PHYS 3446 – Lecture #4
Wedmesday, Feb. 1 2012
Dr. Brandt
1.
2.
3.
4.
Rutherford Scattering with Coulomb force
Scattering Cross Section
Differential Cross Section of Rutherford Scattering
Measurement of Cross Sections
Wednesday Feb. 1, 2012
PHYS 3446 Andrew Brandt
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What do we learn from scattering?
• Scattering of a particle in a potential is completely
determined when we know both
2
ZZ 'e
– The impact parameter, b, and
– The energy of the incident particle, E
b
cot
2E
q
2
• 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, q.
– 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 q, we can learn about the
potential or the forces between the target and the projectile
Wednesday Feb. 1, 2012
PHYS 3446 Andrew Brandt
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Scattering Cross Section
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 q and q-dq.
• In other words, they scatter into the solid angle dW (2psinqdq).
• So the number of particles scattered into the solid angle dW per unit
time is 2pN0bdb.
• Note:have assumed thin foil, and large separation between nuclei—
why?
Wednesday Feb. 1, 2012
PHYS 3446 Andrew Brandt
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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
   2 p bdb
• For the particles to scatter into q and q+dq
Wednesday Feb. 1, 2012
PHYS 3446 Andrew Brandt
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Scattering Cross Section
• In more generalized cases,  depends on both q & f.
  ( q , f )  bdbd f  
Why negative?
d
dW
(q , f ) d W
 
d
dW
( q , f ) sin q d q d f
Since the deflection and change of b are in opposite direction!!
• With a spherical symmetry, f can be integrated out:
  (q )  
reorganize
d
dW
d
dW
(q )  
( q ) 2 p sin q d q
b
db
sin q d q
 2 p bdb
Differential
Cross Section
What is the
dimension of
the differential
cross section?
Area!!
Wednesday Feb. 1, 2012
PHYS 3446 Andrew Brandt
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Scattering Cross Section
femptobarn
• For a central potential, measuring the yield as a function of q
(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:
picobarn
1 barn  10
-24
cm
2
Cross sectional area of a typical nucleus!
Wednesday Feb. 1, 2012
PHYS 3446 Andrew Brandt
Where does this
come from?
nanobarn
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Total Cross Section
• Total cross section is the integration of the
differential cross section over the entire solid
angle, W:
 T otal 

4p
0
d
dW
(q , f ) d W  2p 
p
0
d q sin q
d
dW
(q )
• Total cross section represents the effective size of
the scattering center integrated over all possible
impact parameters (and consequently all possible
scattering angles)
Wednesday Feb. 1, 2012
PHYS 3446 Andrew Brandt
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Cross Section of Rutherford Scattering
• The impact parameter in Rutherford scattering is
b
ZZ 'e
2
cot
2E
• Thus,
db
dq
 
2
1 ZZ 'e
2
q
2E
2
cosec
2
q
2
• Differential cross section of Rutherford scattering is
2
 ZZ 'e 
1
 ZZ 'e2 
4 q

 
 
(q )  
 co sec
4E 
4 q
dW
sin q d q
4
E
2

sin


2
d
2
b
2
db
what happened? can you say “trig identity?” sin(2x)=2sin(x) cos(x)
Wednesday Feb. 1, 2012
PHYS 3446 Andrew Brandt
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plot/applets
Measuring Cross Sections
• Rutherford scattering experiment
–
–
–
–
–
Used a collimated beam of a 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 q not f. Why?
• Due to the spherical symmetry, scattering only depends on q not f.
Wednesday Feb. 1, 2012
PHYS 3446 Andrew Brandt
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1.
HW #2 Due Mon. Feb,8 2012
Plot the differential cross section of the Rutherford scattering
as a function of the scattering angle q for three sensible
choices of the lower limit of the angle.
(use ZAu=79, Zhe=2, E=10keV).
2.
3.
4.
•
Compute the total cross section of the Rutherford scattering in
unit of barns for your cut-off angles.
Find a plot of a cross section from a current HEP experiment,
and write a few sentences about what is being measured.
Book problem 1.10
5 points quiz extra credit if you are one of first 10 people to
email an electronic version of a figure showing Rutherford
1/sin^4(x/2) angular dependence
Wednesday Feb. 1, 2012
PHYS 3446 Andrew Brandt
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