Elements of Modern Physics
322
Nuclear Radius
The wave-function description of a particle does not provide an unambiguous
description of the size of a particle. However, since nuclear forces are large
only within a distance of a few fermis (1 fermi = 10
–15
m), it is useful to consider
the size of the nucleus. The nuclear radius may be estimated from the scattering
of neutrons and electrons by the nucleus, or by analysing the effect of the finite
size of the nucleus on nuclear and atomic binding energies.
Fast neutrons of about 100 MeV energy, whose wavelength is small
compared to the size of the nucleus, are scattered by nuclear targets. The fraction
of neutrons scattered at various angles can be used to deduce the nuclear size.
For example, in the scattering of a high energy particle by a hard sphere, V = ∞
for r < R, V = 0 for r > R, all the incident particles within a cross-sectional area
of 2πR
2
are scattered. The factor of 2 is due to the diffraction of the waves at the
edges. The results of these experiments indicate that the radius of a nucleus is
given by
R ≈ r 0 A
1/3
(9.11)
where A is the mass number and r 0 ≈ 1.3 – 1.4 fm. The scattering can be done
with proton beams as well. In this case, however, the effects due to Coulomb
interaction have to be separated out. The observations are in agreement with
the result in Eq. (9.11) with r 0 ≈ 1.3 – 1.4 fm.
The scattering of fast electrons of energy as high as 10
4
MeV, with a
wavelength of about 0.1 fm, has the advantage that it can directly measure the
charge density inside a nucleus. The results of the experiment are in agreement
with Eq. (9.11) but with a somewhat smaller value of r 0 ≈ 1.2 fm. The slight
difference in the value of r 0 may be ascribed to the fact that the electron scattering
measures the charge density whereas the neutron and proton scattering
experiments measure the region of large nuclear potential, which may be
expected to be somewhat larger than the size of the nucleus.
The finite size of the nucleus modifies the atomic potential (– Z/r) at short
distances. This gives rise to a small separation between the spectral lines of
atoms with the same Z value but different A values—this is known as isotope
shift. The shifts can be used to deduce the nuclear ratius. The isotope shift is
much larger in muonic atoms (which have a muon in place of an electron) since
the radii of the muonic orbits are smaller than the electronic orbits by a factor of
about 200 (m µ ≈ 200 m e ). However, the accuracy of measurements is muonic
atoms is lower since the muons have a short lifetime, about 2 × 100
–6
s. Finally,
the measurement of differences in the binding energies of mirror nuclei can
give an estimation of the nuclear radius. The mirror nuclei are nuclei which are
identical except that one proton is replaced by a neutron. They may be
characterized by
2
1
2
1
1 X,
Y
Z
Z
Z
Z
+
+
+
. The difference between their binding energies
322
Nuclear Radius
The wave-function description of a particle does not provide an unambiguous
description of the size of a particle. However, since nuclear forces are large
only within a distance of a few fermis (1 fermi = 10
–15
m), it is useful to consider
the size of the nucleus. The nuclear radius may be estimated from the scattering
of neutrons and electrons by the nucleus, or by analysing the effect of the finite
size of the nucleus on nuclear and atomic binding energies.
Fast neutrons of about 100 MeV energy, whose wavelength is small
compared to the size of the nucleus, are scattered by nuclear targets. The fraction
of neutrons scattered at various angles can be used to deduce the nuclear size.
For example, in the scattering of a high energy particle by a hard sphere, V = ∞
for r < R, V = 0 for r > R, all the incident particles within a cross-sectional area
of 2πR
2
are scattered. The factor of 2 is due to the diffraction of the waves at the
edges. The results of these experiments indicate that the radius of a nucleus is
given by
R ≈ r 0 A
1/3
(9.11)
where A is the mass number and r 0 ≈ 1.3 – 1.4 fm. The scattering can be done
with proton beams as well. In this case, however, the effects due to Coulomb
interaction have to be separated out. The observations are in agreement with
the result in Eq. (9.11) with r 0 ≈ 1.3 – 1.4 fm.
The scattering of fast electrons of energy as high as 10
4
MeV, with a
wavelength of about 0.1 fm, has the advantage that it can directly measure the
charge density inside a nucleus. The results of the experiment are in agreement
with Eq. (9.11) but with a somewhat smaller value of r 0 ≈ 1.2 fm. The slight
difference in the value of r 0 may be ascribed to the fact that the electron scattering
measures the charge density whereas the neutron and proton scattering
experiments measure the region of large nuclear potential, which may be
expected to be somewhat larger than the size of the nucleus.
The finite size of the nucleus modifies the atomic potential (– Z/r) at short
distances. This gives rise to a small separation between the spectral lines of
atoms with the same Z value but different A values—this is known as isotope
shift. The shifts can be used to deduce the nuclear ratius. The isotope shift is
much larger in muonic atoms (which have a muon in place of an electron) since
the radii of the muonic orbits are smaller than the electronic orbits by a factor of
about 200 (m µ ≈ 200 m e ). However, the accuracy of measurements is muonic
atoms is lower since the muons have a short lifetime, about 2 × 100
–6
s. Finally,
the measurement of differences in the binding energies of mirror nuclei can
give an estimation of the nuclear radius. The mirror nuclei are nuclei which are
identical except that one proton is replaced by a neutron. They may be
characterized by
2
1
2
1
1 X,
Y
Z
Z
Z
Z
+
+
+
. The difference between their binding energies