Nuclear Forward Scattering
Quantum Beats
and Hyperfine Splitting
When the nuclei in the target experience
 hyperfine interactions:
 electric monopole,
 electric quadrupole or
 magnetic dipole,
each nuclear energy level can be split into sublevels.
The transition from excited to ground state is split into several components. This is reflected by the scattering amplitude, which consists then of several terms related to different transition frequencies and depending on the polarisation [Hyp.Int. 123/124(1999)31]:
f^{ss}^{¢}(w) µ 
å
m 

[ww_{0m}] 
iG_{0 }/ 2(^{h}/_{2p})


G^{2}(m_{e}, m_{g}, m) P^{ss}^{¢}(m), 

with 

w_{0m} 

the frequencies of the transitions involved,



m_{e}, m_{g}


the projections of the spins
of the excited and ground states, 


m = m_{e}m_{g},





G(m_{e},m_{g},m) 

ClebschGordan coefficients
giving the probabilities of the transitions 


P^{ss}^{¢}(m) 

polarisation factors taking into account the polarisation of the incident (s) and forward scattered (s') radiation 
The summation is done over all allowed nuclear transitions and the interference between such transitions gives rise to the socalled quantum beats (d) in the time dependence of the forward scattered intensity (see figure d).
Hyperfine splitting
(energy levels)

Mössbauer
spectroscopy
(energy scale)

Nuclear Forward
scattering
(time scale)

Comparison between typical Mössbauer and NFS spectra for ^{119}Sn nuclei:
 Unsplit levels in the absence of hyperfine interactions (upper graphs),
 a quadrupole splitting (middle graphs) and
 a magnetic hyperfine splitting (lower graphs) are represented.
The left part of the figure shows the corresponding scheme of the nuclear levels.
 Not all transitions between ground and excited state are allowed, but some selection rules apply, depending on the multipolarity of the transition. For 119Sn, the transition from the first excited state to the ground state is of magnetic dipole (M1) type, and only a change in the magnetic quantum number Dm = 0, ±1 is allowed during the transition.
 We consider a thin absorber with effective thickness 0.5 in order to neglect dynamical beats due to multiple scattering. In general you have to consider them superimposed on the above spectra.
The above figure shows a comparison between typical spectra obtained in Mössbauer spectroscopy (MS) and in Nuclear Forward Scattering (NFS) for different hyperfine interactions in the case of ^{119}Sn.
 When the nuclear levels are neither split nor shifted by hyperfine interactions, only one transition between ground and excited state is possible.
 The MS spectrum shows a single absorption line
centred at v = 0, while
 the NFS spectrum shows an exponential decay
In general you have to consider the superposition of dynamical beats.
 If the isomer shift of source and absorber is different,
 the MS line will have its centre of mass at v ¹ 0.
 The NFS spectrum is not sensitive to the isomer shift of a single absorber, because quantum beats appear in a spectrum as an interference between the radiation fields corresponding to different transitions.
An isomer shift can still be measured relative to a second absorber if both targets are placed behind each other along the exciting xray beam.
 If the nuclei experience an electric quadrupole interaction, the excited state will be split into two sublevels characterised by a different magnitude of the magnetic quantum number, while the ground state remains unsplit.
 The MS spectrum has two absorption lines
(often referred to as quadrupole doublet) corresponding to the possible transitions between the ground state and the two sublevels of the excited state.
 The NFS spectrum shows
quantum beats with a single frequency, corresponding to the energy difference between the sublevels of the excited state:
DE_{hf} = (^{h}/_{2p})·Dw = (^{h}/_{2p})(w_{02}  w_{01}) 
which can be estimated
via the measured quantum beat period U:
DE_{hf} [ 
mm/s 
] · U [ns] » 87 

 When a magnetic hyperfine field is present at the nuclei both the ground and excited states split into sublevels, each having a defined magnitude and sign of the magnetic quantum number m_{I}.
Because of the selection rules, only six transitions between the sublevels of ground state and excited state are allowed.
 The Mössbauer spectrum will therefore show six absorption lines (magnetic sextet) of different relative intensities, for different transitions generally have different probabilities.
 The NFS spectrum has a complicated structure, with a superposition of quantum beats of different frequencies.

