Nuclear Reactions and Radiation
Nuclear Reactions and Radiation NUC ENG 101
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Date Created: 10/22/15
Homework 4 Due September 25 Lilley Chapter 1 Problem 15 Lilley Chapter 2 Problems 25 26 27 28 29 139064 Euuomufl 05510061me 2 He 91Wquot locum m1 Le39f39s 1m 4390 unhrsiwl Aed rm wwg 4 fo A qu We 3D Square Energy 1 NH 0 R 4 P M e Stq bLy a R 7 1 fm 534722 may For low on ne ScabAKA En lopeV g 0 2V Cam w EbCQHLRC 0 eshnde b 039 Z q6 barns Theory 92 BASIC NUCLEAR STRUCTURE Expt HHIIHI IIHH l I 103 104 euuon km 2 energy eV Flgure 46 The neutron proton scattering cross section at low energy Data taken from a review by FL K Adair Rev Mod Phys 22 249 1950 with additional recent results from T L Houk Phys Rev C 3 1886 1970 In Ae m5 SeaSW bvd39 m Sader 3 cWM Ng 5 Sn anquot 5 awe S I Gina l LN MAJLK 9 lo I I h S 0 5145169 L 039 6 Emu 43 4quot g 07 304 Louus 163 3 LN LMQ AMrhc in Cross SECHM hawem Sunglei ant P Su g I mule ne de em quot Wquot r m a Co M Sca kerRm tdquotolos w 4 Sokro hmw Eqa Lv mSts I square we r 3 39D donquott woi 7 3 About Ag tcaxgt THE FORCE BEWEEN NUCLEONS 89 ur R A V kr phfsi Sam Positive phase shift F A kr Phase shi Negative phase shift I lt I V V k 0 free particle e V0 attractive potential VD repu Isive potential Figure 44 The effect of a scattering potential is to shift the phase of the scattered wave at points beyond the scattering regions where the wave fuhction is that of a free particle A t Skaf l39 chS39famasl nuclem mulem w i39emcrm beams Krane Pu Sl39vE 100 Phase shift Triplet 5 wave Neutronproton scattering Singlet 5 wave l A 100 200 400 500 300 p wave Energy MeV Figure 412 The phase shifts from neutron proton scattering at medium en ergies The change in the swave phase shift from positive to negative at about 300 MeV shows that at these energies the incident nucleon is probing a repulsive core in the nucleon nucleon interaction A 3S1 0 1SO 0 1P1 Data from M MacGregor et al Phys Rev 182 1714 1969 Krane V MeV O V3r spin triplet L 0 r fm V1 spin singlet L 0 V r spin triplet L O Vsotr spin triplet L 1 gure 416 Some representative nucleon nucleon potentials Those shown hclude the attractive singlet and triplet terms that contribute to swave scattering he repulsive term that gives one type of pwave L 1 scattering and the attractive tensor and spinorbit terms All potentials have a repulsive cere at r 049 fm These curves are based on an early set of functional forms proposed by T Hamada and I D Johnston Nucl Phys 34 382 1962 other relatively similar forms are in current use Em on 99 a M watering 5km m we mean 4 Cowle was 9 MQCTIM 3 A WlM QM Simme lr E M e91 A Qamp s 50 5k N n 1 6 39demdlhs m MOW 7K9 San I P8 H 3 I Em Mom For gas mm 9 PM huts 39 C 1r9 9 CD Comwo39f JIS39I Mauulgl e C To 9 7 m P L 1r h n a p l 1 quot mannaquot aces mat LanSEJ VE ner bwl39 AE Aquot Z x k 0 C W k eee Rf s 1 335 an 1quot co 39h39owd 339 12 Le c131 335 Natl I Iv MW quot uWm range of nuclear 1 0ch Shell effects in atomic structure and spins of all electrons will cancel The total angular momentum Ior me nr Av i I i i l I I 20 b l 01 H O ionization energy eV 0 l i l I l I I l i J 1i Z Enge Fig 63 Ionization energies From R M Eisberg Fundamentals of Modern Physics New York John Wiley and Sons 1961 p 413 Reproduced by permission Evidence for shell effects in nuclear structure 03 gt L15 7 U 4kg 7 JD Tl mmw 005 39 de N 7 39 GIUW grin 0 a ollg m ll A9337 l 1 g 126 l l l l l l l l l V I 393 4 U 511 III 120 140 Numlmr ul39 odd nIuIonns Fig 518 Quadrupole distortion Q Z2 I ARI39R for addJ mn loi plnllwl the mid lllll h on IIlllI thl Z 0139 X From L W A BIujamin ImIY 1964 chl39odlll39i ll by 17mm 7 n 3 39III cI39 and Purlz rlm New YuI n Enge 120 BASIC NUCLEAR STRUCTURE IO I I I I E e g g a 39 7 I I 39 5 I20 12 13C 135 New number a l l l l l I Lu In a Ag L 2 100 D m I A 3 mo 5 Sr 7 53 pm v an I I o I 1x r I I I l I I I I quota I I I I ll l H l l l l l 1 20 30 AD 5 so 7 50 9 mo 11 120 1 0 New number b 9 I I I I I I I I I I I I I e 2 I rI l I 39 a n e I I l I I I I I II I I I I so so 40 e 5 I an g 100 no New quotmm In i 3 Additional evidence Vor nuclear Shell Struckure a Energles of parhcles emiued by i otope of Rn Note the su den Incr ase hen m daughter as 7 126 he hen t e parent has N 28 If the daughter nucl 5 IS m e hghuy bound rh a decay Is able to carry ay e energy Neutroncaprure cross sec Ions of varlous u I I Note he ecreases y roug Iy we order of magnItude near N 2 2 and 126 c Change In the nuclear charge ram 3 when 7 2 Note the su den at 2 28 82 nd 12 and compare wn FIgure 0 e V V the Slander H Fl ev C 14 731 1 S76 Krane ependence From E era et al Krane More evidence of shell effects in nuclei l T 4 3 zospb 2 39 6 i 4Ni 1140a 184w 1 a l Irgt 1 V E o I 5 asAry v 1102MO V V 1 39 86Kr 2 A a 3 132Te 14C 4 5 25 es e e 8 5 39 4 4 3 2 i Pb 1 l D 3 A V y Hf E O I I 9 A 3e aA 5 1 7 W J 1 Kr Cd 2 gt Ca 3 NI A4 5 J o 1 l O 50 100 150 Nucleon number Figure 52 Top Two proton separation energies of sequences of isotones constant N The lowest 2 member of each sequence is noted Bottom Two neu tron separation energies of sequences of isotopes The sudden changes at the indicated magic numbers are apparent The data plotted are differences between the measured values and the predictions of the semiempirical mass formula Measured values are from the 1977 atomic mass tables A H Wapstra and K Bos Atomic Data and Nuclear Data Tables 19 215 1 977 Recall Muchms an proin W1 39Fermims IZPSfl z39t M5 Com OM l 391 quot a 2 13M qaavd39um 541 For s m39R 0 3 4 mm 2 Hm we aw quotNerd mm 339 mg Q 39l 01w I 393 To39hQ Jaguarch of SME of quotKhan 0 pre x 264 m Krane Simple minded shell models for nuclei ls Infinite we 4s 3d 2g 1i 3p 21 1h 35 2d 1g 2p 1f 25 1d 13 ls Harmonic oscillator lt gtlt gtlt lt gtlt gtH 0GQG 2101826 Figure 54 Shell structure obtained with infinite well and harmonic oscillator potentials The capacity of each level is indicated to its right Large gaps occur between the levels which we associate with closed shells The circled numbers indicate the total number of nucleons at each shell closure More realistic nuclear potentials 5 53 OO 4Aa lt R r Figure 28 The Woodsisaxon or Fermi form of a realistic nuclear potential The distance over which 7 VrVn falls from 90 to 10 of its central value is 4 ln3 times the diffuseness parameter L Lilley Vo o Var exPlQquot K a gmecum I auhus L l ax 3 5pm 44395949 53 v0 3 50 MeV FIGURE 4 2 Comparison of shelltheory potentials for neutrons and protons The dashed curve is the potential for neutrons from 41 and 42 The proton potential shown by the solid curve is the sum of this and the coulomb potential The latter from 4 4 is shown by the dotdash curve V Lame WW coulomb only NW sfK proton potential neutron coulomb l l l 5 r neutron potential 3 lnuawwwmquot Cohen Lilley 45 as man 5 mm 3 u humanquot Cmp Nudem nudecm inMacHrM Aquian 0A velocitj o39F nwcilems I 5 gt gt Spinorbit term In nuclear potential V30 395 99 9 9 9 9 II39Xp SS182 106 BASIC NUCLEAR STRUCTURE Incident nucleon 1 I i assume V is negative SO Incident nucleon 2 Figure 414 Top view of nucleon nucleon scattering experiment All spins point up out of the paper incident nucleon 1 has r x p into the paper and thus S is negative giving a repulsive force and scattering to the left Incident nucleon 2 has r X p out of the paper resulting in an attractive force and again scattering to the left Krane MagWJSSwsaw Sass Hume 997 adJ sph 40f ot39l39 piefo Ifb 2i sumIUZMsau h For a 31m vMaf X 31th arid MM iQw gammaquot 934 orb39c39l quotspu 39l39njquot squ of Su 1 Lu w 34 w jd39 329 5 wJ JL IncaW 1 gt Magic am My of 4 rr 3 WW 39 5M6 w39K J 117 occur 14 gig q W7 4 5M 74 flu Effects of spinorbit interaction on nuclear potential FIGURE 4 4 The effect of the spinorbit Cohen interaction on the shelltheory potential It has no effect for l 0 so that curve is from 41 and 42 The spinorbit inter action introduces added attraction if I and s are parallel and added repulsion if I and S are antiparallel in the surface region Lilley 48 2g 3p 2f 35 1h 2d 2p 1f ZS 1d 1p Nuclear structure 18 30 6 26 14 2 22 10 18 6 14 2 10 6 2 Woods Saxon Woods Saxon plus well spinorbit coupling Chap 2 2g72 8 gg 4Su2 2 lt I 11152 15 1 2g92 10 lt 13 2f52 6 r 1i132 14 11192 10 1 sx 1h 2 12 quot 2quot5132 4 EII G HI x sta 6 all 2p1a 2 2 21332 4 c 16132 4 a l Ids2 u W 11337 4 1312 2 3d32 3d52 1i112 31312 31332 2f72 3512 1g72 1 392 1f52 2812 11312 4 6 12 2 4 8 2 8 10 6 2 2 Figure 29 Sequences of bound single particle states calculated for different forms of the nuclear shell model potential The number of protons and neutrons allowed in each state is indicated in parentheses and the numbers enclosed in circles indicate magic numbers corre sponding to closed shells