Particle properties/characteristics specifically their interactions are often interpreted in terms of CROSS SECTIONS. For elastically scattered projectiles:. The recoiling particles are identical to the incoming particles but are in different quantum states. E f , p f.
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Particle properties/characteristics
specifically their interactions
are often interpreted in terms of
CROSS SECTIONS.
For elastically scattered projectiles:
The recoiling
particles are
identical to
the incoming
particles but
are in different
quantum states
Ef , pf
Ei , pi
EN , pN
The initial
conditions
may be
precisely
knowable
only
classically!
The simple 2body kinematics of scattering
fixes the energy of particles scattered through .
Besides his famous scattering of particles off gold and lead foil,
Rutherford observed the transmutation:
or, if you prefer
Whenever energetic particles
(from a nuclear reactor or an accelerator)
irradiate matter there is the possibility of a nuclear reaction
Classification of Nuclear Reactions
16
8
3
1
15
8
41
20
3
2
40
20
Zr + d Zr + p
90
40
(d,p)
(3He,d)
91
40
Na + He Mg + d
23
11
3
2
24
12
continuity at x=0 requires
solve for2A = C+ D
The cross section is defined by the ratio
rate particles are scattered out of beam
rate of particles focused onto target material/unit area
a “counting” experiment
notice it yields a measure, in units of area
number of scattered particles/sec
incident particles/(unit area sec) target site density × beamspot × target thickness
how tightly focused
or intense the beam is
number of nuclear
targets
With a detector fixed to record data from a particular location ,
we measure the “differential” cross section: d/d.
Incident monoenergetic beam
v Dt
A
dW
N = number density in beam
(particles per unit volume)
Solid angle dWrepresents
detector counting the dN
particles per unit time that
scatter through qinto dW
Nnumber of scattering
centers in target
intercepted by beamspot
FLUX = # of particles crossing through unit cross section per sec
= NvDt A / Dt A = Nv
Notice: qNv we call current, I, measured in Coulombs.
dN NF dW dN = s(q)NF dW dN =NFds

Symmetry arguments allow us to immediately integrate out
and consider
rings defined
by alone
R
Rsind
R
R
R
Nscattered= NFdsTOTAL
Integrated over all solid angles
dNscattered= NFdsTOTAL
The scattering rate
per unit time
Particles IN (per unit time) = FArea(ofbeamspot)
Particles scattered OUT (per unit time) = F NsTOTAL
(to a specific “final state” momentum pf )
Scattering Probability
Depends on “how much alike” the final and initial states are.
assumed merely to be perturbed
as it passes (quickly!) through
the scattering potential
The overlap of these wavefunctions is expressed by the “Matrix element”
Potential perturbs
the initial momentum state
into a state best described as
a linear (series) combination of
possible final states…
each weighted by the probability of that final state
For “free” particles (unbounded in the “continuum”)
the solutions to
Schrödinger’s equation
with no potential
Sorry!…the V at left
is a volume appearing
for normalization
V
q
momentum transfer
the momentum
given up (lost)
by the scattered
particle
pf
q = kikf =(pipf )/ħ
pi