PHY431 Lecture 2
1
2/6/2000
2. Measuring the Nuclear Charge Distribution
The nuclear charge distribution tells us where in the nucleus the protons live, at least on
average. The density of protons gives us information about the strong force.
Several methods exist to study the internal structure of the atomic nucleus, and indeed of the
protons and neutrons that make up the nucleus themselves. With 15 MeV
α
-particles scattering off a
Z=13 Aluminum foil, one notices a departure from the classical Rutherford scattering formula at large
scattering angles; indeed,
alpha particles of high energies
penetrate
the nucleus at small impact
parameters. With studies of this type the nuclear boundary can be measured for many elements.
A second type of measurement is with
high-energy electrons as projectiles
. Of course,
electrons at energies of a MeV or higher are quite relativistic (
E
e
>
m
e
). In that case, the classical
Rutherford derivation is no longer valid: at relativistic speeds the spin of the electron plays a role in the
scattering as well. The description of the process of electron-pointlike-nucleus scattering is being done
using the Dirac equation. The resulting cross section formula is the
Mott formula
:
222
2
2
42
2
22
2
2
cosec
1
sin
1
sin
2
2
Mott
Rutherford
dz
Z
v
d
v
dv
p
c
d
c
σα
θ
σ
=−
=
×
−
ΩΩ
(1.1)
where
p
is the incident momentum of the projectile, and
v
its velocity. Note that for
v
<<
c
the two
formulae are equivalent. The formula can be re-written as function of the momentum-transfer squared
variable
q
2
, where the
q
=
p
’-
p
, whence
q
= |
q
| = 2
p
sin
/
2
. We then use (integrating over the azimuthal
variable
φ
):
2
2
2
4s
i
n
s
i
nc
o
s
2
2
sin
s
i
o
s 2
2
pd
dq
p
ddd
d
θθ
π
==
=
Ω
(1.2)
Then, using sin
/
2
=
q
/(2
p
):
4
2 2
2
2
2
2
4
2
4
2
.
16
4
11
1
44
4
4
Mott
Ruth
d
d d
z
Z
p
vq
d
zZ
dq
dq
d
p
v p
q
c
p
v q
c
p
dq
c
p
σσ
α
Ω
=×=
−
=
−
=
−
Ω
(1.3)
Electrons can be accelerated easily to high energies, and are, as far as we know, truly pointlike
particles, ideal probes for the study of nuclear charge distributions.
A third method for study of the nuclear matter is the
use of stopping negative muons
. Muons
are copiously produced in cosmic ray interactions in the upper atmosphere. High-energy solar and
galactic protons impinging on molecules in the upper atmosphere produce showers of hadrons, mostly
pions and Kaons. Both pions and Kaons are short-lived and decay ultimately in muons, electrons and
neutrinos. Although muons have a mean life of only 2.1
µ
s, because they move with relativistic speeds,
they profit from time dilation and are able to reach sea level in great numbers (
∼
100 m
-2
s
-1
). By
capturing negative muons in materials, the muon first looses energy in electromagnetic collisions and
eventually slows down enough to get captured in atomic orbits. Because of its much larger mass than
the mass of the electron,
m
µ
= 105 MeV
≈
200
m
e
, the muon energy levels (
E
n
=(
1
/
2
n
2
)
Z
2
α
2
mc
2
) are in
the X-ray regime, and the (most probable) orbital radii (
r
n
=
n
2
ħ
c
/
Z
α
mc
2
) are much smaller than
electron orbits. Thus, the radius of the lowest orbit in a