, are
taken to be of the first order of smallness. This assumption will be
justified for the case of photons in § 64. We now have from (43) and.
(42) that ¢k|1) is of the first order of smallness, provided E’ does not
he near one of the discrete set of energy-levels E,, and (pa'|1> is of
the second order. The value of (pa’|2> to the second order will thus
be given, from the first of equations (41), by
(W'—W) = A 2 Ba" |V |k"> plus (pa’|2>, therefore satisfies
(W’—W){< pa’ |1>-+< pa’ |29}
= Apo’ |V |pw>+ p3 {pa’|V |k>/(£’ —E,)}.
This equation is of the type (23), provided «’ is such that W’ > me?,
which means that «’ as a final state for the scatterer is not incon-
sistent with the law of conservation of energy. We can therefore infer
from the general result (37) that the scattering coefficient is
47h Wow' P|, , ‘a! |¥ |A>CR|V | px |?
NE cpa’ pao) +S Pe ey (44)
k
The scattering may now be considered as composed of two parts,
a part that arises from the matrix element of the per-
turbing energy and a part that arises from the matrix elements
{p’a’|V|k> and . The first part, which is the same as our
previously obtained result (38), may be called the direct scattering.
The second part may be considered as arising from an absorption of
the incident particle into some state k, followed immediately by a
re-emission in « different direction, and is like the transitions through
an intermediate state considered in § 44. The fact that we have to
add the two terms before taking the square of the modulus denotes
interference between the two kinds of scattering. There is no experi-
mental way of separating the two kinds, the distinction between
them being only mathematical.
52. Resonance scattering
Suppose the energy of the incident particle to be varied con-
tinuously while the initial state «° of the scatterer is kept fixed, so
that the total energy H’ or H’ varies continuously. The formula (44)
now shows that as HZ’ approaches one of the discrete set of energy-
levels H,, the scattering becomes very large. In fact, according to
formula (44) the scattering should be infinite when H’ is exactly equal
to an #,. An infinite scattering coefficient is, of course, physically
impossible, so that we can infer that the approximations used in
deriving (44) are no longer legitimate when £’ is close to an H,. To
investigate the scattering in this case we must therefore go back to
the exact equation (E’ —B)|H’) = VIB’)
equation (2) of § 43 with #’ written for H’, and use a different method
202 COLLISION PROBLEMS § 52
of approximating to its solution. This exact equation, written in
terms of representatives like (41), becomes
(’ —W)
= >| dp" (p'a"|H) +S ,
(H!~By) Bp" pa") + IVR RY.
Let us take one particular H,, and consider the case when Z’ is close
toit. The large term in the scattering coefficient (44) now arises from
those elements of the matrix representing V that lie in row & or in
column &, i.e. those of the type <&£|V|pa’> or (pa’|V[k>. The scatter-
ing arising from the other matrix elements of V is of a smaller order
of magnitude. This suggests that in our exact equations (45) we should
make the approximation of neglecting all the matrix elements of V
except the important ones, which are those of the type or
, where a’ is a state of the scatterer that has not too much
energy to be disallowed as a final state by the law of conservation of
energy. These equations then reduce to
(W'—W) = = al Ck|V |pa’> dp Cpe’ |H">,
the «’ summation being over those values of «’ for which W’ given
by (18) is > mc. These equations are now sufficiently simple for us
to be able to solve exactly without further approximation.
From the first of equations (46) we obtain by division
(pa’ HE") = |(W'—W)+28(W'—W). (47)
We must choose A, which may be any function of the momentum
pand «’, such that (47) represents the incident particles corresponding
to |0> or A?|p%®) together with only outward moving particles. [The
representative of h!fp°a) is actually of the form Ad(W’— W), since
the conditions «’ = «® and p= p°® for it not to vanish lead to
W’ = E'—H,(o') = EB’ —H,(o°) = W = W.] Thus (47) must be
= Apo" | pa) +
+{1/(W'— W)—in d(W'—W)}, (48)
and from the general formula (37) the scattering coefficient will be
4a? WW" P' het P®. | p'a’ |V kD |? |Ck| A. (49)
(45)
(46)
§ 52 RESONANCE SCATTERING 203
It remains for us to determine the value of . We can do this
by substituting for in the second of equations (46) its value
given by (48). This gives
(B!—B,)k| "> = WARY (pa) +
+ > ) [kV | po’> [2(1/(W’ — W)—in 8(W'— WY) ap
= hi(a—16),
where a= > | KkIV |po’>|? d®p/(W'—W) (50)
Ot
and yaad | {k|V pw >a’ —W) Mp
= 9 > Hy [|?8(W'— W)P2dP sin w dud
= 7 > P We ff [k|V|P’wya'>|?sin w dedy. (51)
Thus ki") = h8Ck|V |p%°)/(E’ —E,—a-+ib). (52)
Note that a and 6 are real and that 0 is positive.
This value for [PF 1Ce|V | pon |?
cp? (H' —H,,—a)?+-6? :
One can obtain the total effective area that the incident particle
must hit in order to be scattered anywhere by integrating (53) over
all directions of scattering, i.e. by integrating over all directions of
the vector p’ with its magnitude kept fixed at P’, and then summing
over all «’ that are to be taken into consideration, ie. for which
W’ > me. This gives, with the help of (51), the result
Ach? W b], and determine the probability that at some later time
the particle shall be on its way to infinity with a definite momentum.
The method of § 46 can now be applied. From the result (39) of that
section we see that the probability per unit time per unit range of w
and y, of the particle being emitted in any direction w’, y’ with the
scatterer being left in state «’ is
Qa CW ew’ ya |V [> |, (55)
provided, of course, that «’ is such that the energy W’, given by (18),
of the particle is greater than mc”. For values of «’ that do not satisfy
this condition there is no emission possible. The matrix element
here must refer to a representation in which W, w, y,
and « are diagonal with the weight function unity. The matrix
elements of V appearing in the three preceding sections refer to a repre-
sentation in which p,, p,, p, are diagonal with the weight function
unity, or P, w, x are diagonal with the weight function P?sinw.
They would thus refer to a representation in which W, w, y are
diagonal with the weight function dP/dW.P*?sinw = WP/c?.sinw.
Thus the matrix element in (55) is equal to
(W'P'/c? sin wv’)? times our previous matrix element
or (p’a'|V|k>, so that (55) is equal to
2a W'P"
hk oe
sin w’||?. (56)
To obtain the total probability per unit time of the particle being
emitted in any direction, with any final state for the scatterer, we
§ 53 EMISSION AND ABSORPTION 205
must integrate (56) over all angles w’, x’ and sum over all states a’
whose energy 4,(«’) is such that H,(a’)+mce? < #,. The result is
just 2b/%, where b is defined by (51). There ts thus this simple rela-
tion between the total emission coefficient and the half-width b of the
absorption line.
Let us now consider absorption. This requires that we shall take
an initial state for which the particle is certainly not absorbed but is
incident with a definite momentum. Thus the ket corresponding to
the initial state must be of the form (19). We must now determine
the probability of the particle being absorbed after time 4. Since our
final state & is not one of a continuous range, we cannot use directly
the result (39) of § 46. If, however, we take
|0> = |p», (57)
as the ket corresponding to the initial state, the analysis of §§ 44 and 46
is still applicable as far as equation (36) and shows us that the proba-
bility of the particle being absorbed into state & after time ¢ is
2| |-[1 —cos{(H,—B")t/B} | E,— EB’).
This corresponds to a distribution of incident particles of density
h-8, owing to the omission of the factor #! from (57), as compared
with (19). The probability of there being an absorption after time
t when there is one incident particle crossing unit area per unit time
is therefore
2h W/c2®P?. [ |?[1—cos{(H,— #’)i/i} (L,Y. (58)
To obtain the absorption coefficient we must consider the incident
particles not all to have exactly the same energy W° = H'—H,(a°),
but to have a distribution of energy values about the correct value
E,,—H,(o®) required for absorptidn. If we take a beam of incident
particles consisting of one crossing unit area per unit time per unit
energy range, the probability of there being an absorption after time
t will be given by the integral of (58) with respect to H’. This integral
may be evaluated in the same way as (37) of § 46 and is equal to
4h We? P. ||?.
The probability per unit time of an absorption taking place with an
incident beam of one particle per unit area per unit time per unit
energy range is therefore
42h? W/e2P?. [Ck|V | p%) |2, (59)
which is the absorption coefficient.
206 COLLISION PROBLEMS § 53
The connexion between the absorption and emission coefficients
(59) and (56) and the resonance scattering coefficients calculated in
the preceding section should be noted. When the incident beam does
not consist of particles all with the same energy, but consists of a unit
distribution of particles per unit energy range crossing unit area per
unit time, the total number of incident particles with energies near
an absorption line that get scattered will be given by the integral
of (54) with respect to #’. If one neglects the dependence of the
numerator of (54) on H’, this integral will, since
r b
| (HB, — ape
have just the value (59). Thus the total number of scattered particles
in the neighbourhood of an absorption line is equal to the total number
absorbed. We can therefore regard all these scattered particles as
absorbed particles that are subsequently re-emitted in a different
direction. Further, the number of particles in the neighbourhood of
the absorption line that get scattered per unit solid angle about a
given direction specified by p’ and then belong to scatterers in state
a’ will be given by the integral with respect to E’ of (53), which
integral has in the same way the value
2h2T70 yy" Pp!
SE Kp'a' IFIED ICEIV tp %>
This is just equal to the absorption coefficient (59) multiplied by the
emission coefficient (56) divided by 26/#, the total emission coefficient.
This is in agreement with the point of view of regarding the resonance
scattered particles as those that are absorbed and then re-emitted,
with the absorption and emission processes governed independently
each by its own probability law, since this point of view would
make the fraction of the total number of absorbed particles that are
re-emitted in a unit solid angle about a given direction just the
emission coefficient for this direction divided by the total emission
coefficient.
df’ = a,