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Nuclear Physics & Radioactivity
What holds a nucleus together?
What drives radioactive decay?
What sets the timescale for radioactive
decay?
What are the statistics of radioactive decay?
Lecture outline:
1) nuclear physics
2) radioactive decay
3) counting statistics
Nuclear Structure:
A
Z E
(or AE or EA); e.g. 12C
Number of protons = Z (Atomic number)
Number protons + neutrons = A (Mass #)
Same Z, different A -- Isotope
a particles in a cloud chamber
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Particles: Proton, Neutron, Alpha, Leptons
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The Four Forces of Nature
Force
Strength
Range
Occurrence
Strong nuclear
1
<<1/r2 (finite, v. short)
inter-nucleon
Electromagnetic
10-2
1/r2 (infinite, but shielded)
nucleus, atom
Weak nuclear
10-13
<<1/r2 (finite, v. short)
-decay,
neutrinos
Gravity
10-39
1/r2 (infinite)
everywhere
Isotopes
330 natural isotopes
25 unstable, long halflife
35 unstable, short halflife
More than 1000 artificial isotopes
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Binding energy
Let’s revisit the fusion of four protons to form a 4He nucleus:
4( 11H )  1( 24 He)  2e  2 e  E
4(1.007277)  1(4.00150)  m
m  0.02761amu
*these masses come
from the table of nuclides
56Fe
We have calculated the mass deficit --> i.e. the whole is less than sum of the parts
The mass deficit is represented by a HUGE energy release, which can be calculated
using Einstein’s famous equation, E=mc2, and is usually expressed in MeV. The energy
difference can be termed a binding energy, often given on a “per nucleon” basis.
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Contributions to Binding Energy
EB = strong nuclear force binding -surface tension binding + spin pairing
+shell binding-Coulomb repulsion
1) strong nuclear force -- the more nucleons the better
2) surface tension -- the less surface/volume the better (U better than He)
3) spin pairing -- neutrons and protons have + and - spins, paired spins better
4) shell binding -- nucleus has quantized shells which prefer to be filled (magic numbers)
5) Coulomb repulsion -- packing more protons into nucleus comes at a cost (although
neutron addition will stabilize high Z nuclei)
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Radioactive Decay
- a radioactive parent nuclide decays to a daughter nuclide
- the probability that one nucleus will decay in a unit time is defined as  (units of s-1, y-1)
- If we have N unstable nuclei, the number of decays in time dt is
dN = - N dt
Thus dN/N = - dt
ln N = - t + Const
N = N0e-t
- where N0 is the number of nuclei present at t = 0
- the decay constant  is time independent; the mean life is defined as  = 1/ 
Define halflife t1/2 :
N  N0 / 2  N0et1/ 2
Note that t1/2 = ln(2) / 
Activity A: Number of decays per unit time. A = |dN/dt| = N = A0e-t
where A0 is the initial activity
Plot of ln A versus t gives a straight line: ln A = ln A0 - t
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Decay plot for 14C
dN
  N
dt
N  N0e

 t
ln(2)
t1/ 2
N0
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Number of C atoms
1000000
900000
800000
t1/2 = 5730y
700000
600000
500000
400000
t1/ 2
t1/ 2
t1/ 2
N 0 
N 0 / 2 
N 0 / 4 
N0 / 8
300000
200000
100000
0
0
5730
10000
20000
30000
40000
50000
Years
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Activity calculations
Activity   N
- usually reported in Bq (disintegrations per second),
example: 14C activity = 0.226 Bq / gram
(Old Unit: Curie (Ci) = = 3.7x1010 Bq
(1 g of radium)
A  A0e
 t
- because activity is linearly proportional to number N,
then A can be substituted for N in the equation
 t
N  N0e
Example activities:
Radium in watch dial: 4 x 104 Bq = 1 Ci
Sealed sources in lab: < 1 Ci
Cancer treatment source: 4 x 1013 Bq = 1000 Ci
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Four types of radioactive decay
1) alpha (a) decay - 4He nucleus (2p + 2n) ejected; Z→Z-2, A→A-4
2) beta () decay – electron ejected; Z→Z+1; no change in A
3) gamma (g) decay - photon emission, no change in A or Z
4) spontaneous fission - for Z=92 and above, generates two smaller nuclei
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a decay
241
95
a
4
Am 
 237
Np

93
2 He
- involves strong and Coloumbic forces
- alpha particle and daughter nucleus have equal and opposite momenta
(i.e. daughter experiences “recoil”)
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 decay - three types
1) - decay
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1

H 
 23 He  e  e
- converts one neutron into a proton and electron
- no change of A, but different element
- release of anti-neutrino (no charge, no mass)
2) + decay

C  115 B  e  e
11
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- converts one proton into a neutron and positron
- no change of A, but different element
- release of neutrino
3) Electron capture
EC
7

7
Be

e



4
3 Li  e
- converts one proton into a neutron
- no change of A, but different element
- release of neutrino
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g decay
3
2
g
He 
 He  g
*
3
2
- conversion of strong to coulombic E
- no change of A or Z (element)
- release of photon
- usually occurs in conjunction with other decay
Spontaneous fission
256
100
sf
112
Fm 
 140
Xe

54
46 Pd  4n
- heavy nuclides split into two daughters
and neutrons
- U most common (fission-track dating)
Fission tracks from 238U fission in old zircon12
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Radiation Dose, Dose Rate
When radiation (electromagnetic or particles) is
absorbed by any material, energy is absorbed by the
material (photon energy or kinetic energy).
The physical unit of absorbed radiation dose is the
gray (Gy), equal to 1 J per kg of mass.
Dose rate is the rate of dosage, the dose per unit time
interval, for example, 10 Gy/h.
This is a physical measurement, and does not correctly
quantify the long-term biological effect.
Biologically Equivalent Dose = Dose x Weight Factor
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Weighting factor (wR)
• The weighting factor (wR) is 1 for beta particles.
• The weighting factor (wR) is also 1 for X-rays and
gamma rays.
• The weighting factor (wR) is 20 for alpha particles
and 5-20 for neutrons, depending on energy.
• Some beta particles may not be much hazard because
they have low energy and will not penetrate the skin
(for example, from tritium).
• Alpha particles also will not penetrate skin.
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Dose = dose rate x time
For example, if dose rate D = 500 Gy/h then the
dose after 2 hours is 1000 Gy = 1 mGy.
The conventional (old) unit of dose is the rad
1 Gray = 1 Gy = 100 rad.
The absorbed physical radiation dose of 1 mGy is
equal to 100 millirad. (1 milli x 100 rad)
The unit of biologically equivalent dose is the Sievert;
conventional (old) unit of equivalent dose is the rem
1 Sievert = 1 Sv = 100 rem.
E.g, 1 mG dose of beta particles (wR = 1), has
biologically equivalent dose 1 mSv = 100 millirem.
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Counting Statistics
• The radioactive decay process follows Poisson statistics (special
case of binomial statistics). If we count decays for time T for a
sample with (parent) average number of decays  in time T, then
the probability of observing x counts in that time is
PP ( x) 
 xe 
x!
• The standard deviation for # counts in T is  = 
• For a large number of decays in time T, the Poisson distribution
approaches a Gaussian distribution with  = 
1
PG ( x) 
e
2
( x )2

2
• The uncertainty in an average (random) count  is 
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Permitted Radiation Dose & Natural Background
Permitted Dose
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Doses Due to Ionizing Radiation
Natural background (cosmic rays & radiation from naturally radioactive material):
San Francisco
120 mrems/yr
New York
135 mrems/yr
Denver
300 mrems/yr
Pocos de Caldos, Brazil
7,000 mrems/yr
Natural radioactive materials in the body:
K-40
Whole body
17 mrems/yr
K-40
Brain
30 mrems/yr
Ra-226
Whole body
2.8 mrems/yr
Ra-226
Bone
28 mrems/yr
U-238
Kidneys
1.2 mrems/yr
Rn-222 (Radon gas) Lung
200-1,100 mrems/yr (depending on location)
Two-week vacation in the mountains 3 mrems
(due to an increase in cosmic rays at higher elevation)
Cross-country jet flight
>1 mrem/hr
Radium-dial watch (close to wrist area) 0.1 mSv/day (10 mrems/day)
Whole-body diagnostic X-ray
up to 250 mSv (25,000 mrems)
Chest X-ray
up to 1 mSv/film (100 mrems/film)
Complete dental X-ray
up to 50 mSv (5,000 mrems)
NRC, NCRP & EPA whole-body limit
General Public
1 mSv/yr (100 mrems/yr)
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How radioactive is your body?
1.2 radioactive atoms of 40K for every 10,000 non-radioactive
atoms of potassium.
There is of the order of 140 g of potassium in an adult who
weighs 70 kg, and 0.0169 g consists of the 40K isotope.
This amount of 40K disintegrates at the rate of 266,000 atoms
per minute.
89% result in the release of beta particles with maximum
energy of 1.33 MeV
11% result in gamma photons with an energy of 1.46 MeV.
All of the beta particles and about 50 percent of the gamma
rays are absorbed in the body, giving annual doses of 16 mrad
from the beta particles and 2 mrad from the gamma rays.
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