Consider Ν atoms of a radioactive substance. The probability that any arbitrarily selected set of exactly m nuclei will decay during an interval t is
(1 _ e-Xf)mie-\y-m = pmqN-m^ (22-29)
where pm is the probability that the m nuclei will decay and qN-m is the probability that the other Ν — m nuclei will not. The probability given by (22-29) must be multiplied by the number of ways of picking m nuclei out of Ν total ones. This is Nlj(N — m)\. However, since the order of picking the m nuclei is immaterial, we must divide by the ways of permuting m objects, ml. The final equation for the probability W(m) is
W(m) = ΊΊΓτ T t Γ pmqN-m (22-30)
(N — m)l ml ^ v 7
We can relate this result to the standard deviation of a measurement of m. In general, the standard deviation σ of a series of η measurements of a quantity χ is given by
°2 = l t ( x i - *)2· (22-31)
n i=l
Expansion of the sums leads to the alternative form
Σ2 = χ2. (22-32)
That is, σ2 is given by the difference between the average value of x2 and the square of the average value of x.
The quantities m and m2 may be evaluated from Eq. (22-30). It turns out that m= N(l - e~x% which reduces to Eq. (22-19) if t is small. Also, (N - 1) Np2 = m2 — m, and insertion of these results into E q . (22-32) gives σ2 = mq. If m is small compared to N, then q is essentially unity, and σ = m1 / 2. Thus the standard deviation of a measurement of the disintegrations occurring in time t is just the square root of the average value.
If a particular sample o f radioactive material registers 1000 disintegrations in 10 min, w e assume that the figure o f 1000 is close t o the true m, and estimate a t o be ( 1 0 0 0 )1/2 or 31.6. The activity is then reported as 100 ± 3.2 dis min"1, or ± 3.2 %. Were 10,000 disintegrations observed over 100 min, σ would be 100 and w e would n o w report 100 ± 1 dis m i n -1, or an uncertainty of 1 %. Thus the more total disintegrations or emitted particles counted, the smaller is the per
centage o f error in the measurement.
G E N E R A L R E F E R E N C E S
F R I E D L A N D E R , G., K E N N E D Y , J . W . , A N D M I L L E R , J . M . ( 1 9 6 4 ) . " N u c l e a r and R a d i o c h e m i s t r y , "
2nd ed. Wiley, N e w York.
SIEGBAHN, K . , E d . ( 1 9 6 5 ) . "Alpha-, Beta-, and G a m m a - R a y Spectroscopy." North-Holland Publ., Amsterdam.
C I T E D R E F E R E N C E S
C H E W , G . F., G E L L - M A N N , M., A N D ROSENFELD, A . H . (1964). Sci. Amer. (February), p. 74.
HOLCOMB, R. (1970). Science 1 6 8 , 853.
E X E R C I S E S
Take a s exact numbers given t o o n e significant figure.
2 2 - 1 What is the field Η required t o cause electrons t o follow a path of radius 2 c m if the compensating electric field to prevent such curvature is ΙΟ4 V c m- 1?
Ans. 170 G . 2 2 - 2 Calculate Ε in volts per centimeter required t o maintain from failing a water droplet
which, in the absence of the field, falls at the rate of 1 0- 4 c m s e c- 1. Assume the drop carries three units o f electronic charge and the viscosity o f air to be 1.85 χ 1 0 ~4P .
Ans. 0.67 V c m "1. 2 2 - 3 Calculate the specific activity o f 2 3 8U in disintegrations per second per gram.
Ans. 1.23 x 1 04d i s s e c "1g -1. 2 2 - 4 A sample o f 3 8C1 (as N a C l ) shows 1 Ci of activity initially. What should be the activity in
disintegrations per second 3 hr later?
Ans. 1.3 x l O ' d i s s e c- 1. 2 2 - 5 Calculate the high-voltage frequency for the cyclotron acceleration o f deuteron in a field of 10,000 G and the m a x i m u m radius achieved by the spiraling deuterons if the final energy is to be 10 M e V (neglect relativistic effects).
Ans. 7 . 6 M H z , 6 5 c m . 2 2 - 6 What is the relativistic increase in mass o f a 10-MeV deuteron, o f a 10-MeV electron?
What is the velocity o f a 1-MeV electron?
Ans. 0.533%, 19.6-fold, 0 . 9 4 c.
2 2 - 7 Explain h o w it is possible that 3 6C 1 m a y decay by either β~ or β + emission. H o w should US decay?
2 2 - 8 Calculate Q for E q . (22-7).
Ans. 3.27 MeV.
2 2 - 9 Calculate Q for E q . ( 2 2 - 6 ) .
Ans. 3.8 MeV.
2 2 - 1 0 Calculate the binding energy o f the last neutron added t o (a) 1 βΟ , (b) 1 70 . C o m m e n t on the difference between the t w o values.
Ans. (a) 15.7 M e V , (b) 4 . 1 4 M e V .
2 2 - 1 1 Calculate the C o u l o m b barrier in the bombardment o f 2 0 9B i with a particles.
Ans. 19.9 M e V . 2 2 - 1 2 T h e cross section for the reaction 5 9C o ( n , y )6 0C o is 20 b. Calculate the number of e oC o
PROBLEMS 951 atoms formed if 2 g of metal foil is exposed to a neutron flux of 1.5 χ 1 01 3 neutrons c m- 2 s e c- 1 for 10 min, and the resulting radioactivity in millicuries.
Ans. 0.415 mCi.
22-13 Calculate the dosage in roentgens per hour 1 m away from a 1-Ci source of 1-MeV γ radiation. [Note: Estimate the absorption coefficient per centimeter of air from Fig.
22-9 t o determine the fraction o f the γ radiation absorbed per centimeter and hence the energy dissipated; this gives the number of ions produced. The number of ions per cubic centimeter 1 m away can then be calculated, and thence the dosage.]
Ans. 1.30 R h r "1. 22-14 3 1S i is formed at the rate of 2 χ 108 atoms per second by the neutron irradiation of silica. H o w many millicuries will be present (a) immediately after a 5-hr irradiation, and (b) 10 hr after the irradiation is over? What is the saturation activity?
Ans. (a) 3.96 m C i ; (b) 0.281 m C i ; 5.41 m C i . 22-15 The total activity is followed as a function of time for a sample suspected of consisting of two or more independently decaying radioisotopes. Find the half-life of each c o m ponent and the number of disintegrations per minute of each component present at zero time.
/ ( h r ) 0 0.5 1.0 1.5 2.0 2.5
D i s m i n- 1 7300 4500 2900 1950 1350 990
' ( h r ) 3.0 3.5 4.0 5.0 6.0 7.0
D i s m i n- 1 740 580 480 370 310 280
' ( h r ) 8 10 12 14
D i s m i n- 1 255 210 180 155
Ans. (a) t1/2 = 0.7 hr, D° = 6800;
(b) tm = 8.2 hr, D° = 4 7 0 . 22-16 1 4 eC e decays by beta emission to 1 4 eP r , with txi% = 14 min; the 1 4 eP r decays in turn to
stable 1 4 eN d with = 25 min. A sample consists initially of 0.5 mCi of pure 1 4 eC e . Calculate the activity of 1 4 eC e and of 1 4 eP r present 30 min later. Is this a case of secular equilibrium?
Ans. £>! = 0.113 m C i , D2 = 0.385 m C i .
PROBLEMS
22-1 The specific activity of a sample of pure " T c is 105 dis s e c- 1 m g- 1. Calculate the half-life of " T c .
22-2 Calculate the isotopic m a s s of 2soHg from Eq. (22-33) a n d also the Q for adding o n e neutron to obtain 2JJHg.
22-3 What is the average binding energy of a proton in 2 8N a ? 22-4 Calculate the Q for the reaction flBe(«, n )1 2C .
22-5 Derive the equation for the total binding energy of an isotope in terms of its mass number for isotopes of optimum charge. U s e the equation
Ε (MeV) = 11.6A - ™(A - 2 Z )2 - 0 . 0 6 Z2 (22-33)
' ( h r ) 0.0 0.5 1.0 1.5 2.0 3.0 4.0 5.0 6.0 8.0 D i s m i n- 1 8000 1875 850 543 410 288 211 168 139 108 / ( h r ) 10 12 14
D i s m i n- 1 90 80 73
Analyze the data graphically to obtain the half-lives of all the radioelements present, and the percentage of the initial total activity due to each.
22-16 A sample of radioactive zirconium is obtained in pure form by an appropriate processing of a fission product mixture. The activity of the sample increased with time, as shown by the tabulated activity (total disintegrations per minute), due to the growth of a niobium daughter. Analyze the data to determine the half-lives of the parent and the daughter.
/ ( d a y s ) 0 20 40 60 100 150 200 300 4 0 0 500
Z )t o t (dis m i n- 1) 2000 2250 2200 2050 1600 1050 666 240 82 28
to find Ζ A = f(A) such that (dE/dZ)A = 0. By means of this result make a plot of the mass defect against A; also compare your result with the plot for stable isotopes as actually found (compare with Fig. 22-5).
22-6 By m e a n s of the equations developed in Problem 22-5, calculate the energy of a emissions for
z ^ z t l l + t H e
when A = 200. Estimate the average number of betas per alpha for a decay series around A = 200.
22-7 When a magnetic field of 5000 G is imposed in a cloud chamber, the electrons and positrons formed by pair production from incident γ radiation are found to follow a path of 2 c m radius of curvature. What is the energy in M e V of the γ ray?
22-8 H o w many centimeters of lead absorber are needed to reduce the intensity of a source of 5-MeV γ radiation 100-fold (absorption coefficient, μ = 0.30 c m- 1) ?
22-9 What is the dosage in roentgens per hour 1 m away from a 100-Ci point source of 1-MeV gamma rays of absorption coefficient μ = 7 χ 1 0- 5 c m- 1 in air? Assume that only singly charged ions are formed and that 30 eV are required to produce each ion pair.
22-10 The high voltage across the dees of a cyclotron has a frequency of 12 M H z . What must the magnetic field be if deuterons are to be accelerated ?
22-11 What is the most energetic decay process for 1 0C , and h o w much energy is liberated?
22-12 A radioelement of half-life Tx yields a daughter of half-life T2. Assuming that at zero time n o daughter is present, derive the expression for the time of maximum combined activity (that is, disintegrations per unit time) of parent plus daughter.
22-13 What is the cross section for the formation of 1 4C by (n, p) reaction if, o n irradiation of 100 liter of 0.1 Μ a m m o n i u m nitrate for 1 wk at an average pile power of 3000 kW, 0.1 mCi of 1 4C is obtained? (Assume 105 neutrons c m- 2 s e c- 1 W- 1. )
22-14 If the cross section for the reaction 3 1P ( n , y )3 2P is 0.32 b, what will be the saturation activity for 1.0 g o f Ρ irradiated in a neutron flux o f 1.2 χ 1 01 2 neutrons c m "2 sec" *?
22-15 The following decay data are obtained o n a sample suspected of containing several radioelements:
SPECIAL TOPICS PROBLEMS 953 22-17 A sample of 1 4 0B a is allowed to c o m e to transient equilibrium with its daughter 1 4 0L a :
i4o
B a_^ i4o
L a + β- half-life 12.5 days,i4o
L ai4o
B a (assume t o be stable) + β- half-life 4 0 hr.A chemical separation is then made, giving two fractions containing the following weight percentages of the total Ba and La: (a) 9 0 % of the barium, 1 0 % of the lanthanum;
(b) 1 0 % of the barium, 9 0 % of the lanthanum. A t the time of the chemical separation the sample contained 3 χ 10β dis m i n- 1 due to the combined Ba and La activities. Calculate the total number of disintegrations per minute for each fraction 2 days and 25 days after the separation.
SPECIAL TOPICS P R O B L E M S
22-1 A meteorite contains 1.5 χ 1 0- 3 % U . Calculate its minimum age in years if 0.066 c m3 of H e (STP) can be extracted from 10 g of the meteorite.
22-2 A counter has a background of about 35 counts m i n- 1. H o w long should a sample of approximately 1000 counts m i n- 1 be counted and h o w long a background count should be taken in order that the net count can be determined in a minimum total counting time with a probable standard deviation of less than 1 % ?
22-3 The age of some pitchblende U308 is estimated by weighing out a small sample of the crushed mineral and determining the alpha activity. The measurement is made o n e week after crushing and gives 21 dis s e c- 1 m g_ 1. It is t o be assumed that n o gases or other materials escape from the mineral during its geological existence, but that o n crushing, the radon and helium present escape freely, but nothing else. Neglecting any complications due to 2 3 5U or 2 3 4U present, calculate the age of the mineral. [Note: The ore is assumed to have been pure U308 originally, although in the course of time its composition m a y well have changed appreciably.]
22-4 The standard deviation of a counts per minute determination o n a sample is 1 0 % . Neglec
ting background, what was the activity in counts per minute after the sample had been counted for 5 m i n ?
22-5 A counter has a measured background rate of 600 counts in 25 min. With a sample in place the total measured rate is 1050 counts in 20 min. Give the net counting rate per minute for the sample and its standard deviation.
22-6 Estimate the age of a rock which is found to contain 4 χ 1 0- 5 c m3 of helium at STP and 3.5 χ 1 0- 7 g of uranium per gram.