bei48482_FM

470 Appendix to Chapter 12

From Eq. (12.35),

R

and so

ln T  

1  2

Z^1 ^2 R 01 ^2  

1  2

ZE^1 ^2 (12.38)

The result of evaluating the various constants in Eq. (12.38) is

lnT2.97Z^1 ^2 R 01 ^2 3.95ZE^1 ^2 (12.39)

where E(the alpha-particle kinetic energy) is expressed in MeV, R 0 (the nuclear radius)

is expressed in fermis (1 fm 10 ^15 m), and Zis the atomic number of the nucleus

minus the alpha particle. Since

log 10 A(log 10 e)(lnA)0.4343 lnA

we have

log 10 T1.29Z^1 ^2 R 01 ^2 1.72ZE^1 ^2 (12.40)

From Eqs. (12.12) and (12.13) the decay constant is given by

T T

where is the alpha-particle velocity. Taking the logarithm of both sides and substi-

tuting for the transmission probability Tgives

log 10 log (^10) 1.29Z^1 ^2 R 01 ^2 1.72ZE^1 ^2 (12.14)
This is the formula quoted at the end of Sec. 12.4 and plotted in Fig. 12.9.


2 R 0
Alpha decay
constant


2 R 0
m

2
e^2

 (^0)
m

(^0)
4 e


2 Ze^2

4

0 E
12.2 Half-Life

1. Tritium (^31 H) has a half-life of 12.5 y against beta decay. What
   fraction of a sample of tritium will remain undecayed after
   25 y?


2. The most probable energy of a thermal neutron is 0.025 eV at
   room temperature. In what distance will half of a beam of
   0.025-eV neutrons have decayed? The half-life of the neutron is
   10.3 min.


3. Find the probability that a particular nucleus of^38 Cl will
   undergo beta decay in any 1.00-s period. The half-life of^38 Cl is
   37.2 min.
   4. The activity of a certain radionuclide decreases to 15 percent of
   its original value in 10 d. Find its half-life.
   5. The half-life of^24 Na is 15.0 h. How long does it take for
   80 percent of a sample of this nuclide to decay?
   6. The radionuclide^24 Na beta-decays with a half-life of 15.0 h.
   A solution that contains 0.0500 Ci of^24 Na is injected into a
   person’s bloodstream. After 4.50 h the activity of a sample of
   the person’s blood is found to be 8.00 pCicm^3. How many
   liters of blood does the person’s body contain?
   7. One g of^226 Ra has an activity of nearly 1 Ci. Determine the
   half-life of^226 Ra.

EXERCISES

What we have to learn to do, we learn by doing. —Aristotle

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