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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. E f , p f.

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Presentation Transcript

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 2-body kinematics of scattering

fixes the energy of particles scattered through .


Nuclear Reactions

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

  • inelastic scattering

    • individual collisions between the incoming projectile and a single target nucleon; the incident particle emerges with reduced energy

  • pickup reactions

  • incident projectile collects additional nucleons from the target

  • O + d  O + H (d, 3H)

  • Ca + He Ca +  (3He,)

16

8

3

1

15

8

41

20

3

2

40

20

  • stripping reactions

  • incident projectile leaves one or more nucleons behind in the target

Zr + d  Zr + p

90

40

(d,p)

(3He,d)

91

40

Na + He Mg + d

23

11

3

2

24

12


[ Ne]*

20

10

Predicting a final outcome is much like

rolling dice…the process is random!


V0

0

1

2

3

x = 0

x = a


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.


scattered particles

Incident mono-energetic 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

-


dN = FNs(q)dWNFds(q)

the “differential” cross section

R

R

R

R

R


R

R

the differential solid angle d for integration is sin d d

Rsind

Rd

Rsind

Rd

Rsin


Symmetry arguments allow us to immediately integrate  out

and consider

rings defined

by  alone

R

Rsind

R

R

R

Nscattered= NFdsTOTAL

Integrated over all solid angles


dNscattered= NFdsTOTAL

The scattering rate

per unit time

Particles IN (per unit time) = FArea(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

q

momentum transfer

the momentum

given up (lost)

by the scattered

particle

pf

q = ki-kf =(pi-pf )/ħ

pi


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