7. Nuclear models. 7.1 Fermi gas model. 7.2 Shell model

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1 7. Nuclear models 7.1 Fermi gas model 7. Shell model

2 Fermi Gas Model Free electron gas: protons and neutrons moving quasi-freely within the nuclear volume different potentials wells for protons and neutrons. Spherical square well potentials with the same radius Statistics of the Fermi distribution Given a volume V the numer of states dn goes like: dn = 4p dpv (h) n = p F 0 4p dpv (h) N = n = Vp F,n h = Vp F 6 h Z = Vp F,p h If T=0 the nucleus is in the ground state and p F (Fermi Momentum) is the maximum possible momentum of the ground state. N= number of neutrons Z= number of protons Spin 1/ particles

3 V = 4 R = 4 R 0 R 0 =1.1fm Z = N = / Fermi Energy = 4 R o p F p h F = p F,n = p F,p = h 9 50MeV /c R 0 8 E F = p F m N Binding Energy: BE/= 7-8 MeV V 0 =E F +B/~ 40MeV MeV Nucleons are rather weakly bound in the nucleus 1 The nucleon moves in the nucleus with a large momentum

4 Potential well in the Fermi-gas model The neutron potential well is deeper that the proton well because of the missing Coulomb repulsion. The Fermi Energy is the same, otherwise the p-->n decay would happen spontaneously. This implies that they are more neutrons states available and hence N>Z the heavier the nuclei become.

5 < E Kin >= p F 0 E Kin p dp p F p dp = 5 p F m N 0MeV 0 E Kin (N,Z) = N < E Kin,N >+Z < E Kin,Z >= p FN = N h = N 9 h V 4R 0 p FZ = Z h = Z 9 h V 4R 0 h 9 h Z E Kin = 19m N R 0 4R 0 E Kin = h R m N 4 N N Z = 10m N h R 0 (Np F,N 10m N + Zp FZ ) 9 5 (N Z) E Kin +... Decay N-Z

6 L espressione precedente si ottiene da: D = N Z N= + " N Z $ ' = 1 # + D Z= " N Z $ ' = 1 # D sostituendo in:!!! N 5 + Z 5 $ ## # = #" " # # " + D ( ) ( ) $ 5! + D $ # " e sviluppando i binomi arrestandosi al secondo ordine: 5 $! N 5 + Z 5 $ # " = 1 5!! $ # " D D D + 10 $ # 18 1 D # # "! N 5 + Z 5 $ # " =! 1 $ # " NOT BENE: 5! # + 10 " 9 D $ =! 1 $ # "! # + 5 " 9 ) Il risultato prova che l energia cinetica totale ha un MINIMO quando N = Z, e dato che E(totale) = T V, con T proporzionale ad, E(totale), che e negativa ed e anche la BINDING ENERGY totale, e proporzionale ad (primo termine dell equazione semiempirica delle masse) B) ggiungendo il secondo termine, come nell equazione semiempirica delle masse, la BINDING ENERGY diminuisce in modulo. parte I valori dei coefficienti, il modello a gas di Fermi predice correttamente ( N Z) EB = T V = c Vol + c Sym D $

7 Calculation of the state density Considering the approximation that the nuclear potential shold have a sharp edge in correspondence of the nuclear radius, one can approximate that to particles trapped in the pot potential h h = m m x + y + = E z h X(x) = E m x X X(x)... (r) = X(x)Y(y)Z(z) YZE X X + XZE y Y + XYE Z Z = EXYZ X(x) x = kx k = 1 h me a a Volume in which the state density is calculated X = e ik x + B e ik x Boundary Conditions : at x = y = z = a : X = Y = Z = 0 X + = a cosk + x X = i a sink x EX = h m k = 1 m h a E = 1 h ( m a x + y + z ) a p k + = + a, =1,,5.. k p = me = h ( a x + y + z ) =, = 0,,4... a

8 We count how many states we have in a spherical volume of radius between and + = x + y + z dn = 1 8 d = dn = a h p d = 4 a h dp = = a h p p = me p dp = m E de dn = m ( h ) 1 vde E = C 1 E de d = a h dp a h p dp = 4p dpv (h) Number of spin 1/ particles which sit in the well: E F n = dn E de de = C F 1v EdE = ( 8m )( h ) 1 ve F 0 0 Momentum volume Space volume Phase-space occupied by one state E F = maximal energy of the particle in the well dn/de E E F = 1 m E F in 4 n h v / Cu atoms : n v = 610 0MeV 9 64 = 810 cm E F 7eV

9 Hierarchy of energy eigenstates of the harmonic oscillator potential N = (n-1) + l E = h(n + /)

10 Skin thickness Shell Model Potential

11 Spin Orbit Coupling It changes the hierarchy of the energy levels: V (r) = V central (r) + V ls (r) < ls > h < ls > = 1 h ( j( j +1) l(l +1) s(s +1)) = l / j = l +1/ (l +1) / j = l 1/ E ls = (l +1) < V ls (r) > Moving below Moving above The energy levels transform into n lj levels.

12 Saxon-Wood Potential Shell model s 1h d 1g p 1f s 1d (l+1) p 6 Spin-orbit coupling 1s Single particle level calculated in the shell model.

13 Single-Particle-Spectrum of the Wood-Saxon Potential for a heavy nucleus

14 Energy of first excited nuclear level in gg-nuclei

15 Energy levels of some nuclei 4 He 16 8 O 40 0 Ca 48 0 Ca 08 8 Pb Excitation Energy

16 μ nucl = μ N Nuclear Magnetic Moments in Shell Model i=1 g l = 1 for p 0 for n (g l l i + g s si)/h g s = 5.58 for p.8 for n < μ nucl >= μ N < nucl g l l i + g s si nucl > /h i=1 Wigner-Eckart Theorem: The expectation value of any vector operator of a system is equal to the projection onto ist angular momentum < μ nucl >= g nucl μ N < J > h g nucl = i=1 < JM J g l l i + g s si JM J > < JM J J JM J > In the case of a single nucleon in addition to the close shell the angular momentum of the closed shell nuclei couples to 0. The nuclear magnetic Momentum of is equal to the nuclear magnetic mom. of the valence nucleon. g nucl = g l ( j( j +1) + l(l +1) s(s +1)) + gs( j( j +1) + s(s +1) l(l +1)) j( j +1) μ nucl μ N = g nucl J = g l ± g g s l J = l ±1/ l +1

17 Magnetic moments: Shell model calculations and experimental values