Range of Beta Rays

There is an ancient bit of wisdom regarding the range of beta's called the Feather relation, which I always remembered as the range in is about half the energy in MeV. Actually, what Feather (1938) derived empirically was

R

This was 'improved' by Glendenin and Coryell in 1946 to



Both of these expressions appear in Fermi's Nuclear Physics notes (Chicago, 1949). The second form is also contained in the Segre's Nuclei and Particles (Addison-Wesley, 2nd ed., 1977), which has a strong overlap with much of Fermi's notes.

Let's look at the actual plot of this improved version

which shows that these empirical results at least have the nice property of being continuous...

For reasons too complicated to detail here (they are detailed there), I found it necessary to integrate the dE/dx formula for energy loss to find the actual range. Below the minimum ionizing energy (for electrons, this is probably around 3 MeV), one has rather accurately



Where C is some constant of order 1.6 ). Assuming this holds all the down, we can then find the range by integrating:

=

⇒ R =

or in terms of electron kinetic energy (which is what is commonly quoted when dealing with low energy betas, and in fact is what appears in all of the above empirical range formulas:



This is the the 'non-empirical' form I've 'derived'; how does it compare to the empirical results? Taking C to be the observed minimum for electron energy loss of about 1.63 MeV / ), I get  

which compares embarrassingly well with the empirical curve:

Thus emboldened, I dedide to compare my result (in black above) with the  another, more 'modern' parameterization given in the lab manual for the undergraduate beta-gamma lab:



(L. Katz and A.S. Penfold, Rev. Mod. Phys. 24, 1 (1952)).

In fact, let's compare this (in green), the blue+red "Feather++", and my result (in black):

At this stage, I have no reason to favor my approach over the result given in the lab manual, but Segre provides a range plot that goes up to 10 MeV (where the range is supposed to be 5 in Al. Let's compare:

OK, I'm not perfect, but doing a lot better than the lab manual result. BTW, the Feather result, which is below mine at 3 MeV, does not fare well at large energies (nor should it; it's advertised as applying only below 3 MeV):

Let's make that plot in Segre (Figure 2-13) using my result

So I would say my curve (in blue) gives a good representation of the data over the entire range. (The reference for this figure in Segre is wrong, I infer that it must be Beta and Gamma Ray Spectroscopy, Kai Siegbahn, North-Holland, Amsterdam, 2nd ed. 1965. It's in the Physics library, and as near as I can tell, the curve given there is simply an empirical one, i.e., a fit to the available data.)

Practical Application:

(Yes, there is one.)
Suppose you want to test the efficiency of a scintillator paddle. The standard way to do so is to place it between two other paddles. and compare the triple coincidence to the flux through the stack given by . You can do this with cosmics if you are very patient, but it's faster to use a source. Just about the only beta source with a good change of getting through three paddles is Ruthenium-106, with an endpoint of 3.5 MeV. The above figure tells you the range for 3.5 MeV beta's is about 2 , which is about 2 cm of scintillator. So it works- on paper. In the real world, the counting rate goes to zero at the endpoint, so it's always a struggle, which is why people contemplate things like 1 millicurie Ruthenium sources (hot!).