1
REVIEW OF NUCLEAR STRUCTURE
Z – atomic number = number of protons
A – mass number = number of nucleons (p+ and n0)
127
53
I 
53 protons
127 53 neutrons
isotopes – nuclides with same atomic number but different mass number
(different # of n0)
RADIOACTIVITY
- Not all combinations of protons and neutrons can be put together as a stable
nucleus.
- Some combinations are not bound together at all.
i. e. 22 He or 123 Li
- Other combinations may be bound but unstable.
i. e. 31 H or 83 Li
- Some combinations are stable; that is, they will not spontaneously
change.
i. e. 42 He or 126 C
- Bound nuclei that are unstable become stable by undergoing spontaneous
nuclear reaction known as radioactivity.
Forms of Radioactivity
Alpha decay
4
- occurs when 2 He nucleus (alpha particle) is emitted.
236
92
U
Th   or also written
232
90
236
92

U
 23290Th
- note: Mass number always decreases by four and atomic number always
decreases by two.
- only nuclei with Z  84 undergo alpha decay.
2
Beta decay – three forms
1.) Electron emission (- decay)
- - particle is an electron, e-.
- when neutron in a nucleus changes into a p+, e- and anti-neutrino.
60
27
Co 
60
28
Ni  e   
 - neutrino
 - antineutrino
-36
- m  2  10 kg
- compare to me- = 9  10-31 kg
- no charge
- In - decay, A stays the same, Z  by one.
2.) Positron emission (+ decay)
- when proton in a nucleus changes into a n0, e+ and neutrino.
11
6
C  115B  e   
- e+ - positron – same as electron only with positive charge
- In + decay, A stays the same, Z  by one.
11
C is an isotope used in positron emission tomography (PET). A biomolecule
with a 11C atom is introduced into the body and allowed to become part of the
body. When the 11C decays, gamma rays from the annihilation of the positron are
detected and show where in the body the molecule was. Cancer cells use more
glucose than healthy cells, so PET can be used in cancer detection.
3. Electron capture (EC)
- when nucleus captures 1s electron, changing proton into neutron.
7
4
Be  e   73 Li  
- A stays the same, Z  by one.
- It can be difficult to distinguish between position emission and electron capture.
Gamma decay
- when nucleus in an excited state, decreases its energy to ground state by
releasing a photon (gamma ray – )
Tc 
99 m
43
Tc  
99
43
-  = 0.143 MeV   = 870 Å
- Note: This a very low energy gamma ray
99m
Tc (as part of the pertechnetate ion, TcO4-) is able to bind to biomolecules and
be spread throughout the body. The gamma rays from the decay are detected and
various organs (especially the thyroid) can be imaged.
3
Spontaneous Fission
- some nuclei will spontaneously break apart into two (or more) smaller nuclei
250
96
Cm 
132
50
0
Sn114
46 Pd  4 n
- fission also can be induced with reactions with other particles such as n0, p+, , etc…
160
72
0
n 0  239
94 Pu  64 Gd  30 Zn  8 n
n 0  235
92 U 
90
38
0
Sr 140
54 Xe  6 n
- large amount of energy is released during fission
- fission is the basis of nuclear reactors and the atomic bomb
- neutrons released cause more nuclei to undergo fission (chain reaction)
235
U and 239Pu are used in atomic weapons.
238
U and 238Pu are used in nuclear reactors.
NUCLEAR FORCES
Strong Force
- glue that keeps nucleons together
- strong enough to overcome electric repulsion of positive charges
- approximately 100 times stronger
- making nuclear bonds is very, very exothermic
- responsible for  decay and fission
- strong force is between nucleons only (electrons are unaffected).
Weak Force
- controls whether nucleon is proton or neutron
- approximately 1/100,000th as strong as strong force
- approximately 1/1000th as strong as electric force
- responsible for  decay
- nucleons as well as electrons experience weak force.
Electric Force
- same force as that responsible for chemical interactions
- responsible for  decay
4
NUCLEAR STABILITY
- for nuclide of low Z, most stable structure is #n0 = #p+
- as Z increases #n0 > #p+
- table of nuclides has band of stability
- nuclides outside of band of stability are radioactive
- if #n0 too high, - possible
- if #p+ too high, +, EC or  decay possible
- not all nuclei in band of stability are stable
- Rules of thumb for nuclear stability
#p+
even
odd
even
odd
#n0
even
even
odd
odd
very stable
mostly stable
mostly stable
unstable
- 21 H , 63 Li, 105 B, 14
7 N are exceptions to odd-odd rule
- Magic numbers
- nuclei have “shells” similar to electrons in atoms
- neutrons and protons have separate shells to fill
- When shells are filled, nucleus is extra stable
- Magic numbers are 2, 8, 20, 28, 50, 82, 126
- Compare to “magic numbers” for electrons
- 2, 10, 18, 36, 54, 86, 118, etc…
Use rules of thumb to predict relative nuclear stability
Example: Which of the following nuclides is more stable?
a) 199 F or 189 F
b)
40
18
Ar or
39
18
Ar
5
Predicting route of radioactivity
Using average atomic weight as a guide, determine if nuclide has too many p+ or
too many n0.
- compare average atomic weight to mass number.
- If mass number is greater, the nuclide has too many n0, and - is possible.
- If mass number is less, the nuclide has too many p+,
then + or EC is possible.
Z < 84
or  is possible.
Z  84
Example: Predict the path of nuclear decay for the following nuclides
a)
228
92
b)
8
5
c)
68
29
U
B
Cu
6
NUCLEAR REACTIONS
Two nuclei (or a nucleus and a particle) can be smashed together to cause a
nuclear reaction.
1. Spontaneous and induced radioactivity has been discussed.
235
92
U  n0 
71
31
131
49
Ga   
In 9943Tc  5 n 0 - induced fission
71
32
Ge    - induced beta decay
2. Fusion – when two nuclei come together to form new nuclei
He 147 N  178 O  p 
86
208
289
0
36 Kr  82 Pb  118 Uuo  5 n
4
2
2
1
H  31 H  42 He  n 0
Definition: Cross Section – The effective size of a target object when a projectile
is trying to react with it.
Cross section is measure of reactivity of nuclei and particles with each other.
Six differences between nuclear reactions and chemical reactions.
Nuclear Reactions
Chemical Reactions
1. Protons and neutrons react
inside nucleus.
1. Electrons react outside
nucleus.
2. Elements transmute into
other elements.
2. The same number of each
kind of atom appears in the
reactants and products.
3. Isotopes react differently.
3. Isotopes react the same.
4. Independent of chemical
combination.
5. Energy changes equal 108
kJ/mol.
6. Mass changes are
detectable.
4. Depend on chemical
combination.
5. Energy changes equal 10 103 kJ/mol.
6. Mass reactants = mass
products.
7
NUCLEAR KINETICS
All nuclear decay processes follow first-order kinetics.
N
N – number of nuclei
  kN
t
k – first-order rate constant (decay constant)
Recall from Chapter 14, for first-order kinetics
N 
  A t 
ln  At  ln  A0  k t 
   k t  ln t    k t
ln
  A 0 
 N0 
Nt – number of nuclei at time t
N0 – initial number of nuclei
Also recall half-life
t1 
2
ln 2 0.693

k
k
Activity – number of decays per unit time.
Activity = k N
Geiger counter registers counts of radioactive decays.
Example:
Rb decays to 86Sr via - emission. The decay constant is 4.610-19 s-1.
What is the half-life of 86Rb in years?
86
8
Example: A sample of 59Fe initially registers 125 counts/s on a Geiger counter.
After 10.0 days, the sample registers 107 counts/s. What is the halflife of 59Fe?
Remember activity is k N.
 k No = 125/s
and
k Nt = 107/s
N 
1 N 
ln t    k t  k   ln t 
t  N0 
 N0 
t1 
2
ln 2
0.693

 44.7 days
k
1.55 102 day 1
Example: If 18.0% of a sample of 65Zn decays in 69.9 days, what is the half-life
of the isotope?
First we need to calculate the decay constant, k.
N 
1 N 
ln  t   k t  k   ln  t 
t  N0 
 N0 
To do so, we need to use a clever substitution for Nt
N t  N 0  0180
.
N 0  0.820 N 0
 0.820 N0 
1 N 
1
3
1
k   ln  t   
ln 
  2.84 10 day
t  N0 
69.9 day  N0 
Now calculate the half-life.
N 
ln  t   k t
 N0 
 t1 
2
ln 2
0.693

 244 days
k
2.84 103 day1
9
NUCLEAR THERMODYNAMICS
Einstein’s relationship (relativity)
Einstein’s famous equation E  m  c 2 states that energy has a mass.
- Boiling water weighs more than cold water
- l L weighs 4.610-12 kg more
It also states that if mass is converted to energy, an enormous amount of energy is
released.
Binding energy
Energy is lost when nucleons are bound together.
- This energy is analogous to the energy lost when chemical bonds are formed.
The energy lost when nucleons form “nuclear bonds” is called the binding
energy.
Binding energies are huge compared with chemical energies.
- so much energy is lost when the nucleons bond together that Einstein’s
relationship becomes relevant.
- the nucleus loses mass when the protons and neutrons bind together!!
Binding energies are often calculated using the energy unit, megaelectron volts (MeV).
1MeV  1.609 1013 J
Repeating what was said above, when nucleons bind together, the mass of the
energy of attraction (binding energy) is subtracted from the total mass.
As more nucleons bind together, the binding energy increases.
Example: Calculate the binding energy of an alpha particle (4He nucleus).
M(n0) = 1.008665 amu
M(p+) = 1.007276 amu
M(4He) = 4.001506 amu
Since mass is another form of energy, we need to calculate difference in mass
between products and reactants.
M(4He)
4.001506 amu
+
0
- 2  M(p ) - 2  M(n )
- 4.031882 amu
- 0.030376 amu
Thus, we need conversion factor for amu to kg.
0.030376 amu 
1.6605 1027 kg
 5.04 1029 kg
amu
10
How much energy is this much mass?
E  mc2  5.04 1029 kg   2.998 108 m s   4.53 1012 J
2
Convert this binding energy in Joules to MeV
4.53 1012 J 
1MeV
 28.3 MeV
1.602 1013 J
Table of nuclides with binding energy and binding energy per nucleon
Nuclide
2
H
He
3
H
4
He
12
C
31
P
40
Ca
55
Fe
56
Mn
56
Fe
57
Fe
56
Co
109
Ag
208
Pb
235
U
268
Mt
3
Note:
Mass
Binding Energy
(amu)
(MeV)
2.0135
2.3195
3.0149
7.7175
3.0155
8.4813
4.0015
28.2989
11.9967
92.1623
30.9655
262.9176
39.9516
342.0539
54.9240
481.0610
55.9252
489.3466
55.9207
492.2594
56.9211
499.9051
55.9250
486.9108
108.8790
931.7271
207.9316
1636.4644
234.9935
1783.8812
268.0790
1948.5449
B.E. per nucleon
(MeV)
1.15973
2.57251
2.82710
7.07472
7.68019
8.48121
8.55135
8.74656
8.73833
8.79034
8.77026
8.69484
8.54795
7.86762
7.59098
7.27069
56
Fe has the highest binding energy per nucleon.
- Consequently, large stars form 56Fe at the end of their lifetimes.
Energy changes in nuclear reactions
Calculating the energy change in a nuclear reaction is the same as a chemical reaction.
- Energy of products – energy of reactants.
What is different about nuclear reactions is that because of Einstein’s relationship,
energy changes can be measured with mass changes.
- Mass of products – mass of reactants.
11
Consider the following example:
For the  decay of 230Th  226Ra +  and the given masses below, calculate the
energy released in one decay. Calculate the energy for one mole of decays.
M(230Th) = 230.033127 amu
M(226Ra) = 226.025402 amu
M() = 4.00150 amu
Since mass is another form of energy, we need to calculate difference in mass
between products and reactants.
M(226Ra) + M()
230.02690 amu
230
- M( Th)
- 230.03313 amu
- 0.00623 amu
- negative sign indicates energy is released
Convert mass in amu to energy in Joules.
1 kg  m2
Recall 1 J 
s2
Thus, we need conversion factor for amu to kg.
16605
.
 1027 kg
0.00623 amu 
 103
.  10 29 kg
amu
How much energy is this much mass?
E  mc2  1.03 1029 kg   2.998 108 m s   9.30 1013 J
2
How much energy is released per mole? Multiply by Avogadro’s number.
E  N A E  6.022 1023 mol1  9.30 1013 J 
 5.60 1011 J  mol1  5.60 108 kJ  mol1
Compare to combustion of a mole of TNT.
H  178
.  103 kJ  mol 1
Alpha decay is 315,000 times more energetic than the explosion of dynamite!!!
12
Example: How much energy is produced in a mole of induced fission events within
235
U? Assume the fission reaction is n0 + 235U  142Ba + 92Kr + 2 n0.
M(235U) = 235.0439 amu
M(n0) = 1.0087 amu
M(142Ba) = 141.9164 amu
M(92Kr) = 91.9263 amu
M(142Ba) + M(92Kr) + 2  M(n0)
- M(235U) - M(n0)
235.8601 amu
- 236.0526 amu
- 0.1925 amu
2
1.6605 1027 kg  2.998 108 m 
11
E  0.1925 amu 

  2.873 10 J
amu
s


23
6.022 10
1 kJ
E  2.873 1011 J 

 1.730  1010 kJ mol
mol
1000 J
Almost 10,000,000 more powerful than TNT!!
NUCLEAR REACTORS
Critical mass – amount of radioactive substance needed to have a self-sustaining
nuclear reaction (usually fission).
Neutrons released may or may not induce fission in another nucleus.
Whether a reaction occurs depends on the cross section. The cross section is
related to a property called the mean free path.
- average length that neutron travels before it reacts with another nucleus.
The size of the fissionable matter must be of the order of the mean free path or
else the neutrons produced within the material go through it without inducing
more reactions.
When the mass of a radioactive substance is above its critical mass, the size of the
substance is large enough to ensure that at least some of the neutrons released
during fission will induce fission in another nucleus.
13
Fission reactor
The concept of a fission reactor is
- assemble a barely critical mass of a 238U or 239Pu to have a sustained fission
reaction.
- energy released is used to heat water to steam
- steam is used to drive steam turbines which create electricity with
electromagnets.
Moderator – substance used in a nuclear reactor to slow down the speed of neutrons.
Moderators are necessary since fission in 238U can be induced only with ‘slow’
neutrons.
Moderators include ‘heavy’ water, D2O and graphite.
Control Rods – used in a nuclear reactor to control number of neutrons.
Control rods control criticality of reaction. That is, they ensure that each fission
event yields only one neutron usable for another fission event.
Control rods are made with nuclei with high neutron cross sections such as 10B
and 113Cd.
Fusion reactor
The concept of a fusion reactor is
- assemble a large density of deuterium 2H = D and tritium 3H = T
- create conditions for a fusion nuclear reaction:
D + T = 4He + n0
- energy released is used to heat water to steam
- fusion reactors are not yet economically feasible
- they use more energy than they produce
Consider energy released in fusion reaction D + T = 4He + n0
M(D) = 2.0140 amu
M(T) = 3.01605 amu
M(n0) = 1.0087 amu
M(4He) = 4.00260 amu
M(4He) + M(n0)
- M(D) - M(T)

5.0113 amu
- 5.0301 amu
- 0.0188 amu
E  169
.  109 kJ mol
14
If we can bring deuterium and tritium close enough, we can get a lot of energy in
a thermonuclear reaction.
Getting nuclei close enough is the problem.
- Need to have temperature about 40,000,000 K for fusion reaction
- At the present, it takes too much energy to produce this temperature in a
controlled way.
NUCLEAR WEAPONS
Atomic Bomb
An atomic bomb is an uncontrolled induced fission chain reaction.
Two isotopes are used to build a bomb.
1. 235U
2. 239Pu
- induced fission occurs in both isotopes with ‘fast neutrons’
- both isotopes undergo spontaneous fission and release neutrons to induce
further fission
Supercritical mass will cause uncontrolled nuclear reaction.
Thus the key to controlling when the bomb detonates is to keep two or more
subcritical masses separate. Then for detonation, bring the two masses together
quickly.
Little Boy
- Hiroshima, 6 AUG 45, 8:16 a.m.
- 235U used
- wedge of subcritical mass fired in a cannon at another subcritical mass
- yield: 12.5 kilotons (of TNT)
- 100,000 dead (200,000 after five years)
Fat Man
-Nagasaki, 9 AUG 45, 11:02 a.m.
- 239Pu used
- several subcritical pieces brought together with implosion device
- yield: 22 kilotons
- 70,000 dead (140,000 after five years)
15
Hydrogen Bomb
A hydrogen bomb is a thermonuclear fusion reaction of a thermonuclear fuel such
as of deuterium and tritium (actually, lithium deuteride 6LiD).
An atomic bomb is used “to light” the thermonuclear fuel to begin fusion reaction.
Mike
- First H-bomb test
- Marshall Islands, Eniwetok Atoll, Elugelab Island: 1 NOV 52
- Thermonuclear fuel was liquid deuterium
- Yield: 10.4 megatons
- 832 times energy of Little Boy
- The resulting crater was 6240 ft across and 164 ft deep.