Nuclear models: Shell model

Transcription

1 Lectue 3 Nuclea models: Shell model WS0/3: Intoduction to Nuclea and Paticle Physics,, Pat I

2 Nuclea models Nuclea models Models with stong inteaction between the nucleons Liquid dop model α-paticle model Shell model Models of non-inteacting nucleons Femi gas model Optical model Nucleons inteact with the neaest neighbos and pactically don t move: mean fee path λ << R A nuclea adius Nucleons move feely inside the nucleus: mean fee path λ ~ R A nuclea adius

3 III. Shell model 3

4 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:, 8, 0, 8, 50, 8, 6 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 shells within the atomic nucleus 4

5 Excitation enegy fo magicm nuclei 5

6 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. 3) appoximation by the ) Woods-Saxon potential: U U( ) = ectangula potential well with infinite baie enegy : a=0 a>0 + 0 R a e ) a 0: appoximation by the ectangula potential well : U( ) = U 0, 0, < R R U() 0 U( ) = 0,, < R R R 6

7 Schödinge equation Schödinge equation: Single-paticle Hamiltonian opeato: = + sin θ ĤΨ = EΨ H ˆ = h + U( ) M Eigenstates: Ψ() - wave function Eigenvalues: E - enegy U() is a nuclea potential spheically symmetic θ sin θ ϕ ( sinθ ) + () () Conside U() - ectangula potential well with infinite baie enegy U( ) = 0,, < R R Angula pat: ˆ λ = sin θ θ h ˆ λ = L ˆ sin θ ϕ ( sinθ ) + L opeato fo the obital angula momentum = Lˆ h ( θ, ϕ) = h l( l + ) Υ ( θ, ϕ) ˆ Υlm L lm (3) Eigenstates: Y lm spheical hamonics 7

8 Radial pat The wave function of the paticles in the nuclea potential can be decomposed into two pats: a adial one Ψ (), which only depends on the adius, and an angula pat Y lm (θ,ϕ) which only depends on the oientation (this decomposition is possible fo all spheically symmetic potentials): Fom (4) and () => h M Ψ ( θ, ϕ) = Ψ ( ) Υ ( θ, ϕ), lm + => eq. fo the adial pat: ˆ L M Ψ h l( l + ) M + h M ( ) Υ ( θ, ϕ) = E Ψ ( ) Υ ( θ, ϕ) Ψ lm ( ) = E Ψ ( ) lm (4) (5) Substitute in (5): Ψ ( ) = R( ) h M d R( d ) + h l( l + ) R( M ) = E R( ) (6) 8

9 Constaints on E Eq. fo the adial pat: h d R( ) h l( l + ) + E ( ) = 0 (7) R M d M Fom (7) ) Enegy eigenvalues fo obital angula momentum l: E: l=0 s l= p states l= d l=3 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 also have two possible spin diections, this means that the l levels ae (l+) times degeneate if a spin-obit inteaction is neglected. 3) The paity of the wave function is fixed by the spheical wave function Y ml and ˆ l P Ψ, θ, ϕ = P Ψ ( ) Υ θ, ϕ = ( ) Ψ ( ) Υ θ, ϕ eads ( ) l : ( ) ( ) ( ) lm lm s,d,.. - even states; p,f, - odd states 9

10 Main quantum numbe n Eq. fo the adial pat: h M d R( ) h l( l + ) + E ( ) = 0 R d M Solution of diffeential eq: R( ) ( ) + λ( ) R( ) = 0 Spheical Bessel functions j l (x): Ψ A R The wave function: (7) ( θ, ϕ) = j l ( k) Υ ( θ, ϕ), lm A ) = j ( k ) ME k = h ( l U() Bounday condition fo the suface, i.e. at =R: Ψ(R,θ,ϕ) =0 0 R estictions on k in Bessel functions: j l ( kr) = 0 main quantum numbe n coesponds to nodes of the Bessel function : X nl ME h kr = Xnl k R = Xnl R = X nl (8) 0

11 Shell model with ectangula potential well Thus, accoding to Eq. (8) : Xnl h Enl = E nl = Const X nl (9) MR Nodes of Bessel function Enegy states ae quantized stuctue of enegy states E nl l = 0 n = n = n = 3 l = n = n = l = n = l = 3 n = l = 4 n = s states X X X = 3.4 = 6.8 = 9.4 p states X X X = 4.49 = 7.7 d states f X 3 = 5.76 states 4 = 6.99 g states X = 8. j j j j j state s p d s f p g E E E E E E E nl s p d s f p E g = C X nl = C 9.86 = C 0. = C 33. = C 39.5 = C 48.8 = C 59.7 = C 64 degeneacy (l+),l=0 (l+),l= states with E Fist 3 magic numbes ae epoduced, highe not!, 8, 0, 8, 50, 8, 6 Note: this esult coesponds to fo U() = ectangula potential well with infinite baie enegy E nl

12 Shell model with Woods-Saxon potential Woods-Saxon potential: U( ) = + U 0 R a e The fist thee magic numbes (, 8 and 0) can then be undestood as nucleon numbes fo full shells. Thus, 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 shells.

13 Spin-obit inteaction Intoduce the spin-obit inteaction V ls a coupling of the spin and the obital angula momentum: Hˆ = h + U( ) + Vˆ ls M U( ) Vˆ + + ls Ψ (, θ, ϕ) = E Ψ(, θ, ϕ) h M + U( ) Ψ (, θ, ϕ) = ( E Vls ) Ψ(, θ, ϕ) h M whee Vˆ ls Ψ(, θ, ϕ) = V Eigenstates spin-obit inteaction: Vˆ ls = ls Ψ(, θ, ϕ) eigenvalues C ls ( l,s ) - scala poduct of l and s total angula momentum: j = l + s j j = ( l + s)( l + s) = l + s + ls l s = ( j l s ) 3

14 Spin-obit inteaction Eigenvalues: C ls ( j l s ) Ψ(, θ, ϕ) = V ls Ψ(, θ, ϕ) h Vls = Cls + [ j( j + ) l( l + ) s( s ) ] Conside: j = l + : V ls = C ls h ( l + )( l + ) l l 3 = C ls h l j h 3 h = l : Vls = Cls ( l )( l + ) l l = Cls ( l + ) This leads to an enegy splitting E ls which linealy inceases with the angula momentum as l + Els = V ls It is found expeimentally that V ls is negative, which means that the state with j =l+ / is always enegetically below the j = l / level. 4

15 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 levels: 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 shells themselves. Magic numbes appea when the gaps between successive enegy shells ae paticulaly lage:, 8, 0, 8, 50, 8, 6 (0) () (6) (4) (8) (4) () (6) () (4) () 5

16 Collective Nuclea Models 6

17 Collective excitations of nuclei The single-paticle shell model can not popely descibe the excited states of nuclei: the excitation specta of even-even nuclei show chaacteistic band stuctues which can be intepeted as vibations and otations of the nuclea suface low enegy excitations have a collective oigin! The liquid dop model is used fo the desciption of collective excitations of nuclei: the inteio stuctue, i.e., the existence of individual nucleons, is neglected in favo of the pictue of a homogeneous fluid-like nuclea matte. The moving nuclea suface may be descibed quite geneally by an expansion in spheical hamonics with time-dependent shape paametes as coefficients: whee R(θ,φ,t) denotes the nuclea adius in the diection (θ,φ) at time t, and R 0 is the adius of the spheical nucleus, which is ealized when all α λµ =0. The time-dependent amplitudes α λµ (t) descibe the vibations of the nucleus with diffeent multipolaity aound the gound state and thus seve as collective coodinates (tenso). 7 λµ ()

18 Collective excitations of nuclei vibations otations 8

19 Collective coodinates Popeties of the coefficients α λµ λµ : - Complex conjugation: the nuclea adius must be eal, i.e., R(θ,φ,t)=R*(θ,φ,t). () Applying () to () and using the popety of the spheical hamonics (3) () one finds that the α λµ have to fulfill the condition: λµ (4) - The dynamical collective coodinates α λµ (tensos!) define the distotion - vibations - of the nuclea suface elative to the goundstate. - The geneal expansion of the nuclea suface in () allows fo abitay distotions: λ=0,,,. 9

20 Types of Multipole Defomations Goundstate The monopole mode, λ = 0. α Y 00 = 00 R( ϑ, φ,t ) = R 0(+ α00y00 ) = R 0(+ ) 4π 4π The spheical hamonic Y 00 is constant, so that a nonvanishing value of α 00 coesponds to a change of the adius of the sphee. Monopole mode λ=0 The associated excitation is the so-called beathing mode of the nucleus. Because of the lage amount of enegy needed fo the compession of nuclea matte, this mode is fa too high in enegy to be impotant fo the low-enegy specta discussed hee. The defomation paamete α 00 can be used to cancel the oveall density change pesent as a side effect in the othe multipole defomations. The dipole mode, λ =. Y 0 cosθ Dipole defomations, λ = to lowest ode, eally do not coespond to a defomation of the nucleus but athe to a shift of the cente of mass, i.e. a tanslation of the nucleus, and should be disegaded fo nuclea excitations since tanslational shifts ae spuious. 0

21 Types of Multipole Defomations The quadupole mode, λ = The quadupole defomations - the most impotant collective low enegy excitations of the nucleus. The octupole mode, λ = 3 The octupole defomations ae the pincipal asymmetic modes of the nucleus associated with negative-paity bands. The hexadecupole mode, λ = 4 The hexadecupole defomations: this is the highest angula momentum that has been of any impotance in nuclea theoy. While thee is no evidence fo pue hexadecupole excitations in the specta, it seems to play an impotant ole as an admixtue to quadupole excitations and fo the goundstate shape of heavy nuclei.

22 Types of Multipole Defomations

23 Quadupole defomations The quadupole defomations ae the most impotant vibational degees of feedom of the nucleus. Fo the case of pue quadupole defomation (λ = ) the nuclea suface is given by (5) Conside the diffeent components of the quadupole defomation tenso α µ The paametes α µ ae not independent - cf. (4): Fom (4): (6) => α 0 is eal (since α 0 = α 0 ) ; and we ae left with five independent eal degees of feedom: α 0 and the eal and imaginay pats of α and α 0 To investigate the actual fom of the nucleus, it is best to expess this in catesian coodinates by ewiting the spheical hamonics in tems of the catesian components of the unit vecto in the diection (θ,φ) : (6) (7) µ 3

24 Fom spheical to catesian coodinates Spheical coodinates (,θ,φ) Catesian coodinates (x,y,z)=>(ζ,ξ,η) The invention of Catesian coodinates in the 7th centuy by René Descates (Latinized name: Catesius) 4

25 Catesian coodinates Catesian coodinates fulfil subsidiay conditions (8) (9) Substitute (9) in (5): (0) whee the catesian components of the defomation ae elated to the spheical ones by () 5

26 Catesian coodinates In () six independent catesian components appea (all eal), compaed to the five degees of feedom contained in the spheical components. Howeve, the function R(θ,φ) fulfills () Subs. (0) into () and accounting that we obtain: 5 independent catesian components As the catesian defomations ae diectly elated to the steching (o contaction) of the nucleus in the appopiate diection, we can ead off that: α 0 descibes a stetching of the z axis with espect to the у and x axes, α, α descibes the elative length of the x axis compaed to the у axis (eal pat), as well as an oblique defomation in the x-y plane, α, α indicate an oblique defomation of the z axis. () 6