Problems 17.7 Estimate the density of a nucleus by assuming it to be a sphere of radius Ro=1.3 A1/3 fm. What would be the radius of a nucleus with the mass of Earth? 17.8 Pressure can be defined as the rate of change of energy with volume, p=E/av. Similarly, the surface ana- logue of pressure, surface tension, is defined as the rate of change of energy with surface area a, σ = 8E/aa at constant volume. Ignoring the effects of symmetry, Coulomb, and pairing energies, compute the surface tension of a nucleus as described by the empirical parameters of the SEMF. The surface tension acts to keep a nucleus spherical. 17.9 Find the SEMF prediction for Zmin (A = 200) from eq. (17.20). Find the actual value of Z that gives the greatest binding energy for A = 200. How well does the SEMF do? 17.10 [T] Derive eq. (17.20) by differentiating the SEMF expression for b(Z, A) with respect to Z at fixed A. 17.11 [T] Check the assertion in the text that the SEMF pre- dicts that nuclei have positive binding energy up to A 3150. 17.12 [T] The maximum value of A for which nuclei are sta- ble against instantaneous fission can be estimated as follows. When a charged, spherical nucleus, Z., with surface tension is deformed into an ellipsoid leaving its volume constant, it is possible to compute the change in its binding energy as a function of deformation. The result [77] is AB≈- 1 EC Z² 50 A1/3 82 (17.35) where & measures the deformation (the ratio of major to minor axes is (1 + 8)3/2). Note the signs: the sur- face contribution tends to stabilize and the Coulomb contribution tends to destabilize the nucleus. Show that eq. (17.35) predicts that instantaneous fission occurs when 22/A49. The largest A nucleus observed so far, 294 [118], has Z²/A = 47.4. 17.13 Explain how to read from Figure 17.8 the Q-value for a-decay predicted by the SEMF. 17.14 Show that the longest-lived isotope of radium, 22a, is energetically allowed to decay by emission of a 1C nucleus. This decay has been observed with a probability of 3.2 × 10-9. 17.15 235U can decay by emitting a neon nucleus. How many such decays occur per second per mole of 235U? 17.16 Radon gas 22Rn is a serious environmental hazard (see $20). It is a decay product of 238 U, which is a relatively common constituent of rock. 222 Rn under- goes a-decay to 218Po. When 222Rn is created in the foundation of a house, it diffuses out into the base- ment airspace. If its lifetime were very short, it would decay before getting out of the rock. If it were very long, it would diffuse away into the atmosphere. Its half-life is 3.8 days, however, just right for caus- ing a great deal of human radiation exposure. Given this half-life, use Gamow's formula (17.30) to esti- mate the Q-value for this decay and compare with the measured value. Comment on the accuracy of your estimate. 17.17 [T] Examine the derivation of Gamow's formula for a- decay lifetimes, and then derive the equivalent expres- sion for decay by emission of a (small) nucleus of charge z and mass number a. Assume a/A is small enough to ignore recoil. Make sure that the z and a dependences are explicit. Compare your result to an a-decay lifetime with the same Q-value. 17.18 Check that 'He is less tightly bound than 'H. Explain why H decays to He, and not vice versa. 17.19 [T] Verify the expressions for the Q-value in B-decay and electron capture, eq. (17.34). 17.20 Ignoring small electron binding energies and the very small mass of the neutrino, show that the mass of a nucleus increases when it decays by electron cap- ture if the Q-value of the decay is less than m,c² 0.511 MeV. Verify that this is the case for the elec- tron capture decay of the longest-lived isotope of technetium: Te+e Mo+ve. 17.21 97% of naturally occurring calcium is calcium-40, 20Ca. This may seem surprising, since if we use the semi-empirical mass formula to estimate the most sta- ble nuclide with A= 40 we find Z 18. This suggests that Ca might be unstable to electron capture, which would increase its N/Z ratio. Show that Ca cannot electron capture to K. Show, however, that it is possi- ble for Ca to capture two electrons at the same time, making Ar. Look up the information on the lifetime of Ca and comment on its value. Do we have to worry about exposure to radiation from decay of the Ca in our bones? of 18 17.22 [H] When one of the naturally occurring radioac- tive heavy elements like 238U decays, a decay chain follows, leading to the creation of various unstable, radioactive descendant nuclides. Consider a sample known to be pure 238U at time zero. Suppose that none of its decay products migrate away from the original sample. Use the fact that the lifetime of the original nuclide, in this case 238 U, is much longer than the lifetimes of any of its descendants to show that a quasi- equilibrium develops in which the amounts of 238U and all of its descendant are steadily decreasing at a rate determined by the lifetime of 238U. Show that the ratio of the abundance of each descendant nuclide to the abundance of 238U is very nearly constant in time, and 321