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Lesson 8 - Scattering Kinematics

Importance of scattering kinematics

In this lesson, we will be considering the kinematics of collisions between particles and targets.  The term kinematics refers to the study of the physical consequences of a given collision without regard to the probability of its happening.  The study of probabilities and rates of occurrence of various collisions, which is called kinetics, will be covered in the next few lessons, when we will be looking at cross sections.

The study of scattering is very important to shielding; if particles didn't scatter, shielding would be a very easy discipline.  Most particles of interest in real-world shielding problems do not have an absorption event as their first interaction.  Most of them bounce around for dozens or hundreds of collisions, changing direction and losing energy with each one, until they are finally absorbed at an energy far below their original energy.  In so doing, they can penetrate thick shields, bounce around corners, find instrument penetrations through walls, and cause a host of other analysis difficulties.

Since both neutron and photon absorption occurs more readily at lower energies, the general strategy in shielding high energy  sources is to mix effective slowing-down materials with effective absorption materials.

The kinematics of collisions is based on conservation of momentum and energy.  The aim of our study of kinematics is for you to:

  1. Understand the underlying conservation equations for momentum and energy
  2. Understand the "big picture" results of applying these conservation principles to the three situations that we are most interested in:
  • Slowing down of (mass-less) photons by collisions with free electrons;
  • Slowing down of neutrons by elastic and inelastic collisions with target nuclides (which are heavier than the neutrons); and
  • Slowing down of charged particles by elastic collisions with free electrons (which are lighter than the charged particles).

Conservation equations for momentum and energy

Momentum

The equations of conservation of momentum are given in the text in Equations 3.9 through 3.11, which are illustrated in Figure 3.1.  The basic relation involves balancing the three vectors  wpeC2.gif (882 bytes), wpeC6.gif (893 bytes), and wpeCE.gif (897 bytes), which represent, respectively, the momentum vectors of the particle before the collision, the particle after the collision, and the target after the collision.  The laboratory system is used for these vectors.

Energy

The equations of conservation of energy are given in the text in Equations 3.12 and 3.13.  In addition to the three terms that are analogous to the momentum terms -- E, E', and T for the energy of the particle before the collision, the particle after the collision, and the target after the collision -- the energy equation has another term, Q, representing the non-kinetic "extra" energy that is added to or (more often) subtracted from the kinetic energy due to nuclear reactions and excitations.

Reactions that have no Q value are called elastic collisions: they correspond to our classic thinking about elasticity since all of the energy is accounted for in the motion of the particles.  (Don't make the mistake of thinking that elastic means that the original projectile keeps its energy; it doesn't because the target recoils.)

Reactions with a Q value are called  inelastic collisions.   Usually the Q value is negative because the target nucleus is left in an excited state, robbing energy from the motion of the particles.  In the reactions we are examining in this section, only neutron collisions can be inelastic, since free electrons cannot be excited.

Relating momentum and energy

The equation relating energy and momentum is given by Equation 3.8:

wpeA8.gif (1290 bytes)

This is equation is applied directly to particles with relativistic energies, but we are more interested in two simpler types of particles: photons (which have no mass) and non-relativistic neutrons.

For the photons, the equation immediately simplifies (by setting m=0) to:

wpeBB.gif (980 bytes)

or

wpeBD.gif (1009 bytes)

For non-relativistic particles, we can reduce it to a more familiar form by substituting:

wpeAB.gif (978 bytes)

to get:

wpeB2.gif (1297 bytes)

wpeB5.gif (1315 bytes)

and using the first two terms of the Taylor expansion of the square root:

wpeB0.gif (1338 bytes)

to arrive at:

wpeB6.gif (1096 bytes)

 

Kinematic results for collisions of interest

In this course, we are not going to wrestle with the equations directly.   Instead, I think it is more in our interests to concentrate on the results for the three collision types of interest:

  • photon scattering by free electrons

  • neutron scattering

  • heavy particle scattering by free electrons

My goal for each of these is that you will come away with an understanding of the range of energy loss that is possible and an appreciation for the relationship between this energy loss and the particle deflection angle in the laboratory system,   wpeD1.gif (892 bytes).

 

Photon scattering by free electrons

The primary interaction mechanism for photons is not with atomic nuclei as a whole, but interaction with a single atomic electron orbiting the nucleus.  For our kinematic considerations, though, we are going to assume that the interacting electron is a free, unbound electron.  The reason that this is a useful assumption is that the electron binding energies are usually just a few eV, and we are concerned with collisions that generally leave the target electron with much more recoil energy than this.   Therefore, the atomic bounds are easily broken, and the energy needed to break them is negligible for our purposes.

The interaction of photons with free electrons is called Compton scattering.

The basic result of applying the conservation laws to Compton scattering is that the final particle energy is related to the initial particle energy and deflection angle in the laboratory system  wpeD1.gif (892 bytes) by the equation:

wpeD8.gif (1131 bytes)

where wpeD5.gif (869 bytes) is the Compton wave length:

wpeD7.gif (1249 bytes)

A few observations about this result:

1. There is a unique relationship between deflection angle and particle energy loss.

2.  For straight-ahead collisions (no deflection), wpeD9.gif (976 bytes)=1 and there is no particle energy loss.

3.  The maximum energy loss occurs for 180 degree deflection, where wpeD9.gif (976 bytes)=-1 and

wpeDB.gif (1006 bytes)

4. The maximum fractional energy loss depends strongly on the particle energy, but not at all on the mass of the target nucleus.

5. Since wpeD5.gif (869 bytes) decreases as particle energy increases, the previous result means that no matter how energetic the particle before the collision, it is kinematically possible for the particle after the collision to have an energy lower than wpeD5.gif (869 bytes)=2, which corresponds to 0.256 MeV.  Therefore, for high energy particles, Compton scattering can result in a significant energy loss per collision.

6.  The converse is true as well.  The lower the particle energy, the less the maximum fractional energy loss for a single collision.

Neutron scattering

For neutrons, the interaction is with the target nucleus itself.  It is possible, therefore that some of the kinetic energy of the particles will be used to excite the nucleus. This excitation, however, must correspond to particular "states" that are unique for each nuclide; of particular importance to us is the energy of the first excited state -- if there is not enough energy to excite this state, then the scattering collision must be elastic.

The amount of energy that a particle must have to excite a given excited is always slightly greater than the energy of the state itself, because of the energy that must go into the recoil of the target.  This minimum energy is called the threshold energy. Allowing for the negative Q value, the threshold energy is given by:

wpeB7.gif (1162 bytes)  (NOTE: Positive number.)

Table 3.1 gives the energy levels for the first and second excited states for a number of nuclides.


Example: The threshold energy required to excite oxygen-16 is:

wpeC5.gif (2065 bytes)


The basic relationship between the initial particle energy, the final particle energy, the Q value, and the angle of deflection is given in the text by Eq. 3.34.  It is not useful enough to memorize, but you should be aware of two aspects of it:

  1. It relates the initial and final particle energies in the laboratory system to the deflection angle in the center-of-mass system.  This makes it a little cumbersome, but keeps the relationship unique.
  2. The inelastic scattering relationship does not use the Q value itself, but wpeDF.gif (875 bytes), a dimensionless measure of relative excitation energy given by:

wpeD2.gif (1054 bytes)


NOTE: wpeDA.gif (875 bytes) must be between -1 and 0.


What you should memorize (and be able to use in problems) are the equations for the minimum and maximum fractional energy that the neutron can have after a collision:

wpeE3.gif (1644 bytes)

and

wpeEA.gif (1637 bytes)

 


REALITY CHECK when you used these equations:  The remaining energy available for E' cannot be more than the original energy, E, minus the energy that goes into the excitation, -Q.


Note that:

  1. For elastic collisions (wpeDF.gif (875 bytes)=0), the maximum ratio is 1 and the minimum ratio is wpeEB.gif (1171 bytes).

  2. From Equation 3.34, the maximum ratio occurs (like photons) for a 0 degree deflection, the minimum for a head-on collision (180 degree deflection).

  3. For an inelastic collision caused by a neutron with energy just barely over the threshold energy, the above equation for the minimum energy goes negative (before being squared).   This means that for these barely-possible reactions, the energetically allowable deflections that the particle can have (in the laboratory system) from the collision is limited to a cone of directions about the straight-ahead direction..

  4. For the same set of barely-possible reactions, the Eqn. 3.32 relationship between the deflection angle in the laboratory system and the center-of-mass system becomes double-valued.  In other words, instead of there being a unique relationship between post-collision particle energy and direction, the situation is that there are two post-collision energies possible for some directions.

  5. The initial particle energies that can cause the situations noted in (3) and (4) above are from wpeEC.gif (897 bytes) to wpeED.gif (1154 bytes).  Physically, this energy range corresponds to the range for which it is possible for the post-collision velocity of the particle to be less than the recoil velocity of the original compound nucleus.

 

Heavy particle scattering by free electrons

This situation is only one that we are concerned about for which the particle is heavier than the target.  The physical fact of significance in this case is that the deflection angle and energy loss for the particle are severely limited.  The maximum deflection angle possible is that for which:

wpeEE.gif (1146 bytes)

Since the lightest charged particle that we are concerned about -- a proton -- is about 2000 times heavier than an electron, the deflection angle is limited to a fraction of a degree.

 

 

 

 


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