) f(π − θ)

. (6.168)

17Henceforth, we are adopting the normalization constant A = (2π)−3/2.

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This term, which has no classical analog, plays a very important role. It

has a strong spin-dependence, whereas σinc(θ) is independent of the spin.

This point will be discussed in the next section.

Direct inspection of Eqs. (6.154), (6.167) and (6.168), indicates that

the cross sections σinc(θ), σint(θ) and σ±(θ) are symmetric with respect

to θ = π/2. This feature is connected with an important property of the

partial-wave expansion of f±(θ). For simplicity, we discuss the case of

scattering from a short-range spherically symmetric potential.

Inserting Eq. (2.57b) into Eq. (6.166) and using the property of Legendre

Polynomials

Pl(cos(π − θ)) = Pl(− cos θ) = (−)lPl(cos θ),

we get

f±(θ) =1

2ik

∑l

[(2l + 1)Pl(cos θ) (Sl − 1)

]×[1± (−)l

]. (6.169)

This result is analogous to Eq. (2.57b), except for the factor[1± (−)l

]within the partial-wave summation. In boson-boson (fermion-fermion) col-

lisions, this factor doubles the contribution from even-waves (odd-waves)

and eliminates those from odd-waves (even-waves). This is the justification

for excluding odd-waves in the calculations for identical spinless bosons

shown in figures 3.2 and 3.5.

6.3.1.1 Coulomb scattering

Let us apply the results of the previous section to the case of Coulomb

scattering. We first consider in detail collisions of identical spinless bosons.

In this case σ+(θ) is known as the Mott cross section and for this reason

we use the notation σM(θ). We must evaluate

σM(θ) = |fR(θ) + fR(π − θ)|2 , (6.170)

using the analytical expression for the Rutherford scattering amplitude, fR

(Eq. (3.29)). Writing the cross section as in Eq. (6.167) and normalizing it

with respect to the Rutherford cross section at θ = 90o, we get18

σM(θ) ≡ σM(θ)

σR(π/2)= σinc(θ) + σint(θ), (6.171)

with

σinc(θ) ≡ σinc(θ)

σR(90o)=

1

4

[1

sin4 (θ/2)+

1

cos4 (θ/2)

](6.172)

18We use the notation σ(θ) for the cross section normalized with respect to σR(90o).

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30 60 90 120 150(deg)

10-1

100

101

102

103R

30 60 90 120 150(deg)

10-1

100

101

102

103

Fig. 6.1 Comparison between the Mott (solid lines) and the incoherent (dashed lines)

cross sections, for two values of the Sommerfeld parameter. The cross sections are

normalized with respect to the Rutherford cross section at θ = 90o.

and

σint(θ) ≡σint(θ)

σR(90o)=

1

4

[2

cos [2η ln (tan(θ/2))]

sin2(θ/2) cos2(θ/2)

].

Note that σinc(θ) is energy-independent, whereas σint(θ) depends on

the collision energy through the Sommerfeld parameter. This energy-

dependence is carried over to σM(θ). This is illustrated in figure 6.1. The

normalized cross sections are shown for the values of the Sommerfeld pa-

rameter η = 4.0 and η = 0.4. Inspecting this figure, one notices two inter-

esting points. The first is that the values of the normalized cross sections

at θ = 90o, are independent of η. They are: σM(90o) = 4 and σinc(90o) = 2.

The second is that the shape of σM(θ) depends on the collision energy. For

η = 4.0 it has a maximum at θ = 90o while for η = 0.4 it has a minimum.

This behavior can be understood if we separately analyze the two terms

contributing to σM in Eq. (6.171). The incoherent cross section (dashed

lines in figure 6.1) has always a minimum at θ = 90o. On the other hand,

at this angle the amplitudes f(θ) and f(180o−θ) interfere constructively, so

that the interference term has a maximum. The shape of σM(θ) depends on

which of these trends dominates. If the interference term is slowly varying,

which happens for small η values, the incoherent part dominates and σM(θ)

has a minimum at 90o. The opposite situation occurs when η is large. This

condition can be quantitatively stated if we evaluate the second derivative

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30 50 70 90 110 130 150(deg)

100

101

102

103

/R(90o)

2

Fig. 6.2 Same as figure 6.1 but for the transitional value η =√

2.

of σM(θ) with respect to θ, at θ = 90o. We find19[d2σM(θ)

dθ2

]θ=π/2

= 8(2− η2

).

The above equation indicates that the transition from a minimum to a

maximum occurs at η =√

2. This situation is illustrated in figure 6.2. Note

that the cross section in this case becomes remarkably flat over a broad

angular interval around θ = 90o.

A better insight of the influence of the interference term on the Mott

cross section can be achieved if one plots σM normalized with respect to

σinc(θ), instead of to σR(90o). This is done in figure 6.3. In this case, there

is a maximum at θ = 90o independently of the η-value. Thus, a maximum

of σ/σinc ratio at θ = 90o could be taken as a signature for collisions of

identical charged bosons.

We now extend the results of this section to collisions of identical

fermions with aligned polarizations. In this case, the incoherent cross sec-

tion remains the same but σint changes sign. This change leads to an im-

portant consequence: the cross section associated with the antisymmetric

wave function, which we here denote by σA(θ), has always a minimum at

θ = 90o. It can be readily checked that the cross section at this minimum

is zero. This situation is illustrated in figure 6.4, for the same values of the

Sommerfeld parameter of the previous two figures.

19This lengthy derivative was evaluated with the help of an algebraic computer program.

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30 60 90 120 150(deg)

10-1

100

101

M(

inc(

Fig. 6.3 The ratio between the Mott and the incoherent cross sections for the samecollision energies as in figure 6.1.

30 60 90 120 150(deg)

10-2

10-1

100

101

A(

inc(

Fig. 6.4 The ratio between the Mott and the incoherent cross sections for a fermion-

fermion collision with aligned spins, for two values of the Sommerfeld parameter.

6.3.2 Collisions of identical particles with spin

If the identical particles have spin, the exchange operator is no longer equiv-

alent to the parity (space reflection) operator. In this case we have

PPT = ΠP sPT. (6.173)

An important consequence of this equation is that there are always two ways

to get symmetric or anti-symmetric wave functions. Since the eigenvalue λ

is the product of the parity, π, with the eigenvalue of the exchange operator

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in spin space, λs, we may have:

boson-boson collisions

λ = + ⇒

π = +, λs = +

or

π = −, λs = −

fermion-fermion collisions

λ = − ⇒

π = +, λs = −

or

π = −, λs = +

.

In this way, the parity of the wave function in the position space, which

determines the angular distribution, depends on the symmetry of the wave

function in the spin space. We discuss this point in detail below.

Let us consider the collision of two identical particles with spin s and

call µP and µT the spin projections of the projectile and the target, respec-

tively. The exchange symmetry in spin space is determined by the quantum

number S, associated with the total intrinsic angular momentum operator,

S2 = (sP + sT)2. The eigenstates of the operators S2 and Sz, |χSM 〉 , are

linear combinations of products |χsν〉P |χsν′〉T , where |χsν〉P and |χsν′〉T are

respectively eigenstates of the pairs of operatorss2, sz

P

ands2, sz

T,

with Clebsch-Gordan coefficients. That is,

|χSM 〉 =∑νν′

〈sνsν′|SM〉 |χsν〉P |χsν′〉T . (6.174)

Since the Clebsch-Gordan Coefficients have the property (see, e.g., [Ed-

monds (1974)])

〈sν′sν|SM〉 = (−1)2s+S 〈sνsν′|SM〉 (6.175)

and projectile-target exchange in spin space amounts to exchanging the

dummy indices ν and ν′, the eigenvalues λs are

λs = (−1)2s+S

. (6.176)

Therefore, we have

bosons (2s = even) : (6.177)

⇒ S = even→ λs = +, π = +

⇒ S = odd → λs = −, π = −

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fermions (2s = odd) : (6.178)

⇒ S = even→ λs = −, π = +

⇒ S = odd → λs = +, π = − .

In general, one cannot control the eigenvalue S in an experiment. One

can polarize both the projectile and the target but this usually leads to

admixtures of even and odd values of S. There is however one exception.

When the polarization is such that ν = ν′ = ±s, the only non-vanishing

Clebsch-Gordan coefficient has M = ±2s. Therefore the total spin quantum

number is S = 2s and, according to Eqs. (6.177) and (6.178), λs = +1. In

this way, one must have π = + for bosons and π = − for fermions. This

simple case was considered in section (6.3.1).

If the system is not polarized as above, proper averages must be taken.

Let us first consider the inclusive cross section in the scattering of unpolar-

ized identical particles. In this case, the initial spins are randomly oriented

and the cross section is summed over final spins. In this way, each of the N

possible spin state |χSM 〉 , with S = 0, .., 2s and M = −S, .., S, contributes

to the cross section with the same weight, 1/N . As we have seen, such con-

tributions are independent of M and depends on S exclusively through

the parity. That is, states with eigenvalue λs = +1 have π = λ while those

with λs = −1 have π = −λ. Calling respectively N+ and N− the number of

states with eigenvalues λs = +1 and λs = −1 and using Eqs. (6.167) and

(6.168), the cross section is given by the average

〈σλ(θ)〉 = σinc(θ) + λ

(N+ −N−

N

)σint(θ) . (6.179)

Since above the exchange symmetry, λ, appears explicitly, the interference

cross section should be that of Eq. (6.168) with positive sign.

The total number of states can easily be calculated. It is given by the

sum of the arithmetic series

N = 1 + 3 + 5 + ....+ (2Smax + 1) = 1 + 3 + 5 + ....+ (4s+ 1) = (2s+ 1)2.

(6.180)

N+ and N− are also sums of arithmetic series. That is

N+ = 1 + 5 + ....+ (4s+ 1); for λ = +1

= 3 + 7 + ....+ (4s+ 1) ; for λ = −1

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30 60 90 120 150(deg)

100

A(

inc(

Fig. 6.5 Unpolarized proton-proton cross sections, normalized as in figure 6.3.

and

N− = 3 + 7 + ....+ (4s− 1); for λ = +1

= 1 + 5 + ....+ (4s− 1); for λ = −1

Evaluating the series for N+ and N−, we obtain

N+

N=

s+ 1

2s+ 1;

N−N

=s

2s+ 1, (6.181)

and the cross section of Eq. (6.179) becomes

〈σλ(θ)〉 = σinc(θ) + λσint(θ)

2s+ 1. (6.182)

As an illustration, we give in figure 6.5 the cross section for the collision

of unpolarized protons for the same values of the Sommerfeld parameter as

those in figure 6.4. The results are normalized with respect to the incoherent

cross section. In this case, there are two possible values of the total spin.

The singlet, with S = 0, and the triplet, with S = 1. The former, with λs =

−1 and positive parity, contributes with the weight 1/4 (see Eq. (6.181)).

The latter, with λs = 1 and negative parity, has weight 3/4. Since the

cross section is dominated by the triplet, which has larger weight, it has

a minimum at θ = π/2. However, this fermionic signature is much less

pronounced then in the case of aligned polarization, where only the triplet

contributes. Note that 〈σ−(θ)〉 /σ(θ) remains finite at π/2, owing to the

contribution from the singlet.

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6.3.3 Clusters of identical fermions

In typical atomic and nuclear collisions the projectile and/or the target

are composite particles. Usually, they are clusters of fermions20. In this

way, the Pauli Principle requires that the wave function be anti-symmetric

with respect to the exchange of any two identical fermions, either in the

same cluster or in different ones. If the projectile contains NP fermions with

coordinates r1, r2, ..., rNPand the target contains NT particles with position

vectors rNP+1, rNP+2, ..., rNP+NT , the system’s wave function can be written

Ψ(r1, ..., rNP; rNP+1, ..., rN) = A

ΦP(r1, ..., rNP

) ΦT

(rNP+1, ..., rN

).

(6.183)

Above, ΦP and ΦT are respectively the anti-symmetrized intrinsic wave

functions of the projectile and the target, and N = NP+NT. A is the many-

body anti-symmetrization operator, which accounts for the NP!NT!/N ! dif-

ferent ways of permuting fermions between the two clusters. It can written

A =∑

(−)pα Pα , (6.184)

where the operator Pα exchanges fermion coordinates until the permutation

α is reached. The sign (−)pα is positive for permutations obtained through

an even number of fermion pair exchanges, and negative otherwise.

Solving the Schrodinger equation for the many-body scattering problem

is, in most cases, a hopeless task. However, the effects of Pauli Principle can

be included in approximations like Wheeler’s Resonating Group Method

(RGM) [Wheeler (1937); Friedrich (1981)], or the essentially equivalent

Generator Coordinate Method (GCM) [Hill and Wheeler (1953); Griffin

and Wheeler (1957)]. These methods avoid the complications of a Coupled-

Channel calculation, approximating the intrinsic wave functions of the pro-

jectile and the target by the Slater determinants corresponding to their

ground states when they are far apart. In this way, the intrinsic states of

the projectile and the target are kept frozen along the collision.

To derive the RGM equation, one introduces the coordinates of the

projectile’s and target’s centers of mass,

rP =1

NP

(r1 + ...+ rNP

); rT =

1

NT

(rNP+1 + ...+ rN

), (6.185)

20In the case of nuclei, there are two different fermions: protons and neutrons. However,with the isotopic spin formalism they are treated as different states of the same particle

- the nucleon.

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and the intrinsic coordinates21

ξi = ri − rP ; i = 1, .., NP

ξj = rj − rT ; j = NP + 1, .., N , (6.186)

and then adopts the ansatz

Ψ(r1, ..., rN) = ΨCM(Rc.m.) · Aψ(r) ΦP(ξ1, ..., ξNP

) ΦT

(ξNP+1, ..., ξN

).

(6.187)

Above, Rc.m. and r are respectively the center of mass and the projectile-

target separation vectors, and ψ(r) is the relative wave function, which

contains the relevant information for the scattering cross section. This

wave function is determined by the variational condition

δ

[〈Ψ|H |Ψ〉〈Ψ |Ψ〉

]= 0, (6.188)

where H is the many-body collision Hamiltonian

H =

[− 2

2µ

N∑i=1

∇2ri

]+

N∑i>j=1

v(ri − rj)

− TCM. (6.189)

It should be understood that the variations in Eq. (6.188), are taken through

changes of the relative wave function of Eq. (6.187).

For the implementation of the RGM, one rewrites the ansatz of

Eq. (6.187) in the form,

Ψ =

∫d3r′′ ψ(r′′) Φr′′ , (6.190)

with

Φr′′ = ΨCM(Rc.m.) · Aδ(r− r′′) ΦP(ξ1, ..., ξNP

) ΦT

(ξNP+1, ..., ξN

),

(6.191)

and inserts it into the variational principle. Using the property A2 = A,

one obtains the the RGM equation (for details, see [Friedrich (1981)]),[− 2

2µ∇2

r + VD(r)

]ψ(r) +

∫d3r′ VE(r, r′)ψ(r′) = E ψ(r). (6.192)

21Note that these intrinsic coordinates are not linearly independent as they are relatedby the constraints

∑P ξi =

∑T ξj = 0. This feature must be taken into account in the

evaluation of the RGM potentials which will be introduced below.

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Above, VD(r) is the double-folding potential

VD(r) δ(r− r′) =

∫dξP dξT d

3r′′

ΦP(ξP) ΦT (ξT) δ(r− r′′)

×V(r′′,ξP, ξT)

ΦP(ξP) ΦT (ξT) δ(r′−r′′), (6.193)

where

V(r, ξP, ξT) =

NP∑i=1

N∑j=NP+1

v(ri − rj),

and VE(r, r′) is the non-local energy-dependent potential22

VE(r, r′) =

∫dξP dξT d

3r′′

ΦP(ξP) ΦT (ξT) δ(r− r′′)

× (H − E) (A−1)

ΦP(ξP) ΦT (ξT) δ(r′−r′′), (6.194)

arising from exchange.

Although the direct potential can be calculated without major diffi-

culties, the exchange part is very complicated. The difficulty arises from

the fact that as A exchange particles from the projectile with those from

the target, the relative coordinate in the delta functions becomes different

combinations of the single particle coordinates. To handle this problem,

specific technics have been developed (see, for example, [Friedrich (1981)]

and references therein). In typical situations, the exchange potential has

a short range. However, when the projectile and the target are identical

clusters of fermions, the exchange potential contains a local term. It arises

from the permutation Ptot, where all projectile’s fermions are exchanged

with those of the target. In this case, it is convenient to include this term

in VD(r), replacing in Eq. (6.194) (A−1) → (A−1− Ptot). This leads to

the replacement

ψ(r)→ 1√2

(ψ(r) + λψ(−r)) ,

where λ gives the symmetry for the total projectile-target exchange (λ = +1

for even NP and λ = −1 for odd NP). The relevance of VE(r, r′) is illustrated

in figure 6.6, in an example of Nuclear Physics: the scattering of two alpha

particles. We compare the results of the RGM, which includes both the

local and the non-local potentials, with those obtained approximating the

alpha particles by identical bosons. In the latter case, only the projectile-

target exchange, corresponding to the action of Ptot, is taken into account.22We use the short hand notations ξP ≡ ξ1, ..., ξNP

and ξT ≡ ξNP+1, ..., ξN .

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0 5 10 15 20Ecm (MeV)

0.0

0.5

1.0

1.5

2.0

2.5

3.0

l/

l = 0

Fig. 6.6 Phase shift in α-α scattering, treated as a collision of identical clusters offermions (solid lines) and as a collision of identical structureless bosons (dashed lines).

The results are normalized with respect to π and plotted according to the generalizedLevinson’s theorem (see footnote 8 of chapter 3).

The figure shows the phase shifts arising from the short-range term in the

potential, δl (see chapter 3), for the partial waves l = 0, 2 and 4, and

collision energies in the range 0−20 MeV. The RGM results, for a Volkov’s

V1 [Volkov (1965)] nucleon-nucleon interaction23 and Harmonic Oscillators

orbitals, are taken from [Brink and Canto (1977)]). One notices that the

non-local potential can be neglected for l = 4 (or higher partial-waves), but

it is very important for l = 0 and 2. Collisions at the lowest partial-waves

(l = 0 and 2) are dominated by small projectile-target separations, where

the single particle orbitals in the two clusters becomes very similar. In

this way, the Pauli Principle is violated, unless the wave function if fully

anti-symmetrized. Since the exchange potential, which takes care of this

effect, is neglected in the boson-boson approximation (dashed lines), the

agreement with the RGM calculation becomes very poor.

Exercises

(1) Consider the collision of a spin- 1/2 projectile with a spin-zero target,

interacting through the potential V (r) = V0(r)+Vso(r) L·s, with L and

23The Volkov potential is a sum of Gaussians with parametes adjusted to fit experimental

nuclear data. For details see [Volkov (1965)].

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s representing respectively the orbital and the spin angular momenta,

measured in ~ units. Show that the matrix elements of the interaction

are given by (using the short-hand notation:∣∣Y lJM⟩ = |lJM〉 )

〈lJM |V |l′J ′M ′〉 =

[V0(r) +

l

2Vso(r)

]δll′ δJJ ′ δMM ′ ,

for total angular momentum J = l +1/2, and

〈lJM |V |l′J ′M ′〉 =

[V0(r)− l + 1

2Vso(r)

]δll′ δJJ ′ δMM ′ ,

for J = l −1/2.

(2) Show that, in the collisions of the previous problem,

Λ+l =

1

2l + 1

[l + 1 + σ · L

]and

Λ−l =1

2l + 1

[l − σ · L

]are projection operators onto eigenstates of total angular momentum,

with eigenvalues J = l ± 1/2.

(3) Prove that in collisions of an unpolarized beam of spin=1/2 projectiles

with a spin-zero target, the polarization vector is perpendicular to the

scattering plane.

(4) Show that in collisions of one spin- 1/2 particle with a spin-zero one,

the scattering amplitude in the two-dimensional spinor space can be

written as the matrix

f2x2(θ) = g(θ) I2x2 + i h(θ) σ · n,

with the amplitudes g(θ) and h(θ)given by Eqs. (6.86) and (6.87).

(5) Use the optical theorem to show that the total cross section for colli-

sions of one neutral spin- 1/2 projectile with a spin-zero target can be

written as,

σT (E) =4π

k2

∑l

[(l + 1) sin2 δ+

l + l sin2 δ−l

],

where the δ±l are the phase shift at the lth partial-wave for J = l±1/2.

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(6) Consider a small perturbation to the point charge Coulomb potential,

arising from the dipole polarizability of the target,

Vd(r) = −Cr4.

Assume that this interaction is very weak, so that the scattering ampli-

tude is dominated by its near-side component, and that the semiclas-

sical approximation is applicable.

a) Neglecting the influence of Vd on the classical trajectory and on the

modulus of the cross section, show that the scattering amplitude

can be written in the general form of Eq. (5.295),

f(θ) =

√dσR(θ)

dΩeiα(θ), with α(θ) ' αC(θ) + αd(θ).

Above, αC(θ) is a phase arising purely from the Coulomb field:

αC(θ) ≡ 2σ(λ) − λ θ − π/2 ' 2σ0 − η ln [sin(θ/2)] + φ0, where λ

is the angular momentum associated with θ through the Coulomb

trajectory, σ(λ) is the corresponding Coulomb phase-shift and φ0

is an irrelevant constant phase. The term αd(θ) is the correction

due to the potential Vd, given by

αd(θ) =η C

4Ea4tan4 θ

2

[(cot2 θ

2+ 3

)(π − θ

2

)tan

θ

2− 3

],

where a is one-half of the distance of closest approach in a head-on

collision24.

b) Assuming that the projectile and the target are identical particles,

evaluate the Mott cross section and determine the shifts in the

maxima and minima owing to the presence of Vd25.

24Hint: Evaluate the phase-shift by the WKB approximation, expanding the local wavenumber around Vd = 0. In this way, the contributions from the VC and Vd becomeadditive. Next, use the approximate expressions for σ(λ) and σ0 of chapters 3 and 5.25Hint: From the interference term,

cos [2 (αC(θ) + αd(θ)− αC (π − θ)− αd (π − θ))] = cos ∆(θ) (6.195)

find the period Pθ, as,

∆(θ + Pθ)−∆(θ) = 2π (6.196)

which can be reduced to,

Pθ =2π

d∆(θ)/dθ. (6.197)

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Documents

Scattering Theory of Molecules, Atoms and Nuclei || Spin and Identical Particles