Notes on Elastic Scattering
L. Wimmer

Abstract. An overview of "Elastic Scattering of Identical Spin-Zero Nuclei" by Bromley,Kuehner and Almqvist [1].

Contents
1 Theory
1.1 Classical Coulomb scattering [2] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Coulomb scattering of 2 different particles [3] . . . . . . . . . . . . . . . . . . . . . . . .
1.3 Mott formalism for Coulomb scattering of 2 identical particles [3] . . . . . . . . . . . . .
1.4 Blair formalism for Coulomb scattering of 2 identical particles . . . . . . . . . . . . . . .

2234
52 Experimental results
6
2.1 Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2 Summary of the experimental results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

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1Theory

1.1Classical Coulomb scattering [2]Consider the scattering of two charges, Z1 e and Z2 e, with a reduced mass µ. In the lab frame, suppose
the incident particle has an initial velocity of v0 , while the target particle remains stationary. Thus,
the energy of the system in the CM frame can be expressed as
E=µv02.
2Let the scattering angle in the CM frame be denoted as θ, and the impact parameter as b. The motionof the system in the CM frame is described by (r, φ) and is equivalent to the reduced mass being
scattered from an origin with a potentialgV (r) = ,rafter coming with initial velocity v0 from the infinity, where r represents the distance from the origin
andZ1 Z2 e2g=4πϵ0is a constant.

µααrbϕθ

Figure 1: One-body equivalent scattering in the CM frame.
Angular momentum conservation givesv0 b = φ̇r.
The change of the momentum before and after the scattering isZ ∞g∆p = 2µv0 sin θ/2 =sin (θ/2 + φ)dt.
2r0(1)(2)Plug dt from (1) into (2) and integrate from φ = 0 to π − θ to getb=Z1 Z2 e2
cot θ/2.
8πϵ0 E(3)The above equation relates the impact parameter and the scattering angle. Now, we will derive theclassical distance of closest approach, denoted as D. At the moment of closest approach, we assume
that the velocity of the reduced mass v is perpendicular to the position vector. We can use energy
conservation to getµv 2gµv 2+ .
(4)
E= 0 =22D
Additionally, angular momentum conservation yields the equation:v0 b = vD2(5)By combining equations (3) and (4), we can derive the expression for the distance fo closest approach:θ2

Z1 Z2 e2
=D=b8πϵ0 E
1 − sin θ2cosθ
1 + csc2

(6)Now, we will derive the Rurtherford formula for the Coulomb scattering using the above results. To doso, we now assume there are two beams of particles being scattered via binary collisions. We consider
the particles with impact parameter between b and b + db and scattering angles between θ and θ + dθ,and we have the invertible function b(θ). We havedb =dbdθ,dθ(7)the range of solid angles of scattered particlesdΩ(θ) = 2π sin θdθ(8)and the cross-sectional area of the annulus of incoming particles associated to b + db
dσ = 2πbdb.

(9)Equations (3), (7), (8) and (9) together give the Rutherford formula for the Coulomb scattering:
dσ=
dΩ1.2



Z1 Z2 e2
16πϵ0 E

21
sin4 θ/2(10)Coulomb scattering of 2 different particles [3]In this section, we use partial wave analysis to find a solution for the Schrödinger equation
 2ℏ 2 Z1 Z2 e2
ψ = Eψ− ∇ +2µ4πϵ0 r(11)of the system of 2 different particles with reduced mass µ, charges Z1 e and Z2 e, and energy E. Thewave number is
√
2µE.
(12)
k=
ℏ
Then, we can rewrite the Schrödinger equation using parabolic coordinates (ξ, η, ϕ), assume symmetry
with respect to the z-axis and use separation of variables, i.e. assume that
ψ(ξ, η) = G(ξ)H(η),(13)

to get the equations
 2

d dG(ξ)kξ+ξ − C G(ξ) = 0dξdξ
4 2
d dH(η)
kZ1 Z2 e2 mη+
η+C −H(η) = 0,dηdη
44πϵ0 ℏ2(14)(15)

where C is a constant. Further analysis of boundary conditions for the incoming wave for z → −∞gives
H(η) = eikη/2(16)andG(ξ) = e−ikξ/2 f (ξ)(17)

for some function f (ξ). Plug the above result into the equation for G(ξ) to getξ

d2 fdf+ (1 − ikξ) − γkf = 0,dξdξ
3(18)where we definedγ≡

Z1 Z2 e2 m.
4πϵ0 ℏ2 k(19)But it is known that the solution for the equationzd2 wdw+ (b − z)
− aw = 0dzdz(20)which is constant at z = 0 is Kummer’s (confluent hypergeometric) function
M (a, b, z) =∞Xaza(a + 1)z 2
Γ(a + s)Γ(b)z s=1+++ ··· ,Γ(a)Γ(b + s)Γ(1 + s)
1!b2!b(b + 1)
s=0

(21)

where Γ is the gamma function. Thus, we get(22)

f (ξ) ∼ M (−iγ, 1, ikξ).

Using some properties of the gamma function, we get the equation for f (r) including terms up toO(r−1 ) for z → ∞, which plugged back into ψ(ξ, η) gives1γ2(23)+ fc (θ)ei[kr−γ ln 2kr] ,ψ(r, θ) ∼ ei[kz+γ ln k(r−z)] 1 +ik(r − z)rwherefc (θ) =2γe−i[γ ln (sin θ/2)+π−2η0 ]22k sin θ/2(24)is the Coulomb scattering amplitude andη0 = arg Γ(1 + iγ).

(25)

From the scattering amplitude, we once again get the Rutherford formula for the Coulomb scattering:dσ= |fc (θ)|2 =dΩNote that we used
E=Z1 Z2 e216πϵ0 E

21sin θ/24(26)µv02ℏ2 k 2=,
22µand that the result matches equation (10).

1.3

Mott formalism for Coulomb scattering of 2 identical particles [3]Consider the scattering of two identical particles. Since they are identical, the detector cannot distinguish them and we have to consider the influence of both particles in the differential cross section.
The result from the previous section clearly doesn’t do that. To fix that, consider equation (23). We
can rewrite it asψ(r, θ) ∼ ψ1 (r, θ) + fc (θ)ψ2 (r).
(27)
If we add the wave function of the other particle, the new solution becomesψ ′ (r, θ) ∼ ψ1 (r, θ) ± ψ1 (r, π − θ) + fc (θ)ψ2 (r) ± fc (π − θ)ψ2 (r),(28)where + is for bosons (symmetric wave function) and − is for fermions (antisymmetric wave function).
Thus, we havef ′ (θ) = f (θ) ± f (π − θ)(29)
anddσ= |f (θ) ± f (π − θ)|2 .
dΩ
4(30)

However, if the spin of a particle is denoted by s, the multiplicity of such a state is given by 2s+1. Thisis because the z-component of the spin of the system can take values from −sℏ to sℏ, incrementing
by one unit of ℏ. Therefore, for the case of two identical particles with spin s, we can have (2s + 1)2
possible combinations for the z-component of the spins of the system with the two particles, with one
singlet state (z-component of the spin of the system is zero), three triplet states (z-component of thespin of the system is ℏ), and so on, up to (2s + 1) states, where the (4s + 1)-let has (4s + 1) possiblestates and the z-component of the spin of the system is 2sℏ. Note that
1 + 3 + 5 + · · · + (4s + 1) =2s
X(2t + 1) = (2s + 1)2 .

(31)

t=0

Furthermore, if we consider the (4t + 1)-let state whose scattering amplitude is f4t+1 (θ), where t takes
values 0, 1/2, 1, 3/2, . . . , s, and note that2sX(2t + 1) =

t=0

XX

(4t + 1) +t=0,1,2,...
t≤s(4t + 1),(32)

t=1/2,3/2,...
t≤s

we find that the singlet state is associated to a pair of bosons (t = 0), a triplet is associated to a pairof fermions (t = 1/2), and so on. Therefore, using the result from our previous analysis of the wave
function, we know that, for fermions (t = 1/2, 3/2, . . .),dσ= |f4t+1 (θ) − f4t+1 (π − θ)|2 .
dΩ 4t+1(33)On the other hand, for bosons (t = 0, 1, 2, . . .),dσ= |f4t+1 (θ) + f4t+1 (π − θ)|2 .
dΩ 4t+1(34)Since
f4t+1 (θ) = fc (θ)
for all values of t, the weighted differential cross section becomes222dσ1
2s+3f(θ)−f(π−θ)+···+(4s+1)
f(θ)
+(−1)
f(π+θ),=f(θ)+f(π−θ)ccccccdΩ(2s + 1)2where fc (θ) is the Coulomb scattering amplitude as in (24). After some simplifications, we get
dσ
2= |fc (θ)|2 + |fc (π − θ)|2 + (−1)2s[fc (θ)fc∗ (π − θ) + fc∗ (θ)fc (π − θ)] .
dΩ
2s + 1(35)Plugging (24) into the above equation, we get the Mott differential cross section:dσdΩ=Mott

Z1 Z2 e2
16πϵ0 E

2csc4θ
2(Ze)2θθθθ
+ sec4 + (−1)2scosln tan2
csc2 sec2222s + 1ℏv0222(36)1.4Blair formalism for Coulomb scattering of 2 identical particles

The Blair is a modification of the Mott formalism to consider the nuclear potential between particles,
which becomes relevant at high energies and is not accounted for in the Mott formalism. It introduces
the concept that all partial waves in the incident beam, with an impact parameter corresponding toa classical distance of closest approach smaller than the nuclear interaction radius, are completelyabsorbed. In contrast, partial waves with larger impact parameters undergo pure Coulomb scattering.
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By incorporating these considerations, the Blair formalism provides a more comprehensive descriptionof scattering phenomena at high energies. Blair model’s differential cross section can be expressed as:
dσdΩ=Blair(Ze)216πϵ0 E

22 θexp(2iδ0 ) csc22 θexp −iη ln sinll

l=0l=022 θ± exp(2iδ0 ) sec
exp −iη ln cos
222 θmaxmaxi Xi X−(2l + 1) exp(2iδl )Pl (cos θ) ∓(2l + 1) exp(2iδl )Pl (− cos θ)
ηη

2(37)

Here, θ represents the center-of-mass scattering angle, δl denotes the Coulomb phase of order l, Pl
corresponds to the Legendre polynomials, and η is the Sommerfeld number given by:
η=2

Experimental results

2.1

Figures(Ze)2.
ℏv02

Figure 2: Energy spectra measured at 26° for 26.5 MeV beams of C12 and O16 incident on ∼ 50 µg/cm
2and ∼ 100 µg/cm targets of carbon and silicon monoxide, respectively. Horizontal axis is associatedto energy and vertical axis associated to the number of particles detected.

6Figure 3: Rutherford formula for the relation between differential cross section and scattering angle
agrees with experimental data for elastic scattering of O16 ions on gold.

Figure 4: Relation between differential cross section and scattering angle for the elastic scattering ofO16 +C12 with 11.57 MeV and 13.67 MeV. In the forwards angles, the experimental data does notalign with the Rutherford formula. However, the oscillatory structure after the decrease and the small
exceeding of the Rutherford formula before the decrease are characteristics predicted by the Blairmodel.

7Figure 5: Angular distributions of O16 ions on magnesium and aluminum, demonstrating agreementwith the Rutherford formula even at forward angles.

8Figure 6: Relation between differential cross section and the CM energy for the elastic scattering ofO16 +C12 for ϕCM =42 and 90°. Dashes lines are the E −2 predictions from Rutherford formula. The
experimental data deviates from the Rutherford formula at higher energies due to the Coulomb barrier.
As the incident particle’s energy increases, it can overcome the Coulomb repulsion and approach thetarget nucleus more closely, where the nuclear potential is effective. This leads to a deviation from thepredictions of the Rutherford formula for higher energies. The 42°data shows less oscillatory structure
because it has larger classical distance of closest approach (equation (6)) and thus is less affected bysome nuclear potential.

9Figure 7: Relation between differential cross section and energy for O16 +O16 elastic scattering fordifferent angles. We can use Mott formalism to check the experimental data since it is an elastic
scattering of 2 identical particles. For ϕ = 90°, the energy modulation (energy dependence of thethird term of the Mott differential cross section) disappears and data aligns considerably well with
the Mott formalism, except for extreme energies. The data for all three angles conforms to the Mott
formalism for energies below 10.5 MeV. However, beyond this energy threshold, the graph exhibits arapid decline, primarily attributed to nuclear interactions.

Figure 8: Relation between differential cross section and scattering angle for O16 +O16 elastic scattering
at 7 MeV and 8.8 MeV. The solid curve represents a sketch of the data curve, while the dashedcurve represents the Rutherford prediction. The experimental data oscillates around the Rutherford
prediction, but it does not match Mott or Blair predictions.
10Figure 9: Relation between differential cross section and scattering angle for O16 +O16 elastic scattering
at 13 MeV and 15 MeV.The solid curve represents a sketch of the data curve, while the dashed anddot-dashed curves represent the Mott prediction and the Blair prediction for lmax = 6, respectively.
At 13 MeV, Blair aligns better with data than Mott, notably at around 60° and 110°. At 15 MeV, data
is below Mott for most angles and doesn’t match either prediction.

Figure 10: Relation between differential cross section and energy for the elastic scattering of C12 +C12
for different angles. As before, the Mott predictions are in accord with the experimental data before
the Coulomb barrier. However, in contrast to the oxygen results, for energies above the barrier, thecross sections do not decrease in roughly exponential fashion and unexpectedly show marked resonancestructure.

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Figure 11: Angular distributions for C12 +C12 . The solid and dashed curves are the Rutherford andMott predictions, respectively. The data aligns perfectly with the Mott prediction.

Figure 12: Angular distributions for C12 +C12 . Data curve (dark solid) is predominantly shifted below
the Mott prediction (dashed), while the Rutherford prediction (light solid) is also depicted.

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Figure 13: Blair model predictions for C12 +C12 . A and E are the Mott and experimental curves,respectively. In the left figure, B and C correspond to Blair’s model for lmax = 4 and 6, respectively.
In the right figure, C corresponds to lmax = 8, B corresponds to lmax = 6 + l = 10, and D corresponds
to lmax = 4+l = 8. In the case of 11.25 MeV, reasonable qualitative agreement is obtained for lmax = 4
or 6, better than the Mott model. As for the 12.50 MeV data, no qualitative agreement is found forany given lmax up to 16.

Figure 14: Blair model predictions for the 13.0 MeV O16 +O16 angular distribution. The dashed,broken, and full curves are the Mott and Blair predictions and the experimental data, respectively.
The data aligns with the Blair model for lmax = 6.

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Figure 15: Comparison of the O+O, O+C, and C+C 90°excitation curves. The cross sections for
all three reactions appear to be approaching a common value of ∼ 10 mb/steradian at energies significantly in excess of the barrier. This suggests that this cross section is typical of the "compoundelastic processes" including diffraction scattering and potential scattering proceeding through a normalcompound system, and thus relatively independent of the particular nuclides involved (providing thatbarrier penetration effects are small).

2.2

Summary of the experimental results

Experimental data for the elastic scattering of C+C and O+O at different energies was analysed. At
energies below the Coulomb barrier, the measurements of both angle and energy exhibit excellent
agreement with the Mott scattering predictions. However, above the Coulomb barrier, the excitation
curve for O+O drops exponentially below the Mott predictions, reaching a value of 10 mb/sr at 16.5
MeV, and remains relatively constant thereafter. In contrast, the excitation curve for C+C displays
marked resonant interference structures. The Blair model provides improved predictions for the angulardistributions of C+C scattering at 11.25 MeV and O+O scattering at 13 MeV compared to the Mottmodel.

References[1] DA Bromley, JA Kuehner, and E Almqvist. Elastic scattering of identical spin-zero nuclei. PhysicalReview, 123(3):878, 1961.
[2] T Watkins.
The derivation of the rutherford scattering formula and its generalization.
https://www.sjsu.edu/faculty/watkins/scattering.htm. Accessed: 2023-06-15.
[3] 石 飛 由 介、 岡 田 健、 桑 野 将 大、 杉 本 太 郎、 吉 田 登 志 輝.
ラ ザ フ ォ ー ド 散 乱.
http://ppwww.phys.sci.kobe-u.ac.jp/seminar/pdf/Rutherford2016.pdf. Accessed: 2023-06-15.

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