1000 Solved Problems in Modern Physics

7.3 Solutions 423

wherec 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 andc 1 = 36 .5. Using the values ofkandc 1

in (1) 61.46 log 6. 97 +logT= 36. 5

Solving forT, we findT= 4. 79 × 10 −^16 days= 4. 14 × 10 −^11 s

7.105 Forα-decay we use the equation

λ= 1 /τ= 1021 exp(− 2 πzZ/ 137 β)(1)

T 1 / 2 (1)/T 1 / 2 (2)=τ 1 /τ 2 =exp(2πzZ 1 / 137 β 1 )/exp(2πzZ 2 / 137 β 2 )(2)

For 88 Ra^226 decay putZ 1 =86 for the daughter nucleus,z=2forα-particle

andβ 1 =(2E/Mc^2 )^1 /^2 =(2× 4. 9 /3728)^1 /^2 = 0. 05127

For 90 Th^226 decay, putZ 2 =88 for daughter nucleus andz=2forα-particle

andβ 2 =(2× 6. 5 /3728)^1 /^2 = 0. 05905

Using the values in (2), we find

τ 1 /τ 2 = 5. 19 × 107

7.3.11 Beta-Decay ....................................

7.106 The selection rules for allowed transitions inβ-decay are:

ΔI= 0

Ii= 0 →If=0 allowed

Δπ= 0

⎫

⎬

⎭

Fermi Rule

ΔI= 0 ,± 1

Ii= 0 →Ii=0 forbidden

Δπ= 0

⎫

⎬

⎭

G.T.Rule

whereIis the nuclear spin andπis 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 Oin

which the proton number and neutron number are interchanged. The con-

figuration 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 andΔπ=0. The given transition^17 F→^17 O is an example of

superallowed transition.

7.107 If the number of^23 Mg nuclides isN 0 att=0, then at time t the number
decayed will be
N=N 0 [1−exp(−λt)] (1)
Att=t 1 ,N 1 =N 0 [1−exp(−λt)] (2)
Att=t 2 ,N 2 =N 0 [1−exp(−λt)] (3)