Download PHY303 1 TURN OVER PHY303 Data Provided: A formula sheet

PHY303
Data Provided:
A formula sheet and table of physical constants is attached to this paper.
DEPARTMENT OF PHYSICS AND ASTRONOMY
Spring Semester (2014-2015)
NUCLEAR PHYSICS
2 HOURS
Answer question ONE (Compulsory) and TWO other questions.
All questions are marked out of twenty. The breakdown on the right-hand side
of the paper is meant as a guide to the marks that can be obtained from each
part.
Please clearly indicate the question numbers on which you would like to be
examined on the front cover of your answer book.
Cross through any work that you do not wish to be examined.
1
PHY303
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PHY303
COMPULSORY
1
(a) Write down the equations for the energy release Q for the three processes of
electron emission, positron emission and electron capture, carefully defining the
symbols that you use.
[4]
(b) Briefly outline the means by which nuclei may come to have a nuclear magnetic
[4]
moment and an electric quadrupole moment.
(c) Briefly explain the term nuclear force. Include a sketch to illustrate the force
[4]
experienced between two nucleons as a function of distance.
(d) Explain briefly the Saxon-Woods form of charge distribution in nuclei. Include
[4]
suitable sketches and state how the distribution can be experimentally determined.
(e) Outline the concept of an isospin doublet in the context of nuclear physics.
2
PHY303
CONTINUED
[4]
PHY303
2
Many interesting properties of nuclei are revealed by the chart of nuclides and the
curve of average binding energy per nucleon as a function of atomic mass number A.
(a) Sketch a plot of the chart of nuclides. Define the terms isotones, isotopes, isobars
and island of stability, using your plot to illustrate each of these.
[6]
(b) Isodiaphers are nuclides with the same difference between neutrons and protons.
[2]
Illustrate isodiaphers on your plot.
(c) Sketch a plot of the average binding energy per nucleon vs. atomic mass number
A. Show on this plot how fission and fusion occurs and state which isotope has the
[4]
maximum average binding energy.
(d) The semi-empirical formula for the binding energy of a nucleus can be written
B(A,Z)  av A  as A2 / 3  ac Z(Z 1)A1/ 3  aa (A  2Z)2 A1  ap A1/ 2
where values for the constants av, as, ac, aa and ap, can be taken as 14.0, 13.0, 0.6,
19.0 and 12.0 MeV. Although this equation is useful for estimating the binding
energy of most nuclei there is a class of nuclei for which it underestimates B.
Explain the characteristics of these nuclei and where in general they lie on the chart [2]
of nuclides.
(e) If we consider nuclei with the same odd value of A, then the equation for the
binding energy B is a parabola as a function of Z. Find an expression for the Z of the
single stable isobar that occurs in this situation. Find the Z for the stable isobar with [6]
A = 127.
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PHY303
3
9
12
The isotopes 4 Be and 6 C are commonly used in the nuclear industry. Like all nuclei
they show properties that are compatible with a so-called liquid drop model, in which
the nucleons involved behave collectively. However, they also behave as if the
nucleons involved fill specific energy levels, compatible with the so-called shell
model.
(a) Briefly describe, with the aid of suitable plots or sketches, six pieces of empirical
evidence that support the shell model of nuclei.
[6]
(b) Write a simple expression for the radius R of a nucleus in terms of the nucleon
radius and the number of nucleons contained. State what assumption is made in this
expression that is compatible with the liquid drop model. Using this expression give
9
12
estimates of the radii of 4 Be and 6 C .
[4]
(c) In the shell model the energy levels are assigned in order of increasing energy as,
[6]
1s1/ 2,1p3 / 2,1p1/ 2,1d5 / 2,2s1/ 2,1d3 / 2,1f j ,.....

Use this sequence, adding further levels as required, to determine the ground state
9
12
27
43
spin and parity of the following nuclei, 4 Be , 6 C , 13 Al , 21 Sc .
(d) On the basis of the shell model for nuclei that have an unpaired nucleon, a first
excited state can be produced by excitation of the unpaired nucleon into the next sub9
[2]
shell. Determine the spin and parity for the first excited state of 4 Be .
9
(e) 4 Be is used in the nuclear industry as a neutron reflector and can also be used as
a source of neutrons by bombardment with alpha particles. Write out the nuclear [2]
interaction equations that describe these two processes.
4
PHY303
CONTINUED
PHY303
4
The study of nuclear reactions provides an important tool for our understanding of
the excited states of nuclei.
(a) Briefly describe the characteristics of so-called compound nuclear reactions and
of direct reactions. Include a description of the independence hypothesis in
compound interactions. Give an example of a compound reaction.
[6]
(b) Consider the following reaction involving protons and nickel nuclei,
p  64 Ni  p 
64
Ni*
State what specific type of direct reaction this is and explain the meaning of the [2]
notation used.
(c) The figure below shows the excited states of 64Ni. These states can be produced
by bombarding 64Ni with a beam of protons if sufficient energy is transferred from
the incident protons. Under the assumption that all the kinetic energy of the
interaction is taken by the proton and that the nickel nucleus is excited into the first
excited state relative to the ground state, what is the kinetic energy of the proton after
the interaction if the initial proton energy is 11 MeV?
[2]
(d) Now use conservation of momentum to estimate the momentum and kinetic
energy of the recoiling 64Ni nucleus for the situation where the protons recoil at 600
from the beam direction. Re-estimate the proton energy by imposing energy
conservation taking account the recoil of the 64Ni nucleus.
[6]
(e) As can be seen in the figure in part (c) the gap between the ground state and first
excited state is quite large. Briefly explain why this is the case and why the gaps
[4]
between the energy levels above about 3 MeV are much smaller.
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PHY303
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PHY303
5
Nuclear power can be generated either by the process of fission or, hopefully in the
future, fusion reactions.
(a) Consider the fusion of two deuterium nuclei with release of a single neutron.
How much energy is released per fusion? (note: the nuclear masses of deuterium and
3
[5]
He are 3.445 × 1027 kg and 5.008 × 1027 kg respectively).
(b) Briefly describe the process of neutron-induced nuclear fission of 235U and its use
in power generation. Include in your discussion explanations of the terms moderator,
[5]
prompt and delayed neutrons, chain reaction and fission fragments.
(c) Consider neutrons of energy 0.1 eV incident on a piece of natural uranium. The
cross section for fission of 235U at that energy is 250 barns. The amount of 235U in
natural uranium is 0.72% and the density of uranium is 19 g cm3. Estimate the
mean free path length in cm for the neutrons, assuming this is dominated by fission [4]
of 235U.
(d) Taking the flux of neutrons in part (c) to be 1012 neutrons s1 cm2 and given that
each fission produces 165 MeV, estimate the initial nuclear power produced in watts
if the piece of uranium is 1 cm3. State what assumptions you make about the
geometry of the piece of uranium.
[4]
(e) Using your answer to part (a) and (d) estimate the mass of deuterium that must
undergo fusion per second to match the power output for the fission reaction.
[2]
END OF EXAMINATION PAPER
6
PHY303
CONTINUED
PHYSICAL CONSTANTS & MATHEMATICAL FORMULAE
Physical Constants
electron charge
electron mass
proton mass
neutron mass
Planck’s constant
Dirac’s constant (~ = h/2π)
Boltzmann’s constant
speed of light in free space
permittivity of free space
permeability of free space
Avogadro’s constant
gas constant
ideal gas volume (STP)
gravitational constant
Rydberg constant
Rydberg energy of hydrogen
Bohr radius
Bohr magneton
fine structure constant
Wien displacement law constant
Stefan’s constant
radiation density constant
mass of the Sun
radius of the Sun
luminosity of the Sun
mass of the Earth
radius of the Earth
e = 1.60×10−19 C
me = 9.11×10−31 kg = 0.511 MeV c−2
mp = 1.673×10−27 kg = 938.3 MeV c−2
mn = 1.675×10−27 kg = 939.6 MeV c−2
h = 6.63×10−34 J s
~ = 1.05×10−34 J s
kB = 1.38×10−23 J K−1 = 8.62×10−5 eV K−1
c = 299 792 458 m s−1 ≈ 3.00×108 m s−1
ε0 = 8.85×10−12 F m−1
µ0 = 4π×10−7 H m−1
NA = 6.02×1023 mol−1
R = 8.314 J mol−1 K−1
V0 = 22.4 l mol−1
G = 6.67×10−11 N m2 kg−2
R∞ = 1.10×107 m−1
RH = 13.6 eV
a0 = 0.529×10−10 m
µB = 9.27×10−24 J T−1
α ≈ 1/137
b = 2.898×10−3 m K
σ = 5.67×10−8 W m−2 K−4
a = 7.55×10−16 J m−3 K−4
M = 1.99×1030 kg
R = 6.96×108 m
L = 3.85×1026 W
M⊕ = 6.0×1024 kg
R⊕ = 6.4×106 m
Conversion Factors
1 u (atomic mass unit) = 1.66×10−27 kg = 931.5 MeV c−2
1 astronomical unit = 1.50×1011 m
1 eV = 1.60×10−19 J
1 atmosphere = 1.01×105 Pa
1 Å (angstrom) = 10−10 m
1 g (gravity) = 9.81 m s−2
1 parsec = 3.08×1016 m
1 year = 3.16×107 s
Polar Coordinates
x = r cos θ
y = r sin θ
∂
1 ∂2
1 ∂
2
r
+ 2 2
∇ =
r ∂r
∂r
r ∂θ
dA = r dr dθ
Spherical Coordinates
x = r sin θ cos φ
y = r sin θ sin φ
z = r cos θ
dV = r2 sin θ dr dθ dφ
1
∂
1
∂2
1 ∂
∂
2
2 ∂
∇ = 2
r
+ 2
sin θ
+ 2 2
r ∂r
∂r
r sin θ ∂θ
∂θ
r sin θ ∂φ2
Calculus
f (x)
f 0 (x)
f (x)
f 0 (x)
xn
ex
nxn−1
ex
tan x
sin−1
ln x = loge x
1
x
sin x
cos x
cos x
− sin x
cosh x
sinh x
sinh x
cosh x
a
cos−1 xa
tan−1 xa
sinh−1 xa
cosh−1 xa
tanh−1 xa
cosec x
−cosec x cot x
uv
sec x
sec x tan x
u/v
sec2 x
x
√ 1
a2 −x2
− √a21−x2
a
a2 +x2
√ 1
x2 +a2
√ 1
x2 −a2
a
a2 −x2
0
0
u v + uv
u0 v−uv 0
v2
Definite Integrals
Z
∞
xn e−ax dx =
0
Z
+∞
n!
an+1
r
(n ≥ 0 and a > 0)
π
a
−∞
r
Z +∞
1 π
2 −ax2
xe
dx =
2 a3
−∞
Z b
b Z b du(x)
dv(x)
Integration by Parts:
u(x)
dx = u(x)v(x) −
v(x) dx
dx
dx
a
a
a
−ax2
e
dx =
Series Expansions
(x − a) 0
(x − a)2 00
(x − a)3 000
f (a) +
f (a) +
f (a) + · · ·
1!
2!
3!
n X
n n−k k
n
n!
n
Binomial expansion: (x + y) =
x y
and
=
(n − k)!k!
k
k
k=0
Taylor series: f (x) = f (a) +
(1 + x)n = 1 + nx +
ex = 1 + x +
n(n − 1) 2
x + ···
2!
x 2 x3
+ +· · · ,
2! 3!
sin x = x −
ln(1 + x) = loge (1 + x) = x −
Geometric series:
n
X
rk =
k=0
Stirling’s formula:
(|x| < 1)
x3 x5
+ −· · ·
3! 5!
x2 x3
+
− ···
2
3
and
cos x = 1 −
x2 x4
+ −· · ·
2! 4!
(|x| < 1)
1 − rn+1
1−r
loge N ! = N loge N − N
or
ln N ! = N ln N − N
Trigonometry
sin(a ± b) = sin a cos b ± cos a sin b
cos(a ± b) = cos a cos b ∓ sin a sin b
tan a ± tan b
1 ∓ tan a tan b
sin 2a = 2 sin a cos a
tan(a ± b) =
cos 2a = cos2 a − sin2 a = 2 cos2 a − 1 = 1 − 2 sin2 a
sin a + sin b = 2 sin 21 (a + b) cos 12 (a − b)
sin a − sin b = 2 cos 12 (a + b) sin 12 (a − b)
cos a + cos b = 2 cos 12 (a + b) cos 12 (a − b)
cos a − cos b = −2 sin 12 (a + b) sin 21 (a − b)
eiθ = cos θ + i sin θ
1 iθ
1 iθ
cos θ =
e + e−iθ
and
sin θ =
e − e−iθ
2
2i
1 θ
1 θ
cosh θ =
e + e−θ
and
sinh θ =
e − e−θ
2
2
sin a
sin b
sin c
Spherical geometry:
=
=
and cos a = cos b cos c+sin b sin c cos A
sin A
sin B
sin C
Vector Calculus
A · B = Ax Bx + Ay By + Az Bz = Aj Bj
A×B = (Ay Bz − Az By ) î + (Az Bx − Ax Bz ) ĵ + (Ax By − Ay Bx ) k̂ = ijk Aj Bk
A×(B×C) = (A · C)B − (A · B)C
A · (B×C) = B · (C×A) = C · (A×B)
grad φ = ∇φ = ∂ j φ =
∂φ
∂φ
∂φ
î +
ĵ +
k̂
∂x
∂y
∂z
∂Ax ∂Ay ∂Az
+
+
∂x
∂y
∂z
∂Ax ∂Az
∂Ay ∂Ax
∂Az ∂Ay
−
î +
−
ĵ +
−
k̂
curl A = ∇×A = ijk ∂ j Ak =
∂y
∂z
∂z
∂x
∂x
∂y
div A = ∇ · A = ∂ j Aj =
∇ · ∇φ = ∇2 φ =
∂ 2φ ∂ 2φ ∂ 2φ
+
+ 2
∂x2 ∂y 2
∂z
∇×(∇φ) = 0
and
∇ · (∇×A) = 0
∇×(∇×A) = ∇(∇ · A) − ∇2 A