Intro to Nucler nd Prticle Physics (5110) Feb, 009 The Nucler Mss Spectrum The Liquid Drop Model //009 1
E(MeV) n n(n-1)/ E/[ n(n-1)/] (MeV/pir) 1 C 16 O 0 Ne 4 Mg 7.7 14.44 19.17 8.48 4 5 6 6 10 15.4.41 1.9 1.90 1. In fct, if we divide E by the number of α-prticle pirs, given by n(n-1)/, the result is roughly constnt, with vlue round MeV.. This gives us picture tht, t lest for light nuclei, lrge prt of the binding energy lies in forming α-prticle clusters, round 7 MeV per nucleon, s cn be seen from the binding energy of 4 He.. The much smller reminder, round 1 MeV per nucleon or MeV between pir of α-clusters, goes to the binding between clusters. 4. This phenomenon is usully referred to s the sturtion of nucler force. Tht is, nucler force is strongest between the members of group of two protons nd two neutrons, nd s result, nucleons prefer to form α- prticle clusters in nuclei. //009
The liquid drop model detiled theory of nucler binding, bsed on highly sophisticted mthemticl techniques nd physicl concepts, hs been developed by Brueckner nd coworkers (1954-1961). *** much SIMPLER model exists in which the finer fetures of nucler forces re ignored, but the strong inter-nucleon ttrction is stressed. It ws derived by von Weizsäcker (195) on the bsis of the liquid-drop nlogy for nucler mtter, suggested by Bohr. //009
B ve = B/ Over lrge prt of the periodic tble the binding energy per nucleon is roughly constnt. The mss density of nucler mtter is pproximtely constnt throughout most of the periodic tble. These two properties of nucler mtter re very similr to the properties of drop of liquid, nmely constnt binding energy per molecule, prt from surfce tension effect, nd constnt density for incompressible liquids. //009 4
The Essentil ssumptions of the liquid drop model: sphericl nucleus consists of incompressible mtter so tht R ~ 1/. The nucler force is identicl for every nucleon nd in prticulr does not depend on whether it is neutron or proton. V pn = V pp = V nn (V denotes the nucler potentil) The nucler force sturtes. //009 5
The binding energy of nucleus Definition: B(, ) = [ M + NM M (, )]c p n From the liquid drop model Weizsäcker s formul / ( N ) B(, ) = V S C 1/ ± δ + η Your text book s nottion (see eqns.46-.54 pp. 50-5) is similr with the first three terms being identicl V = 1 S = C = Note lso tht your textbook includes electron msses bit nnoying Crl Friedrich von Weizsäcker, 199 Germn physicist (191-007) //009 6
B(, ) = V S ( N / ) C 1/ ± δ +η V is the volume term which ccounts for the binding energy of ll the nucleons s if every one were entirely surrounded by other nucleons. ( Men Field ) S / is the surfce term which corrects the volume energy term for the fct tht not ll the nucleons re surrounded by other nucleons but lie in or ner the surfce. Nucleons in the surfce region re not ttrcted s much s those in the interior of nucleus. term proportionl to the number of nucleons in the surfce region must be subtrcted from the volume term. //009 7
B(, ) = V S ( N / ) C 1/ ± δ +η C 1/ is the Coulomb term which gives the contribution to the energy of the nucleus due to the mechnicl potentil energy of the nucleus chrge. ssume chrged sphere of rdius r hs been built up, s shown in the figure (). The dditionl work required to dd lyer of thickness dr to the sphere cn be clculted by ssuming the chrge (4/) πr ρ of the originl sphere is concentrted t the center of the shell [see figure (b)]. The electricl potentil energy of the nucleus is therefore V Coulomb = 0 R 4 πr 16 = π ρ R 15 ρ (4πr 5 = 5 e R 1 drρ) r ρ = where e 4 π R nd 1/ R ~ //009 8
B(, ) = V S ( N / ) C 1/ ± δ +η Three terms tht were discussed previously re in sense clssicl. The following terms tht re to be discussed re quntum mechnicl. (1) the symmetry term ( N These include ) () the pring term () the shell effect correction term is the symmetry term which ccounts for the preference for = N (giving the most strongly bound nuclei for give ), if ll other fctors were equl, Textbook: - 4 (-/) / nd 4 = but N-=(-)- (N-) =(-) =4(-/) *** 4 = = 4 //009 9
B(, ) = V S ( N / ) C 1/ ± δ +η The Puli exclusion principle sttes tht no two fermions cn occupy exctly the sme quntum stte. t given energy level, there re only finitely mny quntum sttes vilble for prticles. Different system energies due to symmetric configurtions //009 10
1. If = N, then both wells re filled to the sme level (the Fermi level).. If we move one step up wy from tht sitution, sy in the direction of N > (or > N), then one proton must be chnged into neutron. ll other things being equl (including equl proton nd neutron mss), this stte hs energy E greter thn the initil stte, where E is the level spcing t the Fermi level.. second step in the sme direction cuses the energy excess to become E. 4. next step mens moving proton up three rungs s it chnges from proton to neutron nd the excess becomes 5 E. //009 11
Cumultive effect N- 4 6 8 10 1 14 16 18 0 4 6 8 Energy Diff 1 5 8 1 18 5 41 50 61 7 85 98 (N- )^ / 8 0.5 4.5 8 1.5 18 4.5 40.5 50 60.5 7 84.5 98 5. Therefore to chnge from N = 0 to N >, with = N + held constnt, requires n energy of ~ E (N ) /8. 6. This is independent of whether it is N or tht becomes lrger nd it mens tht, if ll other things re equl, nuclei with = N hve less energy nd re therefore more strongly bound thn nucleus with N. 7. The energy levels of prticle in potentil well hve spcing inversely proportionl to the well volume, thus we put E ~ -1. This is the symmetry term. ( N //009 1 )
B(, ) = V S ( N / ) C 1/ ± δ +η ±δ is the piring term which ccounts for the fct tht pir of like nucleons is more strongly bound thn is pir of unlike nucleons. 1. For odd nuclei ( even, N odd or odd, N even) δ = 0.. For even there re two cses; (). odd, N odd (oo) δ (b). even, N even (ee) + δ (, ) = P δ, = 1 MeV 1 / P Textbook: P = 5 //009 1
B(, ) = V S ( N / ) C 1/ ± δ +η η is the term ccounts for the nucler shell effect when or N is some mgic number. This term is much less importnt thn other terms. Therefore this term is not included in most of the pplictions. fvorble set of vlues for the coefficients: 1 = V = 15.560 MeV = S = 17.0MeV = C = 0.6970 MeV 4 /4= =.85 MeV 5 = P = 1.000 MeV //009 14
//009 15 ± δ +η = N B C S V 1/ / ) ( ), ( Contributions from the vrious terms in the Liquid Drop Model semi-empiricl mss formul, in terms of the binding energy per nucleon vs. mss number, except for the piring term.
How well does it work? Dt-Model Comprison Tble of Nuclides: http://tom.keri.re.kr/ton/min.shtml 77 Ge : B = 667.670 MeV (ner stbility line) 5 Mg 1 B = 56.816 MeV (fr from stbility line) 4 U 9 B = 18.669 MeV (close to stbility line) Hyperphysics: Liquid Drop Model Clcultor http://hyperphysics.phy-str.gsu.edu/hbse/nucler/liqdrop.html =77, =: B=667.41 MeV (-0.06%) =5, =1: B=47.558 MeV (-.6%) =4, =9: B=185.81 MeV (+0.17%) //009 16
Mgic Numbers of where binding energy is lrger! //009 17