# Mössbauer Spectroscopy

The basis of the Mössbauer effect is the

recoilless resonance absorption of g-rays:

a g-ray emitted in the de-excitation of a nuclear excited state can be absorbed by another nucleus of the same kind and excite it. By definition, resonance absorption can only take place if

the emission energy matches the absorption energy.

The energy distribution N(E) of the g-radiation emitted by a free nucleus has a Lorentzian shape with a full width at half maximum (FWHM) determined by Heisenberg's uncertainty principle, known as the natural linewidth of the nuclear transition G0 :
 G0 = (h/2p) / t0 ,
(1)
where t0 is the mean lifetime of the nuclear excited state.
Since the g-ray carries a momentum that has to be conserved, this will be transferred to the free emitting nucleus which will recoil with an energy:
 ER = E02 2Mc2 ,
(2)
where E0=Ee-Eg is the energy difference between excited and ground states and M is the mass of the nucleus.
• The energy distribution of the emitted radiation will then be centred at an energy Eem=E0-ER
 N(E) = N0 (G0/2)2 (E-Eem)2 + (G0/2)2 .
(3)
• For the same reason, the absorption cross section sa(E) will be centred at an energy Ea=E0+ER and will have the form:
 sa(E) = s0 (G0/2)2 (E-Ea)2 + (G0/2)2 .
(4)
The quantity s0 is the maximum resonance cross section given by:
 s0 = 2pc2 (h/2p)2 E02 2Ie + 1 2Ig + 1 1 1+a ,
(5)
where Ie and Ig are the spins of the nuclear excited and ground states, respectively, and a is the internal conversion coefficient.

Due to the nuclear recoil, the emission and absorption spectra of free nuclei will be shifted with respect to each other by the amount

Ea-Eem=2ER,

where

• the recoil energy ER
is of the order of 10-3-10-2 eV.

The lifetimes t0 of low-lying nuclear levels can be quite long (several ns), therefore

• the natural linewidth G0
is of the order of 10-9-10-8 eV.

Consequently ER >> G0 and
the

If the emitter and absorber nuclei are bound in a solid, this problem may be circumvented. As
• the bonding energy EB of atoms
is of the order of
eV,

the nuclei cannot recoil freely (ER << EB). They can only exchange energy with the lattice by creating or annihilating phonons while the g-ray transition occurs.

Although the typical energies of phonon excitations are of the same order of magnitude as the recoil energy of a free nucleus, there is a defined probability fLM (the recoilless fraction or Lamb-Mössbauer factor) that no lattice excitations take place, but the phonon state of the crystal stays unaltered during the g-ray emission or absorption. This process is called a zero-phonon g-transition.

The momentum of the recoil is here absorbed by the whole crystal: as its mass is considerably larger than that of the single nucleus (~1020 times), the recoil energy on the emitting or absorbing nucleus is negligible, and the emission takes place at E=E0. A nucleus of the same kind can then absorb the g-ray resonantly.

The recoilless emission and absorption of g-rays
is named the Mössbauer effect
.

## Basic Principle / Experimental setup

The energy spread of the g-rays emitted by a nucleus without recoil, given by the natural linewidth G0 is typically one to two orders of magnitude smaller than the splitting of the nuclear energy levels due to hyperfine interactions. If one can shift continuously the energy of the emitter nuclei over the energy range of these interactions, it is possible to study the

the dependence of resonance absorption
on the energy of the incident radiation.

This is the basic principle of Mössbauer spectroscopy (MS).

In a typical transmission experiment, the energy of the g-rays emitted by a radioactive source containing the Mössbauer nuclei is shifted over a broad energy range (up to ~  meV) via a Doppler shift obtained by moving the source with respect to the absorber. The energy of the g-rays emitted when the source has a relative velocity v is:

 Eg = E0 (1 ± v/c).
(6)
The g-rays transmitted by the absorber at different energies can be counted by a detector placed behind the absorber. A typical transmission spectrum for an absorber whose levels are not split by hyperfine interactions is shown in figure above:

It shows a single absorption line whose minimum is at the velocity at which the energy of the g-rays emitted by the source exactly matches the energy difference between ground and excited state in the absorber. In the approximation of an infinitely thin absorber, the absorption line I(v) is a Lorentzian (like N(E) and s(E) defined in eqs. (3) and (4)), with a full width at half maximum (FWHM) equal to 2G0:

I(v) = 1 -  I0

 1+4 æ è vE0 2G0 c ö ø 2
.
(7)

## Some usefull formulas

The emission spectrum of a single line source
with negligible effective thickness has the shape

 IS(v,w) = fS (G0 / 2p) [w-w0(1 + v/c) ]2 + (G0 / 2)2
 with fS the Lamb-Mössbauer factor of the source v the velocity of the source c the velocity of light w0 the frequency of the emitted g-quanta at v=0 G0 the natural linewidth of the nuclear levels (4.7 neV corresponding to 0.097 mm/s)

The probability for a g-quantum with frequency w
being detected after passing the absorber

 P(w) = exp[-d  nmb  s(w)]  .
 depends on d the thickness of the absorber in the direction of the radiation nmb the density of resonant nuclei in the absorber s(w) the absorption cross section per atom

The cross section per atom contains two additive parts.
The first part is the nuclear cross section which is defined as

 sR(w) = fLM  · (l)2 2p 1 1+a 2 Ie + 1 2 Ig + 1 · (G0 / 2)2 (w-w0 )2 + (G0 / 2)2
 with fLM the Lamb-Mössbauer factor of the absorber l the wavelength of the radiation a the coefficient of internal conversion Ie,Ig the nuclear spin quantum number for the excited and ground state s0 the maximum resonance cross section (w = w0).

The second part of the cross section per atom concerns

• the electron shell (photo effect, Compton effect, scattering) and
• absorption processes where the lattice is involved.

For several G0 around the resonance energy this part is

• nearly energy independent and
• much smaller than the nuclear part at exact resonance energy.

The spectrum for a single line absorber
with natural line-width
is then given by

 I(v) I(¥)
 =
 (1-fS) + ¥ ó õ -¥ IS(v,w) exp[-d n sR(w) ] dw .
 I(v)
 :
 intensity at velocity v
 I(¥)
 :
 intensity at velocity v=¥
 with I(¥)=exp(-d n sel)
The count rate far away from the resonance energy depends only on the electronic part of the cross section since the nuclear part of the cross section is zero.

The effective thickness of the absorber is defined as

 teff
 =
 d  nmb fA s0 .
For a thin absorber (teff << 1) the area of a line is
 A   = p 2 fS  G0 teff.
As the thickness of the absorber increases, the absorption line saturates and the width increases. For a thick absorber the area of a measured line (with width GA) is determined by
 A
 =
p

2
fS GA ·
 ( teff   e-teff /2 é ë I0 æ è teff 2 ö ø + I1 æ è teff 2 ö ø ù û
).
 I0, I1
 :
 modified Bessel function of zeroth and first order

If we neglect the source width, for comparison with nuclear resonance scattering (NRS) where a nearly ideal (impulse) source is provided, then the transmitted intensity can be written as

 I(w)   =  I(¥)  exp[-nmb  d   sR(w)] .

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