# Understanding the concept of range in nuclear reactions

Problem statement: A thick target of 55Mn is irradiated with deutrons with current $I$ during a time $T$. Most of the reactions result in 56Mn, which then decays with half-life $t_{1/2}$. Calculate the number of active 56Mn nuclei at the end of the irradiation under the assumption that the deutrons have range $R$ and that the mean value of the cross-section over this range is $\sigma$.

Following the question there are some numbers given, and the range $R$ is given in units of mg/cm$^2$.

I cannot find a consistent explanation of what this range is defined as. In my understanding it is the maximum distance a particle can go in a material until it runs out of energy. But the units makes no sense.

I've done a similar calculation where I calculated the reaction rate, and solved the ODE $$\frac{dN}{dt} = (\text{reaction rate}) - \lambda N,$$ where $N$ is the number of active nuclei.

Some explanation of this range and how to use it in this calculation is appreciated.

• Another useful word is a "stopping power" of matter. – Vladimir Kalitvianski May 26 '18 at 14:58
• The stopping power is certainly related to range. But, range just tells you how far into the film the beam/individual ions get. So, with a quick calculation you know how many nuclei might interact with the beam in total. – Jon Custer May 26 '18 at 15:02
• I would appreciate if you could elaborate on how to perform the "quick calculation". And why is range expressed in mass per unit area if it is actually a distance? – jock.marshall May 26 '18 at 15:14
• Look at a Wikipedia entry about "stopping power", please. – Vladimir Kalitvianski May 26 '18 at 16:40
• I've added the homework-and-exercises tag. In the future, please use this tag on this type of question. – Ben Crowell May 26 '18 at 17:11

While we usually think of "range" as distance--so dimensions of length and units of meters--that's not totally practical in nuclear physics. When considering the passage of a particle through matter, the areal density of matter per unit distance is important--and that is given by density, $\rho$, (inverse length cubed).

So, if you are given $R$ in mass per unit area (mg/cm^2 is a little unusual):

$$r = \frac R {\rho}$$

which is:

$${\mathrm{meters}} = \frac{\mathrm{kg/m^2}}{ {\mathrm{kg}}/ {\mathrm m^3}}$$.

Since energy loss is a random process, one can't say $R$ is an absolute maximum--but there are many ionizing interactions contributing the energy loss, so it's pretty reliable. Please see this table for your target nuclei:

http://pdg.lbl.gov/2017/AtomicNuclearProperties/HTML/manganese_Mn.html