In this lab, we will do the following:

- study the statistics of random events;
- measure the penetrating ability of alpha, beta, and gamma radiation;
- determine the charge of the beta particle; and
- create a radioactive source by neutron activation and measure its mean lifetime

Before we start today's lab, we are asking all students to complete a short (<5 minute) survey in which you will have a chance to provide feedback on your TA. **Your answers are anonymous and will not affect your grade in any way.** You may access the survey from your personal computer, a lab computer, or your phone.

At the end of the quarter, TAs will receive average scores and comments (without identifying information) from their lab section(s).

Do not include any identifying information in your responses. If you have any feedback to provide to which you would like a response, please send it to David McCowan (mccowan@uchicago.edu).

If you cannot or do not want to complete the survey now, you may complete it at home. **The survey will remain open until Saturday, May 11 at 5:00 pm.**

There are three types of particles that we will observe in this experiment.

- An alpha particle consists of two protons and two neutrons bound together (that is, a helium nucleus).
- A beta particle is an energetic electron or positron (anti-electron) emitted from an unstable nucleus.
- A gamma particle is a quantum of electromagnetic radiation (a photon) emitted from an excited nucleus.

Alphas and betas can be emitted when an unstable isotope of one element decays into a nucleus of a different element, while gammas can be emitted when a nucleus decays from an excited state to its ground state.

An alpha particle (represented by $^4_2\alpha$) consists of two protons and two neutrons bound together to form a helium nucleus. Americium-241 ($^{241}$Am) is an example of a nucleus that decays via alpha emission. The reaction is expressed as

$^{241}_{95}\textrm{Am} \rightarrow ^{237}_{93}\textrm{Np} + ^4_2\alpha + Q$ |

where $Q$ represents the excess energy released in the process. For this reaction, $Q = 5.486$ MeV, most of which is carried away by the alpha particle in the kinetic energy of its motion.

Beta particles are fast moving electrons (represented by $\beta^{-}$) or positrons (also known as anti-electrons, and represented by $\beta^{+}$) emitted from some unstable nuclei when a neutron inside the nucleus converts into a proton, a beta, and a neutrino. (Neutrinos (represented by $\nu$) are very light particles with no charge which are very difficult to detect. We will not observe them in this experiment.)

The beta source used in this lab is strontium-90 ($^{90}$Sr) which decays into yttrium-90 ($^{90}$Y) with the emission of an electron and the release of energy $Q_1 = 0.544$ MeV. (The half-life for this decay is 28 years.) The yttrium-90 in turn decays, again by beta decay, into the stable nucleus zirconium-90 ($^{90}$Zr) with the emission of an electron and $Q_2 = 2.25$ MeV. (The half-life for this decay is 64 hours.) This double decay scheme can be represented as

$^{90}_{38}\textrm{Sr} \rightarrow ^{90}_{39}\textrm{Y} + \beta^{-} + Q_1 +\nu$ |

$^{90}_{39}\textrm{Y} \rightarrow ^{90}_{40}\textrm{Zr} + \beta^{-} + Q_2 +\nu$ |

Note that in each beta decay, the atomic number (the number of protons, $Z$) of the nucleus increases by 1, while the mass number (the number of protons plus the number of neutrons, $A$) stays constant. In each case, the energy $Q$ appears as kinetic energy which is mostly shared between the electron and the neutrino.

Gamma particles are energetic photons emitted from excited nuclei. Gamma photons are identical to visible-light photons except that their energy is typically a few thousand to a million times the energy of a visible-light photon (which is on the order of 1 eV).

Gammas interact with matter by mechanisms different from those of charged particles (such as alpha and beta particles). A charged particle steadily loses energy by ionizing and exciting atoms along its path through matter, while a gamma ray may penetrate far into matter with no effect and then, in one or a few collisions, give all or part of its energy to atomic electrons. Therefore, you will observe that gamma rays penetrate much further into matter than charged particles of the same energy. You will also observe that gamma rays are absorbed in matter such that the number of gammas remaining decreases approximately as an exponential function of the thickness of the absorber.

The Geiger-Müller (GM) counter is one of many devices for detecting energetic particles that make up the various types of nuclear radiation. It consists of a closed, gas-filled metal cylinder with high voltage applied to a central wire and connected electrically as shown in Fig. 2. This device generates a pulse which is counted in a scaler each time radiation deposits enough energy in the counter.

Low energy nuclear radiation may enter the GM tube through the thin window at the end of the tube, while more energetic particles may also enter through the outer metal wall of the tube. Radiation which enters the GM tube may interact with the gas in the tube and give rise to an avalanche of electrons. These electrons are attracted to the positively charged central wire and form an electronic pulse. The pulse is counted by the scaler.

Charged particles lose their energy continuously as they move through the gas of the GM tube. Gammas, on the other hand, rarely interact with the gas. When they do interact, however, they lose much of their energy in relatively few collisions with electrons in the gas or the tube itself. These energetic electrons in turn lose their energy to the gas. The GM counter is therefore much more efficient for counting charged particles than for counting gammas of the same energy.

**CAUTION**:

- The Geiger-Mueller (GM) tube is operated at high voltage. Do not remove the cable from the tube or its power supply while the voltage is on.
- The GM tube has a delicate end window. To protect the tube, please leave the tube in its holder at all times.

One member of the group should click on the link below to start your group lab notebook. (You may be asked to log into your UChicago Google account if you are not already logged in.) Make sure to share the document with everyone in the group (click the “Share” button in the top right corner of the screen) so each member has access to the notebook after you leave lab.

Check that the cable from the Spectech ST-370 counter is plugged into the USB port of the computer. Place a strontium-90 ($^{90}$Sr) radioactive source on the plastic tray and slide it into the top slot of its holder under the GM tube.

Open the *STX* application. Near the left side of the tool bar running along the top of the application window are buttons for the following:

Located on the righthand side of the application window are a column of boxes for setting and viewing the parameters: *Preset Time, Elapsed Time, Runs Remaining*, and *High Voltage*. Boxes for parameters which the user can change contain an up and down arrow as well as a text box for entering new values. Each of these parameters can also be accessed via the *Setup* and *Preset* menus.

Set the high voltage to `950 volts`

(and press *Enter*). This is a good operating voltage for the tube. Click on the *Start Counts* button. Do you see the number increasing in the *Counts* window? Remove the $^{90}$Sr source and verify that the count rate drops almost to zero. The residual counts are due to background radiation, caused by cosmic rays and other natural radioactivity.

One way we study radioactivity is by measuring the number of decays in a sample over a fixed period of time, $\Delta t$. The counting rate observed with the Geiger-Mueller (GM) tube near a radioactive source is proportional to the rate of decays within the sample.

For a short-lived isotope with a half-life on the order of tens of seconds, such as those studied in one part of this experiment, the number of counts per second from the sample decreases significantly over several minutes. This indicates a decrease in the amount of the unstable isotope in the sample. On the other hand, if the counting time is short compared to the half life, the count rate during $\Delta t$ will not decrease noticeably and can be regarded as constant.

You will see, however, that the number of counts in $\Delta t$ is rarely exactly the same for any two measurements (even though $\Delta t$ is the same), and that this number fluctuates about the average number. It turns out that the functional relationship describing the probability that a given counting measurement will yield $x$ counts in the interval $\Delta t$, has a characteristic form which is the same for all types of radiation.

If the number of counts in $\Delta t$ is large, the probability, $P(x)$, is a Gaussian probability distribution and is characterized by its average, or mean value, $\mu$ and its standard deviation, $\sigma$, by

$P(x) = \dfrac{1}{\sqrt{2\pi\sigma^2}} e^{-(x-\mu)^2/2\sigma^2}$. | (5) |

The factor in front of the exponential is chosen so that the function is normalized to make the integral over all values of $x$ equal to unity. A plot of a Gaussian distribution is shown in Fig. 1.

The mean value of the distribution is defined by

$\mu = \dfrac{\sum_{i=1}^{N} x_i}{N}.$ | (6) |

The standard deviation of any distribution reflects the width or spread of the distribution around its mean value and is defined (for large $N$) by

$\sigma = \sqrt{\dfrac{\sum_{i = 1}^N (x_i-\mu)^2}{N-1}}$, | (7) |

where $N$ is a large number of individual measurements of $x$ (the $x_i$ values). For counting experiments with a large mean value, the standard deviation of this Gaussian distribution is simply the square root of the mean number of counts,

$\sigma = \sqrt{\mu}$. | (8) |

Scientists are often interested in the number of counts within a certain interval centered about the mean of a distribution. For example, one may consider the number of counts within one standard deviation of the mean. For a Gaussian distribution, it can be shown that about 68% of all counting measurements fall within one standard deviation of the mean. For example, if we perform an experiment (e.g., counting nuclear decays) 25 times ($N$ = 25), we obtain 25 numbers ($x_1,\dots,x_{25}$). Suppose these numbers range in value from 0 to 200 and suppose their mean value as defined by Eq. (6) is 100. Then, the standard deviation as defined by Eq. (8) would be $\sigma = \sqrt{100} = 10$. That is, about 68% of our experiments would give numbers between 90 and 110.

Even though it is impossible to predict exactly when an unstable atomic nucleus will decay, you will find that when samples are large enough, regular patterns emerge which characterize the decay process. To observe this, gather sufficient data by performing an “experiment” 100 times and analyzing the results. The experiment consists of counting decays in a 2 second time interval. The computer can be helpful here!

It may be observed that the Geiger counter produces pulses even without any radioactive sources nearby. These pulses are due to ionizing radiation from cosmic rays or naturally occurring radioactivity in building materials or in the earth. Thus, the counting rate observed in the laboratory will never drop to zero. This residual rate is called the *background*. To measure the background, one should record the counts after removing the radioactive sources from the proximity of the counter.

The statistical precision in the background count depends on the total number of background counts obtained. Suppose, for example, you want to know the background rate to within $\pm 10\%$. In order to have a measurement with this precision, $N$ counts are required, where

$\sqrt{N}/N = 1/\sqrt{N} = 0.10 = 1/10$. |

Therefore, you would need to count until $N = 10^2 = 100$.

Click on the *Stop Counts* and *Erase All Data* buttons to clear old data. Set the *Preset Time* to 2 (seconds), and *Runs Remaining* to 100. Replace the $^{90}$Sr source in the top slot of the GM holder and click *Start Counts*. Data should begin to accumulate in the data table.

When the runs are complete, save your .tsv file somewhere on the computer where you can locate it.

We'll be using a Google Colab notebook to load, process, and plot the data from the software.

Be sure to include plots of you data in your notebook.

Alpha particles are helium nuclei and consist of two protons and two neutrons bound together. We wish to study the effect of the double charge of the alpha particle on its ability to penetrate matter. Note that a particle which loses energy very quickly in matter will be able to penetrate only a short distance.

You can test penetrating power of alpha particles from an Americium-241 source. Observe count rates from the source itself and with a sheet of paper inserted between the source and GM tube.

Check the penetrating power of the beta particles from a strontium-90 source. To do so, place the Sr-90 source on the plastic tray in the **2nd** slot from the top of the GM holder.

Measure the count rate with a single sheet of paper between the Sr-90 source and the GM tube. Repeat with aluminum of several thicknesses and with one square of lead.

Beta particles are energetic electrons and therefore have a single charge. Charged particles moving through a magnetic field experience a force and are thus deflected by the field. This fact may be used to determine whether the particles from Sr-90 are charged and what the sign of the charge is. Your lab TA will use the apparatus shown in Fig. 4 to demonstrate the charge of the particles from Sr-90.

Question 9: Sketch the magnet, showing the position of its north (N) pole. On your sketch, show the approximate path the beta particles took.

Question 10: From the path the betas took and the direction of the magnetic field (north to south pole), deduce the sign of the charge of the beta particles.

Gamma rays are high energy electromagnetic radiation and have no charge. Test the penetrating power of gammas with paper, aluminum and lead. To do so, place the cesium-137 ($^{137}$Cs) source in the plastic tray in the **4th** slot from the top. The count rate may be lower, so you might need to count for longer times. Set *Preset Time* equal to 60 seconds and *Runs Remaining* equal to 1. Click on the *Stop Counts* and *Erase All Data* buttons before each new data run.

Test how effective paper, aluminum, and lead are at stopping gamma radiation. (You only need to do 1 run each for these.)

Using 0, 1, 2, 3, and 4 lead squares, measure the counting rate as a function of thickness of lead placed between the Cs-137 source and the detector. Plot the count rate vs. number of pieces of lead absorber. Qualitatively, how does the count rate depend on the number of pieces of lead used to absorb the gamma rays?

The neutron howitzer provides a large flux of neutrons. The source of neutrons in the device is a multi-step process. The long-lived radioactive isotope $^{239}$Pu (half-life 24,000 years) decays as follows:

$^{239}_{94}\textrm{Pu} \rightarrow ^{234}{92}\textrm{U} +^{4}_{2}\alpha + Q$. |

The energy of the alpha-particle is 5.1 MeV. The plutonium is mixed with beryllium inside the howitzer. Beryllium absorbs the high energy alpha particles from the plutonium decay and emits neutrons as

$^{9}_{4}\textrm{Be}+^{4}_{2}\alpha \rightarrow ^{12}_{6}\textrm{C} + ^{1}_{0}\textrm{n} + Q$. |

The neutrons leave the beryllium with an energy of several MeV. In the howitzer the plutonium-beryllium neutron source is surrounded by paraffin shielding. Neutrons are slowed down and eventually stopped by collisions with protons (hydrogen nuclei) in the paraffin. When irradiating the samples, it is desirable to have a quantity of protons (paraffin or plastic) between the neutron source and the samples. Slow moving neutrons are more likely to be captured by nuclei than very fast moving neutrons emitted in the plutonium-beryllium reactions. Materials used to slow the neutrons in this manner are referred to as moderators. The thick layer of paraffin between the neutron source and the outer walls of the howitzer is necessary to reduce the flux of neutrons emitted into the room.

Matter may be made radioactive if it is irradiated with neutrons. This process is very important in chemical analyses of trace elements, since very small quantities can be identified through their radioactive decay modes.

The silver isotopes $^{107}$Ag and $^{109}$Ag are both present in naturally-occurring silver and are both stable (i.e., not radioactive). If they are bombarded with neutrons, two activation processes are possible.

In the first, $^{110}_{47}$Ag is produced as

$^{109}_{47}\textrm{Ag} + ^{1}_{0}\textrm{n} \rightarrow ^{110}_{47}\textrm{Ag}$. |

The new unstable isotope decays as

$^{110}_{47}\textrm{Ag} \rightarrow ^{110}_{48}\textrm{Cd} + \beta^{-} + \nu + Q$ |

with a half-life of about 24 seconds. We will ignore this short half-life decay in our experiment.

There is also another decay mode which is longer-lived:

$^{107}_{47}\textrm{Ag} + ^{1}_{0}\textrm{n} \rightarrow ^{108}_{47}\textrm{Ag} + \gamma$. |

$^{108}_{47}\textrm{Ag} \rightarrow ^{108}_{48}\textrm{Cd} + \beta^{-} + \nu + Q$. |

This decay has a half-life of about 2.4 minutes. We shall measure this half-life in our experiment.

The half-life of a radioactive isotope is defined as the time required for the number of nuclei of that isotope present in a sample to be reduced by one-half through radioactive decay. Before describing this procedure in detail, we examine the origin of the exponential relationship for nuclear decay and show that the half-life of an isotope is related to the probability of decay of an individual nucleus.

The decay of a sample of $N$ unstable nuclei can be described by

$\dfrac{dN(t)}{dt} = -N\lambda$, | (1) |

where $N(t)$ is the number of unstable nuclei present at any time $t$, and $\lambda$ is the probability per unit time for a single nucleus to decay. Thus, the total number of decays per unit time is proportional to the number of nuclei present.

The solution to Eq. (1) is

$N(t) = N_0 e^{-\lambda t}$, | (2) |

where $N_0$ the number of nuclei present at time $t = 0$. This exponential decay law can be verified by substituting Eq. (2) directly into Eq. (1).

It can be shown that the mean lifetime of the particle is $1/\lambda$, which is the time required for the number of unstable nuclei to drop to $N_0/e$. The rate of nuclear transitions, however, is most commonly expressed in terms of the *half-life*, $t_{1/2}$, of the isotope, i.e. the time it takes half of the nuclei in a sample to decay. For $t = t_{1/2}$, Eq. (2) gives

$\dfrac{N(t_{1/2})}{N_0} = \dfrac{1}{2} = e^{-\lambda t_{1/2}}$, | (3) |

so that

$\lambda = \dfrac{\ln 2}{t_{1/2}}$. | (4) |

Thus, we see that the decay probability, $\lambda$, is inversely proportional to the half-life.

For some isotopes, a stable nucleus may be made unstable by the addition of a neutron. In particular we shall add a neutron to silver-107 (Ag-107) to create the unstable isotope, Ag-108, the half-life of which you will measure. Your TA will give you a silver foil. Before exposing the foil to neutrons, test it for radioactivity with your detector. Do you observe counts greater than background?

**DO NOT WALK OR STAND IN FRONT OF AN OPEN HOWITZER PORT!**

When the Lucite plug is removed, the howitzer emits an intense flux of neutrons through the open port.

Before activating the silver foil, set up the STX software for a half life measurement.

- From the EXPERIMENT menu select HALF LIFE.
- Set the NUMBER OF RUNS to 30.
- Set the COUNT TIME to 30.
- Set the HIGH VOLTAGE to 900.
- Check the GRAPH RESULTS button.
- Do NOT click the start button until you have activated your silver foil.

Your TA will help you expose your silver foil to an intense source of high energy neutrons for about 3 or 4 minutes. Be prepared to return quickly to the lab and start your measurements as soon as possible.

Plot and fit your decay data using the Google Colab Notebook. How does your estimate of the half life compare to the literature value of 2.42 minutes?

Make sure to submit your lab notebook by the end of the period. Download a copy of your notebook in PDF format and upload it to the appropriate spot on Canvas. **Only one member of the group needs to submit to Canvas, but make sure everyone's name is on the document!**

When you're finished, don't forget to **log out** of both Google and Canvas, and to close all browser windows before leaving!

Write your conclusion in a **new** document (not your lab notebook) and submit that as a PDF to the appropriate assignment on Canvas when you are done. You should write your conclusion *by yourself* (not in a group), though you are welcome to talk to your group mates to ask questions or to discuss.

The **conclusion** is your *interpretation* and *discussion* of your data. In general, you should ask yourself the following questions as you write (though not every question will be appropriate to every experiment):

- What do your data tell you?
- How do your data match the model (or models) you were comparing against, or to your expectations in general? (Sometimes this means using the $t^{\prime}$ test, but other times it means making qualitative comparisons.)
- Were you able to estimate uncertainties well, or do you see room to make changes or improvements in the technique?
- Do your results lead to new questions?
- Can you think of other ways to extend or improve the experiment?

In about *one or two paragraphs*, draw conclusions from the data you collected today. Address both the qualitative and quantitative aspects of the experiment and feel free to use plots, tables or anything else from your notebook to support your words. Don't include throw-away statements like “Looks good” or “Agrees pretty well”; instead, try to be precise.

**REMINDER**: Your post-lab assignment is due 24 hours after your lab concludes. Submit a single PDF on Canvas.