PHYS 3446 – Lecture #4
Monday, Jan. 31, 2005
Dr. Jae Yu
1.
2.
3.
4.
5.
6.
Lab Frame and Center of Mass Frame
Relativistic Treatment
Feynman Diagram
Quantum Treatment of Rutherford Scattering
Nuclear Phenomenology: Properties of Nuclei
A few measurements of differential cross sections
Monday, Jan. 31, 2005
PHYS 3446, Spring 2005
Jae Yu
1
Announcements
• I still have eleven subscribed the distribution list.
• I really hate doing this but since this is the primary class
communication tool, it is critical for all of you to subscribe.
– I will assign -3 points to those of you not registered by the class
Wednesday
• A test message will be sent out Wednesday
• Must take the radiation safety training!!!
– Directly related to your lab score  15% of the total
• Homework extra credit: 5 points if done by the class after
the assignment!
Monday, Jan. 31, 2005
PHYS 3446, Spring 2005
Jae Yu
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Scattering Cross Section
• For a central potential, measuring the yield as a function of q,
or 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
variable
This is equivalent to measuring the probability of certain
process in a specific kinematic phase space
• Cross sections are measured in the unit of barns:
-24
1 barn = 10 cm
Wednesday, Jan. 26, 2005
PHYS 3446, Spring 2005
Jae Yu
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3
Example Cross Section: W(en) +X
• Transverse
mass
distribution of
electrons in
W+X events
Monday, Jan. 31, 2005
PHYS 3446, Spring 2005
Jae Yu
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Example Cross Section: W(ee) +X
• Invariant
mass
distribution of
electrons in
Z+X events
Monday, Jan. 31, 2005
PHYS 3446, Spring 2005
Jae Yu
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Lab Frame and Center of Mass Frame
• We assumed that the target nuclei do not move through
the collision
• In reality, they recoil as a result of scattering
• Sometimes we use two beams of particles for scattering
experiments (target is moving)
• This situation could be complicated but..
• Could be simplified if the motion can be described in
Center of Mass frame under a central potential
Monday, Jan. 31, 2005
PHYS 3446, Spring 2005
Jae Yu
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Lab Frame and CM Frame
• The equations of
motion can be written
m1r1  1V  r1  r2

m2 r2  2V  r1  r2 
where
ˆi
 qˆi 

ˆ
i  ri


; i  1,2
ri ri qi ri sin qi i
Since the potential depends only on relative separation of the particles, we
redefine new variables
r  r1  r2
Monday, Jan. 31, 2005
and
RCM
m1r1  m2 r2

m1  m2
PHYS 3446, Spring 2005
Jae Yu
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Lab Frame and CM Frame
• From the equations in previous slides
V  r 
m1m2
r   r  V  r   
rˆ
m1  m2
r
and
 m1  m2  RCM
Thus
 MRCM  0
Reduced
Mass
RCM  constant Rˆ
• What do we learn from this exercise?
• For a central potential, the motion of the two particles can be
decoupled when re-written in terms of
– a relative coordinate
– The coordinate of center of mass
Monday, Jan. 31, 2005
PHYS 3446, Spring 2005
Jae Yu
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Lab Frame and CM Frame
• The CM is moving at a constant speed in Lab frame
independent of the form of the central potential
• The motion is as if that of a fictitious particle with
reduced mass  and coordinate r.
• In the frame where CM is stationary, the dynamics
becomes equivalent to that of a single particle of mass
 scattering of a fixed, scattering center.
• Frequently we define the Center of Mass frame as the
frame where the sum of the momenta of the
interacting particles vanish.
Monday, Jan. 31, 2005
PHYS 3446, Spring 2005
Jae Yu
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Relationship of variables in Lab and CMS
CMS
• The speed of CM is
vCM  RCM
m1v1

m1  m2
• Speeds of particles in CMS are
v1  v1  vCM
m2v1
m1v1

and v2  vCM 
m1  m2
m1  m2
• The momenta of the two particles are equal and opposite!!
Monday, Jan. 31, 2005
PHYS 3446, Spring 2005
Jae Yu
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Scattering angles in Lab and CMS
 qCM represents the changes in the direction of the relative
position vector r as a result of the collision
• Thus, it must be identical to the scattering angle for the
particle with the reduced mass, .
• Z components of the velocities of particle with m1 in lab and
CMS are:
v cosq Lab  vCM  v1 cosqCM
• The perpendicular components of the velocities are:
v sin q Lab  v1 sin qCM
• Thus, the angles are related, for elastic scattering only, as:
sin qCM
sin qCM
sin qCM
tan q Lab 


sin qCM  vCM v1
sin qCM  m1 m2 sin qCM  
Monday, Jan. 31, 2005
PHYS 3446, Spring 2005
Jae Yu
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Differential cross sections in Lab and CMS
• The particles scatter at an angle qLab into solid angle dWLab
in lab scatters into qCM into solid angle dWCM in CM.
• Since  is invariant, dLab = dCM.
– Why?
  is perpendicular to the direction of boost, thus is invariant.
• Thus, the differential cross section becomes:
d
d
qLab  sin q Lab dq Lab 
qCM  sinqCM dqCM
d WLab
d WCM
d  cosqCM 
d
d
qLab  
qCM 
rewrite
d WLab
d WCM
d  cosq Lab 
Using Eq. 1.53
Monday, Jan. 31, 2005

1  2 cosqCM  
d
d
q Lab  
qCM 
d WLab
d WCM
1   cosqCM
PHYS 3446, Spring 2005
Using
JaeEq.
Yu 1.53
2

3/ 2
12
Relativistic Variables
• Velocity of CM in the scattering of two particles
with rest mass m1 and m2 is:
P1  P2 c
vCM
 CM 
c
E1  E2
• If m1 is the mass of the projectile and m2 is that of
the target, for fixed target we obtain

CM

Pc
Pc
1
1


2 2
2 4
2
E1  E2
P1 c  m1 c  m2 c
Monday, Jan. 31, 2005
PHYS 3446, Spring 2005
Jae Yu
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Relativistic Variables
• At very low energies where m1c2>>P1c, the velocity
reduces to:
m1v1c
m1v1
CM 

2
2
m1c  m2c
 m1  m2  c
• At very high energies where m1c2<<P1c and
m2c2<<P1c , the velocity can be written as:
CM  CM 
1
2
 m1c 
m2 c 2
1 
 
Pc
1
1
 Pc

2
Monday, Jan. 31, 2005
PHYS 3446, Spring 2005
Jae Yu
m2 c 1  m1c 
 1
 

P1
2  P1 
14
2
Relativistic Variables
• For high energies, if m1~m2,
CM
 gCM becomes:

g CM  1   CM
2
• And

1/ 2
 1  CM 1  CM  
2
1  CM

E  m c 
2
• Thus gCM becomes
2
g CM  1   CM
Monday, Jan. 31, 2005
  m2 c  
 2 

P
  1  
1 2

P1
2m2 c
m12c 4  2 E1m2c 2  m22c 4
1

1/ 2
 m2 c 
 1 

P1 

2

1/ 2

2
2
E1  m2 c 2
m12 c 4  2 E1m2 c 2  m22 c 4
PHYS 3446, Spring 2005
Jae Yu
Invariant
15 s
Scalar:
Relativistic Variables
• The invariant scalar, s, is defined as:
2
2
2
s   E1  E2   P1  P2 c



 2E1m2c
• In the CMS frame
s  m12c 4  m22c 4  2E1m2c 2
2
  E1CM  E2CM   P1CM  P2CM
2 4
m1 c
2 4
 m2 c
  E1CM  E2CM 
2
2

 E 
CM
ToT
2

2
c
2
• Thus, s represents the total available energy in
the CMS
Monday, Jan. 31, 2005
PHYS 3446, Spring 2005
Jae Yu
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Useful Invariant Scalar Variables
• Another invariant scalar, t, the momentum transfer, is useful
for scattering:
t

E1f

E1i
 
f
  P
f
2
 P1 
P1i

2
c2
• Since momentum and energy are conserved in all collisions,
t can be expressed in terms of target variables
t

E2f

2
E2i
2

P2i

2
c
2
What does
this variable
look like?
• In CMS frame for an elastic scattering, where
PiCM=PfCM=PCM and EiCM=EfCM:
t

f2
P1CM

Monday, Jan. 31, 2005
2
P1iCM
 2P1
f
 P1i

2
2
c 2 1  cosqCM  .
c 2  2 PCM
PHYS 3446, Spring 2005
Jae Yu
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Feynman Diagram
• The variable t is always negative for an elastic scattering
• The variable t could be viewed as the square of the invariant
mass of a particle with E2f  E2i and P2f  P2i exchanged
in the scattering
t-channel
diagram
Time
• While the virtual particle cannot be detected in the scattering,
the consequence of its exchange can be calculated and
observed!!!
Monday, Jan. 31, 2005
PHYS 3446, Spring 2005
Jae Yu
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Useful Invariant Scalar Variables
• For convenience we define a variable q2, q 2c 2  t
• In the lab frame P2iLab  0, thus we obtain:


 


2
2
f
2
f

q c   E2 Lab  m2c  P2 Lab c 


 2m2 c 2 E2fLab  m2 c 2  2m2 c 2T2fLab
2 2
• In the non-relativistic limit:
f
T2 Lab
1
2
 m2 v2
2
• q2 represents “hardness of the collision”. Small qCM
corresponds to small q2.
Monday, Jan. 31, 2005
PHYS 3446, Spring 2005
Jae Yu
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Relativistic Scattering Angles in Lab and CMS
• For a relativistic scattering, the relationship between the
scattering angles in Lab and CMS is:
tan q Lab 
g CM

 sin qCM
 sin qCM  CM

• For Rutherford scattering (m=m1<<m2, v~v0<<c):
dq 2  2 P 2 d  cos q  
Resulting in a
cross section

d 4 ZZ ' e

2
dq
v2
2

P2d W
2

1
q4
• Divergence at q2~0, a characteristics of a Coulomb field
• There are distribution of q2 in Rutherford scattering which
falls off rapidly
Monday, Jan. 31, 2005
PHYS 3446, Spring 2005
Jae Yu
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Assignments
1.
Derive the following equations:
•
•
•
2.
•
•
3.
4.
Compute the available CMS energy ( s ) for
Fixed target experiment with masses m1 and m2 with incoming
energy E1.
Collider experiment with masses m1 and m2 with incoming
energies E1 and E2.
Reading assignment: Section 1.7
End of chapter problems:
•
•
Eqs. 1.51, 1.53, 1.55, 1.63, 1.71
Prove Eq. 1.45
Derive Eq. 1.51 from Eq. 1.71 in its non-relativistic limit
1.1 and 1.7
These assignments are due next Monday, Feb. 7.
Monday, Jan. 31, 2005
PHYS 3446, Spring 2005
Jae Yu
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