UNIT 102-11: NUCLEAR PHYSICS AND RADIOACTIVITY · Unit 102-11 - Nuclear Physics and Radioactivity Page 3 Author: ... Adapted from Studio Physics I: Unit 2 (c) ... 4 2 He. Because

Name ______________________ Date(YY/MM/DD) ______/_________/_______ St.No. __ __ __ __ __-__ __ __ __ Section__________

UNIT 102-11: NUCLEAR PHYSICS AND RADIOACTIVITY

The TRIUMF Cyclotron, located near the campus of UBC, which is used for research in Nuclear Physics.

OBJECTIVES

1. To learn about nuclear structure and how to determine the binding energy for a given nucleus.

2. To identify and understand the three major types of nuclear radiation.

3. To explore the statistical process of radioactive decay and examine the radioactivity of commonly occurring materials.

© 2009 by S. Johnson & N. Alberding (2013)

SESSION ONE: NUCLEAR PHYSICS AND RADIATION

In the early 1930’s scientists developed a simple model of the nucleus. According to this model. the nucleus is primarily composed of two types of particles, protons and neutrons. A proton has a positive charge +e = +1.6 x 10-19 C and a neutron has no electric charge, hence its name. These two particles are referred to generally as nucleons because they reside in the nucleus of an atom.

The number of protons in a given nucleus is known as the atomic number, Z. The total number of nucleons is given by the atomic mass number, A. You can specify a nucleus with just these two numbers, so a symbol containing them has been developed in nuclear physics:

AZX

where X is the chemical symbol for the element. The number of neutrons, N, can be obtained from N = A - Z.

Experiments have shown that atomic nuclei are roughly spherical in shape. A formula has been developed that estimates the radius of a given nucleus depending upon its atomic mass number as follows:

r ⇡ (1.2⇥ 10�15m)(A13 )

From this formula you can see that the volume of a nucleus is directly proportional to its atomic mass number, which one would expect if nucleons act like solid balls.

✍ Problem Activity 11-1: Nuclear SizesEstimate the diameter of the following nuclei:

(a) 42He

(b) 19779 Au

Page 2 Physics for Life Sciences II Activity Guide SFU

© 2009 by S. Johnson & N. Alberding (2013) Adapted from Studio Physics I: Unit 2

Binding Energy

The total mass of a stable nucleus is always less than the sum of the masses of the nucleons that make up the nucleus. This mass difference occurs because some of the mass goes into binding the nucleus together i.e. the mass converts into the total binding energy. This is the amount of energy that must be put into the nucleus to break it apart. The mass difference �m is defined as:

�m = (mass of nucleons)� (mass of nucleus)

The binding energy can then be calculated according to special relativity as:

Total Binding Energy = �mc2

In order to calculate the binding energy of a nucleus, you must know the masses of the nucleons and the mass of the nucleus in question. In nuclear physics, we specify masses in unified atomic mass units (u). (Your textbook describes the origin of this unit.) The masses of the nucleons in these are units as as follows:

neutron: mn = 1.008665 u

proton: mp = 1.007276 u

Another useful mass as we will see shortly is the mass of a neutral hydrogen atom:

hydrogen: mH = 1.007825 u

✍ Problem Activity 11-2: Determining the Total Binding Energy of Uranium23892 U has an atomic mass of 238.050788 u.

(a) How many neutrons does it have? How many protons? Record the values below.

(b) Determine the mass (in u) of the neutrons in a 23892 Unucleus.

Unit 102-11 - Nuclear Physics and Radioactivity Page 3Author: Sarah Johnson

© 2009 by S. Johnson and N. Alberding (2013) Adapted from Studio Physics I: Unit 2

(c) Because we are using an atomic mass, we need to take into account the electrons that are also contained in the mass along with the nucleus. One way to account for the electrons in calculating the mass difference is to use the mass of a Hydrogen atom instead of the mass of a proton when calculating the mass of the nucleons. You will then automatically include the Z electrons in the mass of the nucleons which will then be cancelled out when you subtract off the atomic mass (which also includes Z electrons) to determine the mass difference. Determine the mass (in u) of the protons and electrons in a

23892 U

atom using the mass of Hydrogen, mH, given above.

(d) Determine the total mass of the nucleons and electrons using your results from (b) and (c). Use this value and the atomic mass given to determine the mass difference for

23892 U. (Make sure that

you keep as many significant figures in your answer as you are allowed.)



(e) The last step is to determine the Total Binding Energy is to calculate�mc2 . The easiest way to do this is to use the fact that 1 u = 931.5 MeV/c2 , therefore 1 uc2 = 931.5 MeV. This will give you a binding energy in MeV.

Radiation

Certain combinations of neutrons and protons combine to form nuclei that are unstable. These unstable nuclei disintegrate or decay into other nuclei and in the process emit radiation. Three distinct types of radiation have been observed: alpha ( ↵), beta ( � ), and gamma ( � ).

Each of the three types of radiation consists of the release of a well known particle. In the case of alpha radiation, the particle is the nucleus of a Helium atom: 4

2He. Because the alpha particle takes protons and neutrons away from the parent nucleus during the decay, the remaining daughter nucleus will be of a different type from the original. In general, alpha decay can be written as:

AZN!A�4

Z�2 N’ +42 He

where N is the parent nucleus and N’ is the daughter. (For some unknown reason, nuclei only have daughters, no sons. :-) )

The total energy released during an alpha decay is called the disintegration energy, Q. It can be determined as follows:

Q = MP c2 � (MD + m↵)c2



where MP is the mass of the parent, MD is the mass of the daughter and m↵ is the mass of the alpha particle. A decay can only occur when Q is positive. The energy released goes into kinetic energy of the alpha particle the recoiling daughter nucleus.

✍ Problem Activity 11-3: Alpha decay of Radium(a) 226

88 Ra is an ↵ emitter. What is the nuclear symbol for the daughter nucleus when it undergoes alpha decay?

(b) Look up the atomic masses for 22688 Ra , its daughter and a

Helium atom in the Appendix of your textbook. Record the values in the space below. (If we use all atomic masses here, including for the alpha particle, the electron masses will cancel out when determining Q.)

(c) Now use the values you found in (b) to calculate the disintegration energy (in MeV) when 226

88 Ra undergoes alpha decay. (Hint: Use the same energy conversion trick you used in Activity 11-2 to go from u to MeV.)



A nucleus can also transmute into another type of nucleus when it undergoes beta decay. In one type of beta decay, known as � - decay, the particle that is emitted is an electron. This electron does not come from the atomic orbitals, but is emitted from the nucleus itself when a neutron transforms into a proton:

n! p + e� + ⌫̄

It was discovered during the analysis of beta decay experiments that in order to conserve energy and momentum a second particle must also be emitted. In the case of � - decay, the second particle is an anti-neutrino, ⌫̄ , which is a small neutral particle composed of anti-matter. (Anti-matter is a complicated topic which we will not delve into here.) In general, in terms of the parent and daughter nuclei, � - decay can be written as follows:

AZN!A

Z+1 N’ + e� + ⌫̄

Some nuclei instead disintegrate by another similar process known as � + decay. In this type of decay a particle known as a positron is emitted. A positron, which is the antimatter equivalent of an electron, has the same mass as the electron but the opposite charge. It is generally written e+. Another difference between �+ and � - decay is that in the former the second particle emitted is a neutrino, rather than an anti-neutrino:

AZN!A

Z�1 N’ + e+ + ⌫

An additional related process in which the nucleus absorbs one of the orbital electrons also occurs. This process is known as electron capture.

✍ Activity 11-4: Beta DecaysWrite out the complete decay process for each of the following nuclei. You will have to look up the identities of the daughter nuclei in your textbook.

(a) � - decay of 4520Ca



(b) � + decay of 4624Cr

✍ Activity 11-5: Energy Released in Beta + Decay(a) Write down an expression for the total energy released, Q, during a � + decay in terms of the masses of parent and daughter nuclei and the mass of the positron, me (the same as the electron mass). Assume the mass of the neutrino is essentially zero.

(b) Most textbooks provide the atomic masses of elements which include the mass of the nucleus and the mass of the orbital electrons. Show that in terms of the atomic mass of the parent MP and the atomic mass of the daughter MD that the energy released is equal to

Q = (MP �MD � 2me)c2



Gamma radiation involves the emission of high energy photons. These photons are emitted when a nucleus goes from a higher energy “excited” state to a lower energy state, similar to the emission of lower energy photons in atomic spectra, only in the case of gamma radiation it is the nucleus that is changing states, not the orbital electrons. One can write gamma decay as:

AZN*!A

Z N + �

where the asterisk denotes the excited nucleus.

One important thing to notice in all three types of decay discussed above is that in each case the total number of nucleons = protons + neutrons is always conserved. This is due to yet another conservation law known as the Law of Conservation of Nucleon Number. Protons may change into neutrons or vice versa, but the number A must remain constant.



SESSION TWO: THE STATISTICS OF RADIOACTIVITY

We’re going to start this session with a radioactive counting experiment that will continue while we do other things.

Natural Radioactivity and Statistics

Radioactivity is a statistical process in which a series of slight disturbances of the nucleus lead to a decay. Any particular unstable atom may remain in its unstable state for many years. However, in a typical sample there are so many such unstable atoms that many decays may occur in a short interval such as a second or a minute. The exact number of decays in an interval varies from interval to interval but the average number of decays per interval can be predicted based on the identity of the radioactive isotopes present in the sample.

The radiation detector we will use is a small tube containing a gas such as helium, argon or neon that can be ionized by an incoming particle of radiation and is called a Geiger-Müller tube. When the gas in the Geiger-Müller tube is ionized it conducts electrical current momentarily. When placed in a circuit the number of radioactive particles received by the tube can be counted and can cause an audible click and can be recorded by a computer.

Background Radiation: They came from Outer Space

There are no obvious radioactive sources in our lab. Nevertheless, a radiation monitor will detect some radiation. What’s going on?

The radiation detected when there are no obvious sources nearby is called “background radiation.” Some of the particles of this radiation come from the sun, other stars and even distance galaxies. There are also radioactive atoms inside the earth such as uranium that decay into radon gas. The radon filters up to the earth’s surface and emits further radiation as it decays. There are some common elements present in the room that have isotopes which are radioactive such as 14C, 57Fe and 40K. These radioactive isotopes of common elements also contribute to the background radiation.

The radiation detector can be used with the Labpro computer interface to allow automatic tracking of the background radiation using the Logger Pro software. For example, the system can be programmed to collect the number of counts in successive one-minute intervals for 80 minutes. The computer can then calculate the average and standard deviation of the counts.

Previously we collected data on the background radiation in the lab. The data can be summarized not only by the average number of counts per minute and the standard deviation of the number of counts per minute but also by its frequency distribution or histogram.



Histogram Radiation vs. Bin

0 10 20 30 40 50

0

5

10

Bin (counts)

His

t-R

adia

tion (

Fre

quency)

Figure 11.1 Histogram of the background radiation in the Physics Studio.

The histogram above shows the results of counting for 80 one-minute intervals. The mean is 11.06 counts and standard deviation is 3.36 counts.

The Frequency Distribution

Suppose you want to know more about the variation from the average number of counts? A graphical approach is to plot a type of graph known as a histogram or frequency distribution and study its shape.

As an example, consider a very simple frequency distribution. Imagine that you have caught ten fish. Of these ten, four are 3" long, two are 4" long, and four are 5" long. The frequency distribution would appear as follows:

4. The horizontal axis of your graph indicates the quantities

whose frequencies you are graphing; the vertical axis of your

graph gives the frequencies. Above each quantity on the

horizontal axis, draw a rectangle whose height corresponds

to the frequency of that quantity. Repeat this step for each

quantity measured.

As an example, consider a very simple frequency

distribution. Imagine that you have caught ten fish.

Of these ten, four are 3" long, two are 4" long, and four

are 5" long. The frequency distribution would appear

as follows:

! Activity 2-7: Frequency Distribution for Your

Time-of-fall Data (a) Draw a frequency diagram (known as a histogram)

representing your time-of-fall data for the ball in the grid

below.

(b) Next, using a different colour

of pen or pencil sketch in the

results of the rest of the class in

the histogram above.

How does the shape of the class

frequency distribution above

compare with the shape shown

in Appendix C on page C-6? !

Does the variation in the time of

fall data seem "normally

distributed"? How does it

compare to your prediction in

Activity 2-6? !Explain.

Page 2-12! Workshop Physics Activity Guide! SFU 1057

© 1990-93 Dept. of Physics & Astronomy, Dickinson College Supported by FIPSE (U.S. Dept. of Ed.) and NSF. Modified for SFU by N. Alberding, 2005.

How to Plot a Frequency Distribution

Since a frequency distribution of values shows how many times you recorded each value, this distribution can be drawn by organizing your data as follows:

1. Load your spreadsheet file with the data to be plotted into the computer.

2. Sort the column of data from the lowest time to the highest time.

3. Count the frequency of occurrence of each quantity that was recorded. For example, if you recorded a time of 0.45 seconds five different times, the frequency of 0.45 seconds is 5.



4. The horizontal axis of your graph indicates the quantities whose frequencies you are graphing; the vertical axis of your graph gives the frequencies. Above each quantity on the horizontal axis, draw a rectangle whose height corresponds to the frequency of that quantity. Repeat this step for each quantity measured.

5. There is a histogram tool in Excel which does a lot of the work for. To use it you must create a column of “bins” representing the x-axis of the histogram. Then select the data for which you want a histogram. Select “Tools..Data Analysis..Histogram” and follow the instructions. (There is a similar feature in Logger Pro that is even easier to use.)

Apparatus to Measure Counts per Interval.

In the exploration of the statistics of radioactivity, we can use a computer set up as a radiation counting system. We will need the following equipment (one set for the whole class):

• Radiation Monitor • a computer interface • a computer • radiation monitoring software (Logger Pro) • a small sample of potasium-rich material.

Doing the Experiment

We will be doing this experiment together as a class. First we will determine the counts/interval for 80 one-minute intervals. The computer plots the frequency distribution histogram automatically as it collects the data. This will allow us to obtain a frequency distribution that can be compared to the background radiation frequency distribution.

1. Plug the radiation monitor into the Dig/Sonic 1 input of the Labpro,Open up the “L01001a (Counts vs Time).cmbl” setup file in Logger Pro. Choose “Radiation Monitor” not “Student Radiation Monitor” as the sensor input.

2. Press the Start button and see if the detector system is working. Attach a small bag of the sample directly against the window of the radiation monitor. The setup file collects data for 20 one-second intervals. This can be changed by choosing “Experiment...Data collection” from the menu. Change the setup to take 80 one-minute intervals. The program displays a table of counts/interval. and a graph of counts vs time.



We will set up the radiation monitor to repeat the 80 one-minute counting experiment done previously, but this time we’ll put a potassum-rich powder near the detector. NuSalt® is a commercial product which is sometimes advised to replace table salt for the diet of people who need to avoid sodium. It consists mainly of KCl with some other flavour-enhancing additives. One in 9,000 atoms of K is the 40K isotope that emits beta radiation. A 311 g NuSalt® container has about 2.8 x 1020 atoms of 40K, of which 4927 decay per second on average.

✍ Activity 11-6: Radiation From A Potassium-rich source

(a) First set up the NuSalt® in a little bag next to the detector

(b) Take a sample one-minute interval and compare the counts to that of the background radiation taken earlier.

Number of background counts in a typical one-minute interval

Number of counts in a one-minute interval with NuSalt

You will probably not be convinced with just one such measurement that NuSalt® has radioactivity significantly above the background level. If the the number of counts is slightly higher, it might just be a random fluctuation and not due to an actual increase in the average radioactive level. (And in fact one reading might be lower than background.) In order to clearly demonstrate the effect it is necessary to continually repeat the measurement 80 times and compare the histogram with that of the background.

While the NuSalt® data are being collected we’ll go ahead with some other activities.

Radioactive Decay - Theory

As stated previously, radioactive decay is a random process. One cannot predict exactly when a given nucleus will decay. We can however predict roughly how many nuclei in a sample will decay over a given time period. The number of decays �N that occur in a short time period �t depends upon the length of the time period and the total number of nuclei present N :

�N = ��N�t

� is known as the decay constant and varies depending on the type of unstable nucleus being examined.



By taking the limit as the time interval approaches zero and applying a little calculus to this equation one arrives at the following relationship:

N = N0e��t

which is known as the radioactive decay law. (A derivation of the law can be found in your textbook.) Here, N0, represents the number of nuclei present at t = 0, and N is the number at time t. As you can see, the number of radioactive nuclei in a sample decreases exponentially with time.

The decay constant � determines how quickly a sample will decay. Another way to specify the decay rate is by a quantity called the half-life, T 1

2,which is defined as being the time it takes for half of a

sample of a given type of unstable nucleus to decay. If one substitutes N = N0 /2 and t = T 1

2 into the radioactive decay law equation, you arrive at an expression for the half-life:

T 1

2=

ln 2

�=

0.693

�

In order to better understand this statistical process, in the next activity we will simulate radioactive decay using another random process, throwing six-sided dice. For the following activities you will need:

• 300 six-sided dice (divided as evenly as possible among the class groups)

✍ Activity 11-7: Theoretical Dice Decay(a) Complete the “Theoretical Decay” table entries on the next page by assuming that exactly 1/6 of all nuclei initially present decay each time period. Do this by taking 1/6 of the “initial number present” and round to the nearest whole number. Enter this numberas the “number decayed.” Note: You may do this either by using an Excel spreadsheet or a calculator. If you use Excel, you do not have to fill out the “Theoretical Decay” table in the activity guide.

(b) Subtract this value from the “initial number present” and record as the “number remaining.” (The “number remaining” now becomes the initial number for the next round.)

(c) Repeat this process until the table is complete or fewer than ten nuclei remain undecayed.

(d) Enter your data for time and initial number present into Excel (if you used a calculator) and make a graph of Initial Number of Radioactive Nuclei Present vs Time. Make sure to save this spreadsheet and graph for the next activity.



Elapsed Time Initial Number Present

Number Decayed Number Remaining

0 300

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23





24

25

26

27

28

29

30

(e) Describe in words what your graph looks like. Does it have the shape you expected?

In the next activity we will simulate radioactive decay by throwing 300 dice 30 times. In order to make this easier, we will share the work by dividing up the dice between the groups. All the groups will have to combine their data after they have finished rolling their set of dice. You cannot proceed with the analysis until all groups are finished.

✍ Activity 11-8: Experimental Dice Decay(a) Enter the number of dice in your set in the first row of initial number present. Toss your set of the dice and remove each one showing the number one. The number you remove will be the “number decayed.” Enter this number into the table below.

(b) Subtract the “number decayed” from the “initial number present” to obtain the “number remaining.” Enter this into the table.

(c) Toss the remaining dice and again remove all those that “decay.”

(d) Repeat this process until the chart is complete or no dice remain.





0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22





23

24

25

26

27

28

29

30

(e) Now combine your data with all of the other groups by adding together the “Initial Number Present” for each time from each group. The number in the t = 0 row should equal 300. Put this data into the table below.


0 300

1

2

3

4

5

6

7

8

9

10

11




12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

(f) Enter your experimental data for initial number present into a new column in your saved spreadsheet next to the theoretical data for the same quantity. Add this data to your theoretical graph of Initial Number of Radioactive Nuclei Present vs Time so that you now have two decay curves. Submit your spreadsheet with graph to WebCT.



(g) How do your two decay curves compare? Would you say that they agree with each other?

(h) Use the graph to estimate the amount of time necessary for your experimental number to go from:

300 to 150 _______

250 to 125 _______

200 to 100 _______

150 to 75 _______ 100 to 50 _______

(i) Based on your answers above, what is the approximate half-life of your “radioactive” sample?

Now let’s return to the radioactive counting experiment on NuSalt® that we started at the beginning of the session.

The histogram of the number of counts in 80 one-minute intervals will be displayed and distributed to the class.

✍ Activity 11-9: Finale: The Radioactivity of NuSalt®: Is it significantly above the background level?

(a) Write down the mean and standard deviation of the NuSalt® data collected in class.



(b) In order to compare this data with the background data we must calculate a quantity known as the standard deviation of the mean, �SDM , which is the standard deviation divided by the square-root of the number of samples. Calculate this quantity for the NuSalt® data.

(c) Calculate �SDM for the background data given earlier in this activity guide.

(d) Now write the mean and the standard deviation of the mean in the following format: Mean ±�SDM in the spaces below.

NuSalt® data: ____________±____________

Background: ____________±____________

(e) Scientists say that the signal is significantly above background (with 68% confidence level) if the signal and the background data do not come within one �SDM of each other. The 95% confidence level occurs when the signal and background do not come within two �SDM of each other. Is the NuSalt® data signal here significantly above background? How sure are you? What is your approximate confidence level?



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UNIT 102-11: NUCLEAR PHYSICS AND RADIOACTIVITY · Unit 102-11 - Nuclear Physics and Radioactivity Page 3 Author: ... Adapted from Studio Physics I: Unit 2 (c) ... 4 2 He. Because