1000 Solved Problems in Modern Physics

3.2 Problems 155

3.115 Given the scattering amplitude

f(θ)=(1/ 2 ik)

∑

(2l+1)

[

e^2 iδl− 1

]

Pl(cosθ)

Show that

Im f(0)=kσt/ 4 π

3.116 Obtain the form factorF(q) for electron scattering from an extended nucleus
of radiusRand chargeZewith constant charge density. Show that the minima
occur when the condition
tanqR=qR, is satisfied

3.117 In the Born’s approximation the scattering amplitude is given by

f(θ)=(− 2 μ/q^2 )

∫∞

0

V(r)sin(qr)rdr

whereμis the reduced mass of the target-projectile system, andqis the

momentum transfer. Show that the form factor is given by the expression

F(q)=(4π/q)

∫∞

0

ρ(r)sin(qr)rdr

whereρ(r) is the charge density

3.118 Obtain the differential cross-section for scattering from the shielded Coulomb
potential for a point charge nucleus of the form
V=z 1 z 2 e^2 exp(−r/r 0 )/r
wherer 0 is the shielding radius of the order of atomic dimension. Thence
deduce Rutherford’s scattering law.

3.119 Electrons with momentum 300 MeV/c are elastically scattered through an
angle of 12◦by a nucleus of^64 Cu. If the charge distribution on the nucleus is
assumed to be that of a hard sphere, by what factor would the Mott scattering
be reduced?

3.120 An electron beam of momentum 200 MeV/cis elastically scattered through
an angle of 14◦by a nucleus. It is observed that the differential cross-section
is reduced by 60% compared to that expected from a point charge nucleus.
Calculate the root mean square radius of the nucleus.

3.121 Assuming that the charge distribution in a nucleus is Gaussian,e

−(r^2 /b^2 )

π^3 /^2 b^3 then

show that the form factor is also Gaussian and that the mean square radius is

3 b^2 / 2

3.122 In the Born approximation the scattering amplitude is given by

f(θ)=

(

−

μ

2 π^2

)∫

V(r)eiq.rd^3 r

Show that for spherically symmetric potential it reduces to

f(θ)=

(

−

2 μ

q^2

)∫

rsin(qr)V(r)dr