LectureLecture 33
NuclearNuclear models:models: ShellShell modelmodel
WS2012/13 : ‚Introduction to Nuclear and Particle Physics ‘, Part I 1 NuclearNuclear modelsmodels
Nuclear models
Models with strong interaction between Models of non-interacting the nucleons nucleons
Liquid drop model Fermi gas model ααα-particle model Optical model Shell model … …
Nucleons interact with the nearest Nucleons move freely inside the nucleus: neighbors and practically don‘t move: mean free path λ ~ R A nuclear radius mean free path λ << R A nuclear radius
2 III.III. ShellShell modelmodel
3 ShellShell modelmodel
Magic numbers: Nuclides with certain proton and/or neutron numbers are found to be exceptionally stable. These so-called magic numbers are 2, 8, 20, 28, 50, 82, 126 — The doubly magic nuclei: — Nuclei with magic proton or neutron number have an unusually large number of stable or long lived nuclides . — A nucleus with a magic neutron (proton) number requires a lot of energy to separate a neutron (proton) from it. — A nucleus with one more neutron (proton) than a magic number is very easy to separate. — The first exitation level is very high : a lot of energy is needed to excite such nuclei — The doubly magic nuclei have a spherical form
Nucleons are arranged into complete shells within the atomic nucleus
4 ExcitationExcitation energyenergy forfor magicm nuclei
5 NuclearNuclear potentialpotential
The energy spectrum is defined by the nuclear potential solution of Schrödinger equation for a realistic potential The nuclear force is very short-ranged => the form of the potential follows the density distribution of the nucleons within the nucleus: for very light nuclei (A < 7), the nucleon distribution has Gaussian form (corresponding to a harmonic oscillator potential ) for heavier nuclei it can be parameterised by a Fermi distribution. The latter corresponds to the Woods-Saxon potential e.g. 3) approximation by the U 1) Woods-Saxon potential: U(r) === −−− 0 rectangular potential well r−−−R with infinite barrier energy : 1+++ e a ∞ 2) a 0: U(r) ∞∞ a=0 approximation by the a>0 rectangular potential well : 0 R r
−−−U0 , r <<< R ,0 r <<< R U(r) === U(r) === ,0 r ≥≥≥ R ∞∞∞, r ≥≥≥ R 6 SchrSchr öödingerdinger equationequation
ˆ Schrödinger equation: HΨΨΨ === EΨΨΨ (1) Consider U(r) - Single-particle h∇∇∇2 rectangular potential Hˆ === −−− +++U(r) (2) Hamiltonian operator: 2M well with infinite •Eigenstates: ΨΨΨ(r) - wave function barrier energy •Eigenvalues: E - energy ,0 r <<< R U(r) is a nuclear potential – spherically symmetric U(r) === • ∞∞∞, r ≥≥≥ R 1 ∂∂∂ ∂∂∂ 1 ∂∂∂ 1 ∂∂∂ 2 ∇∇∇2 === r 2 +++ ((()(sin θθθ )))+++ r 2 ∂∂∂r ∂∂∂r r 2 sin θθθ ∂∂∂θθθ r 2 sin 2 θθθ ∂∂∂ϕϕϕ 2
1 ∂∂∂ 1 ∂∂∂2 Angular part: λλλˆ === ((()(sin θθθ)))+++ sin θθθ∂∂∂θθθ sin 2 θθθ∂∂∂ϕϕϕ2 2 1 ∂∂∂ ∂∂∂ 1 Lˆ −−− h2λλλˆ === Lˆ2 ∇∇∇2 === r 2 −−− r 2 ∂∂∂r ∂∂∂r r 2 h2 L – operator for the orbital angular momentum ˆ2 h2 L ΥΥΥlm (((θθθ,ϕϕϕ)))=== (ll +++ )1 ΥΥΥlm (((θθθ,ϕϕϕ))) (3)
Eigenstates: Ylm – spherical harmonics 7 RadialRadial partpart The wave function of the particles in the nuclear potential can be decomposed into two parts: a radial one ΨΨΨ 111(r), which only depends on the radius r, and an angular part Ylm (θ,ϕϕϕ) which only depends on the orientation (this decomposition is possible for all spherically symmetric potentials): (4) ΨΨΨ(((r,θθθ,ϕϕϕ)))=== ΨΨΨ1(r)⋅⋅⋅ ΥΥΥlm (((θθθ,ϕϕϕ))) From (4) and (1) =>
2 2 h 1 ∂∂∂ ∂∂∂ 1 Lˆ 2 −−− 2 r +++ 2 ΨΨΨ1((()(r)))(ΥΥΥlm ((()θθθ,ϕϕϕ)))(=== E ΨΨΨ1((()r)))(ΥΥΥlm ((()θθθ,ϕϕϕ))) 2M r ∂∂∂r ∂∂∂r r 2M => eq. for the radial part: h2 1 ∂∂∂ ∂∂∂ h2 (ll +++ )1 2 (5) −−− 2 r +++ 2 ΨΨΨ1((()(r)))(=== E ΨΨΨ1((()r))) 2M r ∂∂∂r ∂∂∂r r 2M
Substitute in (5): ℜℜℜ )r( ΨΨΨ ((()(r)))=== 1 r h2 d 2ℜℜℜ )r( h2 l(l +++ )1 (6) −−− +++ ℜℜℜ )r( === E ℜℜℜ )r( 2M dr 2 r 2 2M 8 ConstraintsConstraints onon EE
h2 d 2ℜℜℜ )r( h2 l(l +++ )1 Eq. for the radial part: +++ E −−− ℜℜℜ )r( === 0 (7) 2M dr 2 r 2 2M From (7) 1) Energy eigenvalues for orbital angular momentum l: E: l=0 s l=1 p states l=2 d l=3 f … 2) For each l: -l < m < l => (2 l+1) projections m of angular momentum. The energy is independent of the m quantum number, which can be any integer value between ± l. Since nucleons also have two possible spin directions, this means that the l levels are 2 ·(2 l+1) times degenerate if a spin-orbit interaction is neglected.
3) The parity of the wave function is fixed by the spherical wave function Yml and l reads (−1) : ˆ l P ΨΨΨ(((r,θθθ,ϕϕϕ)))=== P ΨΨΨ1(r)⋅⋅⋅ ΥΥΥlm (((θθθ,ϕϕϕ)))=== (−−− )1 ΨΨΨ1(r)⋅⋅⋅ ΥΥΥlm (((θθθ,ϕϕϕ))) ⇒ s,d,.. - even states; p,f,… - odd states 9 MainMain quantumquantum numbernumber nn
h2 d 2ℜℜℜ )r( h2 l(l +++ )1 Eq. for the radial part : +++ E −−− ℜℜℜ )r( === 0 (7) 2M dr 2 r 2 2M A Solution of differential eq: )r( ′′′′′′ )r( )r( )r( 0 ℜℜℜ )r( === l kr(j ) ℜℜℜ +++ λλλ ℜℜℜ === r
Spherical Bessel functions j (x): 2 2ME l k === h2 The wave function: A ΨΨΨ((()(r,θθθ,ϕϕϕ)))=== j (kr )⋅⋅⋅ΥΥΥ ((()(θθθ,ϕϕϕ))) r l lm
∞∞∞ U(r)
Boundary condition for the surface, i.e. at r =R: Ψ(Ψ(Ψ(R,θ,ϕ) =0 0 R r restrictions on k in Bessel functions: l kR(j ) === 0 ⇒ main quantum number n – corresponds to nodes of the Bessel function : Xnl 2ME ⇒⇒⇒ 2 2 2 ⇒⇒⇒ 2 2 (8) kR === Xnl k R === Xnl 2 R === Xnl h 10 ShellShell modelmodel withwith rectangularrectangular potentialpotential wellwell
X 2 h2 Thus, according to Eq. (8) : nl 2 (9) E === Enl === Const ⋅⋅⋅ Xnl nl 2MR 2 Nodes of Bessel function Energy states are quantized structure of energy states Enl
l === 0 s −−− states j0 2 state Enl === C ⋅⋅⋅ Xnl degeneracy states with E ≤≤≤ Enl n === 1 X10 === 14.3 2·(2 l+1 ),l=0 s1 E s1 === C ⋅⋅⋅ 86.9 2 2 n === 2 X20 === 28.6 1p E1 p === C ⋅⋅⋅ 20 2. 6 2·(2 l+1 ),l=1 8 n === 3 X30 === 42.9 d1 E1d === C ⋅⋅⋅ 33 2. 10 18
l === 1 p −−− states j1 s2 E2s === C ⋅⋅⋅ 39 5. 2 20
n === 1 X11 === 49.4 1 f E1 f === C ⋅⋅⋅ 48 8. 14 34
n === 2 X21 === 72.7 2 p E2 p === C ⋅⋅⋅59 7. 6 40 1g E === C ⋅⋅⋅64 18 58 l === 2 d −−− states j2 1g n === 1 X === 76.5 12 First 3 magic numbers are reproduced, higher – not!
l === 3 f −−− states j3 2, 8, 20, 28, 50, 82, 126
n === 1 X13 === 99.6 Note: this result corresponds to for U(r) = rectangular l === 4 g −−− states j 4 potential well with infinite barrier energy 11 n === 1 X14 === 1.8 ShellShell modelmodel withwith WoodsWoods --SaxonSaxon potentialpotential
Woods-Saxon potential:
U0 U(r) === −−− r−−−R 1+++ e a
The first three magic numbers (2, 8 and 20) can then be understood as nucleon numbers for full shells . Thus, this simple model does not work for the higher magic numbers. For them it is necessary to include spin-orbit coupling effects which further split the nl shells.
12 SpinSpin --orbitorbit interactioninteraction
Introduce the spin-orbit interaction Vls – a coupling of the spin and the orbital angular momentum: h∇∇∇2 Hˆ === −−− +++U(r)+++Vˆ 2M ls h 2 ∇∇∇ ˆ −−− +++U(r)+++Vls ΨΨΨ(r,θθθ,ϕϕϕ) === E ΨΨΨ(r,θθθ,ϕϕϕ) 2M h∇∇∇2 −−− +++U(r)ΨΨΨ(r,θθθ,ϕϕϕ) === ((()(E −−−Vls ))) ΨΨΨ(r,θθθ,ϕϕϕ) 2M ˆ where Vls ΨΨΨ(r,θθθ,ϕϕϕ) ===Vls ΨΨΨ(r,θθθ,ϕϕϕ) Eigenstates eigenvalues
Vˆ C )s,l( - scalar product of l and s •spin-orbit interaction: ls === ls total angular momentum: j === l +++ s
2 2 j ⋅⋅⋅ j === (l +++ s)( l +++ s) === l +++ s +++ 2 sl
1 2 2 2 l ⋅⋅⋅ s === ( j −−− l −−− s ) 2 13 SpinSpin --orbitorbit interactioninteraction
1 2 2 2 C ⋅⋅⋅ ( j −−− l −−− s )ΨΨΨ(r,θθθ,ϕϕϕ) ===V ΨΨΨ(r,θθθ,ϕϕϕ) ls 2 ls h2 Eigenvalues: V === C [[[][ j(j +++ )1 −−− l(l +++ )1 −−− s(s +++ )1 ]]] ls ls 2
Consider: h2 h2 1 1 1 2 1 3 j === l +++ : Vls === Cls (l +++ )( l +++ )−−− l −−− l −−− ⋅⋅⋅ === Cls l 2 2 2 2 2 2 2 h2 h2 1 1 1 2 1 3 j === l −−− : Vls === Cls (l −−− )( l +++ )−−− l −−− l −−− ⋅⋅⋅ === −−−Cls (l +++ )1 2 2 2 2 2 2 2
This leads to an energy splitting ∆Els which linearly increases with the angular momentum as 2l +++1 ∆∆∆E === V ls 2 ls
It is found experimentally that Vls is negative , which means that the state with j =l+ 1/2 is always energetically below the j = l − 1/2 level.
14 SpinSpin --orbitorbit interactioninteraction
The total angular momentum quantum Single particle energy levels: number j = l±1/2 of the nucleon is denoted by an extra index j: nl j j (2j+1) e.g., the 1f state splits into a 1f 7/2 and a 1f 5/2 state 1f 1f 5/2
1f 7/2 The nlj level is (2 j + 1) times degenerate
Spin-orbit interaction leads to a sizeable splitting of the energy states which are indeed (10) (2) (6) comparable with the gaps between the nl (4) shells themselves . (8) (4) Magic numbers appear when the gaps (2) (6) between successive energy shells are (2) particularly large: (4) 2, 8, 20, 28, 50, 82, 126 (2) 15 CollectiveCollective NuclearNuclear ModelsModels
16 CollectiveCollective excitationsexcitations ofof nucleinuclei
The single-particle shell model can not properly describe the excited states of nuclei: the excitation spectra of even-even nuclei show characteristic band structures which can be interpreted as vibrations and rotations of the nuclear surface low energy excitations have a collective origin !
The liquid drop model is used for the description of collective excitations of nuclei: the interior structure, i.e., the existence of individual nucleons, is neglected in favor of the picture of a homogeneous fluid-like nuclear matter.
The moving nuclear surface may be described quite generally by an expansion in spherical harmonics with time-dependent shape parameters as coefficients:
(1)
where R( θθθ,φφφ,t) denotes the nuclear radius in the direction (θθθ,φφφ) at time t, and R0 is the radius of the spherical nucleus, which is realized when all αααλµ =0.
The time-dependent amplitudes αααλµλµλµ (t) describe the vibrations of the nucleus with different multipolarity around the ground state and thus serve as collective coordinates (tensor). 17 CollectiveCollective excitationsexcitations ofof nucleinuclei vibrationsvibrations rotationsrotations
18 Collective coordinates
(1) Properties of the coefficients αααλµ ::: - Complex conjugation: the nuclear radius must be real, i.e., R( θθθ,φφφ,t)=R*(θθθ,φφφ,t). (2) Applying (2) to (1) and using the property of the spherical harmonics (3) one finds that the αααλµλµλµ have to fulfill the condition: (4)
- The dynamical collective coordinates αααλµλµλµ (tensors!) define the distortion - vibrations - of the nuclear surface relative to the groundstate . - The general expansion of the nuclear surface in (1) allows for arbitrary distortions: λλλ=0,1,2,….
19 TypesTypes ofof MultipoleMultipole DeformationsDeformations
Groundstate The monopole mode , λλλ = 0. 1 Monopole ααα00 Y00 === (R ϑϑϑ,φφφ )t, === R 1( +++αααY ) === R 1( +++ ) 4πππ 0 00 00 0 4πππ mode λλλ=0 The spherical harmonic Y 00 is constant, so that a nonvanishing value of ααα00 corresponds to a change of the radius of the sphere . The associated excitation is the so-called breathing mode of the nucleus. Because of the large amount of energy needed for the compression of nuclear matter, this mode is far too high in energy to be important for the low-energy spectra discussed here. The deformation parameter ααα00 can be used to cancel the overall density change present as a side effect in the other multipole deformations.
The dipole mode, λλλ = 1.
Y10 ≈≈≈ cos θθθ Dipole deformations, λλλ = 1 to lowest order, really do not correspond to a deformation of the nucleus but rather to a shift of the center of mass, i.e. a translation of the nucleus, and should be disregarded for nuclear excitations since translational shifts are spurious. 20 TypesTypes ofof MultipoleMultipole DeformationsDeformations
The quadrupole mode , λλλ = 2 The quadrupole deformations - the most important collective low energy excitations of the nucleus.
The octupole mode , λλλ = 3 The octupole deformations are the principal asymmetric modes of the nucleus associated with negative-parity bands.
The hexadecupole mode , λλλ = 4 The hexadecupole deformations: this is the highest angular momentum that has been of any importance in nuclear theory. While there is no evidence for pure hexadecupole excitations in the spectra, it seems to play an important role as an admixture to quadrupole excitations and for the groundstate shape of heavy nuclei.
21 TypesTypes ofof MultipoleMultipole DeformationsDeformations
22 Quadrupole deformations
The quadrupole deformations are the most important vibrational degrees of freedom of the nucleus. For the case of pure quadrupole deformation (λλλ = 2) the nuclear surface is given by
(5)
Consider the different components of the quadrupole deformation tensor ααα2µ2µ2µ
The parameters ααα2µ are not independent - cf. (4): From (4): (6)
∗∗∗ (6) => ααα202020 is real (since ααα202020 ===ααα 202020 ))) ; and we are left with five independent real degrees of freedom : ααα202020 and the real and imaginary parts of ααα212121 and ααα222222
To investigate the actual form of the nucleus, it is best to express this in cartesian coordinates by rewriting the spherical harmonics in terms of the cartesian components of the unit vector in the direction (θθθ,φφφ) :
(7)
23 From spherical to cartesian coordinates
Spherical coordinates (r, θ,φθ,φθ,φ) Cartesian coordinates (x,y,z)=> (((ζ,ξ,η)
r
The invention of Cartesian coordinates in the 17th century by René Descartes (Latinized name: Cartesius )
24 Cartesian coordinates
Cartesian coordinates fulfil subsidiary conditions (8)
(9)
Substitute (9) in (5): (10)
where the cartesian components of the deformation are related to the spherical ones by
(11)
25 Cartesian coordinates
In (11) six independent cartesian components appear (all real) , compared to the five degrees of freedom contained in the spherical components . However, the function R( θθθ,φφφ) fulfills (12)
Subs. (10) into (12) and accounting that we obtain: (12)
5 independent cartesian components
As the cartesian deformations are directly related to the streching (or contraction) of the nucleus in the appropriate direction, we can read off that:
ααα202020 describes a stretching of the z axis with respect to the у and x axes,
ααα222222 , ααα222−2−−−2222 describes the relative length of the x axis compared to the у axis (real part), as well as an oblique deformation in the x-y plane ,
ααα212121 , ααα222−2−−−1111 indicate an oblique deformation of the z axis .
26