Data Provided: A formula sheet and table of physical constants is attached to this paper. Answer question ONE (Compulsory) and TWO other questions.

Transcription

1 Dt Provided: A formul sheet nd tble of physicl constnts is ttched to this pper. DEPARTMENT OF PHYSICS AND ASTRONOMY Spring Semester ( ) NUCLEAR PHYSICS 2 HOURS Answer question ONE (Compulsory) nd TWO other questions. All questions re mrked out of twenty. The brekdown on the right-hnd side of the pper is ment s guide to the mrks tht cn be obtined from ech prt. Plese clerly indicte the question numbers on which you would like to be exmined on the front cover of your nswer book. Cross through ny work tht you do not wish to be exmined. 1 TURN OVER

2 1 COMPULSORY () Write down the equtions for the energy relese Q for the three processes of electron emission, positron emission nd electron cpture, crefully defining the symbols tht you use. (b) Briefly outline the mens by which nuclei my come to hve nucler mgnetic moment nd n electric qudrupole moment. (c) Briefly explin the term nucler force. Include sketch to illustrte the force experienced between two nucleons s function of distnce. (d) Explin briefly the Sxon-Woods form of chrge distribution in nuclei. Include suitble sketches nd stte how the distribution cn be experimentlly determined. (e) Outline the concept of n isospin doublet in the context of nucler physics. 2 CONTINUED

3 2 Mny interesting properties of nuclei re reveled by the chrt of nuclides nd the curve of verge binding energy per nucleon s function of tomic mss number A. () Sketch plot of the chrt of nuclides. Define the terms isotones, isotopes, isobrs nd islnd of stbility, using your plot to illustrte ech of these. (b) Isodiphers re nuclides with the sme difference between neutrons nd protons. Illustrte isodiphers on your plot. [6] (c) Sketch plot of the verge binding energy per nucleon vs. tomic mss number A. Show on this plot how fission nd fusion occurs nd stte which isotope hs the mximum verge binding energy. (d) The semi-empiricl formul for the binding energy of nucleus cn be written B(A,Z) v A s A 2/ 3 c Z(Z 1)A 1/ 3 (A 2Z) 2 A 1 p A 1/ 2 where vlues for the constnts v, s, c, nd p, cn be tken s 14.0, 13.0, 0.6, 19.0 nd 12.0 MeV. Although this eqution is useful for estimting the binding energy of most nuclei there is clss of nuclei for which it underestimtes B. Explin the chrcteristics of these nuclei nd where in generl they lie on the chrt of nuclides. (e) If we consider nuclei with the sme odd vlue of A, then the eqution for the binding energy B is prbol s function of Z. Find n expression for the Z of the single stble isobr tht occurs in this sitution. Find the Z for the stble isobr with A = 127. [6] 3 TURN OVER

4 3 The isotopes 9 Be nd 12 C re commonly used in the nucler industry. Like ll nuclei 4 6 they show properties tht re comptible with so-clled liquid drop model, in which the nucleons involved behve collectively. However, they lso behve s if the nucleons involved fill specific energy levels, comptible with the so-clled shell model. () Briefly describe, with the id of suitble plots or sketches, six pieces of empiricl evidence tht support the shell model of nuclei. [6] (b) Write simple expression for the rdius R of nucleus in terms of the nucleon rdius nd the number of nucleons contined. Stte wht ssumption is mde in this expression tht is comptible with the liquid drop model. Using this expression give estimtes of the rdii of 9 4 Be nd 12 6 C. (c) In the shell model the energy levels re ssigned in order of incresing energy s, 1s 1/2,1p 3/2,1p 1/2,1d 5/2,2s 1/2, 1d 3/ 2,1f j,... [6] Use this sequence, dding further levels s required, to determine the ground stte spin nd prity of the following nuclei, 9 Be, 12 C, Al, Sc (d) On the bsis of the shell model for nuclei tht hve n unpired nucleon, first excited stte cn be produced by excittion of the unpired nucleon into the next subshell. Determine the spin nd prity for the first excited stte of 9 4 Be. (e) 9 Be is used in the nucler industry s neutron reflector nd cn lso be used s 4 source of neutrons by bombrdment with lph prticles. Write out the nucler interction equtions tht describe these two processes. 4 CONTINUED

5 4 The study of nucler rections provides n importnt tool for our understnding of the excited sttes of nuclei. () Briefly describe the chrcteristics of so-clled compound nucler rections nd of direct rections. Include description of the independence hypothesis in compound interctions. Give n exmple of compound rection. [6] (b) Consider the following rection involving protons nd nickel nuclei, p Ni p Ni * Stte wht specific type of direct rection this is nd explin the mening of the nottion used. (c) The figure below shows the excited sttes of 64 Ni. These sttes cn be produced by bombrding 64 Ni with bem of protons if sufficient energy is trnsferred from the incident protons. Under the ssumption tht ll the kinetic energy of the interction is tken by the proton nd tht the nickel nucleus is excited into the first excited stte reltive to the ground stte, wht is the kinetic energy of the proton fter the interction if the initil proton energy is 11 MeV? (d) Now use conservtion of momentum to estimte the momentum nd kinetic energy of the recoiling 64 Ni nucleus for the sitution where the protons recoil t 60 0 from the bem direction. Re-estimte the proton energy by imposing energy conservtion tking ccount the recoil of the 64 Ni nucleus. [6] (e) As cn be seen in the figure in prt (c) the gp between the ground stte nd first excited stte is quite lrge. Briefly explin why this is the cse nd why the gps between the energy levels bove bout 3 MeV re much smller. 5 TURN OVER

6 5 Nucler power cn be generted either by the process of fission or, hopefully in the future, fusion rections. () Consider the fusion of two deuterium nuclei with relese of single neutron. How much energy is relesed per fusion? (note: the nucler msses of deuterium nd 3 He re kg nd kg respectively). [5] (b) Briefly describe the process of neutron-induced nucler fission of 235 U nd its use in power genertion. Include in your discussion explntions of the terms modertor, prompt nd delyed neutrons, chin rection nd fission frgments. [5] (c) Consider neutrons of energy 0.1 ev incident on piece of nturl urnium. The cross section for fission of 235 U t tht energy is 250 brns. The mount of 235 U in nturl urnium is 0.72% nd the density of urnium is 19 g cm 3. Estimte the men free pth length in cm for the neutrons, ssuming this is dominted by fission of 235 U. (d) Tking the flux of neutrons in prt (c) to be neutrons s 1 cm 2 nd given tht ech fission produces 165 MeV, estimte the initil nucler power produced in wtts if the piece of urnium is 1 cm 3. Stte wht ssumptions you mke bout the geometry of the piece of urnium. (e) Using your nswer to prt () nd (d) estimte the mss of deuterium tht must undergo fusion per second to mtch the power output for the fission rection. END OF EXAMINATION PAPER 6 CONTINUED

7 PHYSICAL CONSTANTS & MATHEMATICAL FORMULAE Physicl Constnts electron chrge e = C electron mss m e = kg = MeV c 2 proton mss m p = kg = MeV c 2 neutron mss m n = kg = MeV c 2 Plnck s constnt h = J s Dirc s constnt ( = h/2π) = J s Boltzmnn s constnt k B = J K 1 = ev K 1 speed of light in free spce c = m s m s 1 permittivity of free spce ε 0 = F m 1 permebility of free spce µ 0 = 4π 10 7 H m 1 Avogdro s constnt N A = mol 1 gs constnt R = J mol 1 K 1 idel gs volume (STP) V 0 = 22.4 l mol 1 grvittionl constnt G = N m 2 kg 2 Rydberg constnt R = m 1 Rydberg energy of hydrogen R H = 13.6 ev Bohr rdius 0 = m Bohr mgneton µ B = J T 1 fine structure constnt α 1/137 Wien displcement lw constnt b = m K Stefn s constnt σ = W m 2 K 4 rdition density constnt = J m 3 K 4 mss of the Sun M = kg rdius of the Sun R = m luminosity of the Sun L = W mss of the Erth M = kg rdius of the Erth R = m Conversion Fctors 1 u (tomic mss unit) = kg = MeV c 2 1 Å (ngstrom) = m 1 stronomicl unit = m 1 g (grvity) = 9.81 m s 2 1 ev = J 1 prsec = m 1 tmosphere = P 1 yer = s

8 Polr Coordintes x = r cos θ y = r sin θ da = r dr dθ 2 = 1 ( r ) + 1r 2 r r r 2 θ 2 Sphericl Coordintes Clculus x = r sin θ cos φ y = r sin θ sin φ z = r cos θ dv = r 2 sin θ dr dθ dφ 2 = 1 ( r 2 ) + 1 r 2 r r r 2 sin θ ( sin θ ) + θ θ 1 r 2 sin 2 θ 2 φ 2 f(x) f (x) f(x) f (x) x n nx n 1 tn x sec 2 x e x e x sin ( ) 1 x ln x = log e x 1 x cos 1 ( x sin x cos x tn ( 1 x cos x sin x sinh ( ) 1 x cosh x sinh x cosh ( ) 1 x sinh x cosh x tnh ( ) 1 x ) ) 1 2 x x 2 2 +x 2 1 x x x 2 cosec x cosec x cot x uv u v + uv sec x sec x tn x u/v u v uv v 2 Definite Integrls x n e x dx = n! (n 0 nd > 0) n+1 π e x2 dx = π x 2 e x2 dx = 1 2 Integrtion by Prts: 3 b u(x) dv(x) dx dx = u(x)v(x) b b du(x) v(x) dx dx

9 Series Expnsions (x ) Tylor series: f(x) = f() + f () + 1! n Binomil expnsion: (x + y) n = (1 + x) n = 1 + nx + k=0 ( ) n x n k y k k n(n 1) x 2 + ( x < 1) 2! (x )2 f () + 2! nd (x )3 f () + 3! ( ) n n! = k (n k)!k! e x = 1+x+ x2 2! + x3 x3 +, sin x = x 3! 3! + x5 x2 nd cos x = 1 5! 2! + x4 4! ln(1 + x) = log e (1 + x) = x x2 2 + x3 3 n Geometric series: r k = 1 rn+1 1 r k=0 ( x < 1) Stirling s formul: log e N! = N log e N N or ln N! = N ln N N Trigonometry sin( ± b) = sin cos b ± cos sin b cos( ± b) = cos cos b sin sin b tn ± tn b tn( ± b) = 1 tn tn b sin 2 = 2 sin cos cos 2 = cos 2 sin 2 = 2 cos 2 1 = 1 2 sin 2 sin + sin b = 2 sin 1( + b) cos 1 ( b) 2 2 sin sin b = 2 cos 1( + b) sin 1 ( b) 2 2 cos + cos b = 2 cos 1( + b) cos 1 ( b) 2 2 cos cos b = 2 sin 1( + b) sin 1 ( b) 2 2 e iθ = cos θ + i sin θ cos θ = 1 ( e iθ + e iθ) 2 nd sin θ = 1 ( e iθ e iθ) 2i cosh θ = 1 ( e θ + e θ) 2 nd sinh θ = 1 ( e θ e θ) 2 Sphericl geometry: sin sin A = sin b sin B = sin c sin C nd cos = cos b cos c+sin b sin c cos A

10 Vector Clculus A B = A x B x + A y B y + A z B z = A j B j A B = (A y B z A z B y ) î + (A zb x A x B z ) ĵ + (A xb y A y B x ) ˆk = ɛ ijk A j B k A (B C) = (A C)B (A B)C A (B C) = B (C A) = C (A B) grd φ = φ = j φ = φ x î + φ y ĵ + φ z ˆk div A = A = j A j = A x x + A y y + A z z ) curl A = A = ɛ ijk j A k = ( Az y A y z φ = 2 φ = 2 φ x + 2 φ 2 y + 2 φ 2 z 2 ( φ) = 0 nd ( A) = 0 ( A) = ( A) 2 A ( Ax î + z A ) ( z Ay ĵ + x x A ) x y ˆk