WileyPLUS Assignment 5 is available
Chapters 28, 29, 30
Due Friday, April 9 at 11 pm
Friday, April 9
Review - send questions!
PHYS 1030 Final Exam
Friday, April 23, 1:30-4:30 pm
Frank Kennedy Brown Gym,
30 questions, the whole course, formula sheet provided
Wednesday, April 7, 2010
29
β– Decay
Energy released = [mX - mY] x 931.5 MeV
Atomic masses
β+ Decay
Energy released = [mX - (mY + 2me)] x 931.5 MeV
Atomic masses
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30
Prob. 31.31/27: Find the energy released when !+ decay converts
22
Na (Z = 11, atomic mass = 21.994434 u). Notice that the atomic
mass for 22Na includes the mass of 11 electrons, whereas the atomic
mass for 22Ne (Z = 10, atomic mass = 21.991383 u) includes the mass
of only 10 electrons.
Using tabulated atomic masses, the energy released in the decay is,
"
#E = [mNa - (mNe + 2me)] x 931.5 MeV
1.82 MeV
Wednesday, April 7, 2010
31
Beta-decay – a problem
Beta-decay
X→Y+e
Y (daughter)
Expected energy of the e+
X (parent, at rest)
e– or e+
http://hyperphysics.phy-astr.gsu.edu/hbase/nuclear/beta.html
Kinematics: if energy and momentum are conserved, the electron (e– or e+)
should have a well-determined kinetic energy following the beta-decay.
But, the electron does not have a well-determined energy, as seen above.
Is energy conserved?!
Wednesday, April 7, 2010
Yes, energy is conserved...
32
The Neutrino
A third particle, a neutrino, is also emitted in the decay, so that
the released energy is shared between three particles instead
of two:
64
29 Cu
→
64
28 Ni
+ e+ + ν
Expected energy of the e+
The neutrino is very difficult to detect.
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33
A neutrino detector in Japan
Wednesday, April 7, 2010
34
Neutrino detector in Japan - X-Files version
Wednesday, April 7, 2010
35
Gamma (γ) Decay
A ∗
ZX
→
A
ZX
+!
Excited state
of the nucleus
Nuclear
Energy
X*
Gamma rays are produced in the decay
(de-excitation) of a nuclear state.
$ ray photon
X
This is similar to the production of a
photon by an atom, except that the energy
levels are associated with the nucleus
itself, not with electrons in the atom.
Gamma rays are generally of higher energy and are even more
penetrating than x-rays.
Wednesday, April 7, 2010
36
Gamma Knife – to zap a tumour
60
Co gamma ray source:
60
27 Co
→
60 ∗
−
¯
28 Ni + e + !
60 ∗
28 Ni
→
60
28 Ni + !
Tumour
(� 1.2 MeV)
Gamma rays from 60Co sources are
channeled through collimators in a
metal helmet.
Gamma rays are concentrated at the
site of the tumour, to selectively
destroy the malignant tissue.
Half of the 60Co decays away in 5.3 years, so has to be replaced...
Wednesday, April 7, 2010
37
Winnipeg Free Press,
April 4, 2004
Wednesday, April 7, 2010
38
Winnipeg Free Press, March 19, 2008
Wednesday, April 7, 2010
39
N0
Radioactive Decay
Start with N0 unstable nuclei
Observe how many survive to time t
Half-life, T1/2: the time for half of the nuclei to decay.
After each succeeding halflife, half of the remaining
unstable nuclei remain...
N0/2
N0/4
N0/8
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40
Radioactive Decay
� �
1
After time T1/2,
N0 are left
2
� �� �
� �2
1 1
1
After time 2T1/2,
N0 =
N0 are left
2 2
2
� � � �2
� �3
1 1
1
After time 3T1/2,
N0 =
N0 are left
2 2
2
� �n
1
After time nT1/2,
N0 are left
2
� �n
1
t
So, N(t) = N0
= number of half-lives
n=
2
T1/2
Wednesday, April 7, 2010
41
Radioactive Decay
� �n
1
So, N(t) = N0
2
n=
t
T1/2
= number of half-lives
This is the same as an exponential decay:
% = “decay constant”
Natural log, loge
Take logs: −n ln 2 = −!t
So, ! =
and, n =
ln 2 0.693
=
T1/2
T1/2
!=
Wednesday, April 7, 2010
t
T1/2
1
0.693
=
“mean life”
T1/2
42
Radioactive
decay
� �n
1
N(t) = N0
= N0e−!t
2
T1/2
ln 2
0.693
=
=
λ
λ
Some Half Lives
214Po
0.164 ms
Decay
Mode
&, $
89Kr
3.16 min
!–, $
222Rn
3.83 days
&, $
60Co
5.271 y
!–, $
90Sr
29.1 y
!–
226Ra
1600 y
&, $
14C
5730 y
!–
238U
4.47!109 y
115In
4.41!1014 y!!!
Isotope
% = “decay constant”
n = t/T1/2
Wednesday, April 7, 2010
Half Life
&, $
!–
Radioactive Decay, Activity
43
Radioactive Decay, Activity
Activity, A = number of decays per second = –ΔN/Δt, N = no. of nuclei left
The rate of decay is proportional to the number of radioactive nuclei
present.
A=−
!N
= "N(t) = "N0e−"t
!t
Calculus
initial activity, A0 = %N0
dN
d
= − N0e−!t
dt
dt
= !N0e−!t
A=−
Units: Becquerel (Bq): 1 Bq = 1 decay per second
Curie (Ci, old unit): 1 Ci = 3.7 x 1010 Bq = activity of 1 g of pure
radium
Short half life means large decay constant and large activity (% = 0.693/T1/2).
Wednesday, April 7, 2010
44
Radioactive Decay
The number of radioactive nuclei left after time t is:
� �n
1
N(t) = N0
2
n=
t
T1/2
= number of half-lives
or
N(t) = N0e−!t , ! = decay constant
!=
0.693
T1/2
Activity, A(t) = %N(t)
Wednesday, April 7, 2010
45
238
U
A Radioactive
Decay Series
A
238
92 U
222
→
234
90 Th
Rn
→
234
4
90 Th + 2 He
234
91 Pa +
e− + !¯
and so on...
...ending at 206
82 Pb
Radon, of
basement fame
Other decay series:
235
232
Z
Wednesday, April 7, 2010
U→
Th →
207
Pb
208
Pb
All formed in a supernova
explosion about 4.5
billion years ago.
46
Radon in the Basement
222
Rn – produced in the decay chain that starts with 238U.
222
Rn half life is only 3.83 days, but it is generated continually by the
decay of longer-lived nuclei.
Suppose 3!107 radon nuclei are trapped in a basement when the walls
are sealed so no more can enter. How many are left after 31 days?
� �n
1
t
31
N = N0
, n=
=
= 8.094
2
T1/2 3.83
� �8.094
1
N = N0
= 0.00366 N0 = 1.1 × 105
2
The initial activity of the radon is
0.693 N0
0.693 × 3 × 107
A0 = !N0 =
=
= 62.8 Bq
T1/2
3.83 × 24 × 3600 s
Wednesday, April 7, 2010
47
Prob. 31.35/32: Strontium 90Sr has a half-life of 29.1 years. It is
chemically similar to calcium, enters the body through the food
chain and collects in the bones. Consequently, 90Sr is a particularly
serious health hazard.
How long will it take for 99.99% of the 90Sr released in a nuclear
reactor accident to disappear?
387 years
Wednesday, April 7, 2010
48
Radioactive Dating
The best known is radiocarbon dating, based on the decay of 14C.
“Carbon-based life forms” take up carbon in food or as CO2 (plants,
trees, in photosynthesis).
One atom in 8.3!1011 of carbon has a 14C nucleus, the rest are 12C, 13C
(activity 0.232 Bq per gram of naturally occurring C).
14
C has a half-life of 5730 years,
12
C, 13C are stable.
When the organism dies, the uptake of carbon ceases, and the amount
of 14C present decreases, halving every 5730 years.
Measure how much 14C is left
work out how long since organism died.
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49
31.58/40: Material found with a mummy in the arid highlands of
southern Peru has a 14C activity per gram of carbon that is 78.5% of
the activity present initially. How long ago did this individual die?
(half life of 14C is 5730 y).
2000 years ago
Wednesday, April 7, 2010
50
31.46/52: An archeological specimen containing 9.2 g of carbon
has an activity of 1.6 Bq. How old is the specimen?
Present-day activity is 0.232 Bq per gram of carbon.
2370 years
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51
Prob. 31.47/41: The practical limit for radiocarbon dating is about
41,000 years. What fraction of the 14C is left after this time? (halflife = 5730 years)
The number of half-lives that have elapsed in 41,000 years is:
n = 41,000/5,730 = 7.16
and so the fraction of 14C left after 41,000 years is
N/N0 = (1/2)7.16 = 0.007
so, only 0.7% of the 14C is left.
Wednesday, April 7, 2010
52
Radioactive Dating
The amount of 14C left can be measured by:
• Counting the rate of decay (activity) of 14C, activity being
proportional to the number of radioactive nuclei – the accuracy of
the age measurement depends on the size of the sample and on for
how long you are willing to count. The more decays seen, the more
accurate the measurement.
• Counting the number of 14C nuclei directly by vaporizing the
sample and counting the 14C nuclei in a mass spectrometer. You are
no longer waiting for nuclei to decay and can get much higher
precision on the age.
Wednesday, April 7, 2010
53
Radioactive dating – origin of the
The 14C comes from cosmic rays that interact with
atmosphere:
n+
14
7N
→
14
6C
14
14
C
N in the upper
+ p
Stable, the usual form of N
Number of nucleons: 1 + 14 = 14 + 1
Charge: "
0+7=6+1
The 14C combines with O2 to form 14CO2 which mixes with normal
CO2 in a stable proportion (1 in 8.3!1011).
14
C decays back to 14N
with a half-live of 5730 years.
Wednesday, April 7, 2010
14
6C
→
14
7N
+ e− + !¯
54
Radioactive Dating on Geological Timescales
238
U
206
Pb in decay series and T1/2 = 4.5 ! 109 years for 238U.
Number of &-particles produced in the decay series is (238 – 206)/4 = 8.
If the decays occur in rock, the 4He can be trapped. Measure how much
4
He is present in the rock. Each U decay should generate a total of 8 &particles.
Then, the original number of 238U nuclei in the rock when it was formed is:
N!
N(t) = number of U nuclei present now
8 Now
� �n
Now
1
t
Then, N(t) = N0
, n=
→ t
2
T1/2
N0 = N(t) +
Oldest rocks, t = 3.7!109 y; meteorites, moon rocks, t = 4.5!109 y.
+ other methods based on ratios of isotopes.
Wednesday, April 7, 2010
55
A Geiger Counter
Ionizing radiation (high energy charged
particle, gamma ray) enters the counter:
• ionizes a gas atom
• freed electron is accelerated by the
electric field near the wire electrode,
collides with and ionizes another gas atom
• ionization of atoms continues, generating
an avalanche of charge
• pulse of charge produces a click in a
loudspeaker.
Geiger counter does not discriminate
between types of ionizing radiation or its
energy.
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56
A Scintillation Counter
Ionizing radiation excites molecules of the
scintillating material (eg CsI, NaI, plastic
scintillator).
• The amount of excitation is proportional to
the energy deposited by the radiation.
• Flashes of light are emitted when the
molecules return to their ground state.
• The light is detected via the photoelectrons
it liberates from a photocathode, one
photoelectron per photon.
• The current of photoelectrons is amplified by
accelerating the electrons and making them
strike surfaces from which they release more
electrons.
Wednesday, April 7, 2010
57
Nuclei
• Made up of nucleons (neutrons + protons), and held together by the
strong nuclear force, which is of short range.
• Radius r = r0A1/3 , r0 = 1.2!10-15 m. Constant density.
• Binding energy, B = energy to separate neutral atom into neutrons and
hydrogen atoms. B = Δm c2, Δm = mass defect.
• Unstable nuclei decay to objects of lower total mass, converting the
difference in mass to energy:
" &-decay:
" !-decay:
" $-decay:
Wednesday, April 7, 2010
A
A−4
4
Z X → Z−2 Y + 2He + energy
A
A
−
¯ + energy
Z X → Z+1 Y + e + !
A
A
+
Z X → Z−1 Y + e + ! + energy
A ∗
A
Z X → Z X + ! + energy
58
Radioactive Decay
• Half life, T1/2 = time for half of the radioactive nuclei to decay.
• Number of nuclei left after time t:
� �n
1
N(t) = N0
= N0e−!t
2
n = number of half-lives =
t
T1/2
0.693
! = decay constant =
T1/2
• Activity = rate of decay, A = –#N/#t
A = ! N(t)
" 1 Bq = 1 decay/second
Wednesday, April 7, 2010
59