Lecture 26
Radioactivity,Decay
Review
The Nucleus:
•Formed by two kind of particles: protons and neutrons
•Atomic number: Z = number of protons
•Mass number A = number of protons and neutrons
•number of neutrons: N = A - Z
•Binding energy (amount of energy that must be put into the nucleus to
break it apart) BE= (Z mp + N mn - M) c2 where M=nucleus mass
•atomic mass = nucleus mass + Zme
•Nuclear force: Strong Force binds protons and neutrons together.
(Very strong, very short range)
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Radioactivity:
Disintegration or decay of an unstable nucleus, 3 types:
•Alpha decay (emission of α particle = 42He)
•Beta decay (emission of a β particle = e-)
•Gamma decay: (emission of a γ ray = photon)
•Disintegration energy Q=(Mp -MD -m X)c2
Radioactivity was discovered by Henri Becquerel (right),
who noticed that film placed near rock containing U became
fogged. The other famous person in this area is Marie
Sklodowska Curie, the only person ever to win Nobel
Prizes in both chemistry and physics.
(MP > MD)
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Examples:
α decay : the nucleus changes to another one (transmutation)
232 U
92
= 22890Th+ 42He +Q
Q=(Mp -MD -mα )c2 = 5.4MeV
Q becomes KE of the daughter
β decay : (transmutation)
∗ β−:
14 C
6
= 147N + e- + ν (n
p)
e- is created in the nucleus
antineutrino (m~0,q=0,s=1/2)
n
p + e- + ν
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∗ β+:
19 Ne
10
= 199F + e+ + ν (p
* Electron Capture:
7
4Be
n)
+e- = 73Li + ν (p
n)
γ decay : an excited nucleus decays to a lower energy state by
emitting a γ ray.
Energy of γ ray = Energy difference between the two states
Q ~ MeV
4
Conservation Laws:
• Energy
• momentum
• angular momentum
• charge
• number of nucleons
Steps to calculate Q
0- Check conservation laws of charge and nucleons.
1- Write the equation Q=(Mp -MD -mx)c2
2- Replace nuclear masses by "atomic mass -Z me"
3- Reduce the expression as much as possible (cancel almost all me)
4- Look for the atomic masses in appendix F (write all the digits)
5- Do the calculation.
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Example:
232 U
92
1- Q /c2= M (23292U)
228 Th+ 4 He
90
2
-( M( 22890Th) + M( 42He ) )
2- Q/c = AM (23292U) -92 me -( AM( 22890Th) -90 me+ AM( 42He ) -2 me)
3- Q/c2 = AM (23292U) -92 me -( AM( 22890Th) -90 me+ AM( 42He ) -2 me)
4- Q/c2 =( 232.037131 -( 228.028716+ 4.002602)) x 931.5 Mev/c2= 5.4 MeV
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Example: How much energy is needed to extract a proton from 147N ?
14
7N
6
13C
+p
We calculate what Q (the energy needed or released)
Q/c2 = M(147N ) - ( M(
13
6 C)
+ M(p) )
= AM(147N ) -7me- ( AM(
13
6 C)-6me
+ M(p) )
= AM(147N ) -7me- ( AM(
13
6 C)-7me
+me + M(p) )
= AM(147N )- ( AM(
+ AM(11H) )
13
6 C)
= 14.003074-(13.003355 + 1.007825) = -0.008106 x 931.5 Mev/c2
Q = -7.55 Mev
Negative Q means that the decay will not be spontaneous, the energy
needed to make extract the proton is 7.55 Mev.
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Example: In order to extract a proton from 147N it is given an energy
of 10 Mev. What is the KE of the proton?
+p
We first calculate what Q (the energy needed or released)
14
7N
6
13C
Q = -7.55 Mev
The KE of the proton will be:
10Mev-7.55 Mev = 2.45 Mev
8
•Half life: Rate of decay
•Radioactive decay Law: N = N0 e-λ t
N: number of remaining nucleus
N0 : number at t=0
e=2.718
λ = decay constant
•Decay rate (activity) R= |∆N/∆t| = λ N = (∆N/∆t )0 e-λ t = R0 e-λ t
units of R: Ci (Curie) = 3.7 x 1010 decay/s
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•Half Life: T1/2: time when N = N0/2
N/N0 = exp(-λ T1/2) = 1/2
λ T1/2 = ln ( 2)
-λ T1/2 = ln (1/2)
T1/2 = 0.693/ λ
Example:
Radioactive 146C has a half life of 5730 years. If you start with a
sample of 1000 nuclei, how many will still be around in 17190
years?
λ = 0.693 /5730 =0.00012 yr -1
N = 1000 e-0.00012 x 17190 = 127
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Example:
The half life of radioactive 22688Ra is 1.6 x 103yr .
If a sample contains 3 x 1016 such nuclei, determine the
activity at this time.
R=λN
T1/2 = 1600 yr x 3.16 x 107 s/yr = 5 x 1010 s
λ = 0.693/T1/2 = 1.4 x 10-11 s-1
R = (1.4 x 10-11 s-1 )(3 x 1016) = 4.1 x 105 decays/s
= 11.1 µCi
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Quiz
What fraction of a radioactive sample has decayed
after two half-lives have elapsed?
a)1/4
b)1/2
c)3/4
d)1
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Example: How old is he?
A 50 g sample of carbon is taken from the pelvis bone
of a skeleton, and it's found to have a 14C decay rate
of 200 decays/min. It is known that carbon from a
living organism has a decay rate of 15 decays/min.g.
Find the age of the skeleton.
R=R0 e-λ t
t = - ln (R/R0) / λ
λ=??
λ = 0.693/ T1/2 = 2.3 x 10-10 min
t = -ln (200d/min)/ (50x15 d/min) /(2.3 x 10-10 min)
= 5.74 x 109 min = 10900 yr
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