3.3 Nuclear states: nuclear spin, parity and excitation ... · WS09/10 Mahnke 12.1.10 3.3 Nuclear...
Transcript of 3.3 Nuclear states: nuclear spin, parity and excitation ... · WS09/10 Mahnke 12.1.10 3.3 Nuclear...
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3.3 Nuclear states: nuclear spin, parity and excitationenergy
Nuclear spin I:Nucleons (proton, neutron) are Fermions with s=1/2
Total angular momentum of the nucleon j = l + s (vector sum)
Total angular momentum of the nucleus II = ∑i j i = ∑i (l i + s i)
gg-nucleus I=0uu-nucleus I≠0, but also I=0 ug-,gu-nuclei I half integer
Total angular momentum of the atom („hyperfine structure“):nuclear spin I + electronic shell J = atomic spin F
splitting of atomic J leads to 2I+1 (I≤J) or 2J+1 (J‹I) sublevelsHHfs = A I·J with EHfs = A/2 [F(F+1) – I(I+1) – J(J+1)]
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Fine structure and hyperfine structure of the „yellow“Na-line
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Laser spectroscopy
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Parity π (Iπ):
symmetry behavior of the wave function under reflection (in space)ΠopΨ(r) = Ψ(-r) = π Ψ(r) if Πop H Πop
-1 = H with HΨ=EΨ
2-fold application = identity operationtherefore: eigenvalues for parity operation +1 or -1.
Parity is a multiplicative quantum number
Parity even (+1): with even orbital angular momentumParity odd (-1): with odd orbital angular momentum
characterizing energy levels
0E1
E2
E3 Iπ3
Iπ2
Iπ0
Iπ1
AZX
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Properties associated with angular momentum3.4 Nuclear moments
a) Magnetic dipole moment classical
quantum mechanical
dimensionles g-factor
Bohr-magneton
nuclear magneton
maximum component:electron
proton
neutron
µ=-e/mc L→ →
µB=eħ/2mec=5.78 10-11 MeV/Tesla
µk=eħ/2mpc=3.152 10-14 MeV/Tesla
=5.05 10-27 A m2
µe=gµBs s=1/2, gs=-2
µp=gµkI I=1/2, gs=+5.585
µn=gµkI I=1/2, gs=-3.826
µ=gµBI/ħ→ →
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j l
s
µlµj
µs
The sum of spin and orbital angular momentum is no longer parallel to j
(a)
measurable is the component in the direction of j:
(b)
µ = <IM|µop|IM>M=I
1 proton0 neutrongl = {
µ* = gl l + gs s→→→
µj = µ*·j/|j|·j/|j|= gµkj/ħ→ →→ → →
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Schmidt-lines
(from (b) with (a))
(May 84)
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Experimental determination of magnetic moments
Apply a magnetic field H (defining the z-axis):
equidistant sublevel splitting
Larmor frequency – Precession frequency
ħω = g µk H independent of m
∆E = µ·H = gµk·jz/ħ ·H = gµkmj H→ →
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b) Electric quadrupole moment
electric energy Eel = eqΦ = ∫V(r)ρ(r)d3r
∆V=0 (outside) (∆=∇2 )
expansion into a Taylor series, i.e. into eigen functions („multipole“)
V(r) = ∑LM VLM rL YLM* (Θ,Φ)
eqΦ = e∑LM VLM 1/e ∫ρ(r) rL YLM* (Θ,Φ) d3r
QLM
L=0 charge eZ = ∫ρ(r) d3r
L=1 dipole moment eQ1 = ∫ρ(r) r d3r
L=2 quadrupole moment eQ2 = ∫ρ(r) (3z2-r2) d3r
Π QLM Π-1 = QLM (-r) = (-1)L QLM QLM ≠0 for L even
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with ρ(r) = e∑i δ(r - ri)
QLM=∑i riL YLM* (i)
quantum mechanics (Wigner-Eckart-theorem)
QLM=<IMI|QLM|IMI>=(IMILM|IMI)<I||QLM||I>
i.e. M=0 and 2I≥L
QLM = (IMILM|IMI)/(IIL0|II) QL
E = ¼ eVzz QL [3MI2-I(I+1)]/[3I2- I(I+1)]
non-equidistant, +/-M -degeneracy
Q:=Q2 >0 prolate (cigar), <0 oblate (lentil)
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Experimental quadrupole moments
quadrupole moments are a measure for the deviationfrom a sphere(„deformation“)
Q ~ ZR2 δ
(May 84)
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3.5 Gamma-decay
a) generalized nuclear moments (static)
analoguous to ∆V=0 with the complete solution rLYLM* (Θ,Φ), thereexists a complete set of solution for the vector potential ∇2 ā = 0,
3 independent sets, relevant only āM
magnetic energy Emag = -1/c ∫j(r) ā d3rwith the current density
j(r)= eρv + c ∇ x Mwith M= ρp µk gp s + ρn µk gn s
expansion into a Taylor series or eigenfunctions („multipoles“), nowML = <II| ∑i µk(2/(L+1) Li+gp si )∇(rL PL(i)) + ∑i µkgnsi∇(rL PL(i))|II>
for L=1µ=M1 = µk <II| ∑i (Li + gp si ) + ∑i gn si |II>
Π MLM Π-1 = MLM (-r) = (-1)L+1 MLM MLM ≠ 0 L odd
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b) Generalization to dynamic moments
Ti→f = 2π/ħ |<γ,f|Hγ-Kern|i,0>|2 dN/dE
from Maxwell equations ∆ ā – 1/c2·∂2/∂t2 ā =0 exp(-iωt): ∆ ā + k2 ā =0
solutions yield the same multipole operators: instead of rL now Bessel functions jL~(kr)L/(2L+1)!!
with 2 sets of solutions („transversal wave“): electric and magnetic
Ti → f ~ k2L+1 ~ Eγ2L+1
H= (p-e/c·a)2/2m + V(r) + e/2mc·Ĥ s
H= Ho + Hγ-Kern Φ=0 Coulomb gauge transformation
∇·ā =0 radiation gauge transf.
→→→ →
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Selection rules
<i | ELM | j> static and dynamic
static dynamici=j i≠j
electric QLM L=0, 2, … πi·πj = (-1)Lmagnetic MLM L=1, 3,… πi·πj = (-1)L+1
Mi + M = Mi Mj + M = MiM=0
L ≤2Ii |Ij-Ii|≤L≤Ii+IjL≠0 für γ
Ii + L = Ii Ij + L = Ii→→ →→ → →
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Half lives for electric and magnetic multipole radiation
Ti→f (EL)/Ti→f (ML) ≈ 100 Ti→f (L)/Ti→f (L+1) ≈ 104..8
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Experimental determination of half lives
Doppler shift
decay in-flight versus at rest
order of magnitude 10-12 s
longer half lives:measure electronically
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Angular distribution of multipole radiation
classical: Hertz dipole(antenna)
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Alternative decay mode: internal conversion(coupling to the electronic shell)
(cf. x-ray – Auger-electron)
Ee= Eγ - Be(0 → 0 possible)
λ = λγ + λe= λγ (1+α)
conversion coefficient
α = Ne/Nγ
αK ~ Z3 Eγ-(L+5/2)
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Internal pair production
Eγ> 2 mec2
no influence by the electronic shell, therefore only weak Z dependence, increase with decay energy,0+ → 0+ - transition possible
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3.6 Hyperfine interaction
- measure energies (high precision needed)resonance absorption – Mössbauer-spectroscopy
- high-resolution Laser spectroscopy
- high-frequency absorption
- spin precession
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Mössbauer-effect
Resonance absorptionrecoilless emission and absorption
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Comparing energies
(Aus Mayer-Kuckuk 84)
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57Fe
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Pound and Rebka experiment (1960):Energy shift by gravitational field (57Fe)
nuclear resonance fluorescence with synchrotron radiation
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Isotopic shift
HHfs = A I•J
EHfs = A/2 [F(F+1) – I(I+1) – J(J+1)]
Laser spectroscopy
→ →
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nuclear magnetic resonance
Larmor frequency – precession frequency hν = g µk H
chemical shift (in biomolecules, protein-structure determination), in the order of ppmmagnetic resonance-tomography)
(Aus Schatz, Weidinger 96)
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Angularcorrelation
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Metal physics – point-defect production by neutrino-recoil
(Metzner, Sielemann et al., PRL53(1984)290)
In in Kupfer
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Surface science - Coordination and magnetic field
(Bertschat, Potzger et al., PRL 88(2002)247201)
In on nickel
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(Lany, Wichert et al.,Phys.Rev. B52 (1995)11884,
Koteski, Mahnke et al.,Phys.Scr. T115 (2005)369)
Doping of semiconductors – donor-acceptor-pair
Angular correlation (PAC)X-ray absorption (EXAFS)
In-As in CdTe