Unstable Nuclei Radioactivity

1
Unstable Nuclei Radioactivity
Radioactivity The spontaneous emission of
radiation and/or particles from a nucleus a
pure quantum physics phenomenon.

• ? rays
• photon emitted
• A A
• Z Z

• ? particle
• He2 emitted
• A A4
• Z Z-2

Z,A
Z,A
a
?
Z,A
Z-2,A-4
2
The Decay Law

• The Transition Probability or Decay Constant, ?,
  depends on
• the specific interaction process and (b) nuclear
  wave functions
• for the initial and final state. No macroscopic
  effects like Pressure,
• Temperature or electrodynamic fields can change ?.

A single decay is a completely independent
quantum process, i.e. can not be predicted. Only
an ensemble of decays displays the statistical
behaviour determined by the constant decay
probability. (Analogy single photon in front of
double-slit experiment many photons exhibit
interference pattern hit-location on screen for
single photon is unpredictable)
3
The Decay Law
Radioactive Decay rate of depletion of an
original quantity of nuclei is (a) constant and
(b) depends on the present amount.
reduction of present quantity
Decay Constant
Integration gives the Decay Law
Why (b)? Independent alternatives with
probability p for a statistical process add
up, i.e. each single nucleus has ? to decay in
dt, so for N possibilities its ?N ?N ?.
4
Decay Characteristics
N(t)/N0
t1/2
t1/?
e-1
t
ln2
Half life of a decay t1/2ln2/? Mean lifetime
t1/? Activity A?N
Activity Unit 1Bq 1 s-1 Older Unit 1 Ci 37
GBq
5
Decay Chains
For multiple decays, A ? B ? C ? etc., each with
different decay constant, get a set of
differential equations
where i(A,B,C,...) labels the nucleus in the
chain q is the growth rate of i by the
mother, i-1, when the daughter nucleus, i,
decays.
The nucleus Ni does not need to originate from
another decaying nucleus but might be produced
artificially, for example in a reactor.
6
Decay Chains
The general solution of the differential equation
is
using
Example radioactive daughter, N2, of a
radioactive mother, N1
Put
Get
7
Natural Decay Chains
8
Decay Processes a-Decay
AZXN ? A-4Z-2YN-2 42He2
Strong interaction process Form a Helium
nucleus inside a heavy nucleus Tunnelling through
barrier Gamow Factor ? explains Geiger-Nuttall
rule
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Decay Processes a-Decay
Energetics mXc2 mYc2 TY mac2 Ta Q
TY Ta (mX mY ma)c2
kinetic Energies TY ,Ta use momentum
conservation pYpa
Get
10
Decay Processes a-Decay
Estimate for Q-value using SEMF KR page 250
Q M(A,Z) M(A-4,Z-2) M(4,2)c2
-B(Z,A) B(Z-2,A-4) B(4,2)
Simplify using 4/A ? 1 and 2/Z ? 1 and ignoring
pairing term
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a-Decay Mechanism
Tunnel through the Coulomb Barrier Gamow Factor
Barrier Height above a-particle energy Q
12
a-Decay Mechanism
Geiger-Nuttall rule is really just a rule
Gamow derives ln t1/2 ? -1/?Q as the
general trend. Exact computations are far
beyond syllabus here.
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Decay Processes ß-Decay
AZXN ? AZ1YN-1 e- ?e AZXN ? AZ-1YN1 e
?e AZXN e- ? AZ-1YN1 ?e
e-
Weak interaction process
?e
W-
u
d
neutron
proton
u
u
d
d
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Decay Processes ß-Decay
Energetics
Q M(A,Z) M(A,Z1)c2, for ß- Q M(A,Z)
M(A,Z-1) 2mec2, for ß neglecting electron
binding energy
Tricky bit for Q Three-body Decay for ß?
Q EY Ee E?
where E total energy (rest mass kinetic) for
Y final nucleus e electron(positron),
?(anti)neutrino
Result for maximum kin. energy of, e.g. the
recoiling final nucleus Y
m nuclear masses evaluated in rest frame of
nucleus X
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Decay Processes ß-Decay
Numeric example Free neutron decay
mnc2939.573 MeV mpc2938.28 MeV mec20.511 MeV
Emaxp - mpc2?0.75 keV
Tiny proton recoil energy !
Combined with small upper limit for the neutrino
mass (lt2.3eV/c2)
Electron-neutrino system practically shares
the total Energy Q available.
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ß-Decay Energy Spectrum
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Mini-Introduction to theoretical Nuclear Physics
Fermi Theory of ß-Decay

• Treat the interaction causing the Decay as weak
  perturbation
• can use Fermis Golden Rule (originating from
  Dirac, though).
• (b) Determine the density of final states.
• (c) Determine the Transition Matrix element.

Comments (a) is quite accurate for ß-Decays of
any sort good starting point. For (b) it
turns out experimentally that many Energy Spectra
can be explained with a constant Matrix element,
hence (b) determines the shape of many ß-Spectra
quite accurately. Point (c) is well beyond
syllabus.
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Fermis Theory of ß-Decay
Assuming point (a)
N(p)dp number of electrons in momentum interval
p,pdp per unit time. dn/dEf density of final
states per energy interval Vfi Transition Matrix
element
Assume Vfi constant and work out dn/dEf, gives
19
Selection Rules
Physics in the Transition Matrix Element
Forbidden and Allowed transitions
Assumption made Zero-order approximation for
particle wavefunctions (as
constants) Hence, no orbital angular momentum
for particles ? Particles carry only Spin angular
momentum !
Fermi-Decay
Gamow-Teller-Decay
Electron-Neutrino Pair in Singlet
Electron-Neutrino Pair in Triplet
?I If Ii 0,1 (no 0 ? 0)
?I If Ii 0
-

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Selection Rules
The Parity Quantum Number
Symmetries and conserved quantities C Charge
conjugation q ? -q P Parity or reflection r ?
-r T Time reversal t ? -t
A Physical System invariant to the above discreet
transformations C, P and T has good quantum
numbers, i.e. conserved in interactions.
? (-1)l
Here (for nuclear beta decays)
Additional selection rule
?? 0, no parity change
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Another Quantum Number
The last one for ß-decay Isospin
Assumption Complete Charge-independence of
Strong Force (the symmetry)
Consequence No difference between Neutron and
Proton
Assign a new Quantum Number to the 2
possibilities for each Nucleon Isospin T½ Then
a Proton gets projection value Tz1/2 A Neutron
gets Tz1/2
A Nucleus then has Tz1/2 (Z-N)
Nucleus with TTzMax can form (2T1) states ?
Isospin Multiplet
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Notation Quantum States of Nuclei
T1
0
2mec2
T0
Notation SpinParity or Ip
1
99.4
T1
ß
0
2.312
?
0.6
0
T1
0.156
1
ß-
T0
14C Tz-1
14N Tz0
14O Tz1
Note 99.4 for Fermi transition only 0.6 for
G-T !
Isospin Triplet T1 and Singlet T0 for A14