Wednesday, Sept. 3, 2008PHYS 3446, Fall 2008 Andrew Brandt 1 PHYS 3446 – Lecture #3 Wednesday, Sept. 3, 2008 Dr. Andrew Brandt 1.Rutherford Scattering

Wednesday, Sept. 3, 2008 PHYS 3446, Fall 2008Andrew Brandt

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PHYS 3446 – Lecture #3Wednesday, Sept. 3, 2008

Dr. Andrew Brandt

1. Rutherford Scattering with Coulomb force2. Scattering Cross Section3. Differential Cross Section of Rutherford Scattering4. Measurement of Cross Sections***Don’t forget radiation training!!! ***Please turn in HW


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Rutherford Scattering

• From the solution for b, we can learn the following 1. For fixed b, E and Z’

– The scattering is larger for a larger value of Z.– Since Coulomb potential is stronger with larger Z.

2. For fixed b, Z and Z’– The scattering angle is larger when E is smaller.

– Since the speed of the low energy particle is smaller– The particle spends more time in the potential, suffering greater deflection

3. For fixed Z, Z’, and E– The scattering angle is larger for smaller impact parameter b

– Since the closer the incident particle is to the nucleus, the stronger Coulomb force it feels

2'cot

2 2

ZZ eb

E


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What do we learn from scattering?• Scattering of a particle in a potential is completely determined

when we know both – The impact parameter, b, and – The energy of the incident particle, E

• For a fixed energy, the deflection is defined by – The impact parameter, b.

• What do we need to perform a scattering experiment?– Incident flux of beam particles with known E– Device that can measure number of scattered particles at various

angle, .– Measurements of the number of scattered particles reflect

• Impact parameters of the incident particles • The effective size of the scattering center

• By measuring the scattering angle , we can learn about the potential or the forces between the target and the projectile

2'cot

2 2

ZZ eb

E


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All these land here

• N0: The number of particles incident on the target foil per unit area per unit time.

• Any incident particles entering with impact parameter b and b+db will scatter to the angle and -d

• In other words, they scatter into the solid angle d2sind• So the number of particles scattered into the solid angle d per unit time

is 2N0bdb.• Note:have assumed thin foil, and large separation between nuclei—why?

Scattering Cross Section


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Scattering Cross Section• For a central potential

– Such as Coulomb potential– Which has spherical symmetry

• The scattering center presents an effective transverse cross-sectional area of

• For the particles to scatter into and +d

2 bdb


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Scattering Cross Section• In more generalized cases, depends on both & .

• With a spherical symmetry, can be integrated out:

,

d

d

Differential Cross Section

What is the dimension of the differential cross section?

Area!!

reorganize

bdbd ,d

dd

, sind

d dd

2 sind

dd

2 bdb

Why negative? Since the deflection and change of b are in opposite direction!!

sin

b db

d


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Scattering Cross Section• For a central potential, measuring the yield as a function of

the differential cross section) is equivalent to measuring the entire effect of the scattering

• So what is the physical meaning of the differential cross section?

Measurement of yield as a function of specific experimental variables

This is equivalent to measuring the probability of occurrence of a physical process in a specific kinematic phase space

• Cross sections are measured in the unit of barns:

1 barn Where does this come from?

-24 210 cmCross sectional area of a typical nucleus! nanobarn

picobarn

femptobarn


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Total Cross Section• Total cross section is the integration of the differential

cross section over the entire solid angle, :

• Total cross section represents the effective size of the scattering center integrated over all possible impact parameters (and consequently all possible scattering angles)

Total 4

0,

dd

d

02 sin

dd

d


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Cross Section of Rutherford Scattering• The impact parameter in Rutherford scattering is

• Thus,

• Differential cross section of Rutherford scattering is

2'cot

2 2

ZZ eb

E

db

d

d

d

224'

cosec2 2

ZZ e

E

22

4

' 1

2 sin2

ZZ e

E

221 '

cosec2 2 2

ZZ e

E

sin

b db

d

what happened? can you say “trig identity?” sin(2x)=2sin(x) cos(x)plot/applets


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Rutherford Scattering Cross Section • Let’s plug in the numbers

– ZAu=79

– ZHe=2– For E=10keV

• Differential cross section of Rutherford scattering

d

d

18

4

1.59 10

sin2

barns

22

4

' 1

2 sin2

ZZ e

E

2219

3 194

79 2 1.6 10 1

2 10 10 1.6 10 sin2

422

4

1.59 10

sin2

cm


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Total X-Section of Rutherford Scattering• To obtain the total cross section of Rutherford scattering, one

integrates the differential cross section over all :

• What is the result of this integration?– Infinity!!

• Does this make sense?– Yes

• Why?– Since the Coulomb force’s range is infinite (particle with very large impact

parameter still contributes to integral through very small scattering angle)• What would be the sensible thing to do?

– Integrate to a cut-off angle since after certain distance the force is too weak to impact the scattering. (0>0); note this is sensible since alpha particles far away don’t even see charge of nucleus due to screening effects.

Total 0

2 sind

dd

22 1

0 3

' 18 sin

2 2 sin2

ZZ ed

E


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Measuring Cross Sections

• Rutherford scattering experiment– Used a collimated beam of particles emitted from Radon– A thin Au foil target– A scintillating glass screen with ZnS phosphor deposit– Telescope to view limited area of solid angle– Telescope only needs to move along not . Why?

• Due to the spherical symmetry, scattering only depends on not .


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Measuring Cross Sections• With the flux of N0 per unit area per second• Any particles in range b to b+db will be

scattered into to -d• The telescope aperture limits the measurable

area to

• How could they have increased the rate of measurement?– By constructing an annular telescope– By how much would it increase?

TeleA

2/d

sinRd R d 2R d


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Measuring Cross Sections• Fraction of incident particles (N0) approaching the target in

the small area =bddb at impact parameter b is dn/N0. – so dn particles scatter into R2d, the aperture of the telescope

• This fraction is the same as – The sum of over all N nuclear centers throughout the foil

divided by the total area (S) of the foil.– Or, in other words, the probability for incident particles to enter

within the N areas divided by the probability of hitting the foil. This ratio can be expressed as

0

dn

N Nbd d

S

,N

S Eq. 1.39


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Measuring Cross Sections• For a foil with thickness t, mass density , atomic weight A:

• Since from what we have learned previously

• The number of scattered into the detector angle () is

N A0: Avogadro’s number of atoms per mol

dn

0

tSA

A

0 0N tA

A

0

,dNN dS d

,d

dd

0

dn

N

Nbd d

S

Eq. 1.40


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Measuring Cross Sections

dn

• This is a general expression for any scattering process, independent of the theory

• This gives an observed counts per second

0 0N tA

A

0

,dNN dS d

,d

dd

Number of detected particles

Projectile particle flux

Density of the target particles

Scattering cross section

Detector acceptance


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• Inclusive jet production cross section as a function of transverse energy

Example Cross Section: Jet +X

triumph ofQCD

what if there werean excess at highenergy?


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Assignment #21. Plot the differential cross section of the Rutherford

scattering as a function of the scattering angle for three sensible choices of the lower limit of the angle

2. Compute the total cross section of the Rutherford scattering in unit of barns for your cut-off angles

3. 5 points extra credit if you are one of first three people to email me an electronic version of a figure showing Rutherford 1/sin^4(x/2) angular dependence.

• Assignment # 2 is due Wednesday, Sept. 10• Must have rad training done by then as well

Documents

Wednesday, Sept. 3, 2008PHYS 3446, Fall 2008 Andrew Brandt 1 PHYS 3446 – Lecture #3 Wednesday, Sept. 3, 2008 Dr. Andrew Brandt 1.Rutherford Scattering