# Nuclear Forward Scattering

• The forward scattered amplitude

## Energy dependence of the forward scattered amplitude

A pulse of SR (with a typical duration of ~10-10 s)
can be represented by a continuous set of coherent harmonics:

 Ei(w,t) = ew0 eiwt,

with equal amplitudes ew=ew0 within the energy band selected by the monochromator at the central energy w0.

If the target has a single resonance at w0 the form of each of the harmonics in the set after transmission through the target is given by:

 Etr(w)
 = Ei(w) ·
 exp é ë i teff G0/4(h/2p) (w-w0) - i G0/2(h/2p) ù û
 = Ei(w) ·
 exp é ë - teff/2 u2+1 ù û ·exp[i·F(w)]
(?)
 with teff = d nmb fA s0 the effective thickness F(w) = (teff /2) ·u / (u2+1) the phase shift u = (h/2p)(w-w0)/(G0/2) frequency dependent factor

 The figure shows the energy distribution of the amplitudes of the forward scattered radiation at time zero normalized by the amplitude of incident radiation. The curves correspond to different target thicknesses of teff = 1,10,30.

In the case of a thin target
the energy distribution of the forward scattered radiation
• is peaked at resonance
• its form is close to the Lorentzian one
• with an always smaller amplitude than that of the incident.

As the thickness of the target increases,

• The energy distribution
is
• multiple scattering processes
cause a
double hump structure

• at resonance the forward scattered amplitude
cannot exceed that of the incident field
• but symmetrically around the resonance
• the magnitude and
• the separation
of the field amplitude increases
with increasing thickness of the target

## Time dependence of the forward scattered amplitude:

Since in nuclear resonant scattering experiments we measure the response of the nuclei as a function of time after excitation, we need to determine the time dependence of the transmitted field. This is done by integration of all of the components given by equation

 Etr(t)
 =
 ew0 G0/(h/2p) · exp[-(1/2)sel nmb d ]
 ·
 é ë d(t)-exp(iw0tt0 - (t/ 2)) (teff/2)  [ J1(tefft)1/2 / (tefft)1/2 ] ù û
 with J1 the Bessel function of first order teff = sph n d the attenuation factor due to photoelectric absorption t = t / t0 the time in units of the mean life time

The transmitted wave packet is composed of

• the incident (prompt) part,
represented by the delta function and
• the forward scattered (delayed) part
as the coherent response of the nuclear target

## Time Dependence: Dynamical beat

The envelope of the forward scattered amplitude has the form:

 Efwd(teff,t)
 =
 -ew0 ·(G0/2(h/2p)) ·exp[-(1/2)selnmbd]
 ·
 exp(-t /2) ·teff ·[ J1(teff t)1/2) / (teff t)1/2 ]

• Due to the presence of the Bessel function J1,
the
forward scattered amplitude
• oscillates around zero
• while decaying exponentially:
These oscillations can be interpreted as a beat between the two groups of harmonics distributed symmetrically on both sides of the resonance frequency (see fig. above).

Such a beat is called a dynamical beat.

• The appearent beat frequency (in the forward scattered amplitude) is then determined by the distance of the two maxima (of the double humb structure of the energy distribution) and increases accordingly with the effective thickness teff of the target. It can be directly determined via the minima of the beat structure:

Noting that the first zero points of
J1(xi) occur at x1=3.8, x2=7.0 and x3=10.2 an approximation of the effective thickness for a single transition for 57Fe via the minima of the beats is
teff · t [ns]    »    ì
ï
í
ï
î
 2100
 [ns]
 1st  minimum
 6900
 [ns]
 2nd   minimum
 14600
 [ns]
 3rd   minimum
 ¼

## Time Dependence: Speed up

In the first stage of the target's response (t » teff / 3), the envelope of the forward scattered amplitude can be written as:

 Efwd
µ
 ·
exp[-(1/2)selnmbd]
 (teff / 2) · exp[-(t / 2) (1+ (teff / 4))]

• The field amplitude at t = 0 is proportional to
the
effective thickness teff of the target,
i.e. every nucleus of the ensemble
contributes constructively to it.

• The amplitude decays faster than that of
the natural decay of the nuclear de-excitation
because of the term
(1+ (teff /4)) proportional to
the
effective thickness teff of the target.

This
speed-up effect is connected to
the
of the forward scattered amplitude.

 Mössbauer spectroscopy (energy scale) compared to Nuclear Forward scattering (time scale)

Simulations of the NFS time evolution in case of different sample thicknesses teff compared with conventional Mössbauer spectroscopy.

• The upper part figures (a) and (b) show the energy (a) and time (b) spectra for a thick (teff = 25) and a thin (teff = 1) absorber. The thin target leads to an exponential decay and the thick target to an exponential decay modulated by the Bessel function.
• The lower part figures c and d compare the energy (c) and time (d) spectra of absorber with one and two transition lines. The effective thickness per transition line is teff = 25. The minima of the dynamical beats occur at the same time after excitation for both cases (correspond to the equal effective thickness of 25). The sample with the split energy line shows in addition a fast quantum beat structure. See next chapter!

Remark:

In conventional Mössbauer spectroscopy (MS) it is difficult to distinguish thickness broadening from resonance broadening created for example by the distribution of hyperfine fields over the whole sample or by fluctuating hyperfine fields with a flip rate close to the life time of the excited state of the nucleus. In contrast to MS Nuclear forward scattering (NFS) reflects the thickness as an additional beating and a speed-up of the decay and resonance broadening as an accelerated coherent response (exp[- (1+(D / G))t] ) of the nuclei if the distribution is also along the beam direction. This can be interpreted as a faster dephasing of the field amplitudes of the scattered radiation.

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