7.2 Problems
383
between θ L and the angle of deflection θ M , in the center of mass frame is
tan θ L = sin θ M /[m 1 /m 2 + cos θ M ]
Show also that θ L can not be greater than about 15
◦ if m 1 /m 2 = 4.
[University of London]
7.7 Show that the maximum velocity that can be imparted to a proton at rest by
non-relativistic alpha particle is 1.6 times the velocity of the incident alpha
particle.
7.8 Show that the differential cross section σ (θ ) for scattering of protons by protons in the Lab system is related to σ (θ
∗ ) corresponding to the CMS by the
formula σ (θ ) = 4 cos(θ
∗
/2) σ (θ
∗ ).
7.9 If E 0 is the neutron energy and σ the total cross-section for low energy n–p
scattering assumed to be isotropic in the CMS, then show that in the LS, the
proton energy distribution is given by dσ p /dE p = σ/E 0 = constant.
7.10 Particles of mass m are elastically scattered off target nuclei of mass M initially at rest. Assuming that the scattering in The CMS is isotropic show that
the angular distribution of M in the LS has cos ϕ dependence.
7.11 A beam of particles of negligible size is elastically scattered from an infinitely
heavy hard sphere of radius R. Assuming that the angle of reflection is equal
to the angle of incidence in any encounter, show that σ (θ ) is constant, that is
scattering is isotropic and that the total cross-section is equal to the geometric
cross-section, πR
2 . (Osmania University)
7.2.2 Rutherford Scattering
7.12 Show that for the Rutherford scattering the differential cross section for the
recoil nucleus in the Lab system is given by σ (ϕ) = (z Ze
2
/2T )
2
/ cos
3
ϕ
7.13 A beam of α-particles of kinetic energy 5 MeV passes through a thin foil of
4 Be
9 . The number of alphas scattered between 60
◦ and 90
◦ and between 90
◦
and 120
◦ is measured. What would be the ratio of these numbers?
7.14 If the probability of α-particles of energy 10 MeV to be scattered through an
angle greater than θ on passing through a thin foil is 10
−3 , what is it for 5 MeV
protons passing through the same foil?
[University of Bristol]
7.15 What α-particle energy would be necessary in order to explore the field of
force within a radius of 10
−12 cm of the center of nucleus of atomic number
60, assuming classical mechanics to be adequate?
[University of London]
7.16 In an elastic collision with a heavy nucleus when the impact parameter b is
just equal to the collision radius R 0 /2, what is the value of the scattering angle
θ
∗ in the CMS?
383
between θ L and the angle of deflection θ M , in the center of mass frame is
tan θ L = sin θ M /[m 1 /m 2 + cos θ M ]
Show also that θ L can not be greater than about 15
◦ if m 1 /m 2 = 4.
[University of London]
7.7 Show that the maximum velocity that can be imparted to a proton at rest by
non-relativistic alpha particle is 1.6 times the velocity of the incident alpha
particle.
7.8 Show that the differential cross section σ (θ ) for scattering of protons by protons in the Lab system is related to σ (θ
∗ ) corresponding to the CMS by the
formula σ (θ ) = 4 cos(θ
∗
/2) σ (θ
∗ ).
7.9 If E 0 is the neutron energy and σ the total cross-section for low energy n–p
scattering assumed to be isotropic in the CMS, then show that in the LS, the
proton energy distribution is given by dσ p /dE p = σ/E 0 = constant.
7.10 Particles of mass m are elastically scattered off target nuclei of mass M initially at rest. Assuming that the scattering in The CMS is isotropic show that
the angular distribution of M in the LS has cos ϕ dependence.
7.11 A beam of particles of negligible size is elastically scattered from an infinitely
heavy hard sphere of radius R. Assuming that the angle of reflection is equal
to the angle of incidence in any encounter, show that σ (θ ) is constant, that is
scattering is isotropic and that the total cross-section is equal to the geometric
cross-section, πR
2 . (Osmania University)
7.2.2 Rutherford Scattering
7.12 Show that for the Rutherford scattering the differential cross section for the
recoil nucleus in the Lab system is given by σ (ϕ) = (z Ze
2
/2T )
2
/ cos
3
ϕ
7.13 A beam of α-particles of kinetic energy 5 MeV passes through a thin foil of
4 Be
9 . The number of alphas scattered between 60
◦ and 90
◦ and between 90
◦
and 120
◦ is measured. What would be the ratio of these numbers?
7.14 If the probability of α-particles of energy 10 MeV to be scattered through an
angle greater than θ on passing through a thin foil is 10
−3 , what is it for 5 MeV
protons passing through the same foil?
[University of Bristol]
7.15 What α-particle energy would be necessary in order to explore the field of
force within a radius of 10
−12 cm of the center of nucleus of atomic number
60, assuming classical mechanics to be adequate?
[University of London]
7.16 In an elastic collision with a heavy nucleus when the impact parameter b is
just equal to the collision radius R 0 /2, what is the value of the scattering angle
θ
∗ in the CMS?