1000 Solved Problems in Modern Physics

3.3 Solutions 239

tankR≈kR+

(kR)^3

3

+···

We therefore have

tanδ 0 ≈

k

[

D−R−

(

(kR)^3

3 k

)]

1 +k^2 DR

If the inequalitieskR<<1 andk^2 DR<<1 are satisfied we can still

further simplify the expression for tanδ 0.

tanδ 0 ≈k(D−R)=kR

(

tank 1 R

k 1 R

− 1

)

(8)

The total cross-section is then

σ=

4 π

k^2

sin^2 δ 0 ≈ 4 π(D−R)^2 = 4 πR^2

(

1 −

tank 1 R

k 1 R

) 2

(9)

It follows that if the condition

tank 1 R=k 1 R (10)

is satisfied, the phase shift and the scattering cross-section both vanish. This

phenomenon is known as the Ramsauer-Townsend effect. The field of the

inert gas atoms decreases appreciably faster with distance than the field of

any other atom, so that to a first approximation, we can replace this field by a

rectangular spherical well with sharply defined range and use Equation (10)

to evaluate the cross-section for slow electrons.

Physically, the Ramasuer – Townsend effect is explained as the diffraction

of the electron around the rare-gas atom, in which the wave function inside

the atom is distorted in such a way that it fits on smoothly to an undistorted

function outside.

Here the partial wave wave withl=0 has exactly a half cycle more of

oscillation inside the atomic potential then the wave in the force-free field,

and the wavelength of the electron is large enough in comparision withRso

that higherlphase-shifts are negligible.

3.112 In order that the Schrodinger equation is reduced to the given form is that the
potentialV(r) does not depend on time. From Problem 3.104 the total wave
function

ψ=Σil(2l+1)

[

eiδl

kr

]

sin

(

kr−

1

2

πl+δl

)

pl(cosθ)

For slow neutrons only the first term (l=0) in the summation is important.

Asp 0 (cosθ)= 1

ψ=

exp(iδ 0 )

kr

sin(kr+δ 0 )

u=ψr=

exp(iδ 0

k

sin(kr+δ 0 )=const.sin(kr+δ 0 )

We assume that the wave function inside the well is identical with that

in the deuteron problem. This is justifiable since the total energy inside the

potential well is raised by little over 2 MeV corresponding to the