WS09/10 Mahnke 12.1.10

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)]

WS09/10 Mahnke 12.1.10

Fine structure and hyperfine structure of the „yellow“Na-line

WS09/10 Mahnke 12.1.10

Laser spectroscopy

WS09/10 Mahnke 12.1.10

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

WS09/10 Mahnke 12.1.10

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→ →

WS09/10 Mahnke 12.1.10

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→ →→ → →

WS09/10 Mahnke 12.1.10

Schmidt-lines

(from (b) with (a))

(May 84)

WS09/10 Mahnke 12.1.10

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→ →

WS09/10 Mahnke 12.1.10

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

WS09/10 Mahnke 12.1.10

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)

__

__

WS09/10 Mahnke 12.1.10

Experimental quadrupole moments

quadrupole moments are a measure for the deviationfrom a sphere(„deformation“)

Q ~ ZR2 δ

(May 84)

WS09/10 Mahnke 12.1.10

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

WS09/10 Mahnke 12.1.10

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.

→→→ →

WS09/10 Mahnke 12.1.10

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→→ →→ → →

WS09/10 Mahnke 12.1.10

Half lives for electric and magnetic multipole radiation

Ti→f (EL)/Ti→f (ML) ≈ 100 Ti→f (L)/Ti→f (L+1) ≈ 104..8

WS09/10 Mahnke 12.1.10

Experimental determination of half lives

Doppler shift

decay in-flight versus at rest

order of magnitude 10-12 s

longer half lives:measure electronically

WS09/10 Mahnke 12.1.10

Angular distribution of multipole radiation

classical: Hertz dipole(antenna)

WS09/10 Mahnke 12.1.10

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)

WS09/10 Mahnke 12.1.10

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

WS09/10 Mahnke 12.1.10

3.6 Hyperfine interaction

- measure energies (high precision needed)resonance absorption – Mössbauer-spectroscopy

- high-resolution Laser spectroscopy

- high-frequency absorption

- spin precession

WS09/10 Mahnke 12.1.10

Mössbauer-effect

Resonance absorptionrecoilless emission and absorption

WS09/10 Mahnke 12.1.10

Comparing energies

(Aus Mayer-Kuckuk 84)

WS09/10 Mahnke 12.1.10

57Fe

WS09/10 Mahnke 12.1.10

Pound and Rebka experiment (1960):Energy shift by gravitational field (57Fe)

nuclear resonance fluorescence with synchrotron radiation

WS09/10 Mahnke 12.1.10

Isotopic shift

HHfs = A I•J

EHfs = A/2 [F(F+1) – I(I+1) – J(J+1)]

Laser spectroscopy

→ →

WS09/10 Mahnke 12.1.10

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)

WS09/10 Mahnke 12.1.10

Angularcorrelation

WS09/10 Mahnke 12.1.10

Metal physics – point-defect production by neutrino-recoil

(Metzner, Sielemann et al., PRL53(1984)290)

In in Kupfer

WS09/10 Mahnke 12.1.10

Surface science - Coordination and magnetic field

(Bertschat, Potzger et al., PRL 88(2002)247201)

In on nickel

WS09/10 Mahnke 12.1.10

(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