Appendixes Appendix I Physical Constants and Conversion Factors

Numerical values for the constants are from B. N. Taylor, W. H. Parker, and D. N. Langenberg, Revs. Mod. Phys., 41, 375 (1969). Units are quoted in mksa (Systeme International) and in cgs.

Physical Quantity Symbol Value Units (51) Units (cgs)

I. Speed of light c 2.9979250 108 meters sec-I 10 10 cm sec-I 2. Electron charge e 1.6021918 10-19 C 10-20 emu 4.803250 10-10 esu 3. Planck's constant h 6.6261965 10-34 joule sec 10-27 erg sec n = h/27T 1.05459198 10-34 joule sec I 0-27 erg sec 4. Avogadro's number No 6.022169 1026 kmole-' 10 23 mole-I 5. Electron rest mass me 9.109559 10-31 kg 10-28 gm 0.5110041 MeV 6. Proton rest mass mp 1.672614 10-27 kg 10-24 gm 938.2592 MeV 7. Neutron rest mass mn 1.674920 10-27 kg 10-24 gm 939.5528 MeV 8. Faraday's constant F 9.648671 107 coul kmole-' 10 3 emu mole-I 2.892599 1014 esu mole-I 9. Gas constant R 8.31434 103 joule kmole-' K-' 10 7 erg mole-I K-I 10. Boltzmann's constant k 1.380623 10-23 joule K-I 10-16 erg K-' II. Gravitational constant G 6.6732 IO- llnewt meter2 kgm-2 10-8 dyne cm 2 gm-2

Clusters of Constants

In many atomic and nuclear calculations the above constants appear in clusters, the evaluation of which can become tedious. We provide here a list of some of the more frequently encountered combinations. The serious student is urged to be• come familiar with the clusters listed here so that he will recognize them when they occur in a problem and use them in preference to looking up the constants separately, followed by several multiplications or divisions. 649 650 Appendix I

In these clusters the symbols in brackets apply for calculations in mksa units. For cgs the symbols in parentheses are correct if e is in esu.

Physical Quantity Symbol Value Units (Sf) Units (cgs)

1. Fine-structure a = [tLt,c 2/41T] (eWtc) 7.297351 10-3 10-3 constant a-I 137.03602 2. Ratio of proton mass to mp/me 1836.109 electron mass 3. Electron charge-to- elm, 1.7588029 lO" coul kg- I 107 emu gm-I mass ratio 4. Rydberg's constant [tLt,c 2/41T F (mee4/41r1'1"C) 1.09737312 107 meter-I 105 cm-I 5. nc 1.9732891 lO-ll MeV cm 6. Bohr radius ao [tLt,c2/41T]-1 (n2/mee2) 5.2917716 10-ll meter 10-9 cm 7. Classical electron [tLt,c 2/41T] (e 2/ mec2) 2.817939 10-15 meter 10-13 cm radius 8. Compton wave- length of the h/mec 2.4263097 10-12 meter 10-10 em electron 9. Compton wave- length of the h/mpc 1.3214410 10-15 meter 10-13 em proton 10. Compton wave- length of the h/mnc 1.3196218 10-15 meter 10-13 em neutron II. Bohr magneton [c] (en/2m eC) 9.274097 10-24 jouleltesla 10-21 erg/gauss 12. Nuclear magneton [c) (eI1/2mpc) 5.050952 10-27 joule/tesla 10-24 erg/ gauss

Conversion Factors

1 atomic mass unit (12C = 12) = 1.660531 x 10-27 kg = 1.660531 X 10-24 gm = 931.4812 MeV 1 hertz = 1 cycle seC 1 1 tesla = 104 gauss 1 eV = 1.6021918 x 10-19 joule = 1.6021918 x 10-12 erg 1 kg = 5.609538 X 1029 MeV 1 coulomb = 10-1 emu = c x 10-1 esu 1 volt = 108 emu = 108 c-1 esu 1/47TEo = J-toc 2/47T J-to/47T = 10-7 henry meter-1 300K = 410 eV 1 year = 7T X 107 sec 1 torr = 1 mm of Hg pressure The Greek Alphabet

Lower-Case Letter Capital Letter Name of Letter a A alpha f3 B beta 'Y r gamma 0 ~ delta E E epsilon l; Z zeta 'Y/ H eta () 0 theta iota K K kappa A A lambda J.t M mu v N nu g - xi 0 0 omicron 7T IT pi p P rho as ~ sigma T T tau v Y upsilon cf> phi X X chi 1/1 'I' psi w n omega

651 Appendix II Atomic Weights of the Elements

Atomic Atomic Atomic Weight Atomic Weight Symbol Number (amu) Symbol Number (amu)

Actinium Ac 89 Germanium Ge 32 72.5 9 Aluminum AI 13 26.9815" Gold Au 79 196.9665"

Americium Am 95 Hafnium Hf 72 178.49 Antimony Sb 51 121.75 Helium He 2 4.00260b Argon Ar 18 39.94.b., Holmium Ho 67 164.9303" b Arsenic As 33 74.9216" Hydrogen H 1.0080 •c Astatine At 85 Indium In 49 114.82

Barium Ba 56 137.34 Iodine I 53 126.9045"

Berkelium Bk 97 Iridium Ir 77 192.22

Beryllium Be 4 9.01218" Iron Fe 26 55.847 Bismuth Bi 83 208.9806" Krypton Kr 36 83.80 b Boron B 5 10.8F Lanthanium La 57 138.9055 Bromine Br 35 79.904 Lawrencium Lr 103 Cadmium Cd 48 112.40 Lead Pb 82 207.2'

Calcium Ca 20 40.08 Lithium Li 3 6.941' Californium Cf 98 Lutetium Lu 71 174.97 Carbon C 6 12.011 b,c Magnesium Mg 12 24.305 Cerium Ce 58 140.12 Manganese Mn 25 54.9380" Cesium Cs 55 132.9055" Mendelevium Md 101

Chlorine CI 17 35.453 Mercury Hg 80 200.59

Chromium Cr 24 51.996 Molybdenum Mo 42 95.94

Cobalt Co 27 58.9332" Neodymium Nd 60 144.24 c Copper Cu 29 63.546 Neon Ne 10 20.179 Curium Cm 96 Neptunium Np 93 237.0482"

Dysprosium Dy 66 162.50 Nickel Ni 28 58.71 Einsteinium Es 99 Niobium Nb 41 92.9064" Erbium Er 68 167.26 Nitrogen N 7 14.0067b Europium Eu 63 151.96 Nobelium No 102 Fermium Fm 100 Osmium Os 76 190.2 Fluorine F 9 18.9984" O_xygen 0 8 15.999/'c Francium Fr 87 Palladium Pd 46 106.4 Gadolinium Gd 64 157.25 Phosphorus P 15 30.9738" Gallium Ga 31 69.72 Platinum Pt 78 195.0. 652 Atomic Weights of the Elements 653

Atomic Atomic Atomic Weight Atomic Weight Symbol Number (({IIlIl) Symbol Number (amu)

Plutonium Pu 94 Sulfur S 16 32.06c Polonium Po 84 Tantalum Ta 73 180.947/

Potassium K 19 39.102 Technetium Tc 43 98.9062d

Praseodymium Pr 59 140.0977" Tellurium Te 52 127.60 Promethium Pm 61 Terbium Tb 65 158.9254" Protactinium Pa 91 231.0359- Thallium TI 81 204.3, Radium Ra 88 226.0254"·d Thorium Th 90 232.0381" Radon Rn 86 Thulium Tm 69 168.9342"

Rhenium Re 75 186.2 Tin Sn 50 118.69

Rhodium Rh 45 102.9055" Titanium Ti 22 47.90

Rubidium Rb 37 85.4678 Tungsten W 74 183.85 Rutheium Ru 44 101.0, Uranium U 92 238.029b Samarium Sm 62 150.4 Vanadium V 23 50.941/ Scandium Sc 21 44.9559" Xenon Xe 54 131.30

Selenium Se 34 78.96 Ytterbium Yb 70 173.0.

Silicon Si 14 28.086 c Yttrium Y 39 88.9059" Silver Ag 47 107.868 Zinc Zn 30 65.3, Sodium Na II 22.9898" Zirconium Zr 40 91.22 Strontium Sr 38 87.62 "Elements with no isotopes. These elements exist as one type of atom only. b Elements with one predominant isotope (around 99 to 100% one isotope). c Elements for which the variation in isotopic abundance in earth samples limits the precision of the atomic weight. d Most commonly available isotope (radioactive). » PERIODIC CHART OF THE ELEMENTS 1"0 "0 CD I I ,, ~ :::l II __ ::___ ~I! ~ L .-'L- __- i1f-~""'-.-"1;J c..-- METALS X 1. II ~ lliIJ~1J[1 _J~~ - ~r ~~Ik T_ti~_~______--':J 1. E.If l~ .:).1 1.11~1 - 3J "-;- l ll36 As Se ~Sr i Kr '~Jl-~j~ ~]l. fl~[;J~IJLg -,'- ="Il- ~-r:J~ ]~ : ~ '! !I""""" - ~ _ _L, T"':":'"JL.~~ ~ -....~_~.~ ,J c.w ,_rw...... - ~ """'__ 11•..- ""'_ I 40 41 42 43 44 45 46 47 48 49 50 . 61 62 63 64 37"- 38 - 39 r l lr -ll --- r - -- Te ; I Rb , Sr Y I :lmr! ~G(1Jlf II· -r-- ,___ " JlX. I Zr : Nb ~!~tT~:'~~~":' : ,!!. !! J ;=-_ ~~~ ~I· ~ - --. 72 II 55 II 56 73 11-7475 ~i-7 rn-;78l "-80 1181 ';~-82 83 84 se 11 1i7-71 I -'T- 8 lf -;r --ii lB5'lr- c. II Ba Hf :1 Ta W II Ra Os Ir ' Pt Au r He TI . Pb Bi II Po At Rn r..- ...... ~ 0.-. .... I ~ c.w I ...... "..... - tall II ..... ,......

65 Il 66 I 67 : 6B II 69 ~lJ;~ ~ Th ~ il ~ - & ~

r7 Ir.----~:--lr7 11~ ~~~ II 101 ~ I ~ :...:.. c.ro.:!..r ~ I F~':'Met Appendix IV Table of Nuclear Properties

We provide here a listing of a few of the properties of some of the more impor• tant nuclides. We list Z (the atomic number), the chemical symbol for each element, and the mass number A. The mass excess is the amount of energy by which the mass of an atom exceeds an integer multiple of atomic mass units. In symbols

(mass excess) = (atomic mass)c2 - 931.4812 A The numbers quoted here are from J. H. E. Mattauch, W. Thiele, and A. H. Wapstra, N ucl. Phys., 67, 1 (1965) and are based on the mass of 12C being exactly 12 u. Note that all masses listed refer to the neutral atom. They may be used directly for most nuclear calculations without regard for the masses and binding energies of the associated atomic electrons. Positron decay, however, is an ex• ample of a process for which it is necessary to allow explicitly for electron masses (see Section 15-8). The spin and parity indicated here refer to the gound state of the nucleus. Per• centage abundances are quoted for those that occur in nature. Half-lives are quoted for unstable isotopes. Data mentioned in this paragraph are from Chart of the Nuclides, General Electric Company (1965).

Mass Spin Atomic Excess and Percent Z Element Mass No. (MeV) Parity Abundance Lifetime 1+ 0 n 8.072 '2 10.8 min (fr) 1+ H 1 7.289 '2 99.985 2 13.136 1+ 0.015 1+ 3 14.950 '2 12.26 yr (fr) 1+ 2 He 3 14.931 '2 0.00013 4 2.425 0+ -100 3- 5 11.454 '2 2 x 10-21 sec (n + a) 6 17.598 0+ 0.81 sec (rr) 655 656 Appendix IV

Mass Spin Atomic Excess and Percent Z Element Mass No. (MeV) Parity Abundance Lifetime 3 3 Li 5 11.679 2 - 10-21 sec (p) 6 14.088 1+ 7.42 3- 7 14.907 2 92.58 8 20.946 2+ 0.85 sec (fr) 3- 4 Be 7 15.769 2 53 days (e capt.) 8 4.944 0+ - 3 x 10-16 sec (2a) 3- 9 11.350 2 100 10 12.607 0+ 2.7 x 106 yr (,lr)

5 B 8 22.923 0.78 sec (f3+) 9 12.419 ;;:: 3 x 10-19 sec (p + 2a) 10 12.052 3+ 19.78 3- 11 8.668 2 80.22 12 13.370 1+ 0.020 sec (W)

6 C 10 15.658 0+ 19 sec (f3+) 3- 11 10.648 2 20.5 min (f3+) 12 0.000 0+ 98.89 1- 13 3.125 2 1.11 14 3.020 0+ 5730 yr (f3-) 1+ 15 9.873 2 2.25 sec (f3-) 7 N 12 17.364 1+ 0.0 II sec (f3+) 1- 13 5.345 2" 9.96 min (f3+) 14 2.864 1+ 99.63 1- 15 0.100 2 0.37 16 5.685 2- 7.35 sec (f3-) 17 7.871 4.14 sec (f3-)

8 0 14 8.008 0+ 71 sec (f3+) 1- 15 2.860 2 124 sec (f3+) 16 -4.737 0+ 99.759 5+ 17 -0.808 2 0.037 18 -0.782 0+ 0.204 5+ 19 3.333 2" 29 sec (f3-)

9 F 17 1.952 66 sec (f3+) 18 0.872 1+ 110 min (f3+, e capt.) 1+ 19 -1.486 2 100 20 -0.012 II sec (f3-) 21 -0.046 4.4 sec (f3-)

10 Ne 18 5.319 0+ 1.46 sec (f3+) 1+ 19 1.752 2 18 sec (f3+) 20 -7.041 0+ 90.92 3+ 21 -5.730 2 0.257 22 -8.025 0+ 8.82 23 -5.148 38 sec (f3-) Table of Nuclear Properties 657

Mass Spin Atomic Excess and Percent Z Element Mass No. (MeV) Parity Abundance Lifetime 3+ 11 Na 21 -2.185 "2 23 sec ({3+) 22 -5.182 3+ 2.58 yr ({3+, e capt.) 3+ 23 -9.528 "2 100 24 -8.418 4+ 15.0 hr ({3-) 3+ 12 Mg 23 -5.472 "2 12 sec ({3+) 24 -13.933 0+ 78.70 5+ 25 -13.191 "2 10.13 26 -16.214 0+ 11.17 1+ 27 -14.583 "2 9.5 min ({3-) 5+ 13 Al 27 -17.196 "2 100 28 -16.855 2.30 min ({3-) 14 Si 28 -21.490 0+ 92.21 1+ 29 -21.894 "2 4.70 30 -24.439 0+ 3.09 3+ 31 -22.962 "2 2.62 hr ({3-) 15 P 30 -20.197 1+ 2.5 min ({3+) 1+ 31 -24.438 "2 100 32 -24.303 1+ 14.3 days ({3-) 33 -26.335 25 days ({3-) 16 S 32 -26.013 0+ 95.0 3+ 33 -26.583 "2 0.76 34 -29.933 0+ 4.22 3+ 35 -28.847 2 86.7 days ({3-) 36 -30.655 0+ 0.014 3+ 17 CI 35 -29.014 "2 75.53 36 -29.520 2+ 3 x 105 yr ({3-, e capt.) 3+ 37 -31.765 "2 24.47 18 Ar 36 -30.232 0+ 0.337 3+ 37 -30.951 "2 35.1 days (e capt.) 38 -34.718 0+ 0.063 7- 39 -33.238 "2 270 yr ({3-) 40 -35.038 0+ 99.60 3+ 19 K 39 -33.803 "2 93.10 40 -33.533 4- 0.0118 1.3 x 10 9 yr ({3-, e capt.) 3+ 41 -35.552 "2 6.88 20 Ca 40 -34.848 0+ 96.97 7- 41 -35.140 2 7.7 x 104 yr (e capt.) 42 -38.540 0+ 0.64 7- 43 -38.396 2 0.145 44 -41.460 0+ 2.06 46 -43.138 0+ 0.0033 48 -44.216 0+ 0.18 658 Appendix IV

Mass Spin Atomic Excess and Percent Z Element Mass No. (MeV) Parity Abundance Lifetime r 21 Sc 41 -28.645 2" 0.55 sec (f3+) 7- 45 -41.061 2" 100 22 Ti 46 -44.123 0+ 7.93 5- 47 -44.927 2" 7.28 48 -48.483 0+ 73.94 7- 49 -48.558 2" 5.51 50 -51.431 0+ 5.34

23 V 48 -44.470 16.1 days (f3+, e capt.) 50 -49.216 6+ 0.24 ~ 6 x 10 15 yr (rr, e capt.) 7- 51 -52.199 2" 99.76 24 Cr 48 -43.070 0+ 23 hr (e capt.) 50 -50.249 0+ 4.31 52 -55.411 0+ 83.76 3- 53 -55.281 2" 9.55 54 -56.930 0+ 2.38

25 Mn 54 -55.552 3+ 303 days (e capt.) 5- 55 -57.705 2" 100 26 Fe 54 -56.245 0+ 5.82 56 -60.605 0+ 91.66 1- 57 -60.175 2" 2.19 58 -62.146 0+ 0.33 7- 27 Co 59 -62.233 2" 100 60 -61.651 5+ 5.26 yr (f3-) 28 Ni 58 -60.228 0+ 67.88 60 -64.471 0+ 26.23 3- 61 -64.220 2" l.l9 62 -66.748 0+ 3.66 64 -67.106 0+ 1.08 3- 29 Cu 63 -65.583 2" 69.09 3- 65 -67.266 2" 30.91 30 Zn 64 -66.000 0+ 48.89 66 -68.881 0+ 27.81 5- 67 -67.863 2" 4.11 68 -69.994 0+ 18.57 70 -69.550 0+ 0.62 3- 31 Ga 69 -69.326 2" 60.4 3- 71 -70.135 2 39.6 32 Ge 70 -70.558 0+ 20.52 Table of Nuclear Properties 659

Mass Spin Atomic Excess and Percent Z Element Mass No. (MeV) Parity Abundance Lifetime 72 -72.579 0+ 27.43 9+ 73 -71.293 "2 7.76 74 -73.418 0+ 36.54 76 -73.209 0+ 7.76 3- 33 As 75 -73.031 "2 100 34 Se 74 -72.212 0+ 0.87 76 -75.257 0+ 9.02 1- 77 -74.601 "2 7.58 78 -77.020 0+ 23.52 80 -77.753 0+ 49.82 82 -77.586 0+ 9.19 3- 35 Br 79 -76.075 "2 50.54 3- 81 -77.972 "2 49.46 36 Kr 78 -74.143 0+ 0.35 80 -77.891 0+ 2.27 82 -80.589 0+ 11.56 9+ 83 -79.985 "2 11.55 84 -82.433 0+ 56.90 86 -83.259 0+ 17.37 3- 37 Rb 85 -82.156 "2 72.15 3- 87 -84.591 "2 27.85 4.7 x 10 10 yr (/r) 38 Sr 84 -80.638 0+ 0.56 86 -84.499 0+ 9.86 9+ 87 -84.865 "2 7.02 88 -87.894 0+ 82.56 1- 39 Y 89 -87.678 "2 100 40 Zr 90 -88.770 0+ 51.46 5+ 91 -87.893 "2 11.23 92 -88.462 0+ 17.11 94 -87.267 0+ 17.40 96 -85.430 0+ 2.80 9+ 41 Nb 93 -87.203 "2 100 42 Mo 92 -86.804 0+ 15.84 94 -88.406 0+ 9.04 5+ 95 -87.709 2" 15.72 96 -88.794 0+ 16.53 5+ 97 -87.539 "2 9.46 98 -88.110 0+ 23.78 100 -86.185 0+ 9.63 660 Appendix IV

Mass Spin Atomic Excess and Percent Z Element Mass No. (MeV) Parity Abundance Lifetime

43 Tc 97 -87.240 2.6 x 106 yr (e capt.) 98 -86.520 1.5 x 106 yr (In 9+ 99 -87.327 2" 2.1 x 105 yr (In 44 Ru 96 -86.071 0+ 5.51 98 -88.221 0+ 1.87 5+ 99 -87.619 2" 12.72 100 -89.219 0+ 12.62 5+ 101 -87.953 2" 17.07 102 -89.098 0+ 31.61 104 -88.090 0+ 18.58 1- 45 Rh 103 -88.014 2" 100 46 Pd 102 -87.923 0+ 0.96 104 -89.411 0+ 10.97 5+ 105 -88.431 2" 22.23 106 -89.907 0+ 27.33 108 -89.524 0+ 26.71 110 -88.338 0+ 11.81 1- 47 Ag 107 -88.403 2" 51.82 1- 109 -88.717 '2 48.18 48 Cd 106 -87.128 0+ 1.22 108 -89.248 0+ 0.88 110 -90.342 0+ 12.39 1+ III -89.246 2" 12.75 112 -90.575 0+ 24.07 1+ 113 -89.041 2" 12.26 114 -90.018 0+ 28.86 116 -88.712 0+ 7.58 9+ 49 In 113 -89.339 2" 4.28 9+ 115 -89.542 2" 95.72 50 Sn 112 -88.644 0+ 0.96 114 -90.565 0+ 0.66 1+ 115 -90.031 2" 0.35 116 -91.523 0+ 14.30 1+ 117 -90.392 2" 7.61 118 -91.652 0+ 24.03 1+ 119 -90.062 2" 8.58 120 -91.100 0+ 32.85 122 -89.942 0+ 4.72 124 -88.237 0+ 5.94 Table of Nuclear Properties 661

Mass Spin Atomic Excess and Percent Z Element Mass No. (MeV) Parity Abundance Lifetime 5+ 51 Sb 121 -89.593 "2 57.25 7+ 123 -89.224 "2 42.75 52 Te 120 -89.400 0+ 0.089 122 -90.291 0+ 2.46 1+ 123 -89.163 "2 0.87 1.2 x 10 13 yr (e capt.) 124 -90.500 0+ 4.61 1+ 125 -89.032 "2 6.99 126 -90.053 0+ 18.71 128 -88.978 0+ 31.79 130 -87.337 0+ 34.48 5+ 53 127 -88.984 "2 100 54 Xe 124 -87.450 0+ 0.096 126 -89.154 0+ 0.090 128 -89.850 0+ 1.92 1+ 129 -88.692 "2 26.44 130 -89.880 0+ 4.08 3+ 131 -88.411 "2 21.18 132 -89.272 01- 26.89 134 -88.120 0+ 10.44 136 -86.422 0+ 8.87 7+ 55 Cs 133 -88.160 "2 100 56 Ba 130 -87.331 0+ 0.101 132 -88.380 0+ 0.097 134 -88.852 0+ 2.42 3+ 135 -87.980 "2 6.59 136 -89.140 0+ 7.81 3+ 137 -88.020 "2 11.32 138 -88.490 0+ 71.66

5~ 57 La 138 -86.710 "2 0.089 1.1 x 10 11 yr (e capt., (r) 7+ 139 -87.428 "2 99.911 58 Ce 136 -86.550 0+ 0.193 138 -87.720 0+ 0.250 140 -88.125 0+ 88.48 142 -84.631 0+ 11.07 5 x 10 15 yr (0') 5+ 59 Pr 141 -86.072 "2 100 60 Nd 142 -86.010 0+ 27.11 7~ 143 -84.039 "2 12.17 144 -83.797 0+ 23.85 2.4 x 10 15 yr (0') 662 Appendix IV

Mass Spin Atomic Excess and Percent Z Element Mass No. (MeV) Parity Abundance Lifetime 7- 145 -81.469 2 8.30 146 -80.959 0+ 17.22 148 -77.435 0+ 5.73 150 -73.666 0+ 5.62

61 Pm 143 -82.910 265 days (e capt.) 62 Sm 144 -81.980 0+ 3.09 r 147 -79.300 2 14.97 1.06 x 1011 yr (a) 148 -79.371 0+ 11.24 1.2 x 10 13 yr (a) 7- 149 -77.145 2 13.83 ~ 4 X 10 14 yr? (a) 150 -77.056 0+ 7.44 152 -74.746 0+ 26.72 154 -72.393 0+ 22.71 5+ 63 Eu 151 -74.670 2 47.82 5+ 153 -73.361 2 52.18

64 Gd 152 -74.710 0+ 0.20 1.1 X 10 14 yr (a) 154 -73.653 0+ 2.15 3- 155 -72.037 2" 14.73 156 -72.493 0+ 20.47 3- 157 -70.769 2 15.68 158 -70.627 0+ 24.87 160 -67.891 0+ 21.90 3+ 65 Tb 159 -69.534 2" 100

66 Dy 156 -70.860 0+ 0.052 2 X 10 14 yr (a) 158 -70.374 0+ 0.090 160 -69.673 0+ 2.29 5+ 161 -68.049 2" 18.88 162 -68.182 0+ 25.53 5- 163 -66.363 2" 24.97 164 -65.949 0+ 28.18 7- 67 Ho 165 -64.811 2" 100 68 Er 162 -66.370 0+ 0.136 164 -65.867 0+ 1.56 166 -64.918 0+ 33.41 7+ 167 -63.285 2" 22.94 168 -62.983 0+ 27.07 170 -60.020 0+ 14.88 1+ 69 Tm 169 -61.249 2" 100 70 Yb 168 -61.330 0+ 0.135 170 -60.530 0+ 3.03 Table of Nuclear Properties 663

Mass Spin Atomic Excess and Percent Z Element Mass No. (MeV) Parity Abundance Lifetime 1- 171 -59.220 2" 14.31 172 -59.280 0+ 21.82 5- 173 -57.690 2" 16.13 174 -57.060 0+ 31.84 176 -53.390 0+ 12.73 7+ 71 Lu 175 -55.290 2" 97.41 176 -53.410 7- 2.59 2.2 x 10 10 yr (In

72 Hf 174 -55.550 0+ 0.18 4.3 x 10 15 yr (a) 176 -54.430 0+ 5.20 7- 177 -52.720 2" 18.50 178 -52.270 0+ 27.14 9+ 179 -50.270 2" 13.75 180 -49.530 0+ 35.24 73 Ta 180 -48.862 0.0123 7+ 181 -48.430 2" 99.988 74 W 180 -49.365 0+ 0.14 182 -48.156 0+ 26.41 1- 183 -46.272 2" 14.40 184 -45.619 0+ 30.64 186 -42.438 0+ 28.41 5+ 75 Re 185 -43.725 2" 37.07 5+ 187 -41.140 2" 62.93 4 X 10 10 yr (In 76 Os 184 -44.010 0+ 0.018 186 -42.970 0+ 1.59 1- 187 -4l.l41 2" 1.64 188 -40.909 0+ 13.3 3- 189 -38.840 2" 16.1 190 -38.540 0+ 26.4 192 -35.910 0+ 41.0 3+ 77 Ir 191 -36.670 2" 37.3 3+ 193 -34.454 2 62.7 78 Pt 190 -37.300 0+ 0.0127 6 x ]011 yr (a) 192 -36.190 0+ 0.78 194 -34.721 0+ 32.9 1- 195 -32.776 2" 33.8 196 -32.633 0+ 25.3 198 -29.905 0+ 7.21 3+ 79 Au 197 -31.166 2 100 80 Hg 196 -31.838 0+ 0.146 664 Appendix IV

Mass Spin Atomic Excess and Percent Z Element Mass No. (MeV) Parity Abundance Lifetime 198 -30.966 0+ 10.02 1- 199 -29.547 "2 16.84 200 -29.503 0+ 23.13 3- 201 -27.658 "2 13.22 202 -27.346 0+ 29.80 204 -24.689 0+ 6.85 1+ 81 TI 203 -25.753 "2 29.50 1+ 205 -23.807 "2 70.50 206 -22.259 4.3 min (/r) 207 -21.005 4.78 min (rr) 208 -16.754 5+ 3.1 min (rn 209 -13.697 2.2 min (rr) 210 - 9.264 1.3 min

82 Pb 204 -25.109 0+ 1.48 1.4 x 10 17 yr (ex) 205 -23.772 3 x 107 yr (e capt.) 206 -23.783 0+ 23.6 1- 207 -22.446 "2 22.6 208 -21.750 0+ 52.3 209 -17.622 3.3 hr (rr) 210 -14.730 0+ 22 yr (/r) 211 -10.486 36.1 min (In 212 - 7.541 0+ 10.64 hr (/r) 214 - 0.218 0+ 26.8 min (In

83 Bi 208 -18.880 3.7 x 105 yr (e capt.) g- 209 -18.262 "2 100 210 -14.791 1- 5.0 days (In 211 -11.830 2.15 min (In 212 -8.124 1- 60.6 min (In 213 -5.294 4.7 min (In 214 -1.224 19.7 min (In I- 84 Po 209 -16.370 "2 103 yr (ex, e capt.) 210 -15.950 0+ 138.40 days (ex) 211 -12.429 0.52 sec (ex) 212 -10.371 0+ 0.30 x 10-6 sec (ex) 213 -6.683 4 x 10-6 sec (ex) 214 -4.470 0+ 0.164 x 10-3 sec (ex) 215 -0.537 0.0018 sec (ex, Ir) 216 1.790 0+ 0.15 sec (ex) 218 8.318 0+ 3.05 min (ex, fn 85 At 212 -8.641 0.31 sec (ex) 214 -3.410 < 5 sec (ex) 215 -1.245 - 10-4 sec (ex) Table of Nuclear Properties 665

Mass Spin Atomic Excess and Percent Z Element Mass No. (MeV) Parity Abundance Lifetime

216 2.246 - 3.0 x 10-4 sec (a) 217 4.329 0.032 sec (a) 218 8.017 1.3 sec (a)

86 Rn 216 0.254 0+ 45 X 10-6 sec (a) 217 3.629 500 X 10-6 sec (a) 218 5.219 0+ 0.035 sec (a) 219 8.832 4.0 sec (a) 220 10.620 0+ 56 sec (a) 222 16.329 0+ 3.823 days (a) 87 Fr 218 7.020 < 5 sec (a) 219 8.622 0.02 sec (a) 220 11.492 28 sec (a) 221 13.211 4.8 min (a) 223 18.383 22 min (In

88 Ra 220 10.274 0+ 0.02 sec (a) 221 12.940 30 sec (a) 222 14.322 0+ 37 sec (a) 1+ 223 17.233 "2 II. 7 days (a) 224 18.832 0+ 3.64 days (a) 225 21.916 14.8 days (In 226 23.623 0+ 100 1620 yr (a) 227 27.161 41 min (/r) 228 29.005 0+ 5.7 yr (fn 89 Ac 222 16.540 5 sec (a) 223 17.832 2.2 min (a, e capt.) 224 20.200 2.9 hr (e capt., a) 225 21.566 10.0 days (a) 3+ 227 25.851 "2 21.2 yr (tr, a) 228 28.950 6.13 hr (tr)

90 Th 224 20.005 0+ - I sec (a) 225 22.288 8 min (a, e capt.) 226 23.195 0+ 31 min (a) 227 25.807 18.17 days (a) 228 26.780 0+ 1.91 yr (a) 5+ 229 29.483 "2 7300 yr (a) 230 30.820 0+ 76000 yr (a) 5+ 231 33.804 "2 25.6 hr err) 232 35.512 0+ 1.41 X 10 10 yr (a) 233 38.628 22.1 min (rn 234 40.596 0+ 24.10 days (rr) 91 Pa 226 25.900 1.8 min (a) 5- 227 26.837 "2 38.3 min (a, e capt.) 666 Appendix IV

Mass Spin Atomic Excess and Percent Z Element Mass No. (MeV) Parity Abundance Lifetime

228 28.880 22 hr (e capt., a) 5- 229 29.828 "2 1.5 days (e capt., a) 230 32.074 2- 17 days (e capt., rr, (3+, a) 3- 231 33.419 "2 32480 yr (a) 232 35.966 1.32 days (In 3- 233 37.383 "2 27.4 days (rr) 234 40.332 6.66 hr ({3-)

92 U 228 29.236 0+ 9.3 min (a, e capt.) 229 31.187 58 min (e capt., a) 230 31.612 0+ 20.8 days (a) 232 34.621 0+ 72 yr (a) 5+ 233 36.814 "2 1.62 X 105 yr (a) 234 38.102 0+ 0.0057 2.48 X 105 yr (a) 7- 235 40.906 "2 0.72 7.13 X 108 yr (a) 237 45.277 6.75 days ({3-) 238 47.291 0+ 99.27 4.51 X 109 yr (a) 239 50.579 23.5 min ({3-) 93 Np 231 35.660 - 50 min (e capt., a) 5+ 237 44.763 "2 2.14 X 106 yr (a) 238 47.408 2+ 2.10 days ({3-) 5+ 239 49.297 "2 2.35 days ({r) 94 Pu 232 38.360 0+ 36 min (e capt.) 234 40.347 0+ 9 hr (e capt.) 236 42.914 0+ 2.85 yr (a) 238 46.118 0+ 89 yr (a) 1+ 239 48.573 "2 24360 yr (a) 5+ 241 52.849 "2 13 yr (In 95 Am 239 49.383 12 hr (e capt., a) 5- 241 52.828 "2 458 yr (a) 242 55.425 1- 16 hr (/r, e capt., a) 96 em 240 51.739 0+ 26.8 days (a) 242 54.760 0+ 163 days (a) Appendix V-1 Review of Vector Notation

In this appendix we review the notation that is employed in the manipulation of vector quantities. A vector quantity is printed in boldface type, as in the vector r. The same symbol in italic type, r, denotes the magnitude of the vector; by defi• nition, Irl =r (1) The sum or difference of two vectors is readily obtained by the parallelogram method: the sum of two vectors is the diagonal of the parallelogram, as seen in Figure A-I. Two kinds of product of vector are defined. The scalar product (also dot prod• uct, inner product) of two vectors a and b is defined by a . b = ab cos () (2) where () is the angle between the two vectors. As its name indicates, the result of a • b is a scalar quantity. The vector product (also cross product, outer product) is defined by

a X b = ab sin ()E (3)

where () is defined as before and E is a vector of unit magnitude which is perpen• dicular to both a and b and chosen according to a righthand rule. Clearly b·a=a·b

b X a = -a X b (4) Products involving three vectors can also be defined. One of these is written

Figure A-1 The sum of two vectors, C = a + b. 667 668 Appendix V-1

a . b X r. The meaning of this expression is unambiguously a . (b X c); the result is a scalar. The interpretation (a . b) X c is without meaning, since the cross prod• uct of a scalar with a vector is not defined. The following identities are readily derived by use of components of the vectors involved: a·bXc=axb·c =cXa·b =bXc'a = -b . a X c (5)

Expressed in words, the quantity a . b X c is invariant under interchange of the dot and cross but changes sign under an odd number of permutations of two of the vectors. The triple product a X (b X c) can also be defined. In this instance the result is a vector. Its meaning is sensitive to the grouping and accounts for our inclusion of parentheses. Again, the use of components leads to a useful result: a X (b X c) = b (a . c) - c (a . b) (6) Appendix V-2 Displacement Equation for Brownian Motion

The second type of experiment performed by Perrin for the determination of A vogadro's number consisted in observing the displacement of a particle in a given time interval. The fundamental assumption used in the derivation of the equation for the displacement of such particles is that the particles suspended in the fluid have a mean kinetic energy equal to the mean kinetic energy of gas mole• cules at the same temperature. The equation of state for one mole of an ideal gas is

Pu=RT (I) and on the basis of the kinetic theory of gases it can be shown that this equation takes the form 1 - Pu = "3 Nmc2 (2)

where c2 is the average of the squares of the velocities of the molecules of the gas at temperature T. Thus the mean kinetic energy of a gas molecule is

1 - 3 R & =zmc2 =-- T (3) 2N In general, a particle has three degrees of freedom of translatory motion; but if the observations are confined to a single direction, say the x direction, the average kinetic energy due to the motion in this direction will be one third of its total kinetic energy, or

tm (~~r =~=~ ~ T (4) where dx/dt is the instantaneous velocity of the molecule in the x direction and (dx/dt)2 is the average of the squares of its velocity in the x direction. The motion of a Brownian particle in a horizontal plane is determined by two forces: (a) that due to the bombardment of the molecules of the medium giving an unbalanced force whose component in the x direction i~ X and (b) the resistance to the motion due to the viscosity of the medium. From Stokes' law the resistance to the motion is proportional to the velocity; hence the force on the particle in the x direction at any instant is, frorr: Newton's second law, 669 670 Appendix V-2

(5)

where K is a factor of proportionality determined by the viscosity of the medium and is given by (6)

where 1) is the coefficient of viscosity of the fluid and a is radius of the particle. In studying Brownian motion it is easier to consider the magnitude only of the dis• placement; hence the equation of motion is modified to yield x 2 instead of x. Multiplying through by x, we get dx d2x Xx-Kx-=mx- dt dt 2 Now

and dx 1 d x-=--(x2) dt 2 dt so that

2 2 Xx _ t K d(x ) = m d2(x ) _ m (dX)2 (7) dt 2 dt 2 dt We are interested only in average values of these quantities that can be observed in a time interval t. A bar over a symbol will represent its average value. Let z = d(X2)/dt and note that Xx = 0 since Xx is probably positive as often as it is nega• tive; then Equation (7) becomes

- ~z = ; ~~ - m (~;r (8)

or

(9)

from Equation (4). Separating variables, we get

dz = _!i. dt (10) z - (2RT/KN) m and, integrating from 0 to t, this yields

z - 2RT =A exp (_ Kt) (11) NK m where A is a constant of integration. For any reasonable time il}terval required for observation, the exponential term becomes negligible so that Displacement Equation for Brownian Motion 671

(12) and - 2RT X 2 =--t (13) NK where t is the time interval for observation. This equation states that the average square of the displacements of a particle depends on the time interval t and on the temperature of the medium. This equation was derived by Einstein (1905). Appendix V-3 Path of an Alpha Particle in a Coulomb Field of Force

Consider a nucleus of charge Ze stationary at point C and an alpha particle of mass M and charge Q approaching it along the line AB (Fig. A-2). The original velocity of the alpha particle in the direction of A B is V. There will be a force of repulsion between the two charges given by Coulomb's law.

(1)

where r is the distance of the alpha particle from the nucleus. The alpha particle will be deflected from its original direction by this force of repulsion and its path will be a conic section, since the motion is governed by an inverse square law of force. This conic section will be one branch of a hyperbola with the nucleus at the focus on the convex side of this branch. To derive the expression for the path of the alpha particle let us choose polar coordinates with the nucleus at the pole. The acceleration of the particle may be resolved into two components, one along the radius, ar , and the other transverse

c x

Figure A-2 The hyperbolic path of an alpha particle in the field of force of a nucleus. 672 Path of an Alpha Particle in a Coulomb Field of Force 673 to the radius, a. In texts on mechanics it is shown that these components are given by the equations

_ d2r (d1» 2 aT - dt2 - r dt (2)

a =!!!.. (r2 d1» (3) r dt dt Since the force of repulsion is along the radius vector r, the radial component of the acceleration is F kZeQ J a (4) T =-=--=-M Mr2 r2 where

J= kZeQ M Since there is no force transverse to the radius vector,

(5)

Integrating Equation (5) yields

(6)

Equation (6) is a statement of Kepler's law of areas and also expresses the prin• ciple of conservation of angular momentum, for, multiplying both sides by M, we get

Mr2 dcf> = MK = MVp (7) dt where M r2 is the moment of inertia of the alpha particle with respect to an axis through C, d1>fdt is the angular velocity of the particle, and MVp is the initial angular momentum of the particle. The distance p from the nucleus to AB is called the impact parameter. The differential equation of the path of the particle may be obtained by combin• ing Equations (2) and (4); this yields

d2r (d1» 2 J dt2 - r di =~ (8)

Before integrating this equation, let us change the independent variable from t to 1>, and also, for convenience, let u = l/r. To carry out this transformation note that dr dr d1> dt d1> dt dr 1 du -=--- d¢ u2 d¢ and from Equation (6) Appendix V-5 Evaluation 0lf Ic!ltegrals of the Form vn exp (_AV2)dv o

Integrals of the form In = L" V" exp (_AV2) dv arise in varied applications, most

notably in the quantum-mechanical treatment of the harmonic oscillator and in the classical statistical description of an ideal gas. The easiest of these integrals is the one for which n = I.

= -~ [exp (-AV2)]'" 2A 0

(I)

If we now take ~~, differentiating both the first and third lines of (I), we get

(2)

Clearly this process can be repeated to supply the result for all odd values of n. To evaluate 10 it is necessary to use a trick.

(3)

2/0 = L: exp (_AV2) dv (4)

We next square both sides, remembering that the variable of integration for a definite integral is a dummy variable for which the symbol may be changed without affecting the result. 681 674 Appendix V-3

and

Equation (8) now becomes

or

(9)

This is a well-known differential equation and its solution is 1 u + K2 = A cos (¢ - 8) (10)

where A and 0 are constants of integration. To verify that this is the solution of the differential equation, we need only to differentiate Equation (10) twice. By the proper choice of the x axis the phase angle 0 can be set equal to zero. Further, by setting AK2 E=-- (11) 1 Equation (10) becomes

from which I 1 u=-=--(1-ECOS ¢) (12) r K2

Equation (12) is one form of the equation of a conic section in polar coordinates; E is the eccentricity of this conic section. When E is greater than unity, the conic section is a hyperbola. The eccentricity of this path can be determined with the aid of the principle of conservation of energy, which yields

1 1 kZeQ 1 1M 2"MV2 = 2"MV2 + --= 2"MV2 + - (13) r r where kZeQlr is the potential energy of the particle at any point in its path. Re• solving the velocity v into two components, one along the radius and one trans• verse to the radius, we get

(14)

From Equation (12) we get Path of an Alpha Particle in a Coulomb Field of Force 675

and since

Equation (14) becomes

(15)

Substituting this value of v2 in Equation (13) and the value for r from Equation (12) and simplifying, we get

from which

or

(16)

The eccentricity of the orbit is greater than unity and is expressed in terms of the initial energy of the alpha particle, the impact parameter p, and the nuclear charge Ze; thus

e -_ ~ 1 + (Ml5z.)2kZeQ (17) The orbit is therefore a hyperbola with the nucleus at the focus outside the branch of the curve followed by the alpha particle. The asymptote AB represents the direc• tion of the initial velocity of the alpha particle, and the second asymptote OD represents the direction of the final velocity of the alpha particle. The angle (J be• tween the two asymptotes is the angle through which the particle has been deflected and is the angle of scattering. This angle can be expressed in terms of the eccen• tricity from a consideration of the properties of the hyperbola. The equation of the hyperbola in rectangular coordinates is

(18) where

ae=OC and

b 2 = a2(e 2 - I) The equations of the asymptotes are obtained by setting the left-hand side of 676 Appendix V-3

Equation (18) equal to zero, yielding b y=±-x a From the figure

b 1T-(} () - = tan --= cot - a 2 2 hence

() e2 - 1 = cot2 - (19) 2 Using Equation (16), we get () MV2 cot 2"= kZeQ P (20) Appendix V-4 Derivation of the Equations for the Compton Effect

The three equations derived in Section 5-19 are hI! = hI!' + me 2 (y - I) (I) hI! hI!' - = - cos 1> + ymv cos e (2) e e hl!'. . O = - sm 1> - ymv sm e (3) e where

To solve these equations, let hI! h a=-=- me 2 meA

,hI!' h a =-=-- me2 meA'

b = -v?-=-T = yv e or

y= Vi + b2 It = cos 1>

nt = sin 1> 12 = cos e

n2 = sin e

By dividing Equation (I) by mc2 we get

a = a' + VI + b2 - I (4) 677 678 Appendix V-4

and by dividing Equations (2) and (3) by me we get

a = a'/\ + bl2 (5) and

(6)

From Equation (5) (7)

and from Equation (6) (8)

Adding Equations (7) and (8), we get

or

(9)

since

From Equation (4),

b2 = a2 - 2aa' + a'2 + 2a - 2a' (10) Subtracting Equation (10) from Equation (9), we get 0= 2aa'(I - 1\) - 2a + 2a' from which

or

(II)

from which A' h --I =-(l-cos cf» A mel.. or h A' - 1..=-(1 -cos cf» (12) me which is the same as Equation (5-34) in the text. To get the expression for the kinetic energy of the recoil electron solve Equation (II) for a' to obtain Derivation of the Equations for the Compton Effect 679

, a (13) a = 1 + a( 1 - II)

Substitute this value in Equation (4) to obtain ,/-- a 2 v I + b - I = a-I + a( 1 - II)

a 2(1 -II) I + a(l -II) Now the kinetic energy & is given by

& = mc2(y - I) = mc2(Yl + b2 - I)

2 = mc2 a (1 - cos 1» I + a( I - cos 1» But hv mc2 =• a therefore

& = hv a( I - cos 1» (14) 1+ a(1 -cos 1»

The energy of the recoil electron can also be put in terms of the angle (J by noting that, from Equations (5) and (6),

.!a. = a - a'/l =..!... (.!!.. -I ) (15) n2 a'nl nl a' I and eliminating a' with the aid of Equation (II) we get

12 1 - = - (I + a)(I - II) (16) n2 nl

Substituting the values of Ip 12 , nl , and n2 , we get

I - cos 1> cot (J = (I + a) -7"".-""• (17) sm 1> Now

I - cos 1> 1> . =tan- sm 1> 2 therefore

cot (J = (I + a) tan ! (18) also

cot (1)12) tan (J= I (19) +a 680 Appendix V-4

which is Equation (5-35) of the text. By dividing Equation (18) by Equation (19) we get

cos 2 () cos2 () cp --= =(1 + a)2tan2 - sin2 () I - cos2 () 2

Now

2 cp _I_-_co_s_cp-,• tan - =-:- 2 I + cos cp

hence

2 cos () = (I + a)2 I - cos cp (20) I - cos2 () I + cos cp

Solving Equation (20) for cos cp and then for (I - cos cp), we get

2 cos2 () 1 - cos cp = -,---:-:---::----,:------:---:-• (21) (I + a)2 - 2a cos 2 () - a::2 cos2 () Substituting this value in Equation (14) yields

2 Ii> = hv 2a:: cos () (22) (I + a)2 - a 2 cos 2 () which is Equation (5-36) of the text. 682 Appendix V-5

(5)

Next we write the product of integrals as a double integral over all x - y space.

(6)

The two-dimensional integral can be performed in polar coordinates, for which r2 = x2 + y2, B = tan-I (y/x), and the element of area is r dB dr. " f27T 4/0 2= f0 0 exp(-Ar2)rdBdr (7)

U sing (I), we get

or (8)

1 - 10 = ·{Y 7T A-1/2 By taking partial derivatives with respect to A we can now generate the integrals corresponding to even values of n.

(9)

(10)

i00 v4 exp (- AV2) dv = t v:;:;: A-5/2 (II)

and so on. These results are listed in tabular form.

n

o t v:;:;: A-1/2 t A-I

2 t v:;:;: A-3/2

3 t A- 2

4 t v:;:;: A-5/2 Appendix V-6 Quantum Mechanical Solution of the Harmonic Oscillator

In this appendix we examine the solutions of the Hermite differential equation (7-66). We shall employ the method of factorization, a method used by Schroe• dinger. We begin by applying this method to the solution of the classical equation (7-59). The classical oscillator is described by

d2x -+w2x=O (1) dt2 If we use the symbol D to represent the operator d/dx, the equation becomes (D2 + w2)x = 0 The quantity in parentheses is easily factored to yield

(D + iw)(D - iw)x = 0 (2)

But any solution of (D - iw)x = 0 is also a solution of Equation (2), and it is easily verified that x = A exp (iwt) is a solution. Similarly, we can reverse the order of the factors in Equation (2) and obtain

(D - iw)(D + iw)x = 0

This reversal is possible only because w is a constant. Any solution of (D + iw)x = 0 is now also a solution of Equation (2), and the result is x = B exp (-iwtkSo the general solution of Equation (I) is

x = A exp (iwt) + B exp-(--iwt) where A and B are arbitrary constants. We now proceed to work with the Hermite equation,

d21jJ (1) - y2)1jJ = 0 (3) -dy2 +

Again, we define D = d/dy to obtain [D2 + (1) - y2)]1jJ = 0 (4) 683 684 Appendix V-6

But the factorization is not so straightforward this time because the coefficient (71 - y2) is not constant. To prepare for factoring we need to recall the rule for the differentiation of a product:

D(yl/l) = 1/1 + yDI/I = (1 + yD)I/I (5) By use of this rule we can derive three useful identities:

(D + y)(D - y)I/I = [D2 - (1 + yD) + yD - y2]1/1 = (D2 - y2 - 1)1/1 (6a) Similarly,

(D - y)(D + y)I/I = (D2 - y2 + 1)1/1 (6b) Subtracting these two, we get

[(D - y)(D + y)]1/1 - [(D + y)(D - y)]1/1 = 21/1 (6c) It is worthwhile writing these three identities as operator equations:

(D + y)(D - y) = D2 - y2 - I (7a)

(D - y)(D + y) = D2 - y2 + I (7b)

(D - y)(D + y) = (D + y)(D - y) + 2 (7c) Using Identity (7b), we now rewrite the Hermite equation in the form

[(D - y)(D + y) - I] 1/1 = -711/1 (8) It might seem that little has been gained, but it will be noticed that the equation has been factored. It is also apparent that if 71 = I we can solve the equation. Let 1/10 represent the solution for which 71 = I and for which

(D - y)(D + Y)I/Io = o. (9)

'f'hen any solution of (D + Y)I/Io = 0 will also be a solution. We separate variables as follows:

y2 In 1/10 =-'2+ In C

1/10 = C exp (-t) (10) As we shall see, this is the wave function for the ground state of the oscillator. The energy of the ground state is obtained by recalling from Equation (7-65) that 71 = 2&Fliw. For the ground state 710 = I, so (11) At this point it is reasonable to ask why we did not use Identity (7a) instead of Quantum Mechanical Solution of the Harmonic Oscillator 685

(7b). Then the Hermite equation would have read

[(D + y)(D - y) + I]t/I = -1/t/I

To obtain the ground state we would have chosen 1/ = -I and then we would have proceeded to solve the equation (D - Y)t/lo = 0; but the solution to this equation is proportional to exp (+y2/2), a result that would not allow us to calculateL: t/l2 dy.

The correct result t/lo = C exp (-y2/2) can be used in integrals over all values of y. Furthermore, the choice of 1/ = -I implies a negative total energy for the oscillator; but both the kinetic energy (p2/2m) and the potential energy (tkX2) are positive, implying that the total energy must be positive. We conclude then that Identity (7a) does not lead to a useful solution. Next we shall show that if t/lo is a solution of the equation

[(D - y)(D + y) - I] t/lo = -1/ot/lo (8) the function

t/ll = (D - y)t/lo (12) is a solution of the equation

[(D - y)(D + y) - l]t/ll = -1/lt/il (13) This fact is easily proved by starting with Equation (8) and operating from the left with the operator (D - y).

[(D - y)(D - y)(D + y) - (D - y)Jt/lo = -(D - Y)1/ot/lo N ext we refer to Identity (7c) for use in rewriting the first term on the left. We get

{(D - y)[(D + y)(D - y) + 2] - (D - Y)}I/Io = -rMD - Y)I/Io Regrouping terms, we get

[(D - y)(D + y) - I] (D - y)l/Io = -(1/0 + 2)(D - y)t/lo This equation has the desired form if

1/11 = (D - y)l/Io (14) and

1/1 = 1/0 + 2 Notice that this result does not depend on the fact that 1/0 refers to the ground state; so the proof that we used is general enough to enable us to write immediately

% = (D - y)t/ll

1/2 = 1/1 + 2 or more generally

I/In+t = (D - y)IjJn

I/In+1 = 1/n + 2 686

Since we know "'0 and 1)0' it is now possible to generate an infinite number of solutions to the Hermite equation just by repeated application of the operator (d/dy - y). The general form of 1) is

1)n = 2n + 1)0 (16)

Since 1)0 = I and 1)n = 2&n/(nw), we get

&n = (n + t)nw, (17) the general expression for the energy of the nth state of the harmonic oscillator. A few concluding comments are appropriate. We have derived an infinte set of solutions for the harmonic oscillator, but we have not proved that we have found all of the possible solutions. In fact, there are many others but they all have the prop-

erty that L: ",2 dy is infinite; those solutions that we have found are the only ones for which the integral remains finite, allowing us to normalize by appropriate choice of the constant C in Equation (10). Instead of using mathematics to prove that we have found all the useful solutions, it is perhaps better to use a physical argument. If the curves for the infinite square well in Figure 7-3 are studied, it will be seen that the ground-state wave function has no zeros in the interior of the physical region. The first-excited-state wave function has one zero and so on. Comparison with Figure 7-7 shows that the same statements apply to the wave functions for the harmonic oscillator. For the in• finite square well we proved mathematically that we found all the physically useful solutions. By analogy we infer that we have done the same for the oscillator, a fact that can be proved. The approach we have used to solve this problem is not the one that is ordinarily used. The more conventional treatment can be studied in nearly any of the refer• ences at the end of Chapter 7. The method of factorization has the advantages of compactness and a certain amount of elegance. Another attribute is the fact that it is useful in the study of the quantum mechanical nature of angular momentum. Unfortunately the method is not so general as was hoped when it was first dis• covered. There are numerous problems in quantum mechanics for which factori• zation is not a useful technique. Appendix V-7 Evaluation of fPrdr

We wish to evaluate

(I)

Now dr PT = m dt (2) is the momentum along the radius. Changing the independent variable from t to cjJ by the relationship dr dr dcjJ (3) dt dcjJ dt we get

(4) and noting that

(5) and

(6) we may write the integral as

(7) or

(8) P~ f ~ (:~r dcjJ = nTh The equation of the ellipse may be written as

I + E cos cjJ (9) r a(l - E2) 687 688 Appendix V-7

where a is the semimajor axis and E is the eccentricity. Therefore

dr a(l - E2)E sin 4> (10) d4> (I + E cos 4>)2 and

1 dr E sin 4> (II) r d4> 1 + E cos 4>

The integral equation now becomes

2" E2 sin 2 4> (12) peJ> 10 (I + E cos 4»2 d4> = nrh

The integral

(13)

can be integrated by parts. In the usual standard form

JU dv = uv - Jv du (14)

let

U = E sin 4>

du = € cos e/> de/>

dv = E sin 4> d (I + E cos 4»2 e/> so that 1 u=---- 1 + E cos e/> then 2 1= Esin 4> ]2" - 1" Ecos 4> d4> (15) 1 + E cos 4> 0 0 1 + E cos 4>

On substitution of the limits of integration, the first term on the right-hand side becomes zero. The value of the integral is

2" E cos 4> 1=-1 d (16) o 1 + E cos 4> 4>

which may be written in the form

I = 2" (I -I ) d (17) Jo 1 + E cos 4> 4>

yielding Evaluation of #Prdr = nrh 689

21T 1=(1 _ e2)112 - 21T (18)

(see Peirce's Table of Integrals, page 41). Putting this back in the integral equa• tion, we get

(19) or

(20) since

Hence

(21)

so that

(22)

which is the expression for the eccentricity of the ellipse in terms of the azimuthal and radial quantum numbers. Of great interest is the expression for the total energy 6, of an elliptic orbit. The potential energy is -kZe2/r. The kinetic energy can be written as

2 2m1 [(dr)- (dcf»2]r- dt + dt where dr/dt is the radial component of the velocity and r(dcf>/dt) is the transverse component of the velocity. Now, using Equations (3) and (5), we get dr p.p dr (23) dt = mr2 dcf> and

(24)

The expression for the total energy then becomes

6, = l.m [p.p2 (dr)2 + p.p2] _ kZe2 2 m2,.-t dcf> m2r2 r 2 =K [(! dr)2 + kZe (25) 2mr2 r dcf> 1] _ r

Solving this for G;~r yields 690

2 (.!. dr)2 = 2m&r2 + 2mkZe r _ I (26) r dcjJ p4>2 p4>2

From Equation (II) we get

I dr)2 e2 sin 2 cjJ ( (27) -;: dcjJ = (1 + e cos cjJ)2 Eliminating the angle cjJ between this equation and the equation of the ellipse (9) yields I dr)2 r2 2r ( (28) -; dcjJ = - a 2(1 - e2) + a(1 - e2) - I By equating coefficients of like powers of r in Equations (26) and (28), we get 2mlf, (29) p4>2 = - a2(1 - e2) and mkZe2 (30) --p;r = a(1 - e2) from which kZe2 If,=-- (31) 2a The total energy depends only on the major axis of the ellipse and is the same as that for a circle of radius a. Eliminating a from Equations (29) and (30) yields

If, = _ mkZ2e4 (1 - e2) (32) 2p4>2 Appendix V-8 Derivation of the Fermi-Dirac and Bose-Einstein Distributions

We assume a collection of identical indistinguishable particles; we also assume that any forces of interaction between pairs (or among groups) of particles are not very strong. Suppose that each particle has a number of energy levels available to it, and that the energies of these levels are {1,1' {1,2' {1,3' •.• , {1", .... Further, suppose

that the ground state contains n1 particles, the state of energy {1" contains n, par• ticles, and so on. The key assumption we make concerns [({1", I), the probability of finding one particle in the jth state. A rigorous derivation of the form ofJis beyond the scope of this work. Even a nonrigorous derivation is lengthy and will not be given here. Qualitatively, we expect thatJ({1", I) should decrease as {1" increases, and one of the simplest forms that satisfies this condition happens to be correct; so we shall assume this form: IL - (1,) J({1", I) = C 1 exp (U (1)

The quantity C1 is a constant, k is Boltzmann's constant, T is the absolute tem• perature, and IL is a constant for which the interpretation depends on the specific problem. Since the particles do not have much interaction with each other, the probability of finding two of them in the jth state will be the square of the one-body expression.

_ [ (IL - 2 J({1", 2) -C2 exp ~(1,,)] (2)

Clearly, we can generalize to n, particles in the jth state.

[({1", n) = C [ exp (UIL - {1, )]"' (3)

The next task is the evaluation of the constant C. The principle is simply that there must be some integer number of particles in the state, and so the sum of all the probabilities must add up to unity:

~J({1", n) = I (4) n 691 692 Appendix v-a

Using the explicit form in Equation (3), c ~ [exp (~;r~,) r= 1 (5) nj

It is convenient to denote the quantity in the brackets by y. Then 1 (6) c=--I ynj nJ

The probability of finding nJ particles becomes

(7)

The statistical distribution function is, by definition, the average number of particles in the state. This quantity can now be calculated.

f(~;) = ~ nJ f(~j' n) nj

(8)

The details. of the summation depend on the kind of particle. For Fermi particles the Pauli exclusion principle ensures that no two identical particles may occupy the same quantum state. Therefore nj will assume only the values 0 and I. y 1 fFD (&J) = I + y = I/y + I (9) 1 fFD(~J) = exp [(~J - ~)/(k1)] + I

For Bose particles nJ can be any integer from zero to infinity. So the distribution has the form fBE (~) = In = 0 n,ynJ (10) In=o yn J The infinite series in the numerator and denominator are most easily summed by long division. The results are

00 I ~ yn J = I + y + y2 + y3 + ... = I-y (11) nJ=o

Then the Bose-Einstein distribution becomes

fBE =y(l - y) =_y_=_I• (12) (I_y)2 I-y I/y-I

I fBE = exp [(&, - ~)/(k1)] _ I Appendix V-9 Probability Density Functions

In both quantum mechanics and statistical mechanics the idea of a probability density function (or statistical distribution function) has great importance. If a variable x can take on any value in some range, the probability of measuring x and getting Xo as an answer is zero, but if we ask for the probability that the result will lie between Xl and X2' the result is not zero in general. Typically there exists a function f(x) such that

(I)

where P means "the probability that .... " Since competent measurement must result in some answer, it is clear that f(x) satisfies the normalization condition

L: f(x) dx = I (2) The situation often occurs when we know f(x) but when x is not really the rele• vant physical quantity. Some other quantity y is the interesting parameter, and y(x) is a known monotonic function. The problem is to determine g(y), the probability density function for the new variable. The solution to this problem is contained in the integral property of a probability density, expressed in Equation (I). Since the

[(x) g(y)

L------~~~--x Y2

Figure A-3 Example of a transformed probability density function. 693 694 Appendix V-9

function y(x) is monotonic, there exists a one-to-one correspondence between values of x and values of y. Further, the probability of a certain physical result must be independent of whether it was measured as x or y. Therefore we expect that

P(xl < X < x2) = P(YI < Y < Y2) (3)

where Yl = y(xl) and Y2 = y(x2)· Using Equation (I), we get

(4)

This equation is illustrated in Figure A-3. The shaded areas are equal, as im• plied by the equality of the integrals in the equation. Since this equality must be true for every choice of Xl and X2' it follows that

f(x) dx = g(y) dy (5) This equation is the statement of the solution to our problem. The differentials dx and dy are essential to the result and must not be ignored. Equation (5) is correct whenever y is a monotonically increasing function of x; however, if y(x) is a monotonically decreasing function, then dx and dy will have opposite signs. Since f and g must be positive (to ensure that any probability is positive which results from integration), we see that an extra minus sign must be inserted into the equation when y(x) is decreasing. We illustrate the transformation of probability density functions by an example. Suppose x is the cosine of a scattering angle and it is observed that for many scat• terings the distribution of x is uniform between x =-I and x=+ I; that is,f(x) =t. We may inquire about the distribution of y where y is the scattering angle itself, bounded by zero and 1T radians. The functional relation between x and y is

x = cos y from which we obtain

dx = -sin y dy Here y is a monotonically decreasing function of x over the interval under con• sideration, so we suppress the minus sign that appears above. Then

f(x) sin y dy = g(y) dy or g(y) = t sin y Index 705

muonic,607 Hall, D., 520 nuclear, 550 Hallwachs, W., 93 Gaillard, J. M., 612 Halpern, J., 572 Galilean relativity, 28 Hamiltonian, 178 Galilean transformation, 29 Hanna, G. c., 458, 525 (")', d) reaction, 492 Hansen, W. W., 555 (")', n) reaction, 491 Harmonic oscillator, 199-203,330,683-686 (")', p) reaction, 492 Hart, E. L., 597 (")', t) reaction, 492 Harvey, B. G., 525 Gamma decay, 447-462 Haxby, R. 0., 523 Gamma ray, 50 Haxel, 0., 564 resonance at absorption, 452-454 Hayward, R. W., 501, 601 wavelength of, 124 Heisenberg, W., 171,549 Gamow, G., 440 Helicity, 447, 612 Gamow-Teller selection rules, 570 Helmholtz, A. C., 576 Gardner, E., 589 Henderson, J. E., 412 Gardner, F. T., 623 Hermite's equation, 200 Garwin, R. L., 605 Hertz, G., 228 Gasteiger, E. L., 524 Hertz, H., 25, 75, 93 Geiger, H., 57, 60, 147 Herzberg, G., 220 Geiger counter, 53, 112,404 Hess, V. F., 592 Geiger-Nuttall law, 439 hfs,548 Geiger's rule, 62 Hill, D. L., 518 Gell-Mann, M., 627, 635, 638, 645 Hill, R. D., 623 Gell-Mann-Nishijima formula, 628 Hillier, J., 181 General relativity, 458-462 Hoenigschmidt, 0., 50 Germer, L. H., 155-157 Hofstadter, R., 148,489,578-580,633 GeV,71 Holloway, M. G., 439 Ghiorso, A., 536, 537 Holography, 304 Glaser, D. A., 414 Hoppes, D. D., 501, 601 Goldhaber, Gerson, 640 Hoyle, F., 540 Goldhaber, M., 491, 498, 610 Hudson, A. M., 524 Gondsmit, S., 261 Hudson, R. P., 601 Goulianos, K., 612 Hughes, D. J., 520 Grating, ruled, 132 Hull, A. W., 101, 127 Grating space, 118 Hunt, F. L., 123 Gravitation, 45, 583 Hunt, S. E., 563 Gravitational mass, 46 Huygens, C., 75 Gray, J. A., 448 Hydrogen, 219-255 Grodzins, L., 610 Bohr's theory, 222 -228 Grosse, A. V., 495, 512 emission spectrum, 220 Ground state, 191 energy-level diagram, 226 atoms, 271-273 fine structure, 296 nuclei, 548 mass, 22 Group velocity, 169 -171 reduced mass, 229-233 Gurney, R. W., 440 Schroedinger equation, 233 -242 Gyromagnetic ratio, 290-292 selection rules, 242 muonic,607 spectral series, 224 - 226 nuclear, 550 wave functions, 238 Hypercharge, 636 Haga, H., 109 Hyperfine structure, 457, 548 Hagenow, C. F., 142 Hyperon, 616-619 Hahn, 0., 434, 508, 509 Half-life, 426 Ideal gas, 336-339 Answers to Selected Problems

Chapter 1 2-23. (K -13K ) -13K K I-I. 1.59 x 10-19 coul 1-2. (b) 1.84 x lOll coul/kg; (c) 1.83 x Chapter 3 1011 coul/kg 1-3. (a) 4.8 X 10-10 erg; (b) 1.03 x 109 3-1. (c) 55 min cm/sec 3-2. 8.5 x 10-6 1-4. 1.2 cm 3-3. (a) 11.4 x 10-12 cm; (b) 95 rp; 1-5. 2.64 x 101 m/sec (c) 11.4 X 10-12 cm 1-6. (a) Helical trajectories; axes paral• 3-4. 2.3 x 1014 gm/cm3 lel to B 3-5. (a) 8.80 MeV; (b) 2.06 x 109 R = mu sin (J cm/sec Be 3-6. (a) 13.27 em; (b) 5.30 MeV 3-7 (a) 1.17 mm; (b) 6.059 MeV; Chapter 2 6.098 MeV 2-1. 0.986 c 3-8. 8.82 em; 9.53 em 2-2. 0.94 c 3-9. (a) 0.315 kilogauss; (b) 313 em, . Wx + u 315 em 2-3. x= 3-10. 0.007781 u; 7.248 MeV I + uwx /c 2 . Wy 3-11. u=E/B y= 3-12. (a) R = MV/qB K(1 + uwxfc 2) 3-13. 5.93 x lOB reactions/sec Z = W Z 2 K(1 + uwx /c ) Chapter 4 2-5. (a) 2.3 mc2; (b) 1.3 mc2 2-8. 0.42 c 4-1. (a) 1.17 X 1023 cm/sec2; (b) 22.5 X 2-9.2xI03 c 10-10 nt/coul; 7.5 X 10-14 stvolt/ 2-10. u < 0.38 c cm; (c) 2.24 X 10-29 joule/m3; 2-11. u < 0.995 c 2.24 X 10-28 erg/cm3 2-15. 156 (mu6/c4) 4-2. 7.8 X 10-5 erg/sec 2-16. (a) u = 0.0199 c; (b) u = 0.99 c 4-3. (a) 2668 ft.; (b) 11.2 eV; (e) 15.83 2-17. 0.995c eV 2-18. 0.80 c 4-4. &:::s 1.5 eV 2-19. 1.18 MeV 4-5. (a) 1.8 X 1034 erg/sec; (b) 1.73 X 695 696 Answers to Selected Problems

10 15 kg/day; (c) 8.95 cal/cmz min; Chapter 7 4.5 times solar constant 7-2. a = nA/2 4-6. 4829 A 7-3. a = nA/2 4-7. 0.375 x 10-14 volt-sec 7-4. (a) e/a - 1/27Tn sin (27Tne/a) 4-8. (a) 1.222 keV; (b) 0.069 c (b) e/a 4-9. 59°C 2 (c) "3nz 7T Z(e/a)3 4-10. (a) 3.5 x 10-5 erg/cm3; (b) 2.96 x 7-5. (b) &n = nZ7T zhz/2maz IO-z dyne/stcoul; (c) 3.18 x 1017 tjJn = VfIa sin (n7Tx/a) photons/sec 7-11. R = I 4-11. (b) 1.9 x 10 18 photons/sec 7-12. &/Vo = 1.03 7-15. 1.64 x 10-34 joule Chapter 5 7-16. n = 3.74 x 1030 3 7-17. = 0; Ix2 > = '22 5-1. (a) 6.81 x lO-z1 cmz/atom; 1.97 x z 10-zl cmz/atom 7-18.

= 0; = tliza 7-19. 3 5-2. (a) 2.06 x IO-zo cmz/atom; 1.73 x tn 7-21. (a) ground state; (b) A = 5/Z; 10-19 cmZ/atom v'3Oia z 5-3. (a) 0.27 cmz/gm; (b) 1.22 x lO-z3 (c) = a/2; (d) = 2a /7; (e) more localized, i1x = 0.19 a cmz/atom 7-22. (a) T = 17 = kwAz/87T; WZ = kim; 5-4. AI: 0.051 cm; Cu: 0.00155 em (b) = = za z/4m, a4 = 5-5. (a) 4.07 x 10-15 joule sec/coul; Ii mk/liz (b) 6.6 x 10-34 joule sec 7-23. lim Prob = 4 i1x/a 5-6. 3.035 x 10-8 em n->OO 5-7. (a) 0.309 A; (b) 2° 55.4' 7-24. Transmission = 3.22 x 10-7 per- 5-8. (a) 5° 19'; (b) 10° 42'; (c) 10th cent order 7-25. 0.49,3.7,10.0,18.8,0.96,7.5,15 5-9. (a) 1.363 A; (b) () = I - /-L = 4.9 x 7-26. a = 1.94. X 10-13 em 10-7; (c) 3.42 min 7-27. ka = n7T; a = nA/2 5-10. (b) 13.5 min 7-28. 1~lz + 1~lz = I; therefore probabil- 5-12. (a) 0.024 A; (b) 28 min 5-13. (a) -43° 59'; (b) 579 eV ity is conserved. 5-14. (a) 0.172 A; (b) 2.79 x 104 eV 7-30. (a) t; (b) = t z 7-31. a ~ b, allowed; a ~ c, forbidden; 5-17. (b) = 4(hv)2 + 4hvmc p 4 hvc + 2 mc3 b ~ c, allowed 5-18. 0.136 liZ dZ liz d2 7-32. (a) [--+--+ 2ml dx lz 2mz dxzz

(&1 +&2)] tjJ= 0; (b) tjJI tjJz; (c) &1 +&z Chapter 6 7-33. (a) = 0; (b)

=lik; (c) A = 6-1. 6.11 x 10-13 em -lili 6-3. (a) 1.025; (b) 29° II'; (c) 9.56 x 7-34. Harmonic Oscillator 1011 em/sec; (d) 9.41 x 108 em/sec nodes n + 2 8 6-4. (a) 59.8 min; (b) 8.75 x 10 m/sec maxima {n/2 + t n odd 6-5. (a) A = hV27T/MkT; (b) 3.14 A n/2 + I n even 6-8. (b) 0.9789 minima {n/2 + t n odd 6-10. (e) u = dv/dk n/2 n even 6-11. 1.15 x 10 8 em/sec points of inflection n + 2 6-12. 65.8 MeV Square Well 6-14. Photon nodes n + I Answers to Selected Problems 697

1 9-9. (a) 1.88 eV, 2.92 eV; (b) 4 3PI - maxima {n/2 + -2 n odd n/2 n even 4 ISO 0 n/2 - -21 n odd 9-10. (a) 2542.6A, 1852.6A; (b) 84405.6 minima { n/2 n even points of inflection n - I 9-11. (c) 5.37 eV; (e) 1.84 eV 9-12. (a) 2; (b) ±4.635 X 10-17 dyne; (c) 62.7 m Chapter 8 9-13. (a) ±4.635 X 10-17 dyne; (b) 590 2n -I m; (c) 1.74 X 10 16 gauss/cm 8-1. ,,= cR --:...-• n4 - 2n3 + n2 9-15. (a) 4,3,2; (b) 1,0; (c) 5,4,3,2, I lim ,,= 2cR/n3 9-16. 2n2 n ~'" 9-17. (a) L = I, S = I,J=O; (b) ISO' ID2 8-3. ~A= 2.38 A 3p 3p 8-5. 1.85 eV; 12.06 eV; 13.94 eV; I 2 1 9-19. = 0, = 0, = 2 14.60 eV 1 8-6. 12,193 cm-I; 8201 A; 1.51 eV 9-20. 0,0, -2 9-24. iS x 8-7. n, = 5 - nf < 5; n, = 4 - nf < 4; etc. 9-25. iS y 8-8. e=O 9-26. 0,0,0 8-9. (a) 303.75 A, 256.3 A, ultraviolet; (b) 1640.25 A, 1180.01 A, ultra- Chapter 10 violet 10-1. 78.1 kV 8-10. 22.6 x 103s 10-3. 3.87 cm 8-11. (a) 40.8 volts; (b) 54.4 volts 10-4. (a) 5.37 keY, 5.93 keY, 7.26 keY; 8-12. 2.49 x 10-11 cm (b) 1.65 cm, 1.73 cm, 1.92 cm; (c) 8-13. (a) 7.29 X 10-3; (b) 4.14 X 10 14 0.16 cm, 0.38 cm rad/sec; (c) 1.26 10-5. (a) 72.68 keY, 94.60 keY, 94.75 8-14. 137 keY; (b) 315.6 cm, 361.34 cm, 8-15. 0.114 A 361.43 cm; (c) 2.88 X 10 10 cm/ 8-16. (c) k = wave number sec 8-19. 6ft2; 2ft 10-6. (b) 16.33 keY 8-20. 0; 0 10-7. (b) 7.05 keY 8-21. Allowed 10-8. 54.41 keY 8-22. Forbidden 8-23. Forbidden Chapter 11 8-24. 3ao/2 8-25. (c) 8 X 10-10 sec 11-2. (a) 2.42 X 10-21 gm; (b) 3.08 X 10-37 gm cm 2; (c) 2.23 X 10-6 eV 39 Chapter 9 11-3. (a) 7.4 X 10- gm cm 2; (b) 9.3 X 10-5 eV 9-1. L = 5, 4, 3, 2, I 11-5. 4 X 10-8 cm 9-2. (a)j= t, t; (b)j= v'63/2, V35/2; 11-7. k=2a2 D (c) 60°, 131 ° 12' 11-I2. Il(u)du= 9-3. (a)J=5, 4,3,2, I; (b)J' = 5, 4,3, cVfri312 u2 exp (-tmu2/ kT)dv 2, I 11-13. V8kT/m1T 11 9 7 5 3 1 9-4. (a) J = -2-, 2, 2, 2, 2' 2; 11-14. V3kT/m 9 7 5 3 1 (b) J = 2, 2, 2, 2, 2 11-15. (a) 99 percent; (b) 12~ eV 9-6. 2.65 X 1011 sec-1 11-23. (a) I joule (approx); (b) 2 X 104 9-7. 2.77 X 10-16 erg joule 698 Answers to Selected Problems

11-25. 0.494 (c) 7.3 x 1020 protons/cm 3 ; (d) 3.2 11-28. &F/k protons/cm3 11-32. (a) Ug = 13-12. 2.22 meters 31 2 r& L sin kL - (2k/ ( 2) sin aL 13-13. 8.25 x 10- cm 2 'V 2m L sin aL + (2kL/ a) cos aL' 13-14. n 3 1.371 21T 1& . kL 13-15. (a) 2 x 10 10 cm/sec; (b) 0.174 (b) Ug = -5 'V2m sm ; MeV; (c) 0° 13-16. 39° 6' (c)u =-21T~-sm & . kL g 5 2m 13-17. 0.525 MeV 13-18. (a) 0.566 c; (b) 11.0° Chapter 12 13-19. (a) 0.557 c; (b) 191 MeV 13-20. 295 MeV 12-1. (b) 17.2 MeV 13-21. 7.47 x 106 Hz for deuterons; 12-4. (a) 412 MeV; (b) 970 MeV/c; 7.37 x 106 Hz for protons (c) 0.72 c; (d) 191 cm 13-22. 1.2 MeV 12-5. (a) 153 MeV; (b) 766 MeV/c; (c) 0.42 c; (d) 165 cm 12-6. (a) 1.07 x 109 rad/sec; (b) 1.71 x Chapter 14 lOs Hz; (c) 8350 gauss 14-1. l.ll x lOs alpha particles/sec; 12-7. (a) 2.5 x 109 rad/sec; (b) 3.96 x 3 mCi lOs Hz; (c) 32.4 MeV 14-2. 1.126 x 10 14 alpha particles/sec; 12-8. 15 e V /cycle 3.045 x 106 mCi 12-10. (a) 4.4 x 107 m2/weber sec; (b) 14-3. l.ll x 10-6 gm 7 x 106 Hz; (c) 2.24 meters 14-6. (a) 1.69 X 10-17 sec-I; (b) 2.54 x l2-11. (a) 12.1 GeV/sec; (b) 70.7 eV/ 105 beta particles/sec; (c) 29.97 cycle; (c) 0.865 beta particles/sec 12-12. (a) 34 GeV/sec; (b) 714 eV/ 14-7. (a) 4.88 x 10-18 sec-I; (b) 1.41 x cycle; (c) 0.566 10 4 alpha particles/sec 12-13. (a) 17.5 eV/sec; (b) 1.06 x 10-7 14-8. (a) 2.098 X 10-6 sec-I; (b) 5.6 x eV/cycle; (c) 0.69c, 1.035 x lOs 10 13 alpha particles/sec rad/sec 14-9. 8.625 x 10 4 beta particles/sec 12-14. (a) 8 x.IO-3 amp; (b) 4.8 x 10-7 14-10. 91.2 x 103 beta particles/min amp; (c) 8.0 megawatts 14-11. (c) 3.13 cm 15 12-15. 6.25 x 10 protons/sec 14-15. (a) 5.40 MeV; (b) 0.102 MeV 12-16. 5 percent 14-16. (a) 10.74 MeV; (b) 282.9 MeV/c; 12-17. 16.5 cm (c) 0.20 MeV 12-18. RI = R2/e 300(Z - 2)e x 2 . 12-19. (a) 22.9 MHz; (b) 15.7 MeV 14-17. (a) & = , r m cm, r 12-20. 5.09 x 109 wavelengths e = 4.8 x 10-10 stcoul, & in eV; (b) 35.63 MeV; (c) 5.43 x 10-12 Chapter 13 cm 13-1. (b) 290 me; (c) 0.915 c 14-18. (a) 4.869 MeV; (b) 4.782 MeV 13-2. (a) 874 MeV; (b) 0.355 c 14-19. (a) 624.75 keV: (b) 662.19 keV 13-3. (a) 15 percent; (b) 5 percent 14-20. (a) 545.67 keV: (b) 661.26 keV 13-4. (a) 0.66; (b) 0.29; (c) 0.052 14-21. (a) 47 keV/c; (b) 5.65 x 1O-3 eV 13-8. (a) - 0.9 rad; (b) 0.09 rad 14-22. (a) 2.62 MeV/c; (b) 17.7 eV 13-11. (a) 0.36 x 1023 protons/cm3; (b) 14-23. = I/A 0.48 x 1023 protons/cm3; 14-24. 0.978 gm Answers to Selected Problems 699

14-25. (a) 1.067 x 10 12 alpha particles/ Chapter 16 day; (b) 3.416 x 10-7 coul 16-1. 193.5 MeV 14-26. 0.135 decay 16-2. Smaller mass 14-27. p = mA/2(TBn, n = no. of atoms/ 16-3. (a) 6.48 X 106 V; (b) 291.6 MeV cm3 in target 16-4. 171.8 MeV 14-28. (a) t (hV)2\ (b) 1.7 e V 16-5. (a) 213.6 MeV; (b) 260 MeV me 16-6. 0.02585 eV; 1.2 keV; 8.617 keV 2 14-29. (a) t P2 ; (b) e (BR)2; (c) 4 eV 16-7. (a) 0.151 MeV; (b) 0.094 MeV M 2 M 16-8. 1.465 x 10-12 cm; 1.27 rl 14-30. 1.22 X 10-7 sec 16-9. (a) 39.97 MeV; (b) -24.15 MeV; 3 14-31. 5.02 X 10- eV (c) 15.82 MeV _v'[1 + (v/e)] 14-32. () 16-10. (a) 71.06 MeV; (b) 3.25 MeV; a v - (v/e)2 VI _ (c) 74.31 MeV 14-33. 122.13 yr 16-11. (a) 96.62 MeV; (b)-11.64MeV; 14-34. t = In(A I/A 2)/(1/T2 - I/TI) with T= (c) 84.98 MeV T/ln 2 = mean life 16-12. About 17 16-13. 19.27 MeV Chapter 15 16-14. (b) 168.04 MeV 16-15. 3.125 x 10 10 fissions/sec 15-3. 3.361 MeV 16-16. (c) 0.192, 2.097 15-4. 14.1 MeV 16-17. 0.809 15-5. 8.00479 u 16-19. (a) 4.56 millisec; 15-6. 0.627 MeV (b) 0.721 microsec 15-7. 5.021 MeV 16-20. L ~ 9.8 meters 15-8. (b 1.0; (c) 8/9; (d) 0.284 16-22. 0.72 MeV 15-9. 5.063 MeV 16-23. C = V2/7Te 15-10. 3.367 MeV 16-24. 1.385 percent 15-11. (b) 10 min 16-25. (a) 1.2 MeV; (b) 9.28 x 109 K 15-12. (a) nCl (n,y) ~ ~gA + {3- + v; 16-26. (a) 2.75 x 10-13 em; (b) 3.145 (c) 36 min MeV; (c) 3.65 x 10 10 K 15-13. 2.970 MeV 16-27. Eq. (16-39) 1.442 MeV 15-14. (a) -4.036 MeV; (b) 4.278 MeV Eq. (16-40) 5.494 MeV 15-15. (a) -3.869 MeV; (b) 4.062 MeV Eq. (16-41) 12.859 MeV 15-16. (a) 3.982 MeV; (b) 4.964 MeV; Eq. (16-42) 19.795 MeV (c) 3.383 MeV; (d) 4.033 MeV; 16-28. In the order of Eq. (16-44): 1.944 (e) -15.669 MeV; (f) -12.126 MeV; 2.220 MeV; 7.550 MeV; MeV 7.293 MeV; 2.760 MeV; 4.964 15-17. (a) 2~~Th; (b) nNa; (c) ~H; (d) v+ MeV i8Ne; (e) I~O; (f) a 15-19. (a) dN = -AN dt + R dt; (b) N = Chapter 17 !i [1 - exp (-At)] A 17-1. (a) 7.275 MeV; (b) 2.425 MeV 15-20. 1.113 MeV; 7.976 MeV; 8.790 17-2. (a) 23Na; (b) 23Ne; (c) 2°F MeV; 8.548 MeV; 8.445 MeV; 17-3. (a) 1.802 A; (b) 6.374 x 10-12 cm 7.916 MeV; 7.570 MeV 17-4. (a) 11.84 MeV; (b) 23.67 MeV 15-23. (a) eu; (b) Zn 17-5. (a) 61. 7 f, (b) 54.3 f 15-24. 1;lC; I~; 19B; I~N; I~N; I~C 17-6. (a) 57 eV; (b) 9.6 eV; (c) 3.96 x 15-25 63.54 u 106 em/sec, 7.08 x 105 cm/sec 700 Answers to Selected Problems

17-7. (a) 7.0 eV; (b) Pe =0.160 MeV/c; 18-7. (a) 1.09 x 10-12 cm, 4.36 x 10-12 Pv = 0.13 MeV/c; PR = 0.177 cm; (b) 1.04 MeV MeV/c 18-8. (a) 4.8 x 10-12 cm, 19.2 x 10-12 17-8. (a) 1.42 keY; (b) Pe= 1.94 MeV/c, cm; (b) 45.6 keY Pv = 2.0 MeV/c,PR =0.48 MeV/c 18-9. (a) 1.4 x 1O- 12 cm,5.7 x 1O- 12 cm; 17-9. (b) 0.862 MeV; (c) 57.7 eV (b) 152 keY 17-10. (b) 1.710 MeV; (c) 4700 gauss cm 18-10. (a) 2.92 x 10-13 cm; (b) 0.41 17-11. (a) 0.1806; (b) 1.88 x 10-7 eV; 18-11. I05Mo (c) 0.393 cm/sec 18-12. (a) 7f ~ e + 1/; (b) 139.07 MeV 17-12. (b) -0.102; (c) 1.07 x 10-8 eV; 18-13. (a) 214.67 MeV; (b) 75.09 MeV; (d) 2.24 x 10-2 cm/sec (c) 388.16 MeV 17-16. (a) 380 MeV; (b) 0.207 MeV 18-14. (a) 5.62 GeV; (b) 4.03 GeV 17-17. 1.24 x 1036 18-15. 3.04 x 1017 cm; 0.323 light years 17-19. 0.04 18-16. 29.9 MeV/c; 4.5 MeV 17-22. (a) 28Si 18-18. 52.5 MeV 18-20. 6.8 eV Chapter 18 18-22. (a) 870 MeV 18-1. (a) 954 MeV/c; (b) 2.14 x 108 18-27. (a) C and T; (b) P and T m/sec 18-28. None 18-2. (a) 0.282 c; (b) 31 MeV/c; (c) 18-31. (a) 303.5 MeV/c; (b) 1236 MeV 20.6 cm 18-33. (a) 1= 1; (b) 1= 1, (c) 1=0 18-3. (a) 0.36 c; (b) 54 MeV/c; (c) 30.0 18-36. (a) QA = -"3,1 Qp ="3, 2 Qn = -"3;1 cm (b) ppn; (c) nH 18-4. (a) 157 MeV/c; (b) 259.8 MeV 18-37. (a) K+; (b) pn 18-5. (b) 14.3 meters, 6.5 meters 18-38. (a) 0.121 A; (b) t 18-6. KO ~ 7f+ + 7f- 18-41. (a) 75 MeV; (b) 4.15 X IQ-14 em Index

Abelson, P. H., 522 Antiproton, 620-625 Absorption coefficient, 134-136 Arago, F., 75 Absorption limit, 3 13 - 317 Aston, F. W., 66, 232 Absorption of x-rays, 134-136 Atom, nuclear theory, 57 Absorptivity, 85 Atomic absorption coefficient, 136 Accelerated charge, 76-79 Atomic bomb, 528 Accelerators, 363-393 Atomic mass number A, 69 Acceptor semiconductor, 254 Atomic mass unit, 9, 70 Acoustic spark chamber, 408 Atomic number, 310 Actinide, 537 measurement with x-rays, 139-141 Actinium series, 431 Atomic number Z, 68 AGS, 386-388 Atomic weight, 652 Allen, K. W., 525 Auger, P., 321 Allen, J. S., 572 Auger effect, 321 Allowed bands, 352 Aurora, 592 Alpha decay, 426, 435 -442 A verage lifetime, 426 barrier penetration, 211 A verage value of observable, 187 (lX, y) reaction, 482 -484 Avogadro's number, 4-8, 14, 134 (lX, n) reaction, 576 Azimuthal quantum number, 242, 248 (lX, p) reaction, 470-473 Alpha particle, discovery, 50 Bainbridge, K. T., 66 mass, 54 Baker, W. B., 478 nucleus of 4He, 55 Baldwin, G. C., 491, 523, 524 path in Coulomb field, 672-676 Balmer, J. J., 219 Q/M,51 Balmer series, 219-221, 224-226 range, 436-440 Band theory, 348-354 scattering, 58-64 Bardeen, J., 354 velocity measurement, 56 Barkla, C. G., 109, 141 Alpha-particle model, 558 Bam, 65 Alvarez, L. W., 373, 553, 632 Barrier penetration 208-212, 440-442 Ambler, E., 601 Baryon, 617 Anderson, C. D., 478, 585 Basov, N., 300 Anderson, H. L., 509 Batzel, R. E., 527 Andrade, E. N. da c., 124 Beam transport, 394-402 Angstrom, 120 Bearden, J. A., 120, 313 Angular momentum, diatomic molecule, 327 Becker, H., 474 hydrogen atom, 240-243 Becquerel, H., 3, 49, 402 orbital, 259 Bell, R. E., 497 of radiation, 103-105 Bending magnets, 395-397 spin, 261 Bergmann series, 257 - 259 total,262-264 Beta decay, 442-447,567-571 Angular quantum number, 248 induced, 481 Anomalous Zeeman effect, 292-294 non conservation of parity, 601-605 Antibaryon, 620-625 Beta particles, discovery, 50 Antilambda, 624 Betatron, III, 369-373 Antineutrino, 446 Betatron synchrotron, 380-3 84 Antiomega, 640 Beth, R., 103 Antiparticle, 594-599, 620-625 Bethe, H., 419, 538, 586 702 Index

Be V (see GeV) Chadwick,J., 57, 62, 63, 468, 474-476, 491, Bevatron, 384 498 Binding energy, 69 Chain reaction, 509 Binomial theorem, 43 Chamberlain, 0., 621 Birge, R. T., 232 Charge conjugation, 599-605 Blackbody radiation, 3, 85-92, 339-342 Charge of electron, 134 Blackett, P. M. S., 569 Charge independence, 557, 625 Bloch, F., 350, 419, 449, 553-555 Chemical potential, 334 Bohr, A., 559 Chow, C. N., 380 Bohr, N., 3, 62, 76, 172,220-228,246, Christophilos, N., 386 419,470,511,520,558 Cleavage planes, 114 Bohr magneton, 265 Closed shell, 270 Bohr radius, 223, 239 Cloud chamber, 413 muonic atoms, 586, 591 photographs of, 437, 443, 469, 478, 479, Bohr's frequency condition, 256, 312 493,494,510,588,596 Boltzmann's constant, 164, 334 Cloudy crystal ball, 559 Bondi, H., 540 Cluster model, 558 Bonner, T. W., 518 Cobalt 60 experiment, 601-605 Booth, E. T., 512, 520 Cocconi, G., 597 Born, M., 346 Cockcroft, J. D., 364,484 Bose, S. N., 261 Cockcroft-Walton generator, 364, 386 Bose-Einstein statistics, 332-335, 691 Coherent radiation, 300-304 Boson, 261, 332 Collective model, 559 Bothe, W., 147,474 Collins, G. B., 412 Brackett series, 224-226 Compound nucleus, 470, 558, 560-563 Bragg, W. L., 114, 124, 126 Compton, A. H., 142, 147 Bragg equation, 116 Compton effect, 143-149, 677-61SU Bragg's law, 127-129, 314 Compton wavelength, 144 for electrons, 157-159 Condon, E. U., 440 Branching, 433 Conduction band, 352 Bremsstrahlung, 122 Conduction electron, 97 - 99 inner, 449 Conductivity of solids, 348-354 outer, 448 Conservation laws, 4, 628 Brickwedde, F. G., 232 energy, 40 Brown, R. H., 614 four-momentum, 41 Brownian motion, 9-14, 669-671 momentum, 40 Bubble chamber, 414-418 Constants of nature, 649-651 Bubble-chamber photographs, 415, 618, 619, Contact potential, 94 623,624,639,640 Conversi, M., 586 Burcham, W. E., 485 Cook, C. F., 487 Butler, C. C., 614 Cook, C. S., 568 Coolidge tube, 110 Cahn, A., 520 Cooper, L. N., 354 Calcium spectrum, 297 - 300 Cooper pair, 354 Callihan, D., 532 Correspondence principle, 246 Camerini, U., 614 Cosmic ray, 478, 592-594 Carbon, 12C mass scale, 9 Cosmotron, 384 Carbon cycle, 538 Coulomb barrier, 562 Carbon dating, 495 Coulomb scattering, 58-65 Cascade hyperon, 619 Counter, Cerenkov, 411-413 Cathode rays, 19, 20 Geiger-Mueller, 404 Cauchois, Y., 125 proportional, 404 Cerenkov, P. A., 411 scintillation, 408-410 Cerenkov counter, 411-413 solid-state, 410 Index 703

Coupling, Russell-Saunders, 263 magnetic moment, 553-557 Courant, E. D., 386 orbital angular momentum, 556 Cowan, C. L., Jr., 573-575 Deutsch, M., 598 CPT theorem, 601, 605 Dewan, J. T., 525 Crane, H. R., 443, 572, 596 Diatomic molecule, 325 - 332 Crewe, A., 182 Dicke, R. H., 46 Critical angle, 130 Diffraction, atoms and molecules, 161 Critical mass, 529-533 electrons, 155 -160, 173 Critical voltage, 317 neutrons, 163 -166 Crookes dark space, 19 photons, 175 Cross, W. G., 147, 148 x-rays, 112-120 Cross section, 65 Diffuse series, 257 - 259 slow neutrons, 561 Diffusion cloud chamber, 414 Crystal, as x-ray spectrometer, 117 photograph of, 597 Curie, I., 508 Dirac, P. A. M., 76, 221, 261, 296, 620 Curie, M. S., 50 Direct reaction, 559 Curie, P., 50 Disintegration energy, 470 Curie (unit of radioactivity), 428 Disintegration of nuclei, artificial, 468 Curie-Joliot, 474, 479 natural (see Radioactivity) Curved crystal spectrometer, 125 Donor semiconductor, 353 Cyclotron, 367 - 369 Doppler effect, 452 -454 Cyclotron, FM, 377-380 gravitational, 460 Drift tube, 374 (d, a) reaction, 488 Duane, W., 123 (d, a) reaction, 488 Du Mond, J. W. M., 125 (d, p) reaction, 487 Dunning, J. R., 512, 520 Dabbs, J., 520 Dynode, 99 Daltiz, R. H., 632, 634 Dalton, J., 4 Eccentricity, 248 Danby, G., 612 Edge focusing, 397 Dash, J. G., 113 Effective multiplication factor, 529 Davisson, c., 155-157 Einstein, A., 3, 10, 13,25,31,45,69,76,90, Dead time, 405 96,261,300,342,671 De Benedetti, S., 599 Einstein-Lorentz transformation, 31-33 Debieme, A., 50 Einstein's A and B coefficients, 91 De Broglie, L., 76, 153, 192, 228 Electrochemical equivalent, 7 De Broglie wave in Bohr theory, 228 Electrolysis, 6 Debye, P., 143, 342 Electromagnetic force, 583 Debye frequency, 344 Electron, absent from nucleus, 549 Debye temperature, 345 charge, 8,14-18,134 Dee, 367 charge-to-mass ratio, 282, 286 Dee, P. I., 484 Compton recoil, 147 Deflecting magnets, 395 - 397 Compton wavelength, 144 Deformed nuclei, 559 conduction, 97 -99 Degeneracy, 205, 240, 250-252 diffraction, 155 -160, 173 Delayed neutron, 511, 515, 520 distribution in atoms, 270-275 Delta resonance, 630 elh, 355 Dempster, A. J., 66 elm, 20-22, 93, 282, 286 Denenstein, A., 96, 356 magnetic moment, 264-266 Density of states, 334 mass, 22 Detectors, 402-418 in solids, 346-354 Deuterium, 232 spin, 249, 261, 294-296 mass, 66 Electron capture, 322, 500-504 Deuteron, 233 Electron microscope, 179 -182 704 Index

Electron optics, 179 -182 Faraday's constant, 134 Electronic orbits, 243 - 246 Fermi, E., 261, 498, 508, 509, 568, 586, 594, Electron synchrotron, 380-384 629 Electron volt, 70 Fermi-Dirac statistics, 332-335, 691 Electrostatic accelerator, 364 - 366 Fermi energy, 97, 347, 348, 352 Electrostatic separator, 400 Fermi-gas model, 558 Elliott, L. G., 497 Fermi sea, 348 Elliptic orbits, 221, 247 -252 Fermi selection rule, 570 Elsasser, W., 563 Fermion, 97, 261, 332 Emissivity, 85 Ferrell, R. A., 518 Emulsion, nuclear, 402 Ferrite, 388 End-point energy, 444-447 Fine structure, 276 neutron decay, 500 of hydrogen, 221 Energy, alpha decay, 435 x-rays, 314 binding, 69 Fine-structure constant, 252 disintegration (Q), 470 Firestone, A., 640 end-point, 444-447, 500 Fission, 508-537 equipartition, 89 discovery, 508 equivalent to mass, 69 products, 515 -520 rest, 41 ternary, 524 stellar, 538 of uranium, 509-513 units, 70 Fitzgerald, G. F., 34 zero-point, 201 Fizeau, A. H. L., 75 Energy-level diagram, 434, 451, 455, 486, Flerov, G. N., 525 488,502,503,610 Fluorescence, 135 anomalous Zeeman effect, 293 x-ray, 322 calcium atom, 298 Flux, 198 decay of 226Ra, 448 Flux bar, 381 harmonic oscillator, 200 Focal circle, 125 He-Ne laser, 303 Focusing, magnetic, 396-399 hydrogen atom, 226, 296 Forbidden bands, 352 isomers of 234Pa, 434 Force, nuclear, 557 normal Zeeman effect, 288 Force laws, 583 rigid rotator, 328 Form factor, 579 sodium atom, 277 Foucault, J. B. L., 75 square well, 190, 207 Four-vector, 39 x-rays, 311-314 Fowler, P. H., 614 Energy loss, 419-421 Fowler, W. A., 513 Eotvos experiment, 46, 459, 462 Franck, J., 228 Equipartition of energy, 89, 338 Frank, I. M., 411 Equivalence, principle of, 45 Frauenfelder, H., 608 Eta meson, 634 Free particle, 194 -196 Ether, luminiferous, 25, 75 Freeman, J. M., 485 Euler's identity, 237 Fresnel, A., 75 eV,70 Fretter, W. B., 585 Evans, G. R., 501 Friedrich, W., 114 Even-even nuclei, 546 Frustrated total reflection, 211 Excitation energy, 548 Fry, W. F., 380 Exclusion principle, 270, 348 iT value, 571 Expectation value, 187 Fundamental particles, 583 -645 Exponential decay, 426 Fundamental series, 257 - 259 Fusion, 538-541 Factorization, 683 Faraday, M., 7 g factor, 290-292 706 Index

Identical particles, 332 Knipping, P., 114 Identification of particles, 419 -421 Koch, H. W., 491, 523, 524 Impact parameter, 673 Kolhoerster, W., 592 Impurity semiconductor, 353 Kowarski, L., 517 Independent particle model, 559 Kronig-Penney model, 349 Index of refraction, x-rays, 129 Kurie, N., 568 Induction, nuclear, 553 -556 Kusch, P., 553 Inertial mass, 46 Inertial system, 28 Lamb, W. E., 221, 297 Infinite square well, 188 -194 Lambda hyperon, 616 Infrared, 326 Lande g factor, 290-292 Inner bremsstrahlung, 449 Langenberg, D. N., 96, 356 Interaction, forces, 583 Langer, L. M., 571 Intermediate-energy nuclear physics, 576-578 Langmore, J., 182 Internal conversion, 321, 450 Langsdorf, A., 414 Internal energy, 337 Lanthanide, 537 Ion source, 367 Laplacian, 203, 234 Ionization, 419-421 Larmor's theorem, 284, 287, 550 Ionization chamber, III, 404 Laser, 300-304 Ionization potential, 228 Lattes, C. M. G., 586, 589 table of, 271-273 Laue, M., 109, I I3 Isobar, 445 Laue pattern, 116, 166 Isobaric spin, 625 Lawrence, E. 0., 367 Isomer, 433, 450-452 Leachman, R., 513 Isospin, 625 Lebedew, 101 Isotone, 564 Lederman, L. M., 605, 612 Isotope, 9 Lee, T. D., 601, 605, 615 mass, 66-71 Legendre's equation, 237 Isotope shift, 229-23, Lepton, 590, 641 Isotopic spin, 625 Libby, W. F., 495 I-spin, 625 Light, angular momentum of, 103 -105 circularly polarized, 82 - 84 Jensen, J. H. D., 564 corpuscular vs. wave theory, 75 Johansson, S. D., 623 linearly polarized, 79 Johnson, T. H., 162 momentum of, 101 - 103 Johnston, 495 polarized, 80-84 Joliot, F., 517 pressure of, 101 - I 03 Jones, W. M., 563 quantum, 76 Josephson, B., 212 speed,31 Josephson effect, 354-357 Lillie, A. B., 493 Linac, 373 - 377, 387 Ka line, 311 Lindner, M., 577 K capture, 500-504 Linear accelerator, 373 - 377, 387 K meson, 614-616 Liquid drop model, 511, 558 Kaon, 614-616 Lissauer, D., 640 Kaufmann, 44 Livingston, M. S., 367, 386,439 Kellogg, 553 Lord, J. J., 380 Kelvin temperature, 86 Lorentz, H. A., 34, 282 Kennedy, 522 Lorentz transformation, 3 I - 3 3 Kepler, J., 248, 673 Lorentz unit, 292 Kerst, D. W., 111,370 Luminiferous ether, 25, 75 KeY, 71 Lyman series, 224-226 Kinetic energy, 43 Knipp, 449 Magnet, bending, 395 - 3 97 Index 707

quadrupole, 397 - 3 99 Metastable state, 299, 301,451 Magnetic dipole moment, deuteron, 553 -557 MeV, 71 electron, 264-266 Meyer, N. c., 520 7Li, 553 Michelson-Morley experiment, 26-28, 31, 34 neutron, 553 Microwaves, 326 nuclei, 549-557 Miller, D. C., 28 proton, 553 Millikan, R. A., 14 -18, 95 Magnetic monopole, 641 Millman, S., 553 Magnetic quantum number, 242, 266-269 Mirror nuclides, 565 Magnetic resonance, 550 Mistry, N., 612 Magnetic spectrograph, 3 18 Model, liquid drop, 511, 558 Magneton, Bohr, 265 Models of nuclei, 557-567 nuclear, 550 Moderator, 531 Maimon,300 Moessbauer, R. L., 45, 454 Marney, M. c., 166 Moessbauer effect, 454-462 Marquez, L., 578 Moffat, R. D., 571 Marsden, E., 57, 60 Mole, 5 Marshak, R. E., 586 Molecular physics, 325 - 332 Maser, 28, 300 Moment of inertia, 327 Mason, F. D., 124 Momentum of radiation, 101-103 Mass, atomic, 8 Momentum space, 334-337, 340 equivalent to energy, 69 Morse, P., 330 gravitational, 46 Moseley, H. G. J., 309-311 inertial, 46 Moseley diagram, 310 neutron, 491 Moss, N., 525 nuclei from reactions, 473 Mottelson, B., 559 relativistic, 44 Muirhead, H., 586, 614 Mass absorption coefficient, 136 MukheIjee, N. R., 160 Mass excess, 471, 655-666 Multipole radiation, 450 Mass spectrometer, 66 Muon, 35, 585 -592 Matrix element, 206, 299 decay, 589 Mattauch, J. H. E., 66, 655 spin, 590, 607 Maxwell, J. C., 25, 75, 101,220 Murphy, G. M., 232 Maxwell-Boltzmann statistics, 334 Myers, W. D., 510 Mayer, M. G., 563, 564 McDaniel, B. D., 485 (n, a) reaction, 493 McElhinney, J., 524 (n, y) reaction, 497 Mcintyre, J. A., 148,489 (n, p) reaction, 495 McMillan, E. M., 378, 522, 576 Neddermeyer, S. H., 585 Mean life, 427 Ne'eman, Y., 635 Meitner, L., 508 Neptunium, 521-523 Menzel, D. H., 232 Neptunium series, 432 Meson, definition, 590 Neutrino, 446, 571-576, 609-613 discovery, 584-587 direct observation, 573 - 57 5 eta, 634 mass, 575 K,614-616 muon, 612 nonconservation of parity, 605 -607 Neutron, decay, 447 omega, 634 delayed, 511, 515, 520 pion, 586-591 diffraction, 163 -166 rho, 633 discovery, 474-476 tau, 614 economy in a reactor, 529-531 theta, 614 lifetime, 498-500 vector, 633 magnetic moment, 553 Mesotron, 585 mass, 68, 491 708 Index

Neutron (continued) Pancini, E., 586 prompt, 511 Parastatistics, 333 thermal, 164, 493, 517 Parity, 449, 570, 599-610 Newton, A. S., 526 square well functions, 207 N ewton, I., 75 Parker, W. H., 96, 356 Newtonian relativity, 28 Particle physics, 583 -645 Nichols, E. F., 101 Paschen series, 224 - 226 Nicol prism, 81, 103 Pauli, W., 270, 446, 548, 601 Nier, A. 0., 66, 512 Pauli exclusion principle, 97, 348 Nishijima, K., 627 Penetration of a barrier, 208 - 212 Noddack, I., 508 Periodic potential, 349 Notation, spectral, 275 Periodic table, 654 Nuclear binding energy, 69 Perkins, D. H., 586 Nuclear cross section, 65 Perlman, I., 523, 577 Nuclear emulsion, 402 Perrin, J., 10-14, 669 Nuclear fission, 508-537 Petrzhak, K. A., 525 See also Fission Pevsner, A., 634 Nuclear force, 557 Pfund series, 224-226 Nuclear fusion, 538-541 Phase stability, 377 Nuclear g factor, 550 Phonon, 343 Nuclear induction, 553 -556 Photodisintegration, 490 -492 Nuclear magneton, 550 Photoelectric effect, 93 -101 Nuclear models, 557 -567 Photofission, 523 Nuclear radius, 64, 560 Photomultiplier, 99 -10 I, 409 Nuclear reaction, 467-504 Photon, 3, 76, 96 energy, 470 diffraction, 175 notation, 472 Piccioni, 0., 586 Nuclear reactor, 509 Pickup reaction, 577 See also Reactor Pile (see Reactor) Nuclear spin, 548 Pion, decay scheme, 587 Nucleon, 68 discovery, 586 Nucleus, charge distribution, 579 lifetime, 590 mass, 69 neutral, 590 Pirenne, J., 598 Observable, 187 Pitchblende, 523 OcchiaJini, G. P. S., 586 Planck, M., 3, 76, 88, 96 O'Connor, P. R., 526 Planck radiation law, 87-92 Odd-odd nuclei, 546 Planck's constant,1{, 103 Okubo, S., 638 h, 76, 123 Omega meson, 634 Plasma, 332, 541 Omega minus, 638-640 Plutonium, 521-523 Operator, 178, 187 Polarization, beta particles, 607 -609 Oppenheimer-Phillips process, 577 neutrino, 612 Optical model, 559 x-rays, 141 Orbit, electronic, 243 - 246 Polonium, 50 elliptic, 247 -252 Positron, 594-599 Ore, A., 598 decay, 480 Outer, bremsstrahlung, 448 discovery, 478 Owen, G. E., 568 production in linac, 376 (p, ex) reaction, 484 Positronium, 597 (p, d) reaction, 485 Postulates, Bohr theory, 220-223 (p, 'Y) reaction, 485 quantum mechanics, 186 - 188 Packard, M., 555 special relativity, 30 Pair production, 595 Potential, contact, 94 Index 709

stopping, 95 Radium, 50 Pound, R. V., 461, 555 Radius of nucleus, 64 Powder, diffraction of x-rays from, 126, 165 Ramsey, N. F., 147, 148,553 Powell, C. F., 586, 614 Range, alpha particles, 436 -440 Powell, J. L., 598 Range-energy curves, 420 Poynting, J. H., 10:' Rayleigh-Jeans law, 87 -90 Precession, 267, 550 Reaction, notation, 472 Present, R., 524 nuclear, 467-504 Pressure of radiation, 101-103 Reactor, 509, 528-535 Principal quantum number, 240 breeder, 532 Principal series, 257 - 259 critical, 529 Principle of equivalence, 45 heterogeneous, 533 Probability density function, 693 homogeneous, 532 Probability in quantum theory, 175, 187 swimming-pool, 534 Prokhorov, A. M., 300 Rebka, G. A., Jr., 461 Prompt neutron, 511 Recoil-less emission, 455 Proper time, 38 Red-shift, 45, 458 Proportional counter, 404 Reduced mass, 326 Proton, magnetic moment, 553 definition, 230 mass, 68 hydrogen, 229-233 Proton-proton cycle, 538 Reflection, from barrier, 208 Proton synchrotron, 384-390 frustrated total, 211 Purcell, E. M., 555 from step potential, 197 pythagoras, 42 x-rays, 116, 130-133 Refraction of x-rays, 127-132 Q value, 470 Reiling, V. G., 412 Quadrupole magnet, 397 - 399 Reines, F., 573 -575 Quantum, 96 Relativistic mass, 44 Quantum of light, 76 Relativity, energy, 38-44 Quantum mechanics, 3, 186-215 fundamental postulates, 30 postulates of, 186 -188 general theory, 45 Quantum number, 191 kinetic energy, 43 angular, 248 length, 33 azimuthal, 242, 248 momentum, 38-40 harmonic oscillator, 200 theory, 3, 25-48 hydrogen atom, 239 time, 35 magnetic, 242, 266-269 velocity, 36 principal, 240 Resonance, 629-635 radial,248 Resonance absorption, gamma rays, 452 -454 three dimensions, 205 Resonance capture, 483 Quark,645 Resonance potential, 227, 281 Resonance radiation, sodium, 280 Rabi, I. I., 550, 553 Rest energy, 41 Radial quantum number, 248 Retherford, R. C., 221, 297 Radiation, from accelerated charge, 76-79 RF separator, 401 angular mOlJlentum, 103 -105 Rho meson, 633 blackbody, 3 Rigid rotator, 326-330 momentum, 10 I - 103 Rinehart, M. c., 518 Planck's law, 87-92 Ritson, D. M., 614 pressure, 101 -103 Roberts, R. B., 520 Radiative capture, 482 -484 Robinson, H. R., 318,442 Radioactive series, 428-433 Robson, J. M., 498 Radioactivity, 425 -452 Rochester, G. D., 614 discovery, 49 Rodeback, G. W., 572 710 Index

Roentgen, W. C., 109 Shoupp, W. E., 523 Rogers, E. H., 555 Shull, C. G., 165 Rosen, L., 513, 524 Siegbahn, K., 319 Rotator, 326-330 Siegel, R., 599 Row, 0.,160 Sigma hyperon, 617 Russell-Saunders coupling, 263 Sikkeland, T., 537 Rutherford, E., 50, 51, 54-64, 124,330,467, Simon, A. W., 147 468 Simultaneity, 35 Rydberg, J. R., 219, 257 Slack, F. G., 520 Rydberg's constant, 219, 231, 232 Smith, J. R., 487 Smoluchowski, M., 10 Salpeter, E. E., 540 Snell, A. H., 498, 521 Scattering, x-rays, 137-149 Snell's law, x-rays, 128 Schrieffer, J. R., 354 Snyder, H. S., 386 Schroedinger, E., 76, 221 Soddy, F., 9 Schroedinger equation, l76-179, 188 Sodium spectrum, 276-282 harmonic oscillator, 330, 683 -686 Solid-state detectllr, 410 hydrogen, 233 - 242 Solid-state physics, 342 - 357 rigid rotator, 327 Sommerfeld, A., 221, 247, 251 three dimensions, 203 Sound, propagation in solids, 343 Schwartz, M., 612 Space quantization, 269 Schwinger, J., 383 Spallation, 526, 577 Scintillation counter, 112, 408 -41 0 Spark chamber, 405 -408 Seaborg, G. T., 522, 523, 526, 527, 536, 537 Special relativity, hydrogen atom, 252 Secondary emission, 100 Specific heat, ideal gas, 338 Secular equilibrium, 462 solid, 342 - 346 Segre, E., 525, 62 I Spectral notation, 275 Selection rules, 205 - 208, 641 Spectral term, 219, 256 atoms, 299 Spectrograph, magnetic, 318 beta decay, 570 Spectrometer, rf, 401 dipole, 206 velocity, 400-402 gamma decay, 449 Spectrum, beta-ray, 442 -445 harmonic oscillator, 330 calcium, 297 - 300 hydrogen, 242 sodium, 276-280 particles, 628 x-ray, 120-124,309-324 rigid rotator, 328 Spin, 97 square well, 207 connection with statistics, 332 x-rays, 319-321 electron, 249, 261, 294-296 Sellin, J. M., 597 even-even nuclei, 549 Semiconductor, 353 muon, 590 Semi empirical mass formula, 558 nuclei, 548, 655 -666 Separation of variables, 203, 235 phonon, 343 Serber, R., 370, 578 Square well, 188 -194 Series, K and Lx-rays, 309 three dimensions, 203 - 205 optical, 257 Stability of nuclei, 546-548 radioactive, 428 -433 Statistical mechanics, 332 - 339 spectral, 224-·226, 257-259 distributions, 691 Sewell, D., 577 Fermi-Dirac, 97, 98 Shapiro, S., 356 Statistical model, 558 Sharp series, 257 - 259 Staub, H. H., 555 Shawlow, A. L., 300 Stefan-Boltzmann law, 86 Sheldon, B. M., 640 Steinberger, J., 612 Shell, electron, 270 Stellar energy, 538 Shell model, 559, 563 -567 Stellar evolution, 539 Index 711

Stenstrom, W., 127 step potential, 197 Step potential, 196 -199 Transmutation of elements, 468 Stephens, W. E., 523 Transuranic elements, 433, 521-523, 535- Stephenson, S. T., 124 537 Stem-Gerlach experiment, 294-296 discovery, 508 Stevenson, E. C., 585 Triggering, 413 Stimulated emission, 91, 300 Trilling, G. H., 640 Stokes'law, 12, 15, 17,669 Tritium, 487, 495 Stopping potential, 95 Triton, 492 Storage rings, 389 Tunnicliffe, P. R., 525 Strangeness, 627 Strassman. F., 508, 509 Uhlenbeck, G. E., 261, 449 Streamer chamber, 407 Uncertainty principle, 172 Street, J. c., 585 harmonic oscillator, 203 Street, K., Jr., 536 square well, 194 Stripping, deuteron, 577 Uniform particle model, 558 Strong focusing, 386, 399 Unitary symmetry, 635 -640 Strong interaction, 584 Uranium, 49, 50 SU(3), 635 -640 fission, 509 - 513 Suess, H. E., 564 Uranium series, 429 Sunyar, A. W., 610 Urey, H. c., 232 Superconductivity, 354-357, 390 Swiatecki, W. J., 510 Valence, 6 Synchrocyclotron, 377 - 380 Van Allen, J. A., 592 Synchrotron, 380-390 Van de Graaff, F. R., 364-366 Synchrotron radiation, 383 Van de Graaff accelerator, 385 Szilard, L., 517 Vector meson, 633 Vector model of the atom, 259-264 Tamm, I. E., 411 Vector notation, 667 Tandem accelerator, 366 Veksler, V., 378 Tau meson, 614 Velocity, de Broglie waves, 167 -171 Taylor, B. N., 96, 356 group, 169-171 Technetium, 310 wave, 169-171 Teller, E., 586 x-rays, 132 Term, spectral, 219, 256 Velocity selector, 73 Ternary fission, 524 Velocity spectrometer, 400-402 Terrell, J., 34 Von Halban, H., 517 Thermal neutron, 493, 517 Von Karman, T., 346 Theta meson, 614 Thiele, W., 655 Wahl. A. C., 522, 523 Thompson, S. G., 536 Walke, H., 501 Thomson, G. P., 159 Walker, R. L., 485 Thomson, J. J., 3, 9, 20 Walker, W. D., 633 Thorium series, 430 Wall, J., 182 Threshold, 491 Walton, E. T. S., 364, 484 Threshold frequency, 95 Walton, J. R., 537 Time, proper, 38 Wang, P., 520 Time-reversal, 599 -605 Wapstra, A. H., 655 Titterton, E. W., 524, 525 Watt, B. E., 519 Torr, 110 Wave function, 176-179, 186 Torrey, H. C., 555 harmonic oscillator, 202 Townes, C. H., 28, 300 symmetry under interchange, 333 Tracer, radioactive, 498, 522 Wavelength, Compton, 144 Transmission, barrier, 209 - 211 De Broglie, 154 712 Index

Wave mechanics (see Quantum mechanics) discovery, 3, 109 Wave number, 154, 194, 219 emission spectra, 120 -124 Wave packet, 168-171 energy-level diagram, 311 - 314 Wave velocity, 169-171 fine structure, 314 Weak interaction, 584 fluorescent spectra, 322 Wedge focusing, 396 index of refraction for, 129 Weinrich, M., 605 Kline,311 Weisskopf, V. F., 586 K and L series, 309 Weizsaecker, C. F. von, 558 mesic atom, 591 Wells, W. H., 523 polarization of, 141 Wertheim, G. K., 458 powder diffraction, 126 Weyl, H., 610 produced by betatron, 370 Wheeler, J. A., 511, 520, 598 production of, 109 - 111 Whitmore, B. G., 478 reflection, 116, 130 -13 3 Wiegand, C. E., 621 refraction, 127 -132 Wien filter, 73, 401 ruled gratings, 132 Wien's displacement law, 87, 341 scattering, 137- 149 Wigner, E., 558 selection rules, 319-321 Williams, E. J., 501 spectra, 309-324 Wilson, C. T. R., 15,413 tubes, 110 Wilson, H. A., 15 velocity of, 132 Wind, C. H., 109 wavelength standard, 120 Wolfgang, R. L., 495 Wollan, E. 0., 165 Yang, C. N., 601, 60S, 615 Wood, R. W., 279 Yield, 472 Wu, C. S., 449, 601 Young, T., 75 Wyckoff, H. 0., 412 Ypsilantis, T., 621 Wyman, L. L., 127 Yukawa, H., 584 Xi hyperon, 619 X-rays, 109 -152 Zacharias, J. R., 553 absorption, 134-136 Zeeman, P., 282 absorption limit, 313 - 317 Zeeman effect, 282 -289, 299 angular distribution, 142 anomalous, 292 -294 critical voltage, 317 Zero-point energy, 201 detection of, I II Zmn, W. H., 164,509,517 diffraction of, 112 - 120 Zweig, G., 645 CONVERSION FACTORS

1 atomic mass unit = 1.66 X 1O-27kg = 931 MeV 1 hertz = 1 cycle sec-1 1 tesla = 104 gauss 1 A = 10-10 m 1 eV = 1.6 X 1O-19 joule

300K=_1 eV 40 1 torr = 1 mm of Hg 1 curie = 3.7 X 1010sec-1 1 fermi = 1O-15 m

1 year ~ 1T X 107 sec

MASSES OF PARTICLES Particle Symbol mc2 (MeV) Photon '1 o Neutrino v o Electron e 0.511

Muon 106 Pion 140 135 Kaon 494 498 Proton p 938 Neutron n 940 lambda hyperon A 1116 Deuteron d 1876 PHYSICAL CONSTANTS FOR RAPID CALCULATION (for better values see Appendix I, p. 649)

Speed of light c = 3 X 108m sec-1 Electron charge e= 1.6X 1O-19C Planck's constant h = 6.63 X lO-34joule sec fI = 1.05 X lO-34joule sec Gas constant R = 8.31 X 103 joule kmole-1 K-1 Avogadro's number No= 6.02 X 1026 kmole-1 Boltzmann's constant k = 1.38 X 1O-23 joule K-1 Gravitational constant G = 6.67 X 10-11 newt m2 kg-1

CLUSTERS OF CONSTANTS

Electron charge-to-mass ratio elm = 1.76 X 1011 C kg-1 Bohr radius fl2/mke2 = 5.29 X 10-11 m Electron Compton wavelength h/mc = 2.43 X 1O-12 m Classical electron radius ke 2/mc 2 = 2.82 X 1O-15 m Bohr magneton eh/2m = 9.27 X 1O-24joule tesla-1 = 5.79 X 10-geV gauss-1

Fine-structure constant Q = ke2/flc = 1/137 flc= 1.97 X 10-11 MeV cm