ne.gif (2791 bytes) NE406 Radiation Protection and Shielding

Lesson 16 - Response functions

Definition of response function

In our concentration on the detector side of our shielding problem, the ultimate goal of this chapter is to reduce all of the data for a given detector into a single flux-weighting function that can be used to get the detector response when used in an equation of the form:

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where is called the "response function", and has units of response/unit flux. This form of the response function is not the most general, because it does not include angular dependence that the text includes in Equation 5.6. This is because most detectors are not directionally dependent.

Note: Generally, is a function and; is a distribution. Therefore, their product is a distribution with units of response/MeV/cc, which integrates to give response. With this formalism, the flux can include Dirac deltas (which are distributions). (A good example of this is the Dirac delta in energy that the uncollided flux "inherits" from a monoenergetic source.)

With this mathematical form, we can create various simple response functions.

Example 1: R = Total flux. For a desired response equivalent to the total flux -- integrated over all space and all energies -- we have:

From which we must have .

Example 2: R = Total absorption. As an example of a response equivalent to the total number of reactions of a given type, we would have:

which gives us .

Example 3: R = Thermal fission rate over a sub-volume, . As an example of a partial response equivalent to the total number of reactions of a given type, we would have:

which gives us:

.

Point response functions

In the previous mathematical form, the response function is a true function. It is possible, mathematically, to create a point response function using a combination of the Dirac delta function in space and a nominal volume:

(The is necessary to balance the units, since the Dirac delta would have units of per unit volume.)

In use, the basic equation changes to be energy only:

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Relation of response function to cross sections

The response functions that are of interest to health physics are based on the biological "response" of dose (i.e., energy deposition per unit mass) times quality factor. The DOSE part of these response functions can be computed using our knowledge of material properties of the medium that the detector is composed of. As described in the text, the only piece of the puzzle that is needed to compute the response function from cross sections is , which is the energy transfered to the material medium from reaction type j of isotope i. Using this, the response function can be computed from:

This response function is sensitive to the flux inside the detector; this flux is, of course, is perturbed by the presence of the detector itself. We will later relax this strict physical relationship in favor of an "point" response function that is sensitive to the "unperturbed" flux that would exist if the detector was not there. For the next couple of lessons, though, we will physically tie the response function to material densities, cross sections, and fluxes associated with a physical medium.