General Introduction to Mössbauer Spectroscopy - Mossbauer Spectroscopy - page 6

From ( 1.8 ), the product of lifetime and width is

s C ¼ h ð 1 : 15 Þ

This relation can be interpreted as a Heisenberg uncertainty relation,
Dt DE h. To measure the energy of the state within an uncertainty DE ¼ C,a
time Dt ¼ s is needed. For 57 Fe, the lifetime of the first excited state is 141 ns and
the width of this state given by C ¼ h = s = 4.67 9 10 -9 eV. The ratio of decay
energy 14.4 keV of the first excited state to width is E = C = 3.1 9 10 12 which is
extremely large, that can make the physical measurements in very high precession.

1.1.3 Recoilless Emission and Recoilless Resonant
Absorption of c Photons

Figure 1.1 shows the essence of the Mössbauer effect and the widths of the excited
states of source and absorber are extremely small which is the order of 10 -9 eV for
57 Fe. The width of the emitted c-ray is the order of 10 -9 eV. When the c-ray
emission and absorption processes do not need any energy and momentum transfer
to nuclei, the resonant absorption of c photons can be expected. However, the
energy and momentum conservation law is always important for the emission and
absorption process of c photons. Consider the c-ray emission process by the decay
from first excite state to ground state of 57 Fe, momentum conservation gives
p nucleus =-p photon . The magnitude of the photon momentum is connected with the
photon energy by p photon = E photon /c. There will be a recoil energy loss R in this
process. To calculate R, we assume that the photon is emitted by a nucleus of mass
M that is at rest before the decay. Since the nuclei are vey heavy, we can use the
nonrelativistic approximation to connect the magnitude of the momentum with the
recoil energy R that is the recoil energy of the free nucleus and given by

R ¼ p nucleus = 2M ¼ E 0 = 2Mc 2 : ð 1 : 16 Þ

For the absorption process, the same conservation law should be satisfied. For the
transition E 0 = 14.4 keV in 57 Fe, R is 1.9 9 10 -3 eV which is 10 6 times large
compared to the natural line width of the excited state and no resonance between
source and absorber for the free nucleus can be expected. When the nucleus is
bounded into the solid, the recoil energy can be dispersed by the excitation of the
solid. When the source and absorber are the nuclei embedded into the solid, recoil
energy R may be used for the excitation of phonon that is the vibration state of solid.
Phonon is quantized as discrete value in solid and in usual metals the excitation
energy of phonon states is the order of 10 -1 -10 -2 eV and there is rather large
probability to have a zero phonon excitation in emission and absorption process; in
other word, the recoilless emission and recoilless resonant absorption of photon.
This is the most important characteristic feature of the Mössbauer effect. As a
consequence, the c photon emitted by the decay from the first excited state that has a