LECTURE 10 NUCLEAR REACTIONS PHY492 Nuclear and Elementary Particle Physics

LECTURE 10
NUCLEAR REACTIONS
PHY492 Nuclear and Elementary Particle Physics
Direct Reactions
Direct reactions:
Nuclear reactions that occur in a time comparable to a
transit time of an incident particle across the nucleus
de Broglie wavelength
λ =
h ≤ 1fm
p ( Energy ≥ 20 MeV for proton ) Elastic Scattering :
a+ A → a+ A
A(a,a)A
Inelastic Scattering : a + A → a’ + A*
A(a,a’)A*
Transfer reaction :
January 31, 2014 p+
( a: beam, A: target ) 16O
→ d + 15O ( pickup reaction )
16O(p,d)15O
d + 16O → p + 17O ( stripping reaction )
16O(d,p)17O
PHY492, Lecture 10 2 Q value
Q value:
the change in the kinetic energy in the reaction
Q > 0 (kinetic energy increases) : exothermic reaction
Q < 0 (kinetic energy decreases) : endothermic reaction
In two-body reactions, 1 + 2 → 3 + 4　,
The Q value is given by;
Q = [ ( M1
2(1,3)4 + M2 ) – ( M3 + M4 ) ] c2 The energy threshold for the reaction is given by
M1 + M2 ( – Q)
M2 January 31, 2014 PHY492, Lecture 10 3 Lab
CM transformation
In many cases,
the physics is simpler in the Center of Mass (CM) frame
rather than the laboratory (Lab) frame the center-of-mass moves along the z-axis with a speed Vcm,
(m1+m2) · Vcm = m1 · V1
Vcm =
January 31, 2014 m1
V1 m1+m2 PHY492, Lecture 10 4 Lab
CM transformation (2)
Consider elastic scattering 1 + 2 → 1 + 2 in non-relativistic kinematics
P1Lab P1CM θLab m1 P1Lab · sin(θLab) = P1CM · sin(θCM)
P1Lab · cos(θLab) = P1CM · cos(θCM) + PCM tan(θLab) =
sin(θCM)
cos(θCM) + PCM / P1CM PCM
= m1·VCM
P1CM θCM m12
=
V1 m1+m2 = sin(θCM)
cos(θCM) + (m1/m2) P1CM = m1 · ( V1 - VCM )
=
January 31, 2014 m1·m2
V1 m1+m2 PHY492, Lecture 10 5 Lab
CM transformation (3)
Consider elastic scattering 1 + 2 → 1 + 2 in relativistic kinematics
P1Lab P1CM θLab m1 - Vcm Lorentz transformation from CM to Lab with (-Vcm),
θCM P1Lab · sin(θLab) = P1CM · sin(θCM)
E1CM
P1Lab · cos(θLab) = γ(Vcm) · P1CM · cos(θCM) + β(Vcm)·γ(Vcm)· c sin(θCM)
1
tan(θLab) = γ(Vcm) ·
cos(θCM) + β(Vcm)·E1CM / P1CM·c sin(θCM)
1
= γ(Vcm) ·
cos(θCM) + Vcm/ u E1CM = m1·c2·γ(u)
P1CM = m1·u·γ(u)
u : velocity of 1 in CM =(m1/m2) (non-relativistic limit) January 31, 2014 PHY492, Lecture 10 6 Lab
CM transformation (4)
Intuitive image
for the Lab
CM transformation in relativistic kinematics
P1Lab P1CM θLab PCM
m1 = m1·VCM
P1CM θCM m12
=
V1 m1+m2 γ = 1 relativistic
(forward focusing) γ > > 1 January 31, 2014 non relativistic PHY492, Lecture 10 7 Inverse kinematics
In (normal) kinematics,
beam is provided as a probe to investigate a target
p + 16O → p + 16O (16O is a nucleus of interest)
In many cases,
m1 << m2
PCM << P1CM
P1Lab m1·m2
=
V1 P
1
CM
θLab ≈ θCM θLab m1+m2 PCM
=
θCM m12 V1 m1+m2 In scattering experiments with rare-isotope beams,
an unstable nucleus of interest is produced in nuclear reactions
and provided as a beam. In this case, a target works as a probe.
24O + p → 24O + p (24O is a beam and a nucleus of interest)
This is called “ inverse kinematics “.
θLab P1Lab P1CM In this case,
m1 >> m2
m1 forward focusing θCM January 31, 2014 PHY492, Lecture 10 8 Compound Reactions
Compound reactions:
Beam and target nuclei make a highly excited compound
system (fusion), which stays together sufficiently long for its
excitation energy to be shared more or less uniformly by all
its constituent nucleons.
a + A → C* → b + B
→ C
January 31, 2014 PHY492, Lecture 10 9 DR vs CR : energy dependence
Difference between Direct reactions (DR) and Compound reactions (CR) Typical spectrum of energies of the nucleons emitted at a fixed angle
in inelastic nucleon-nucleus reactions. January 31, 2014 PHY492, Lecture 10 10 DR vs CR :differential cross sections
Difference between Direct reactions (DR) and Compound reactions (CR) 25Mg(p,p)25Mg
Ep = 6MeV (CR) + (DR) (CR) January 31, 2014 DR:
peaked in the forward angles,
falling rapidly with angle and
with oscillations
CR:
Istropic and
symmetric about 90 degrees
(DR) PHY492, Lecture 10 11