PHYS 3446 – Lecture #4
Monday, Sept. 11, 2006
Dr. Jae Yu
1.
2.
3.
4.
Lab Frame and Center of Mass Frame
Relativistic Treatment
Feynman Diagram
Invariant kinematic variables
Monday, Sept. 11, 2006
PHYS 3446, Fall 2006
Jae Yu
1
Announcements
• I now have 9 on the distribution list.
– Only three more to go!! Whoopie!!
– Please come and check to see if your request has been
successful
• I will send a test message this week
• No lecture this Wednesday but
• The class time should be devoted for your project
group meetings in preparation for the workshop
Monday, Sept. 11, 2006
PHYS 3446, Fall 2006
Jae Yu
2
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
Monday, Sept. 11, 2006
PHYS 3446, Fall 2006
Jae Yu
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3
Lab Frame and Center of Mass Frame
• We assumed that the target nuclei do not move through
the collision in Rutherford Scattering
• 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..
• If the motion can be described in the Center of Mass
frame under a central potential, it can be simplified
Monday, Sept. 11, 2006
PHYS 3446, Fall 2006
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 


 i  rˆi
ri ri qi ri sin q i i
i  1,2
Since the potential depends only on relative separation of the particles, we
redefine new variables, relative coordinates & coordinate of CM
r  r1  r2
Monday, Sept. 11, 2006
and
RCM
PHYS 3446, Fall 2006
Jae Yu
m1r1  m2 r2

m1  m2
5
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, Sept. 11, 2006
PHYS 3446, Fall 2006
Jae Yu
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Now with some simple arithmatics
• From the equations of motion, we obtain


m1r1  m2r2  m1r1  m2 r1  r   m1  m2  r1  m2 r
 
 

ˆ
   rˆ1
 rˆ2
V
r


r
Vr
  
r2 
r
 r1

• Since the momentum of the system is conserved:
m1r1  m2 r2
 RCM  0  m1r1  m2 r2   m2 r1  r
m1  m2


• Rearranging the terms, we obtain
m2
 m1  m2  r1  m2r  r1 
r
 m1  m2 
m2
m1m2

r  m2 r 
r  r̂ V  r
 m1  m2 
r
 m1  m2 
 m1  m2 
Monday, Sept. 11, 2006
PHYS 3446, Fall 2006
Jae Yu

7
Lab Frame and CM Frame
• The CM is moving at a constant velocity in the lab
frame independent of the form of the central potential
• The motion is as if that of a fictitious particle with mass
 (the 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 off of a fixed scattering center.
• Frequently we define the Center of Mass frame as the
frame where the sum of the momenta of all the
interacting particles is 0.
Monday, Sept. 11, 2006
PHYS 3446, Fall 2006
Jae Yu
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Relationship of variables in Lab and CMS
CMS
v1
q Lab
VCM
• The speed of CM is
vCM  RCM
• Speeds of the particles in CMS are
v1  v1  vCM
q CM
v1
m1v1

m1  m2
m2v1
m1v1

and v2  vCM 
m1  m2
m1  m2
• The momenta of the two particles are equal and opposite!!
Monday, Sept. 11, 2006
PHYS 3446, Fall 2006
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 in CMS
• 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 


cosqCM  vCM v1
cosqCM  m1 m2 cosqCM  
Monday, Sept. 11, 2006
PHYS 3446, Fall 2006
Jae Yu
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Differential cross sections in Lab and CMS
• The particles that scatter in lab at an angle qLab into solid
angle dWLab scatter at 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
q Lab  sin q Lab dq Lab 
qCM  sin qCM dqCM
d WLab
d WCM
d
d  cosqCM 
d
qCM 
q Lab  

reorganize
d WLab
d WCM
d  cosq Lab 
Using Eq. 1.53
Monday, Sept. 11, 2006

1  2 cosqCM  
d
d
qCM 
q Lab  
d WCM
1   cosqCM
d WLab
PHYS 3446, Fall 2006
Jae Yu
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
3/ 2
11
Some Quantities in Special Relativity
• Fractional velocity   v c
1
• Lorentz g factor g 
1  2
• Relative momentum and the total energy of the
particle moving at a velocity v   c is
P  M g v  M g c
E  T 2  E 2  P 2c 2  M 2c 4  M g c 2
Re st
• Square of four momentum P=(E,pc), rest mass E
P  Mc  E  p c
2
Monday, Sept. 11, 2006
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PHYS 3446, Fall 2006
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Relativistic Variables
• Velocity of CM in the scattering of two particles
with rest mass m1 and m2 is:
P1  P2 c
vCM
CM 

E1  E2
c
• If m1 is the mass of the projectile and m2 is that of
the target, for a fixed target we obtain

CM

Pc
Pc
1
1


2 2
2 4
2
E1  E2
P1 c  m1 c  m2 c
Monday, Sept. 11, 2006
PHYS 3446, Fall 2006
Jae Yu
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Relativistic Variables
• At very low energies where m1c2>>P1c, the velocity
reduces to:
m1v1
m1v1c
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, Sept. 11, 2006
PHYS 3446, Fall 2006
Jae Yu
m2 c 1  m1c 
 1
 

P1
2  P1 
Expansion
14
2
Relativistic Variables
• For high energies, if m1~m2,
•
CM
gCM becomes:

2
g CM  1   CM
• And

1/ 2
1 
 1  CM 1  CM  
2
CM

E  m c 
2
• Thus gCM becomes
2
g CM  1   CM
Monday, Sept. 11, 2006
  m2c  
 2 

P
  1  
1 2
P1

2m2c
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, Fall 2006
Jae Yu
Invariant
15 s
Scalar:
Relativistic Variables
• The invariant scalar, s, is defined as:
2
2
s   P1  P2    E1  E2   P1  P2



 E 

c
2

 2E1m2c
• So what is this the CMS frame?
0
s  m12c 4  m22c 4  2E1m2c 2
2
2
2
  E1CM  E2CM   P1CM  P2CM c
2 4
m1 c
2 4
 m2 c
2
  E1CM  E2CM 
2
2
CM
ToT
2
• Thus, s represents the total available energy in
the CMS
Monday, Sept. 11, 2006
PHYS 3446, Fall 2006
Jae Yu
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Useful Invariant Scalar Variables
• Another invariant scalar, t, the momentum transfer
(differences in four momenta), is useful for scattering:

t  P1 f  P1i
 
2

E1f
 E1i
 
2
 P1
f
 P1i

2
c2
• Since momentum and total energy are conserved in all
collisions, t can be expressed in terms of target variables
t

E2f

E2i
 
2
 P2 
f
P2i

2
c2
• In CMS frame for an elastic scattering, where
PiCM=PfCM=PCM and EiCM=EfCM:


f2
i2
f
i
2
t   PCM
 PCM
 2 PCM
 PCM
c 2  2 PCM
c 2 1  cos qCM  .
Monday, Sept. 11, 2006
PHYS 3446, Fall 2006
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
Momentum of the carrier is
the difference between the
two particles.
• While the virtual particle cannot be detected in the scattering,
the consequence of its exchange can be calculated and
observed!!!
– A virtual particle is a particle whose mass is different than the rest
mass
Monday,
Sept. 11, 2006
PHYS 3446, Fall 2006
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Jae Yu
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 Labc 


 2m2 c 2 E2fLab  m2 c 2  2m2 c 2T2fLab
2 2
• In the non-relativistic limit:
f
T2 Lab
1
2
 m2v2
2
• q2 represents “hardness of the collision”. Small qCM
corresponds to small q2.
Monday, Sept. 11, 2006
PHYS 3446, Fall 2006
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 P 2 d  cos q  
2
Resulting in a
cross section
d

2
dq
P 2d W

4 kZZ ' e 2
v2


2
1
q4
• Divergence at q2~0, a characteristics of a Coulomb field
Monday, Sept. 11, 2006
PHYS 3446, Fall 2006
Jae Yu
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Assignments
1.
Extension of previous w/ the following (due 9/18):
•
2.
3.
•
•
4.
5.
•
Plot the differential cross section of the Rutherford scattering as a
function of the scattering angle q with some sensible lower limit
of the angle + express your opinion on the sensibility of the cross
section, along with good physical reasons
Derive Eq. 1.55 starting from 1.48 and 1.49
Derive the formulae for 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 problem 1.7
These assignments are due next Wednesday, Sept. 20.
Monday, Sept. 11, 2006
PHYS 3446, Fall 2006
Jae Yu
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