1. PHYS 3446 – Lecture #4 Monday, Sept. 11, 2006 Dr. Jae Yu
Lab Frame and Center of Mass Frame
Relativistic Treatment
Feynman Diagram
Invariant kinematic variables
Monday, Sept. 11, 2006
PHYS 3446, Fall 2006
Jae Yu

2. 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

3. 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:
Monday, Sept. 11, 2006
PHYS 3446, Fall 2006
Jae Yu

4. 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

5. Lab Frame and CM Frame The equations of motion can be written where
Since the potential depends only on relative separation of the particles, we redefine new variables, relative coordinates & coordinate of CM
and
Monday, Sept. 11, 2006
PHYS 3446, Fall 2006
Jae Yu

6. Lab Frame and CM Frame From the equations in previous slides and Thus
Reduced Mass
and
Thus
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

7. Now with some simple arithmatics
From the equations of motion, we obtain
Since the momentum of the system is conserved:
Rearranging the terms, we obtain
Monday, Sept. 11, 2006
PHYS 3446, Fall 2006
Jae Yu

8. 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 m (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 m 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

9. Relationship of variables in Lab and CMS
The speed of CM is
Speeds of the particles in CMS are
The momenta of the two particles are equal and opposite!!
and
Monday, Sept. 11, 2006
PHYS 3446, Fall 2006
Jae Yu

10. 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, m.
Z components of the velocities of particle with m1 in lab and CMS are:
The perpendicular components of the velocities are:
Thus, the angles are related, for elastic scattering only, as:
Monday, Sept. 11, 2006
PHYS 3446, Fall 2006
Jae Yu

11. 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 f is invariant, dfLab = dfCM.
Why?
f is perpendicular to the direction of boost, thus is invariant.
Thus, the differential cross section becomes:
reorganize
Using Eq. 1.53
Monday, Sept. 11, 2006
PHYS 3446, Fall 2006
Jae Yu

12. Some Quantities in Special Relativity
Fractional velocity
Lorentz g factor
Relative momentum and the total energy of the particle moving at a velocity is
Square of four momentum P=(E,pc), rest mass E
Monday, Sept. 11, 2006
PHYS 3446, Fall 2006
Jae Yu

13. Relativistic Variables
Velocity of CM in the scattering of two particles with rest mass m1 and m2 is:
If m1 is the mass of the projectile and m2 is that of the target, for a fixed target we obtain
Monday, Sept. 11, 2006
PHYS 3446, Fall 2006
Jae Yu

14. Relativistic Variables
At very low energies where m1c2>>P1c, the velocity reduces to:
At very high energies where m1c2<<P1c and m2c2<<P1c , the velocity can be written as:
Expansion
Monday, Sept. 11, 2006
PHYS 3446, Fall 2006
Jae Yu

15. Relativistic Variables
For high energies, if m1~m2,
gCM becomes:
And
Thus gCM becomes
Invariant Scalar: s
Monday, Sept. 11, 2006
PHYS 3446, Fall 2006
Jae Yu

16. Relativistic Variables
The invariant scalar, s, is defined as:
So what is this the CMS frame?
Thus, represents the total available energy in the CMS
Monday, Sept. 11, 2006
PHYS 3446, Fall 2006
Jae Yu

17. Useful Invariant Scalar Variables
Another invariant scalar, t, the momentum transfer (differences in four momenta), is useful for scattering:
Since momentum and total energy are conserved in all collisions, t can be expressed in terms of target variables
In CMS frame for an elastic scattering, where PiCM=PfCM=PCM and EiCM=EfCM:
Monday, Sept. 11, 2006
PHYS 3446, Fall 2006
Jae Yu

18. 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 and exchanged in the scattering
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
t-channel diagram
Momentum of the carrier is the difference between the two particles.
Time
Monday, Sept. 11, 2006
PHYS 3446, Fall 2006
Jae Yu

19. Useful Invariant Scalar Variables
For convenience we define a variable q2,
In the lab frame, , thus we obtain:
In the non-relativistic limit:
q2 represents “hardness of the collision”. Small qCM corresponds to small q2.
Monday, Sept. 11, 2006
PHYS 3446, Fall 2006
Jae Yu

20. Relativistic Scattering Angles in Lab and CMS
For a relativistic scattering, the relationship between the scattering angles in Lab and CMS is:
For Rutherford scattering (m=m1<<m2, v~v0<<c):
Divergence at q2~0, a characteristics of a Coulomb field
Resulting in a
cross section
Monday, Sept. 11, 2006
PHYS 3446, Fall 2006
Jae Yu

21. Assignments Extension of previous w/ the following (due 9/18):
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 starting from 1.48 and 1.49
Derive the formulae for the available CMS energy ( ) 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