Nuclear models: Fermi-Gas Model Shell Model

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1 Lectue Nuclea mode: Femi-Gas Model Shell Model WS/: Intoduction to Nuclea and Paticle Physics

2 The basic concept of the Femi-gas model The theoetical concept of a Femi-gas may be applied fo systems of weakly inteacting femions i.e. paticles obeying Femi-Diac statistics leading to the Pauli exclusion pinciple Simple pictue of the nucleus: Potons and neutons ae consideed as moving feely within the nuclea volume. The binding potential is geneated by all nucleons In a fist appoximation these nuclea potential wel ae consideed as ectangula: it is constant inside the nucleus and stops shaply at its edge Neutons and potons ae distinguishable femions and ae theefoe situated in two sepaate potential wel ach enegy state can be ocupied by two nucleons with diffeent spin pojections All available enegy states ae filled by the pais of nucleons no fee states no tansitions between the states The enegy of the highest occupied state is the Femi enegy F The diffeence B between the top of the well and the Femi level is constant fo most nuclei and is just the aveage binding enegy pe nucleon B /A 7 8 MeV.

3 n Numbe of nucleon states Heisenbeg Uncetainty Pinciple: The volume of one paticle in phase space: The numbe of nucleon states in a volume V: V 4π p F p dp V 4π p F π d d p n π π At tempeatue T i.e. fo the nucleus in its gound state the lowest states will be filled up to a maximum momentum called the Femi momentum p F. The numbe of these states follows fom integating eq. fom to p max p F : V 4π pmax π π 6π Since an enegy state can contain two femions of the same species we can have Neutons: N n V π p F p Potons: p Fn is the femi momentum fo neutons p Fp fo potons dp n Z V p F p V π p F

4 Use R R. A / fm n Femi momentum V V p 6π F 4 π 4 R π R The density of nucleons in a nucleus numbe of nucleons in a volume V: Femi momentum p F : p F 6π V n / 4π R two spin states 9π 4A R n / A A pf 6π 9π n 4A 4A R p 9π / R Afte assuming that the poton and neuton potential wel have the same adius we find fo a nucleus with nzn A/ the Femi momentum p F: / n p 9π pf pf pf 5 MeV 8 R Femi enegy: pf F M MeV c F The nucleons move feely inside the nucleus with lage momenta. M 98 MeV- the mass of nucleon 4 4

5 Nucleon potential V The diffeence B between the top of the well and the Femi level is constant fo most nuclei and is just the aveage binding enegy pe nucleon B/A 7 8 MeV. The depth of the potential V and the Femi enegy ae independent of the mass numbe A: ' V F B 4 MeV Heavy nuclei have a suplus of neutons. Since the Femi level of the potons and neutons in a stable nucleus have to be equal othewise the nucleus would ente a moe enegetically favouable state though -decay this implies that the depth of the potential well as it is expeienced by the neuton gas has to be lage than of the poton gas cf Fig.. Potons ae theefoe on aveage less stongly bound in nuclei than neutons. This may be undestood as a consequence of the Coulomb epuion of the chaged potons and leads to an exta tem in the potential: 5

6 Kinetic enegy The dependence of the binding enegy on the suplus of neutons may be calculated within the Femi gas model. Fist we find the aveage kinetic enegy pe nucleon: F F dn d dn d d d p F dn dp dp F dn dp dp p whee dn dp Const p V n 6π p F distibution function of the nucleons The total kinetic enegy of the nucleus is theefoe 5 whee the adii of the poton and the neuton potential well have again been taken the same. 6

7 Binding enegy This aveage kinetic enegy has a minimum at N Z fo fixed mass numbe A but vaying N o equivalently Z. Hence the binding enegy gets maximal fo N Z. If we expand 5 in the diffeence N Z we obtain The fist tem coesponds to the volume enegy in the Weizsäcke mass fomula the second one to the asymmety enegy. The asymmety enegy gows with the neuton o poton suplus theeby educing the binding enegy Note: this consideation neglected the change of the nuclea potential connected to a change of N on cost of Z. This additional coection tuns out to be as impotant as the change in kinetic enegy. 7

8 Shell model Magic numbes: Nuclides with cetain poton and/o neuton numbes ae found to be exceptionally stable. These so-called magic numbes ae The doubly magic nuclei: Nuclei with magic poton o neuton numbe have an unusually lage numbe of stable o long lived nuclides. A nucleus with a magic neuton poton numbe equies a lot of enegy to sepaate a neuton poton fom it. A nucleus with one moe neuton poton than a magic numbe is vey easy to sepaate. The fist exitation level is vey high: a lot of enegy is needed to excite such nuclei The doubly magic nuclei have a spheical fom nucleons ae aanged into complete shel within the atomic nucleus 8

9 xcitation enegy fo magicm nuclei 9

10 Nuclea potential The enegy spectum is defined by the nuclea potential solution of Schödinge equation fo a ealistic potential The nuclea foce is vey shot-anged > the fom of the potential follows the density distibution of the nucleons within the nucleus: fo vey light nuclei A < 7 the nucleon distibution has Gaussian fom coesponding to a hamonic oscillato potential fo heavie nuclei it can be paameteised by a Femi distibution. The latte coesponds to the Woods-Saxon potential e.g. appoximation by the Woods-Saxon potential: U U ectangula potential well with infinite baie enegy : a a> R a e a: appoximation by the ectangula potential well : U U < R R U U R < R R

11 Schödinge equation Schödinge equation: Single-paticle Hamiltonian opeato: Ĥ H ˆ U M igenstates: - wave function igenvalues: - enegy U is a nuclea potential spheically symmetic sin θ θ sin θ ϕ sinθ Angula pat: ˆ λ sin θ θ ˆ λ L ˆ sin θ ϕ sinθ L opeato fo the obital angula momentum Lˆ θ ϕ l l Υ θ ϕ ˆ Υlm L lm igenstates: Y lm spheical hamonics

12 M Radial pat The wave function of the paticles in the nuclea potential can be decomposed into two pats: a adial one which only depends on the adius and an angula pat Y lm ϕ which only depends on the oientation this decomposition is possible fo all spheically symmetic potentia: Fom 4 and > θ ϕ Υ θ ϕ lm > eq. fo the adial pat: ˆ L M l l M M Υ θ ϕ Υ θ ϕ lm lm 4 5 Substitute in 5: R d R l l 6 R R M d M

13 Constaints on q. fo the adial pat: d R l l R 7 M d M Fom 7 negy eigenvalues fo obital angula momentum l: : l s l p states l d l f Fo each l: -l< m <l > l pojections m of angula momentum. The enegy is independent of the m quantum numbe which can be any intege value between ± l. Since nucleons ao have two possible spin diections this means that the l leve ae l times degeneate if a spin-obit inteaction is neglected. The paity of the wave function is fixed by the spheical wave function Y ml and ˆ l P θ ϕ P Υ θ ϕ Υ θ ϕ eads l : lm lm sd.. - even states; pf - odd states

14 Main quantum numbe n q. fo the adial pat: M d R l l R d M 7 Solution of diffeential eq: y λ y l l l Bessel functions j l k A θ ϕ j l k Υ θ ϕ lm k M X X X X X X Bounday condition fo the suface i.e. at R: Rθϕ estictions on k in Bessel functions: j l k main quantum numbe n coesponds to nodes of the Bessel function : X nl M k R Xnl k R Xnl R X 8 nm 4

15 Shell model Thus accoding to q. 8 : Xnl nl nl Const X nl 9 MR Nodes of Bessel function negy states ae quantized stuctue of enegy states nl l n n n l n n l n l n l 4 n s states X X X p states X X d states X X 5.76 f states g states X 8. j j j j j 4 state s p d s f p g nl s p d s f p g C X nl C 9.86 C. C. C 9.5 C 48.8 C 59.7 C 64 degeneacy l states with Fist magic numbes ae epoduced highe not! Note: hee fo U ectangula potential well with infinite baie enegy nl

16 Shell model with Woods-Saxon potential Woods-Saxon potential: U U R a e The fist thee magic numbes 8 and can then be undestood as nucleon numbes fo full shel.this simple model does not wok fo the highe magic numbes. Fo them it is necessay to include spin-obit coupling effects which futhe split the nl shel. 6

17 7 Spin Spin-obit inteaction obit inteaction V U M H ˆ ˆ ˆ ϕ θ ϕ θ V U M ϕ θ ϕ θ V U M ˆ ϕ θ ϕ θ V V ˆ s l C V s l j Intoduce the spin-obit inteaction V a coupling of the spin and the obital angula momentum: whee eigenvalues spin-obit inteaction: total angula momentum: s l s l s l j j s l j s l igenstates

18 Spin-obit inteaction C j V l s θ ϕ V C s θ ϕ [ j j l l s ] Conside: j l : V C l l l l C l j l : V C l l l l C l This leads to an enegy splitting which linealy inceases with the angula momentum as l V It is found expeimentally that V is negative which means that the state with j l / is always enegetically below the j l / level. 8

19 Spin-obit inteaction The total angula momentum quantum numbe j l±/ of the nucleon is denoted by an exta index j: nl j Single paticle enegy leve: j j e.g. the f state splits into a f 7/ and a f 5/ state f f 5/ f 7/ The nlj level is j times degeneate Spin-obit inteaction leads to a sizeable splitting of the enegy states which ae indeed compaable with the gaps between the nl shel themselves. Magic numbes appea when the gaps between successive enegy shel ae paticulaly lage

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