Solution of Second Homework/Nuclear Physics
Dr. bodoor
Q1) This question deals with parity. Refer to the paper entitled “parity-violation-paper” for additional
reference, added recently on the “Materials” section of my website. (20 points)
a) Write down an expression showing the effect of the parity operator on the wavefunction of a
particle in three dimensions.
Answer
 

(r )  (r )
(Eq. 1)
b) When the parity operator is applied to a wavefunction TWICE, what is the result? From that,
what are the two possible eigenvalues of the parity operator?
Answer

 




 2  ( r )   ( r )   (  r )   ( r )
(Eq. 2)
The above equation shows that when the parity operator is applied twice to the wavefunction,
The result is the wavefunction itself; i. e., the eigenvalue of the square of the parity operator,

 2 , is +1 and all functions are eigenfunctions of the square of the parity operator. BUT not
all functions are eigenfunctions of
eigenvalue of π, such that
 


 ( r )   (  r )    ( r )


 . If the function is an eigenfunction of  with an
(Eq. 3)
Then



 2 (r )   2  (r ) , (Eq. 4)
that is, the eigenvalue of the square of the parity operator is
above,
 2 should be +1, so that
  1
 2 .But, by definition, from Eq. 2
That is, the possible eigenvalues of the parity operator are +1 and -1, but this is only true if the
function is an eigenfunction of the parity operator.
c) Are all wavefunctions eigenfunctions of the parity operator? What is the condition (on the
potential) for the wavefunction to be an eigenfunction of the parity operator? Can you express
this condition in a mathematical form?
Answer
As we saw above, not all functions are eigenfunctions of the parity operator. For example, a
x
x
function in one dimension such as e or e
is NOT an eigenfunction of the parity operator,
even though it is an eigenfunction of the SQUARE of the parity operator.
If the potential is an eigenfunction of the parity operator, such that
 


V ( r )  V (  r )   V ( r ) ,
d) What do we mean by a polarized nucleus? How can we achieve polarization of a sample of
nuclei?
Answer
A polarized nucleus is one with spin pointing in a definite direction, and a polarized sample of
nuclei is one where the majority of nuclei have spins pointing in a definite direction. The degree
of polarization ranges from zero to one; in the case of polarization of one ALL nuclei spins point
in the same direction, while a polarization of zero means that spins of the nuclei in the sample
have completely random directions and no preferred direction in space.
We can achieve polarization of nuclei in a sample by cooling the sample to very low
temperatures and at the same time subjecting the nuclei to a strong magnetic field.
e) What was the purpose of the experiment done by Wu and her collaborators? When and where
was it conducted?
Answer
The purpose of the experiment was to test whether parity is conserved by the weak process of
neutron decay. The Wu experiment performed at the Bureau of Standards low temperature
laboratory, Washington DC, in 1956.
f)
Explain the principle behind the above experiment, i. e., what exactly was measured in the
experiment and why? Make the connection with the purpose of the experiment.
Answer
The experiment sought to measure the angular distribution of the electrons emitted by the
cobalt nuclei decaying according to
If the rates measured at angles θ and π- θ ( i. e., the rate of electrons emitted in opposite
directions) are not equal, then this is proof that the beta decay does not conserve parity and
that the weak interaction in general does not conserve parity. The angles at which the electrons
are emitted are measured with respect to the direction of polarization; that is why we need to
have the nuclei polarized: to establish an intrinsic reference direction (the direction of
polarization of the nuclei) to measure angles with respect to.
g) Describe the experimental apparatus used in the experiment above, explaining in detail the
purpose of each piece of the apparatus, including detectors, cryostat, sample, magnetic field
source, etc.
Answer
The apparatus consisted of the following:
1) A cobalt source.
2) Apparatus for polarizing the cobalt nuclei, including a cryostat crystal for achieving
low temperatures and a strong magnetic field
3) An anthracene crystal to detect beta particles.
4) A gamma detector to measure the degree of polarization of the nuclei.
5) A vacuum chamber containing the cobalt source and beta detector.
6) A photomultiplier tube.
7) Other parts.
h) Refer to the paper cited above for this part. What is the main experimental result of the
experiment and how is it expressed? (Hint: discuss the the graph from the paper or from Krane
pg. 313)?
The main result is that, upon reversing the direction of the magnetic field, the same beta
detector recorded different counting rates, indicating an asymmetry in the emitted beta rays
and hence a violation of parity conservation. This result is recorded in the graph below.
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Answer
Q2) Calculate an estimate of the nuclear density. Sketch on a graph the variation of the nuclear density
as a function of distance from the nuclear center. Show your work in detail. (5 points)
We can estimate the density of U238, for example, as follows.
Q3) If all nucleons in a nucleus attracted each other, the contribution to the binding energy of the
nucleus would be proportional to the square of the nucleon number. Explain how one could arrive at
this conclusion. But in a nucleus, the binding energy is roughly proportional (in a first approximation) to
the number of nucleons (A). What does that mean for the range of the nuclear force? (5 points)
Answer
If all nucleons attracted each other via the nuclear force, regardless of the distance between them, with
the same strength, then each pair would contribute the same amount to the binding energy of the
nucleus. So, the number of contributions would equal the number of interacting pairs. Now, the
number of interacting pairs in a system of A nucleons (if all nucleons interacted with each other) would
be
½ A (A-1). For example, in a system of three nucleons (A,B,C) there would be three interacting pairs
(AB, AC, and BC) while in a system of four nucleons (A, B, C, D) there would be 6 interacting pairs (AB,
AC, AD, BC, BD, and CD). The net result is that the binding energy would be proportional to ½ A (A-1) or,
in leading order, to A2. In reality, the binding energy is found to be proportional, in leading order, to the
number of nucleons, not its square. This means that not all nucleons pairs interact together via the
nuclear force and, therefore, that it is reasonable to assume that only nucleons that are close together
(i. e., within a certain distance or range of each other) do interact together through the nuclear force.
So, this is equivalent to assuming that the nuclear force has a finite (not infinite) range: two nucleons
can interact together via the nuclear force only if they are within a certain range of each other, the
range of the nuclear force.