130_notes.dvi

Doppler broadening is a simple non-quantum effect. We know that thefrequency of photons is
shifted if the source is moving – shifted higher if the source is moving toward the detector, and
shifted lower if it is moving away.

∆ω=

v‖

c

ω

∆ω

ω

=

√

kT/m

c

=

√

kT

mc^2

This becomes important when the temperature is high.

Finally, we should be aware of the effect of recoil. When an atom emits aphoton, the atom must
recoil to conserve momentum. Because the atom is heavy, it can carry a great deal of momentum
while taking little energy, still the energy shift due to recoil can be bigger than the natural line
width of a state. The photon energy is shifted downward comparedto the energy difference between
initial and final atomic states. This has the consequence that a photon emitted by an atom will
not have the right energy to be absorbed by another atom, raisingit up to the same excited state
that decayed. The same recoil effect shifts the energy need to excite a state upward. Lets do the
calculation for Hydrogen.

~pH=~pγ

pγ≈

E

c

EH=

p^2

2 mp

=

E^2

2 mpc^2

∆E

E

=

E

2 mpc^2

For our 2p to 1s decay in Hydrogen, this is about 10 eV over 1860 MeV, or less than one part in
108. One can see that the effect of recoil becomes more important as the energy radiated increases.
The energy shift due to recoil is more significant for nuclear decays.

29.12Phenomena of Radiation Theory

29.12.1The M ̈ossbauer Effect

In the case of the emission of x-rays from atoms, the recoil of theatom will shift the energy of
the x-ray so that it is not reabsorbed. For some experiments it is useful to be able to measure the
energy of the x-ray by reabsorbing it. One could move the detector at different velocities to find
out when re-absorption was maximum and thus make a very accurate measurement of energy shifts.
One example of this would be to measure the gravitational red (blue)shift of x-rays.

M ̈ossbauer discovered that atoms in a crystal need not recoil significantly. In fact, the whole crystal,
or at least a large part of it may recoil, making the energy shift very small. Basically, the atom
emitting an x-ray is in a harmonic oscillator (ground) state bound to the rest of the crystal. When
the x-ray is emitted, there is a good chance the HO remains in the ground state. An analysis shows
that the probability is approximately

P 0 =e−Erecoil/ ̄hωHO