1000 Solved Problems in Modern Physics

(Tina Meador) #1

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
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