Post on 11-May-2018
Nuclear Decay kinetics : Transient and Secular Equilibrium
What can we say about the plot to the right?
IV. Parent-Daughter Relationships
Key point: The Rate-Determining Step
A. Case of Radioactive Daughter
A (Parent)
B
(Daughter)
t1/2(A)
C
(Grand-
daughter)
t1/2(B)
210Bi
210Po
5.01d -
206Pb
(stable)
138.4d
Same problem as a stepwise chemical reaction!
B. Mathematics of the Problem
1. Parent: A
Decay: t
AAAAA AeNtNN
dt
dN 0)( Nothing
new here
2. Daughter: B
Formation: dNB/dt = ANA
Decay: dNB/dt = BNB
NET: BB
t
AABBAAB NeNNN
dt
dNA
0
This is a linear first-order differential equation
3. Solution
t
B
tt
AB
AAB
BBA eNeeN
N
0
0
If the sample is pure A initially, second term vanishes since 00 BN
4. Daughter: C
CBAA NNNN 0 Conservation of atoms
5. Special Cases: Long time solutions
classification time relationship Rate-determining step
a. No Equilibrium t > tB > tA B C
b. Transient Equilibrium t > tA > tB A B
c. Secular Equilibrium tA >> t > tB A B
(c. is case of U-Th decay series)
C. No Equilibrium
Daughter is rate-determining step
t1/2 (B) > t1/2 (A)
a. Curve a: Total activity of initially PURE
sample of A
NOTE: resembles two-component independent
decay curve. NOT !!!
A(total) = A(A) + A(B)
b. Curve b: Parent activity, A(A)
c. Curve d: Daughter activity, A(B) ; note
growth curve
1. Figure 3-4 (on right)
2. Curve c: Long-term behavior: t > > t1/2 (A)
Therefore, t/t1/2(A) =
0 tAe
t
AB
AAB
BeN
N
0
Note: B < A ; ()() = +
3. Long term curve:All of A has disappeared ; this defines t1/2(B)
D. Transient Equilibrium
Parent decay is rate-determining step
t1/2 (A) > t1/2 (B)
1. Fig. 3-2 (on right)
a. Curve a: Total activity of initially pure
sample of A
A decay curve that initially increases with
time is a signature of transient or secular
equilibrium.
A(total) = A(A) + A(B)
b. Curve b: Parent Activity: A(A)
c. Curve d: Daughter Activity: A(B)
d. Initial growth of A(total) and then decay
according to the half-life of the parent.
2. Maximum Daughter Activity
a. Maximum occurs when dNB/dt = 0
Therefore, differentiate equation for NB and set this equal to zero; this
defines tmax as
NB is maximum when t = tmax
AB
ABt
)/ln(max
b. Importance
Medical isotopes – Milking a cow – how long must one wait before
extracting the daughter activity again?
c. Example: 55137
Cs 1
50137m
Ba
m30 y
2 6.137
Ba (stable)
min6.2/693.0
]min/103.5min)6.2/30ln[(
30
693.0
6.2
693.0
)](/)(ln[)/ln( 5
2/12/1max
yy
ym
BtAtt
AB
AB
tmax = 58 min
3. Long-Time Solution: Curve e
a. For initially pure A, t > > t1/2(B)
0 tBe
t
AB
AAB
AeN
N
0
AB
AAB
NN
AB
A
A
B
N
N
i.e., at long time, ratio NB/NA is CONSTANT with time SYSTEM APPEARS TO BE IN EQUILIBRIUM
4. Consequences
a. Long term decay is governed by parent
b. Activity: multiply equation in 3a. above by cB
cBNB = = AB =
B
A B
AA
cBANA
B - A
c. Half-life of B can be determined by
combining:
long term behavior – t1/2(A)
activity ratio above
E. Secular Equilibrium (Fig. 3-3)
t1/2(A) > > t > > t1/2 (B)
Special Case of Transient Equilibrium; e.g., U-Th Decay Series
Rn gas problem; natural background
from U and Th decay products; dating
1.
a. Curve a – Total activity of
initially PURE Sample
NOTE: at long time, ACTIVITY IS
CONSTANT ; signifies special
case
b. Curve b – Parent activity ; since
t/t1/2 ~ 0 , N/t = N0 = constant
c. Curve d – Daughter activity growing in
2. General Solution
a. Assume NB0 = 0 (i.e., pure A) ; t1/2(A) > > t
10
eetA
BABAB
t
B
AAB
BeN
N
1
0
Growth curve
b. Long-time solution
t >> t1/2(B) ; eBt e = 0
00
AABB
B
AAB NN
NN
c. If cA = cB AB = AA = Atotal/2
d. Example:
How many atoms of 222Rn are present in an initially pure sample
of 226Ra after 3 months? Assume 226 mg of 226Ra;
What is the activity?
t1/2(222Rn) = 3.82 d ; t1/2(
226Ra) = 1620 y
NRn =
Ra
Rn 1/2
N (Ra)Rn)
t Ra)N (Ra) =
(3.82 d)
1620 y (365 d / y)
226 10-3
g
226 g / mole
01 2
0t / (
(
NRn = 3.94 1015
atoms = 6.54 109
moles = 1.46 104
mL @ STP
dNdt
N =3.82 d
3.94 10 atoms
1440 m/ dRn
15
0693.
≅ 5.0 1011
d/min
Solution :
3. Points to keep in mind
a.
b.
c. Half-life of A
.0 constAA AA 0
AtotalB AAA
Weigh and determine from AA = cNA0
weighMeasure
counts
d. Half-life of B
at t = t1/2(B) ; e Bt= 1/2
NN
= ANA0
2 or AB AB =
A A
0
B BA0
( / ) ,1 1 2
1
2
corresponds to t1/2 (B)
Since A(total) = A AA0
B , t1/2 (B) is also time at which
A(total) = (3/2) AA0
IV. Branching Decay
Competitive Decay Modes
for the same nucleus
EC
2NO+O2
N2O4
A. Total Probability = = 1 + 2 + 3 + …
or 1
t 1
t 1
t 1
t1/2 1/2(1) 1/2(2) 1/2(3)
... ; ti = partial half-lives
B. Partial Half-Life
Definition: The half-life a nucleus would have if the competing decay modes were switched off. (but NO switch).
C. Determination of Partial half-lives
1. Branching Ratio: BR (Assume two branches, 1 & 2)
BR =
1 1 1 2
1 2 1total total
t (total)
t /
/ ( )
2. Measurement; if c1 = c2
BR =
1 1 1
c N
c N
A
Atotal total
3. Example: 40
K ; t1/2 = 1.28 109 y
BR(EC) = (0.107) = t
tEC
1 2/ ; tEC = 1.19 1010
y
BR() = 0.893) =
t
t1 2/
; t
= 1.43 10
9 y
NOTE: t1/2 < t1/2 () < t1/2 (EC)
V. Determination of Half-lives
Measurable range: 1023
s to ~ 1030
y (60 orders of magnitude)
A. T1/2 ≳ 1 year: Specific Activity
1. A = cN A = counts/unit time 0.693
N = weight of sample Measure =
c = detection coefficient
e.g., 238 mg of 238
U has a specific activity of ~300 dps c = 1
2. Limit: Natural background radiation; when A(sample – A(bkg) 0,
large errors
B. Decay Curves: 10 y ≳ t1/2 ≳ 1 s
C. Electronic Techniques
D. Doppler Shift
E. Channeling in Crystals
F. Heisenberg Uncertainty Principle
G. Angular Distributions of Emitted Particles
H. Small Angle Correlations
t1/2
1
0.5 t1/2
t
1n A