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Equation (1266) is not particularly useful, as it stands, because
the quantity
depends on the, as yet, unknown wavefunction
[see Eq. (1261)]. Suppose, however, that the scattering is
not particularly strong. In this case, it is reasonable to suppose that the total
wavefunction, , does not differ substantially from the
incident wavefunction,
. Thus, we can
obtain an expression for
by making the
substitution
in Eq. (1261). This procedure is called
the Born approximation.
The Born approximation yields

(1267) 
Thus,
is proportional to the Fourier transform of the scattering potential with respect to the wavevector
.
For a spherically symmetric potential,

(1268) 
giving

(1269) 
Note that
is just a function of for a
spherically symmetric potential.
It is easily demonstrated that

(1270) 
where is the angle subtended between the vectors
and . In other words, is the scattering angle. Recall that the
vectors and have the same length, via energy conservation.
Consider scattering by a Yukawa potential

(1271) 
where is a constant, and measures the ``range'' of the
potential. It follows from Eq. (1269) that

(1272) 
since

(1273) 
Thus, in the Born approximation, the differential crosssection
for scattering by a Yukawa potential is

(1274) 
given that

(1275) 
The Yukawa potential reduces to the familiar Coulomb potential as
, provided that
. In this limit, the Born differential crosssection becomes

(1276) 
Recall that is equivalent to , so the above
equation can be rewritten

(1277) 
where is the kinetic energy of the incident particles.
Of course, Eq. (1277) is the famous Rutherford scattering crosssection formula.
The Born approximation is valid provided that is
not too different from
in the scattering region.
It follows, from Eq. (1258), that the condition for
in the vicinity of
is

(1278) 
Consider the special case of the Yukawa potential. At low energies,
(i.e., ) we can replace
by unity,
giving

(1279) 
as the condition for the validity of the Born approximation.
The condition for the Yukawa potential to develop a bound state
is

(1280) 
where is negative. Thus, if the potential is strong enough to
form a bound state then the Born approximation is likely to break
down. In the high limit, Eq. (1278) yields

(1281) 
This inequality becomes progressively easier to satisfy as increases,
implying that the Born approximation is more accurate at high
incident particle energies.
Next: Partial Waves
Up: Scattering Theory
Previous: Fundamentals
Richard Fitzpatrick
20100720