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

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

Homework Assignment #3 Derive Eq. 1.55 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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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 ~ 10-13 – 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

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

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

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

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

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