The isotope ⁵⁷Co decays slowly (half life t_½  = 270 days) to ⁵⁷Fe*.  ⁵⁷Fe* has a very short t_½ (0.2 μs). Corresponding to this half life, the uncertainty in energy is ΔE  =  ħt_½ =    2 MHz.  A γ - ray of  n = 3.5 x 10¹⁸ Hz (or E = 14.4 keV) is emitted by ⁵⁷Fe* to reach another absorbing ⁵⁷Fe.  Since ΔE/E = 2 MHz/3.5 x 10¹⁸ Hz is only one part in a trillion (1/10¹²), the resonance is very sharp.  If this source γ - ray (Fig 15.1) can be absorbed by a sample ⁵⁷ Fe, a very sharp absorption would occur, except for a major problem due to Doppler broadening (or Doppler effect).  If a stationary ⁵⁷Fe*  emitted a γ - ray of such high energy, it would recoil with velocity  υ of  hν/m_Fec    which is about 100ms^-1.  This recoil velocity υ will cause a Doppler broadening of nυ/c which will spoil the resonance condition entirely.  One way to avoid Doppler broadening would be use large lattices so that the atoms in the lattice can not move.  A better way is to move the source relative to the sample to counter the effect of Doppler broadening.  The source is mounted on a support which can be moved at a few mm/s either by rotating a screw or electromagnetically.  It is a stroke of good fortune that such small speeds are sufficient to match the emitter and the absorber levels.  A motion towards the absorber (positive velocity) implies that the absorber levels have a larger separation than the source levels.

Example 15.1: Calculate the frequency shift between the emitter and absorber for the ⁵⁷Co source if the source is moved towards the sample at a speed of 1 mm/s.

Solution:

The Doppler shift is δν = ν v/c                                                                                                 (15.1)

Frequency  ν corresponding to 14.4 keV = (14400 eV x 8066 eV/cm) x c. Here eV is converted to cm^-1 by using 1 eV = 8066 cm^-1.  The product is multiplied by c (in cm/s) to get frequency in Hz.

δν = ν v/c = ν x 1.16 x 10⁸(c/c) cm^-1

                = 0.1(cm/s) x 1.16 x 10⁸cm^-1 = 11.6 MHz

            So far, we have seen why the sample absorption can occur at a frequency other than source emission frequency.  In Mossbauer spectroscopy, the change in the position of the resonance is often called the isomer shift (rather than chemical shift).  The spectra are recorded as the γ - ray counting rate versus the velocity of the source.  The other reason for isomer shifts is the difference in the s-electron density at the nucleus.  Excited states of nuclei often shrink by a few percent on excitation.  The change in the energy of the nuclear isotope

ΔE α  Ψ_s²(0) δR                                                                                                                   (15.2)

Here  Ψ_s²(0)  is the square of the electron density at the nucleus and  δR the change in the nuclear radius.  If the nuclear spin of the excited state is different from the ground state (eg I =  ½ for ⁵⁷Fe and I = 3/2  for ⁵⁷Fe*), the quadrupole moment (resulting from spins >½)  leads to quadrupolar splitting of the absorption frequency.  The nuclear energies of spins 1/2 and  3/2  are different giving rise to two lines.  In fig 15.2, the Mossbauer spectra of Fe(CN)₆^4-, Fe[(CN)₅NO]^2-  and FeSO₄are shown.