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 sub-levels.

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]:

fss¢(w) µ å
 G0 / 2(h/2p

[w-w0m]- iG0 / 2(h/2p
 G2(me, mg, m) Pss¢(m),
with w0m

the frequencies of the transitions involved,

me, mg

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

m = me-mg,

G(me,mg,m) Clebsch-Gordan coefficients
giving the probabilities of the transitions
Pss¢(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 so-called quantum beats (d) in the time dependence of the forward scattered intensity (see figure d).

Hyperfine splitting
(energy levels)
(energy scale)
Nuclear Forward
(time scale)

Comparison between typical Mössbauer and NFS spectra for 119Sn 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 119Sn.

  • 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 x-ray beam.
  • If the nuclei experience an electric quadrupole interaction, the excited state will be split into two sub-levels characterised by a different magnitude of the magnetic quantum number, while the ground state remains un-split.

    • 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 sub-levels of the excited state.
    • The NFS spectrum shows
      quantum beats with a single frequency, corresponding to the energy difference between the sub-levels of the excited state:
      DEhf = (h/2pDw = (h/2p)(w02 - w01)
      which can be estimated
      via the measured quantum beat period U:
    DEhf [ mm/s ]  ·  U [ns]   »   87
      i.e. the smaller the energy splitting,
      the slower the beating.

      For equal probability of the two transitions the contrast is maximal i.e. the intensity goes to zero for each minimum of the quantum beats when thickness effects are omitted. The contrast vanishes with the asymmetry of the two probabilities.

  • When a magnetic hyperfine field is present at the nuclei both the ground and excited states split into sub-levels, each having a defined magnitude and sign of the magnetic quantum number mI.
    Because of the selection rules, only six transitions between the sub-levels 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.

Based on the PhD thesis
of Alessandro Barla, Herdecke 2001
and Hanne Grünsteudel, Lübeck 1998

Last modified 19/06/02 03:01 PM by Ernst Schreier