Radiation Protection and Shielding
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:
Conservation equations for momentum and energy
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 , , and , 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.
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:
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:
For non-relativistic particles, we can reduce it to a more familiar form by substituting:
and using the first two terms of the Taylor expansion of the square root:
to arrive at:
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:
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, .
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 by the equation:
where is the Compton wave length:
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), =1 and there is no particle energy loss.
3. The maximum energy loss occurs for 180 degree deflection, where =-1 and
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 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 =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.
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:
(NOTE: Positive number.)
Table 3.1 gives the energy levels for the first and second excited states for a number of nuclides.
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:
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:
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:
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.
Return to Course Outline © 1998 by Ronald E. Pevey. All rights reserved.