Lecture 10: Nuclear Potentials and Radioactive Decay
I.
Nuclear Stability and Basic Decay Modes
A.
Schematic Representation:
Synthesis
X + Y + Energy

Equilibration
A Z*

Z
( ≲ 10−20s)
( ~ 10−  10
16
Decay
20
s
Composite Nucleus
(Activated Complex)
B.
Stable Nuclei
1.
N/Z composition: Does not change with time  peak of <BE> curve
Kinetic vs. Thermodynamic stability; detection limit ≲ 1020 y
2.
C.
Total: 266
At least one stable nucleus for all Z=183 EXCEPT 43Tc and 61Pm
Radioactive Nuclei
1.
Definition:
A nucleus that SPONTANEOUSLY alters its
neutron/proton composition or energy
state ⇒ FIRST-ORDER RATE PROCESS
RADIOACTIVE DECAY IS IDENTICAL WITH AN ELEMENTARY
UNIMOLECULAR DISSOCIATION IN CHEMISTRY. ( A  B + C)
Contrast with: nuclear reactions – n/p changes induced by collisions,
2nd order
NUCLEAR REACTIONS HAVE THE SAME FORM AS AN ELEMENTARY
BIMOLECULAR CHEMICAL REACTION (A + B  C + D)
2.
Half-life: t1/2
Definition:
The length of time required for one-half the nuclei in a
0.693
sample to disintegrate (decay): N = N0et ;  =
t1/2
3.
Primary Decay Modes
a. Alpha Decay:
b. Beta Decay:
0   ≡ e ;
-1
4 He emission
2
neutron ⇔ proton conversion
 specifies nuclear origin
SAME PARTICLE
e specifies atomic origin
c. Gamma Decay: 0  , photon emission
0
 = nuclear origin ; x-ray, uv, visible, ir = atomic/molecular origin
d. Exotic decay modes: fission, protons, neutrons, 14C, etc.
4.
Radioactivity in Nature (t1/2 ≳ 108y)
a. U–Th Decay series
EXTINCT:
238 U (4.5  109y) 8
92
6 
206 Pb
82
(24.1%) A = 4n + 2
235 U (7.1  108y) 7
92
4 
207 Pb
82
(22.1%)
232 Th (1.4  109y) 6
90
4 
208 Pb
82
(52.3%) A = 4n
237 Np (2  106y) 7
93
4 
209 Bi
(100%) A = 4n + 1
A = 4n + 3
where n is
an integer
TOTAL: 45 NUCLEI (t1/2 of all daughters < t1/2 of parents.)
b. Lighter Radionuclides in Nature
(1)
(2)
Survivors of Nucleosynthesis ; esp. 40 K , 87 Rb , 147 Sm
19
37
62
TOTAL = 15
Cosmic-Ray-Induced Activity
3
H(12y), 14C(5280y), 7Be(52d), 10Be(~106y), …
c. Natural radioactivities carry history of solar system and its evolution
5.
Synthetic Nuclei (t1/2 ≲ 108y)
Isotopes of all elements:
Z = 0  112 + [114 (??)
6.
Grand Total:≈ 3500 nuclei and still counting
Factors that Govern Decay Rate
1.
Energetics
2.
large Q ⇒ rapid decay (short half-life)
AB+C+Q
Quantum Structure
Spin and Parity: Changes in I between parent and daughter.
Slow down decay rate
e.g.
II.

3 s1/2
1 p3/2
I = 1/2+
5/2+
3/2
 even
even
odd
Alpha Decay
A.
Mechanism:
AX
Z

Alpha
B.

2 d5/2
4 He + A-4 Y + Q

2
Z-2
He2+
Recoil
Y2
 Atomic Ionization State
Energetics
1.
Spectra: Discrete energies
2.
Q = (x)   (y)  ()
3.
Energy systematics
Range of values Q  1.5-12 MeV measured
4.
228
Th Example
228 Th  224 Ra + 4 He + Q

90
88
2
Q = (228Th)  (224Ra)  ()
= 26.758  18.313  2.425 = 5.520 MeV
Measure:
E = 5.423 MeV
WHY?
5.
Disposition of Q
(1)
Kinetic energy of  + recoil : E + ER
(2)
Internal excitation energy ( heat) of recoil nucleus, E*
( has no stable excited states)
a.
Case I: Q  Kinetic Energy Only
  X  Y
X, Y 
all in lowest (ground) energy state
i.e., E* = 0 (T = 0)

Energy Conservation:

Linear Momentum Conservation
RESULT:
E
E

R
(E = 1/2 Mv2)


0 = p + p R  p = 2ME 
Q = E + ER


M
M
R
R
M
A
R Q
Q ≈


A  A
R

M
M
R

M
Q


≈
A
 Q
A  A
R
True for all
2-body
breakup
processes
SPECTRA MUST BE DISCRETE, since A, Q, M are all constants
Tag for nucleus ID
b.
Case II: Decay to Excited States
 
Ea
X 
ER + E*
Y*
E* ⇒ E i.e., system then undergoes -decay
 Energy Conservation:
Q = E + ER + E*
=
E + ER + E
Q ≅ E since M = 0
 Momentum Conservation
0 = p + pR + p
0 = p + pR
p  0 since ;  neglect
 RESULT:
E 
 A
A
R
R
A

 (Q  E )


; E 
R A
A
R

A

 (Q  E )


 NOTE: TOTAL ENERGY MUST BE THE SAME, REGARDLESS
OF PATHWAY
O
E
E
E
Q
C. Alpha Decay Probability
1.
Energetics: Q positive for all A>140 nuclei
2.
Range of Measured Half-Lives (~1044)
1016 y > t1/2 > 1021 s
3.
Why  ?
a. Proton & Neutron Emission: Qp,Qn are negative near valley of
beta stability (peak of peninsula); Thermodynamically forbidden
b.
Other Nuclei ; e.g. 12C, 16O …
 Q(12C), Q(16O) positive ;  possible
 Probability is low (i.e., t1/2) is long)  P(14C)/P ~ P10
232 Th 14 C 218 Po
(Exotic decay mode)
90
6
84
11.7 MeV  particles from 212m Po are the highest energy alphas from a
radioactive source
2.0 MeV alphas from Sm are the among lowest energy alphas from a
radioactive source
Most alpha particles from radioactive sources fall in the range of 4-8 MeV.
Associated with this narrow range in energy is the enormous range in halflife noted above.
4.
Coulomb Barrier Penetration
What is the relevant potential for the nucleus?
We can consider this by examining the approach of an alpha particle
from infinity (microscopic reversibility).
If the collision is head-on (dead center) there are two parts to consider
the attractive nuclear potential and the repulsive Coulomb potential.
V(r) = Vnuclear + VCoulomb
a. Coulomb Barrier  Vcoul
 escapes via quantummechanical tunneling
through the Coulomb
barrier
Vcoul
Q
V(r)o
Qp
b.
Electrostatic Repulsion Energy for Two Charged Spheres:
VCoul =
1.44 Z Z
Z eZ e
1 2 =
1 2
R
R
MeV  fm
,
where Z1 = 2
Z2 = Zrecoil
R= R1+ R2 = r0 (A11/3 + 41/3)
220 No ; let r = 1.4 fm
0
102
(1.44)(2)(100)
= 25.5 MeV  Typical of most heavy nuclei
Vcoul =
(1.4)(41 / 3  2161 / 3 )
c.
Example:
d. Q ≲ 10 MeV ;
∴ Vcoul > > Q and can't escape classically
e. Relative Barriers (approximate)
Vcoul(): Vcoul (12C): Vcoul(16O)  2:6:8 (since Z & R of recoil  constant)
 heavier fragments have much higher (and thicker) hill to punch through
5. Probability  P
a. Tunneling Probability: P  Pformation  e
Q  Vcoul 
; P 
1
t 1/2
b. P favored by:
(1) large Q and low Vcoul or small Q  Vc
 t1/2 ()  1/P()  e(Q Vc)
or log t1/2  Vc  Q (Figure above)
D. Applications/Environment
Am (458y)  smoke detectors
1.
241
2.
238
3.
222
Rn (3.8d)
4.
226
Ra (1620y)
Pu (88y)
 remote sensing devices; power sources
 natural radioactivity  health effects
 cancer therapy
5. (210Po) (138d)  power sources
6. Dating tags  U,Th, Sm
( )