1. PHYS 3446 – Lecture #5 Lab Frame and Center of Mass Frame
Monday, Sept 19, 2016
Dr. Jae Yu
Lab Frame and Center of Mass Frame
Relativistic Treatment
Feynman Diagram
Invariant kinematic variable
Nuclear properties
Mott scattering
Spin and Magnetic Moments
Stability and Instability of Nuclei
Monday, Sept. 19, 2016
PHYS 3446, Fall 2016

2. Announcements Colloquium at 4pm this Wednesday
UTA Physics faculty expo II
A special lecture this Wednesday
Dr. Ben Jones
Monday, Sept. 19, 2016
PHYS 3446, Fall 2016

3. Homework Assignment #3
Derive Eq starting from 1.48 and 1.49 (5 points)
Derive the formulae for the available CMS energy ( ) for
Fixed target experiment with masses m1 and m2 with incoming energy E1. (5points)
Collider experiment with masses m1 and m2 with incoming energies E1 and E2. (5points)
End of chapter problem 1.7 ( 5points)
These assignments are due next Monday, Sept. 26
Reading assignment: Section 1.7
Monday, Sept. 19, 2016
PHYS 3446, Fall 2016

4. Lab Frame and Center of Mass Frame
We assumed that the target nuclei do not move throughout 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. 19, 2016
PHYS 3446, Fall 2016

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. 19, 2016
PHYS 3446, Fall 2016

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 from that of the reference frame when re-written in terms of
a relative coordinate
The coordinate of center of mass
Monday, Sept. 19, 2016
PHYS 3446, Fall 2016

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. 19, 2016
PHYS 3446, Fall 2016

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. 19, 2016
PHYS 3446, Fall 2016

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. 19, 2016
PHYS 3446, Fall 2016

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 wit mass m1 after the scattering in lab and CMS are:
The perpendicular components of the velocities are:
Thus, the angles are related, for elastic scattering only, as:
Monday, Sept. 19, 2016
PHYS 3446, Fall 2016

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 the 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. 19, 2016
PHYS 3446, Fall 2016

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. 19, 2016
PHYS 3446, Fall 2016

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. 19, 2016
PHYS 3446, Fall 2016

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. 19, 2016
PHYS 3446, Fall 2016

15. Relativistic Variables
For high energies, if m1~m2,
gCM becomes:
And
Thus gCM becomes
Invariant Scalar: s
Monday, Sept. 19, 2016
PHYS 3446, Fall 2016

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. 19, 2016
PHYS 3446, Fall 2016

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. 19, 2016
PHYS 3446, Fall 2016

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. 19, 2016
PHYS 3446, Fall 2016

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. 19, 2016
PHYS 3446, Fall 2016

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. 19, 2016
PHYS 3446, Fall 2016

21. Nuclear Properties: Sizes
At relativistic energies the magnetic moment of electron also contributes to the scattering
Neville Mott formulated Rutherford scattering in QM and included the spin effects
R. Hofstadter, et al., discovered the effect of spin, nature of nuclear (& proton) form factor in late 1950s
Mott scattering x-sec (scattering of a point particle) is related to Rutherford x-sec:
Deviation from the distribution expected for point-scattering provides a measure of size (structure)
Monday, Sept. 19, 2016
PHYS 3446, Fall 2016

22. Nuclear Properties: Sizes
Another way is to use the strong nuclear force of sufficiently energetic strongly interacting particles (p mesons, protons, etc)
What is the advantage of using these particles?
If the energy is high, Coulomb interaction can be neglected
These particles readily interact with nuclei, getting “absorbed” into the nucleus
Thus, probe strong interactions directly
These interactions can be treated the same way as the light absorptions resulting in diffraction, similar to that of light passing through gratings or slits
Monday, Sept. 19, 2016
PHYS 3446, Fall 2016

23. Nuclear Properties: Sizes
The size of a nucleus can be inferred from the diffraction pattern
All these phenomenological investigation provided the simple formula for the radius of the nucleus to its number of nucleons or atomic number, A:
How would you interpret this formula?
Monday, Sept. 19, 2016
PHYS 3446, Fall 2016

24. Nuclear Properties: Spins
Both protons and neutrons are fermions with spins
Nucleons inside a nucleus can have orbital angular momentum
In Quantum Mechanics orbital angular momenta are integers
Thus the total angular momentum of a nucleus is
Integers: if even number of nucleons in the nucleus
Half integers: if odd number of nucleons in the nucleus
Interesting facts are
All nucleus with even number of p and n are spin 0.
Large nuclei have very small spins in their ground state
Hypothesis: Nucleon spins in the nucleus are very strongly paired to minimize their overall effect
Monday, Sept. 19, 2016
PHYS 3446, Fall 2016

25. Nuclear Properties: Magnetic Dipole Moments
Every charged particle has a magnetic dipole moment associated with its spin
e, m and S are the charge, mass and the intrinsic spin of the charged particle
The constant g is called Landé factor with its value:
: for a point like particle, such as the electron
: Particle possesses an anomalous magnetic moment, an indication of having a substructure
Monday, Sept. 19, 2016
PHYS 3446, Fall 2016

26. Nuclear Properties: Magnetic Dipole Moments
For electrons, me~mB, where mB is Bohr Magneton
For nucleons, magnetic dipole moment is measured in nuclear magneton, defined using proton mass
Measured magnetic moments of proton and neutron:
Monday, Sept. 19, 2016
PHYS 3446, Fall 2016

27. Nuclear Properties: Magnetic Dipole Moments
What important information do you get from these?
The Landé factors of the nucleons deviate significantly from 2.
Strong indication of substructure
An electrically neutral neutron has a significant magnetic moment
Must have extended charge distributions
Measurements show that mangetic moment of nuclei lie -3mN~10mN
Indication of strong pairing
Electrons cannot reside in nucleus
Monday, Sept. 19, 2016
PHYS 3446, Fall 2016

28. Nuclear Properties: Stability
The number of protons and neutrons inside the stable nuclei are
A<40: Equal (N=Z)
A>40: N~1.7Z
Neutrons outnumber protons
Most are even-p + even–n
See table 2.1
Supports strong pairing
N~1.7Z N Z
Nnucl
Even
156
Odd 48 50 5
N=Z
Monday, Sept. 19, 2016
PHYS 3446, Fall 2016

29. Nuclear Properties: Instability
In 1896 H. Becquerel accidently discovered natural radioactivity
Study of Uranium salts’ fluorescent properties
Nuclear radio activity involves emission of three radiations: a, b, and g
These can be characterized using the device on the right
a: Nucleus of He
b: electrons
g: photons
What do you see from above?
a and b are charged particles while g is neutral.
a is mono-energetic
b has broad spectrum
What else do you see?
Monday, Sept. 19, 2016
PHYS 3446, Fall 2016

30. Nature of the Nuclear Force
Scattering experiments help to
Determine the properties of nuclei
Learn more global information on the characteristics of the nuclear force
From what we have learned, it is clear that there is no classical analog to nuclear force
Gravitational force is too weak to provide the binding
Can’t have an electromagnetic origin
Deuteron nucleus has one neutron and one proton
Coulomb force destabilizes the nucleus
Monday, Sept. 19, 2016
PHYS 3446, Fall 2016

31. Short-range Nature of the Nuclear Force
Atomic structure is well explained by the electromagnetic interaction
Thus the range of nucleus cannot be much greater than the radius of the nucleus
Nuclear force should range ~ – 10-12cm
Binding energy is constant per each nucleon, essentially independent of the size of the nucleus
If the nuclear force is long-ranged, like the Coulomb force
For A nucleons, there would be ½ A(A-1) pair-wise interactions
Thus, the BE which reflects all possible interactions among the nucleons would grow as a function of A
For large A
Monday, Sept. 19, 2016
PHYS 3446, Fall 2016

32. Short-range Nature of the Nuclear Force
If the nuclear force is long-ranged and is independent of the presence of other nucleons, BE per nucleon will increase linearly with A
This is because long-range forces do not saturate
Since any single particle can interact with as many other particle as are available
Binding becomes tighter as the number of interacting objects increases
The size of the interacting region stays fairly constant
Atoms with large number of electrons have the sizes compatible to those with smaller number of electrons
Monday, Sept. 19, 2016
PHYS 3446, Fall 2016

33. Short-range Nature of the Nuclear Force
Long-rangeness of nuclear force is disputed by the experimental measurement that the BE/nucleon stays constant
Nuclear force must saturate
Any given nucleon can only interact with a finite number of nucleons in its vicinity
What does adding more nucleons to a nucleus do?
Only increases the size of the nucleus
Recall that R ~ A1/3
The size of a nucleus grows slowly with A and keeps the nuclear density constant
Another supporting evidence of short-range nature of nuclear force
Monday, Sept. 19, 2016
PHYS 3446, Fall 2016

34. Shape of the Nuclear Potential
Nuclear force keeps the nucleons within the nucleus.
What does this tell you about the nature of the nuclear force?
It must be attractive!!
However, scattering experiments with high energy revealed a repulsive core!!
Below a certain length scale, the nuclear force changes from attractive to repulsive.
What does this tell you?
Nucleons have a substructure….
This feature is good, why?
If the nuclear force were attractive at all distances, the nucleus would collapse in on itself.
Monday, Sept. 19, 2016
PHYS 3446, Fall 2016

35. Shape of the Nuclear Potential
We can turn these behaviors into a square-well potential
For low energy particles, the repulsive core can be ignored, why?
Can’t get there..
This model is too simplistic, since there are too many abrupt changes in potential.
There would be additional effects by the Coulomb force
Repulsive Core
Attractive force
Monday, Sept. 19, 2016
PHYS 3446, Fall 2016

36. Nuclear Potential w/ Coulomb Corrections
Results in
Classically an incident proton with total energy E0 cannot be closer than r=r0. Why?
For R<r<r0, V(r) >E0 and KE<0  Physically impossible
What about a neutron?
Could penetrate into the nuclear center.
Low energy scattering experiment did not provide the exact shape of the potential but the range and height of the potential
The square-well shape provides a good phenomenological description of the nuclear force.
Monday, Sept. 19, 2016
PHYS 3446, Fall 2016