Nuclear Forward Scattering
Synchrotron Radiation as source
 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:
E_{i}(w,t) = e_{w0} e^{iwt}, 


with equal amplitudes e_{w}=e_{w0} within the energy band selected by the monochromator at the central energy w_{0. }
If the target has a single resonance at w_{0 }the form of each of the harmonics in the set after transmission through the target is given by:



exp 
é
ë 
i t_{eff }G_{0}/4(^{h}/_{2p})
(ww_{0})  i G_{0}/2(^{h}/_{2p})

ù
û 




exp 
é
ë 
 
t_{eff}/2
u^{2}+1

ù
û 
·exp[i·F(w)] 



(?) 
with 

t_{eff} = d n_{mb} f_{A} s_{0} 

the effective thickness 


F(w) = (t_{eff }/2) ·u / (u^{2}+1) 

the phase shift 


u = (^{h}/_{2p})(ww_{0})/(G_{0}/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 t_{eff} = 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 raised and broadened and
 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


e_{w0} G_{0}/(^{h}/_{2p}) · exp[(1/2)s_{el} n_{mb} d ] 



é
ë 
d(t)exp(iw_{0}tt_{0}  (t/ 2)) (t_{eff}/2) [ J_{1}(t_{eff}t)^{1/2} / (t_{eff}t)^{1/2} ] 
ù
û 



with 

J_{1} 

the Bessel function of first order 


t_{eff} = s_{ph} n d 

the attenuation factor
due to photoelectric absorption 


t = t / t_{0} 

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:


e_{w0} ·(G_{0}/2(^{h}/_{2p})) ·exp[(1/2)s_{el}n_{mb}d] 



exp(t /2) ·t_{eff} ·[ J_{1}(t_{eff }t)^{1/2}) / (t_{eff }t)^{1/2} ] 


 Due to the presence of the Bessel function J_{1},
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 t_{eff }of the target. It can be directly determined via the minima of the beat structure:
Noting that the first zero points of J_{1}(x_{i}) occur at x_{1}=3.8, x_{2}=7.0 and x_{3}=10.2 an approximation of the effective thickness for a single transition for ^{57}Fe via the minima of the beats is
t_{eff }· t [ns] » 
ì
ï
í
ï
î 


Time Dependence: Speed up
In the first stage of the target's response (t » t_{eff }/ 3), the envelope of the forward scattered amplitude can be written as:



exp[(1/2)s_{el}n_{mb}d]

(t_{eff }/ 2) 
· 
exp[(t / 2) (1+ (t_{eff }/ 4))] 





 The field amplitude at t = 0 is proportional to
the effective thickness t_{eff} 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 deexcitation
because of the term (1+ (t_{eff }/4)) proportional to
the effective thickness t_{eff} of the target.
This speedup effect is connected to
the broadening of the energy distribution
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 t_{eff} compared with conventional Mössbauer spectroscopy.
 The upper part figures (a) and (b) show the energy (a) and time (b) spectra for a thick (t_{eff }= 25) and a thin (t_{eff }= 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 t_{eff }= 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 speedup 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.
