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irodov problems in atomic and nuclear physics

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A plasma turn with the current I = 10 kA is located in
the external uniform magnetic field B z directed normally to the
turn's plane. Assuming that the current flows along the surface of
the turn, find the value of B z at which the turn will be at equilibri-
um, if its great and cross-sectional radii are equal to R = 50 CJn and
r = 1.0 em, The expression for the inductance of the turn is to be
taken from the foregoing problem.
15.69. A circular plasma filament is formed in a toroidal chamber
over which a winding is contrived to produce the longitudinal magnet-
ic field BqJ. When the filament carries the current I, the magnetic
lines of force take, under these conditions, the helical form, so that
the plasma filament can develop instabilities of the helical type.
Such instabilities do not occur though, if the filament length is less
than the lead of the helical lines of force on the filament's surface. Find
the limiting value of the current I, if BfP = 20 kG and the great
and cross-sectional radii of the filament are R = 50 cm and r =
= 1.0 cm.
• In all formulas of this chapter, energy, momentum, and mass are expressed
in energy units: p and m being an abbreviation of pc and me", Whenever the
term "mass" is used, the rest mass is meant.
• Kinetic energy of relative motion is the total kinetic energy of particles
in the C frame.
• Lorentz invariant:
E2 - p2 = inv, (16.1)
where E and p are the total energy and total momentum of a system of particles.
On transition from one inertial frame of reference into another this quantity
remains constant.
• Velocity of the C frame relative to laboratory one (the L frame):
B== vic == piE, (16.2)
where p and E are the.sotal momentum and total energy of a system of particles.
• Lorentz equations transforming the momentum, total energy, and angles
on transition from the L to C frame (Fig. 41):
__ p'y-E~ . E- E-p~~ . -- V1-~2 sin O
p",- l/1-~t' - V1-~2' tan fr cos1't-(E/p)~ ,(16.3)
where Bis the velocity of the C frame relative to the L frame.
• Threshold kinetic energy of a particle m striking a stationary particle M
and activating the reaction m + M -+ L mi
T (~mi)2_(m+M)2 (16.4)
m th= 2M •
When a particle of mass 111 decays into t\VO particles, the momenta of the
generated particles are equal (in the C frame) to
-- 1 ..iP==2M y [M2_(mt+ m2)2)·[M2_(rnl - m 2)2] , (16.5)
where m1 and m2 are their masses.
• Vector diagram of momenta for the decay of a relativistic particle of mass M
into t\VO particles with masses mt and m2 (Fig. 42). The locus of possible loca-
tions of the tip of the momentum vector PI of particle ml is an ellipse for which
where b and a are the semi-minor and semi-major axes, f is the focal distance,
p is the momentum of generated particles in the C frame, ~ is the velocity of the
decaying particle (in units of c).
The centre of the ellipse divides the section AB into t\VO parts al and at
in the ratio at : a2 = Et : E2 , where El and £2 are the total energies of the
generated particles in the C frame.
9-0339 f29
Maximum angle at which the particle mt is ejected is defined by the formula
sin 'fr1mu : = M • P , (16.7)
ml P..v
where P1\1 is the momentum of the decaying particle.
• In particle interactions the conservation laws for lepton and baryon charges
hold. In strong interactions also hold the conservation laws for strangeness S.,
isotopic spin T, and its projection T z.
Fig. 41
Fig. 42
• It follows from the generalized Pauli exclusion principle that for a system
of two particles with identical isotopic spins
l+s+T {-1 for half-integer spin particles(-1) = 1 for zero-spin particles,
where 1 is the orbital moment, s is the spin of the system, T is the isotopic'spin,
16.1. Calculate the momenta (GaV/c) of a proton, muon, and
electron whose kinetic energies are equal to 1.0 GeV.
16.2. A relativistic particle with mass m and kinetic energy T
strikes a stationary particle of the same mass. Find the kinetic ener-
gy of their relative motion, momentum of either particle in the C
frame, and velocity of this system.
16.3. What amount of kinetic energy should be provided to a pro-
ton striking a stationary proton to make the kinetic energy of their
relative motion equal to that effected in a collision of two protons
moving toward each other with kinetic energies T = 30 Ge\T?
16.4. A relativistic particle with mass m 1 and kinetic energy T
strikes a stationary particle with mass mi: Find: (a) the kinetic
energy of their relative motion; (b) the momentum and total energy
of either particle in the C frame.
16.5. Determine the kinetic energies of particles with masses m1
and m 2 in the C· frame, if the kinetic energy of their relative motion
is known to be equal to i:
16.6. One of the particles of a system moves with momentum p
and total energy E at the angle {} (in the L frame) relative to the
velocity vector ~c of the C frame. Find the corresponding angle ~ in
the C frame.
16.7. A relativistic proton with the kinetic energyT is scattered
through the angle 1}1 by a stationary proton. As a result of the colli-
sion, the initially stationary proton is ejected at the angle '6'2.
(a) Demonstrate that cot {}l cot {}2 == 1 + T/2m.
(h) Calculate the minimum possible angle of divergence of the
two particles.
(c) Determine T and kinetic energies of either particle after colli-
sion, if '6'1 == 30° and tt 2 == 45°.
16.8. Demonstrate that when a relativistic particle with mass m l
is elastically scattered by a stationary particle with mass m2 < mit
the maximum scattering angle of the incoming particle is given by
the expression sin {}max == m 21m1 -
16.9. A negative muon with the kinetic energy T == 100 MeV
sustains a head-on collision with a stationary electron. Find the
kinetic energy of the recoil electron.
16.10. Relativistic protons with the kinetic energy T are ~astical-
ly scattered by stationary nuclei of hydrogen atoms. Let '6' be the
proton scattering angle in the C frame corresponding to the angle a
in the L frame. Prove that
(8) tan (,fr/2) == V1 + T/2m tan '6', m is the mass of the proton;
(b) the differential cross-sections of this process in the C and L
frames are related as
,....,,...., (1+asin2 'fr)2
a ({)-) == 4 (1+0:)cos~ (J (1t) cm-/sr; a = T12m;
(c) the scattering in the C frame is anisotropic, if the differential
cross-sections <11 and <1 2 corresponding to angles '6'1 = 15° and '6'2 =
= 30° are equal to 26.8 and 12.5 mb/sr respectively at T = 590 MeV.
16.11. Relativistic protons with the kinetic energy To are elasti-
cally scattered by nuclei of hydrogen atoms.
(a) Demonstrate that the differential cross-section a (T) corre-
sponding to the energy T of the scattered proton in the L frame is defined
by the expression
,...., ,...., 4n
a(T)=a ('l~) --r;cm2/1V1eV,
where a(6) is the differential cross-section in the C frame in which
the angle '.fr corresponds to the kinetic energy T.
(b) Find the energy distribution of the scattered protons in the L
frame, if their angular distribution in the C frame is isotropic.
16.12. A positron whose kinetic energy is equal to its rest energy
and a stationary free electron annihilate. As a result, two 'V-quanta
emerge, the energy of one being n == 2 times that of the other. Cal-
culate the divergence angle between the motion directions of the
16.13. Demonstrate that when relativistic positrons with momen-
tum p and free electrons annihilate, the differential cross-section of
9· 131
-v-quanta production with energy E l' varies inversely with the posi-
trons' momentum, if the angular distribution of y-quanta in the C
frame is isotropic.
16.14. Calculate the threshold energy of a 'V-quantum required for
1£+.n- pair production in the field of stationary proton.