7.3 Solutions
423
where c 1 is another constant.
k log 3.36 + log(1622 × 365) = c 1
(2)
k log 3.85 + log 138 = c 1
(3)
Solving (2) and (3), k = 61.46 and c 1 = 36.5. Using the values of k and c 1
in (1) 61.46 log 6.97 + log T = 36.5
Solving for T , we find T = 4.79 × 10
−16 days = 4.14 × 10
−11 s
7.105 For α-decay we use the equation
λ = 1/ τ = 10
21 exp(−2 π z Z/137 β)
( 1 )
T 1/2 (1)/T 1/2 (2) = τ 1 /τ 2 = exp(2 π z Z 1 /137 β 1 )/ exp(2 π z Z 2 /137 β 2 ) (2)
For 88 Ra
226 decay put Z 1 = 86 for the daughter nucleus, z = 2 for α-particle
and β 1 = (2E/Mc
2 )
1/2
= (2 × 4.9/3728)
1/2
= 0.05127
For 90 Th
226 decay, put Z 2 = 88 for daughter nucleus and z = 2 for α-particle
and β 2 = (2 × 6.5/3728)
1/2
= 0.05905
Using the values in (2), we find
τ 1 /τ 2 = 5.19 × 10
7
7.3.11 Beta-Decay
7.106 The selection rules for allowed transitions in β-decay are:
Δ I = 0
I i = 0 → I f = 0
allowed
Δ π = 0
⎫
⎬
⎭
Fermi Rule
Δ I = 0, ±1
I i = 0 → I i = 0 forbidden
Δ π = 0
⎫
⎬
⎭
G.T. Rule
where I is the nuclear spin and π is the parity. In view of the above selection
rules the first transition is Fermi transition, the second one Gamow–Teller
transition. The third one occurs between two mirror nuclei
17 F and
17 O in
which the proton number and neutron number are interchanged. The configuration of nucleus is very much similar in such nuclei, consequently the
wave functions are nearly identical. This leads to a large value for the overlap
integral. The log ft value for such transitions is small being in the range of
3–3.7. These are characterized by
ΔI = 0,±1 and Δπ = 0. The given transition
17 F →
17 O is an example of
superallowed transition.
7.107 If the number of
23 Mg nuclides is N 0 at t = 0, then at time t the number
decayed will be
N = N 0 [1 − exp(−λ t)]
(1)
At t = t 1 , N 1 = N 0 [1 − exp(−λ t)]
(2)
At t = t 2 , N 2 = N 0 [1 − exp(−λ t)]
(3)
423
where c 1 is another constant.
k log 3.36 + log(1622 × 365) = c 1
(2)
k log 3.85 + log 138 = c 1
(3)
Solving (2) and (3), k = 61.46 and c 1 = 36.5. Using the values of k and c 1
in (1) 61.46 log 6.97 + log T = 36.5
Solving for T , we find T = 4.79 × 10
−16 days = 4.14 × 10
−11 s
7.105 For α-decay we use the equation
λ = 1/ τ = 10
21 exp(−2 π z Z/137 β)
( 1 )
T 1/2 (1)/T 1/2 (2) = τ 1 /τ 2 = exp(2 π z Z 1 /137 β 1 )/ exp(2 π z Z 2 /137 β 2 ) (2)
For 88 Ra
226 decay put Z 1 = 86 for the daughter nucleus, z = 2 for α-particle
and β 1 = (2E/Mc
2 )
1/2
= (2 × 4.9/3728)
1/2
= 0.05127
For 90 Th
226 decay, put Z 2 = 88 for daughter nucleus and z = 2 for α-particle
and β 2 = (2 × 6.5/3728)
1/2
= 0.05905
Using the values in (2), we find
τ 1 /τ 2 = 5.19 × 10
7
7.3.11 Beta-Decay
7.106 The selection rules for allowed transitions in β-decay are:
Δ I = 0
I i = 0 → I f = 0
allowed
Δ π = 0
⎫
⎬
⎭
Fermi Rule
Δ I = 0, ±1
I i = 0 → I i = 0 forbidden
Δ π = 0
⎫
⎬
⎭
G.T. Rule
where I is the nuclear spin and π is the parity. In view of the above selection
rules the first transition is Fermi transition, the second one Gamow–Teller
transition. The third one occurs between two mirror nuclei
17 F and
17 O in
which the proton number and neutron number are interchanged. The configuration of nucleus is very much similar in such nuclei, consequently the
wave functions are nearly identical. This leads to a large value for the overlap
integral. The log ft value for such transitions is small being in the range of
3–3.7. These are characterized by
ΔI = 0,±1 and Δπ = 0. The given transition
17 F →
17 O is an example of
superallowed transition.
7.107 If the number of
23 Mg nuclides is N 0 at t = 0, then at time t the number
decayed will be
N = N 0 [1 − exp(−λ t)]
(1)
At t = t 1 , N 1 = N 0 [1 − exp(−λ t)]
(2)
At t = t 2 , N 2 = N 0 [1 − exp(−λ t)]
(3)