PHGN 422: Homework 1

PHGN 422: Homework 1
Exercise 1 – Manipulations of Units
Express in [MeV.fm] the following constants: hc, !c, e2/4πε0 with:
h = 6.626 x 10-34 J.s
c = 2.998 x 108 m/s
9
2
2
(1/4πε0) = 8.988x10 N.m /C
e = 1.6 x 10-19 C
[Please show the details of the calculations].
Exercise 2 – De Broglie Wavelength
The goal of this exercise is to find out an expression of the De Broglie wavelength for
relativistic particles and apply it in two cases.
1. Show that the momentum p of a relativistic particle can be expressed as a
function of its kinetic energy K and its rest mass energy mc2 by: p =
(1/c)×√(K(K+2mc2)).
2. Deduce an expression of the De Broglie wavelength for relativistic particles.
3. Simplify this expression when K>>mc2. Using this simplification, calculate the De
Broglie Wavelength (in m) of an electron with K = 50MeV.
4. Show that the expression found in question 2 reduces to the expected classical
formula, when K<<mc2. Using this simplification, calculate the De Broglie
wavelength (in m) of a proton with K = 1MeV [mp=938.3MeV/c2].
Exercise 3 – Binding energy, mass excesses, particle separation energies
Consider the following nuclei: 40Ca, 41Ca, 42Ca, 41Sc and 42Ti.
1. Calculate their respective mass excess in [MeV].
2. Calculate the following particle separation energies: Sn(41Ca), S2n(42Ca), Sp(41Sc)
and S2p(42Ti).
3. Compare these values with the average binding energy of 40Ca. Conclusions?
Masses:
M(40Ca)=39.962591u; M(41Ca)=40.962278u; M(42Ca)=41.958618u
M(41Sc)=40.969250u; M(42Ti)=41.973030u
1
M( H)=1.007825u; M(1n)=1.008665u; 1u=939.494 MeV/c2
Exercise 4 – Liquid Drop Model (Computational)
The objective of this exercise is to compare the experimental binding energies per
nucleon B/A across the chart of nuclei with the predictions from the liquid drop model as
defined in class and given again below:
B( ZA X N ) = av A − as A 2/3 − ac Z(Z −1)A −1/3 − asym (A − 2Z )2 A −1 + δ a p A −3/4
with av=15.85MeV, as=18.34MeV, ac=0.71MeV, asym=23.21MeV and ap=12MeV (Note:
δ=+1 for even Z and even N, δ=0 for odd Z and even N or even Z and odd N, and δ=-1
for odd Z and odd N).
1. Download the data file in the homework section of the PHGN422 webpage.
The file is formatted as follows: A <tab> Z <tab> Binding Energy (in keV).
Write a computer program that reads this data file and extract the
experimental binding energy per nucleon Bexp/A for each nucleus. Plot Bexp/A
as a function of A (scatter plot). You should get something very similar to the
picture(s) shown in class (Hint: don’t forget to convert Bexp from keV to MeV!).
2. In the same computer program (for simplicity), compute the binding energy
per nucleon Bliq/A predicted by the liquid drop model for the same set of
nuclei. Plot Bliq/A as a function of A (scatter plot). You should get something
very similar to the Bexp/A distribution. Overlay the two distributions (using two
different colors – please indicate which one is which!).
3. As mentioned in class, we are particularly interested by the difference
between the two quantities as it reveals the fraction of the binding energy that
cannot be predicted by the semi-classical model.
a. Plot A ⋅ (Bexp A) − (Bliq A) as a function of Z (scatter plot). Identify
(
)
Z=28, Z=50 and Z=82.
b. Plot A ⋅ (Bexp A) − (Bliq A)
(
)
as a function of N (scatter plot). Identify
N=28, N=50, N=82 and N=126.
These particular numbers of Z and N are the so-called “Magic Numbers” in
nuclear physics, they correspond to nuclei with closed shell structure in
protons and neutrons conferring these nuclei more overall stability (and
therefore more binding energy) than their neighboring nuclei.
Exercise 5 – When does alpha decay become energetically possible? (Semi
computational)
The objective of this exercise is to determine roughly from which mass (A) does alpha
decay become possible. At the end of this exercise, you will need to use the
experimental binding energy per nucleon Bexp/A as a function of A, determined in
Exercise 5.1.
Consider the following reaction:
A
Z
X→
A−4
Z−2
Y + 24 He
with the binding energy of 4He equal to B(4He)=28.3 MeV.
1. For the alpha decay to occur spontaneously, the Q-value of the reaction
should be greater than 0. Deduce a condition for the alpha decay to occur as
a function of B(4He), B(AX) and B(A-4Y).
2. Show that: B( ZA X) − B( A−4
Z−2Y ) ≈ 4
3. Show that:
dB
dA
dB
d(B A) B
=A
+
dA
dA
A
4. Determine numerically the values of coefficients a and b in the equation of a
B
= a ⋅ A + b , which defines the limit after which alpha decay is possible.
A
d(B A)
In order to do this, you need to know the numerical value of
. Alpha
dA
line
decay occurs in heavier nuclei, where (B/A) drops down fairly constantly as a
function of A. If you completed Exercise 4 question 1, you can probably fit the
slope and find a numerical value for
d(B A)
. Otherwise, use -7.7x10-3 MeV.
dA
5. Find graphically the intersection of the line with (Bexp/A) as a function of A and
determine Alimit, from where alpha decay becomes energetically possible.

Download Report