Exercises for

Nuclear Physics, Particle Physics, and Astrophysics 1

Prof. Dr. Peter Fierlinger, Prof. Dr. Alejandro Ibarra
Winter Semester 2023/2024

Nr. 7 28.11.2022
M. Sc. Anton Riedel, anton.riedel@tum.de

Exercise 21: Form factor of a homogeneously charged sphere

The form factor F (q) is a property of the elastic scatting cross-section for an extended charge
distribution, which describes how the scattering depends on the momentum transfer q. In the Born
approximation (weak potential; incoming and outgoing particles approximately plane waves) the
form factor is the Fourier transform of the normalized charge distribution f (x):

Z
i q·x 3
F (q) = f (x) e d x with q = pin − pout

and
f (x) = ρ(x)/(Ze)
with ρ(x) as charge density.

(a) Prove that the form factor of a homogeneously charged sphere with radius R and a charge
distribution
 3Ze for |x| ≤ R


ρ(x) = 4πR3
0 else


can be written as
3
F (q) = 3 3 [sin(|q|R) − |q|R cos(|q|R)]
|q| R
R −2
Hint: x sin(ax) dx = a [sin(ax) − ax cos(ax)]

(b) For the elastic scatting of electrons on carbon nuclei a minimum of the differential cross-section
−1
is present at |q| = 1.8 fm . Calculate the radius of the carbon nucleus based on this result and
the mentioned charge distribution.

Exercise 22: Charge radius of nuclei

Charge density distributions ρ(⃗r) of light nuclei are (to good approximation) well described by
Gaussian functions

 
2 2
ρ(⃗r) = A exp −r /R

with the width of the Gaussian distribution R and the charge of the nucleus Z.

1
(a) Determine the normalization factor A using

Z
3
ρ(⃗r) d ⃗r = Ze

Hint:

∞
(2n)! √
Z 2
2n −ax −(n+ 21 )
x e dx = π (4a) für n ∈ N.
0 n!

(b) Find the expression for the mean square radius

D E Z
2 2 3
r = r f (⃗r) d ⃗r

as a function of the parameter R, where f (⃗r) ≡ ρ(⃗r)/(Ze) is the normalized charge distribution.

(c) Calculate the charge form factor, given (in natural units ℏ = 1) by the Fourier transform of the
charge distribution

Z
2 q ·⃗
i⃗ r 3
F (q ) = f (⃗r) e d ⃗r

2
Show that the mean square radius can be determined from the slope of F (q ) at small momentum
transfer q via

2
D E
2 dF (q )
r = −6 2
dq q =0
2

Hint:

Z ∞ √ 2
!
2
−ax b π b +
xe sin(bx) dx = 3 exp − für a, b ∈ R .
0 4a 2 4a

rD E
2 4 6
(d) Estimate the mean charge radii r of the nuclei He and Li from the measured form factors
plotted below.

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 6 2 2 2
Hint: For Li the plotted quantity is |F (q )| , not F (q ).

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