. Equat-
v 2 ) = m 2 c
.(52-2).
d quanta,
(52-31),
(52-32).
n contain-
a,ve-length
le incident
It is not
to experi-
f high fre-
natter can
rmula. It
sctron can
e-length is
such as to
ust be just
)f Planck’s
re consider
QUANTUM THEORY
Denote the electric force in the electromagnetic oscillations which
constitute the radiation by X. Acting on an electron of charge — e and
mass m, it will produce an acceleration
r = — eX/m.
hjow according to classical electromagnetic theory an accelerated electron
should radiate energy at the rate
§e 2 r 2 /c 3 = §e 4 Z 2 /w 2 c 3 (53-1).
This radiation is not in the direction of the incident beam and is scattered
radiation. By the conservation of energy it must be supplied at the expense
of the incident beam.
If we have a screen containing N electrons per sq. cm. the radiation
scattered per second will be
2Ne* x -
3 m 2 c 3
where the mean value of X 2 is to be taken. This assumes that the conditions
are such that the electrons scatter independently, and that there is no
systematic phase relation of the wavelets from the separate electrons. The
energy of the incident radiation is Z 2 / 4?7 per cu. cm. (half electric and half
magnetic) ; hence the amount incident on the N electrons in 1 second is
cZ2/47t (53-3).
Dividing (53-2) by (53-3), the fraction of the incident radiation scattered
by the screen is
877 Ne* 8n , t ,
3 ra 2 c 4 3 v n
where b = e 2 jmc 2 = 2-81 . 10 -13 cm.*
The usual definition of a scattering coefficient is the proportion scattered
by a screen containing 1 gm. per sq. cm.; but since we are not likely to
meet with a screen composed wholly of free electrons we prefer to modify
the definition in this case. Instead of a gram of electrons we take the
electrons contained in 1 gm. of matter, assuming (as is roughly true for
all elements except hydrogen) that there is one electron for every two units
of atomic weight, and therefore 3-01 . 10 23 electrons per gm. The scattering
coefficient is then by (53-4)
As with the Compton effect this scattering coefficient has been verified
experimentally by using hard X rays which act on the bound electrons
of the lighter elements practically as if they were free. When the wave
length is comparable with the diameter of the atom the scattering is much