1. PHYS 3446 – Lecture #6 Monday, Sept. 15, 2008 Dr. Andrew Brandt
Relativistic Variables
Invariant Scalars
Feynman Diagram
Monday, Sept. 15, 2008
PHYS 3446, Fall 2008
Andrew Brandt

2. Cross Section of Rutherford Scattering
The impact parameter in Rutherford scattering is
Thus,
Differential cross section of Rutherford scattering is
3 factors
of 2
sin(2x)=2sin(x) cos(x) gives 4th factor so 1/16 is correct
Monday, Sept. 15, 2008
PHYS 3446, Fall 2008
Andrew Brandt

3. Rutherford Scattering Cross Section
Let’s plug in the numbers
ZAu=79
ZHe=2
For E=10keV
Differential cross section of Rutherford scattering
Monday, Sept. 15, 2008
PHYS 3446, Fall 2008
Andrew Brandt

4. 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. 15, 2008
PHYS 3446, Fall 2008
Andrew Brandt

5. Relativistic Variables-Special Cases
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. 15, 2008
PHYS 3446, Fall 2008
Andrew Brandt

6. Relativistic Variables
For high energies, if m1~m2,
gCM becomes:
In general, for fixed target
Thus gCM becomes
Invariant Scalar: s
Monday, Sept. 15, 2008
PHYS 3446, Fall 2008
Andrew Brandt

7. Relativistic Variables
see this
by evaluating
in lab frame
The invariant scalar, s, is defined as:
So what is this in the CM frame?
Thus, represents the total available energy in the CM; At the LHC
Monday, Sept. 15, 2008
PHYS 3446, Fall 2008
Andrew Brandt

8. Useful Invariant Scalar Variables
Another invariant scalar, t, the momentum transfer (difference 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 CM frame for an elastic scattering, where PiCM=PfCM=PCM and EiCM=EfCM:
Monday, Sept. 15, 2008
PHYS 3446, Fall 2008
Andrew Brandt

9. Feynman Diagram
The variable t is always negative for 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 less than the rest mass of an equivalent free particle
t-channel diagram
Momentum of the carrier is the difference between the two particles.
Time
Monday, Sept. 15, 2008
PHYS 3446, Fall 2008
Andrew Brandt

10. 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. 15, 2008
PHYS 3446, Fall 2008
Andrew Brandt

11. Relativistic Scattering Angles in Lab and CM
For a relativistic scattering, the relationship between the scattering angles in Lab and CM 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. 15, 2008
PHYS 3446, Fall 2008
Andrew Brandt

12. Assignment 3 Derive Eq. 1.55 starting from 1.48 and 1.49
Derive the formulae for the available CM 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.
Given m1= m2 =proton, what beam energy would be needed for the fixed target experiment to have the same CM energy ( )as the LHC
End of chapter problem 1.7
Due 9/22/08
Monday, Sept. 15, 2008
PHYS 3446, Fall 2008
Andrew Brandt