# Nuclear Forward Scattering

## Polarization Aspects

The time dependence for the forward scattered intensity can have a complicated form if many nuclear transitions are involved. In order to reduce the number of free parameters in the experiment it is convenient to use the intrinsic polarization of the SR to select transition lines with a certain polarization:

 Only oscillators with their axis parallel to the vector b = kg × e can be stimulated. (e denotes the polarization of the incident beam kg)

In case of 57Fe (l=1) there are three types of oscillators:
 Dm = 0 a linear magnetic dipole oscillator along the quantization axis Dm = +1 a right hand about Dm = - 1 a left hand circular oscillator

### Example: Pure electric quadrupole interaction

For simplicity let us consider a single crystal with pure electric hyperfine interactions. The excited state is split into two levels and the ground-state remains degenerated. This leads to two transitions R1 and R2 from the excited to the ground state
(
DE(R1) < DE(R2) for Vzz > 0 ):

 R1 | 3/2, ± 1/2ñ ® | 3/2, ± 1/2ñ Dm = 0 | 3/2, ± 1/2ñ ® | 3/2, ± 1/2ñ Dm = ±1 R2 | 3/2, ± 1/2ñ ® | 3/2, ± 1/2ñ Dm = ±1 .

From conventional Mössbauer spectroscopy we know that for single crystals the intensity ratio of the two transition lines is depending on the angle between the quantization axis (Vzz of the EFG in this case) and the direction of the incident radiation. The two extreme cases lead to a ratio of

• R1:R2=1:3 for kg || Vzz and
• R1:R2=5:3 for kg ^ Vzz.

Figure 4: The figure compares nuclear time response and nuclear absorption for different polarizations of the incident radiation and for three scattering geometries.

• Column (a) shows the time response (NFS) after exciting the nuclei in the sample with a short synchrotron radiation pulse (linearly polarized in the plane of the synchrotron).
• Column (b) shows the corresponding spectra in the energy domain assuming a source which is polarized in the same way as the synchrotron radiation.
• Column (c) shows the corresponding energy spectra assuming an unpolarized source as it is typically the case for conventional MS.

The scattering geometries of row (1), (2) and (3) are sketched on the left side. Vzz, the main axis of the electric field gradient (EFG) represents the quantization axis of the system. The weight of the nuclear transitions is indicated in the spectra.

In case of NFS (column (a)) additionally the polarization has to be taken into account: The synchrotron radiation incident on the sample is linearly polarized in the plane of the electron orbit in the storage ring. Due to the radiation characteristics of the relevant multipole radiation only certain transitions can be excited for certain geometries.

• In the geometries (1) and (2) with for b ^ Vzz
both transitions (
Dm = ±1) take place and lead to
a quantum beat structure in the time response,
whereas
• in the geometry (3) with b || Vzz
only
Dm = 0 can be excited and
no quantum beats are observed.

In all other cases the polarization of the incoming beam has to be decomposed into parts perpendicular and parallel to the projection of the quantization axis to a plane normal to the incoming beam and for each part the scattering has to be regarded separately.

### Example: Magnetic dipole interaction

The following figure shows the intensity of an a-iron foil [Hyp.Int. 97/98(1996)589] for different scattering geometries:

• The top part of the figure shows an unmagnetized a-iron foil which leads to a beating originating from all six allowed transitions.
• The bottom left part shows the interference pattern of the four linearly polarized Dm=±1 transitions which were selected by magnetizing the sample in the direction of the polarization of the beam.
• Finally, in the bottom right part only the two Dm = 0 transitions were excited due to the magnetization of the foil being perpendicular to the direction and the polarization of the incident beam.

Nuclear forward scattering by a 5mm thick a-iron foil, 80% enriched in 57Fe. The applied field was 0.2 Tesla, the direction relative to the incident beam is indicated in the figure [Hyp.Int. 97/98(1996)589].

The resulting beat pattern (bottom right) is much less complicated
than the two others. It consists out of quantum beats with
• only one frequency and
• a pronounced dynamical beat.

From the minimum of the dynamical beat at 85 ns the effective thickness of each of the two symmetric lines can be estimated to teff /line=25. This result is in perfect agreement with the 5mm thick a-iron foil which was used for the measurement: the weight of one Dm=0 transition in this geometry is a fraction of [ 4/(3+4+1+1+4+3)]=[ 1/4] of the sum of all (theoretically possible) transitions. The effective thickness of the whole sample is then 4·25 = 100 which corresponds to the 5mm thick foil.

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