Part of a Series of Activities in Plasma/Fusion Physics

Simulating Fusion

Part of a Series of Activities in Plasma/Fusion Physics to Accompany the chart Fusion: Physics of a Fundamental Energy Source

Teacher's Notes

Robert Reiland, Shady Side Academy, Pittsburgh, PA Chair, Plasma Activities Development Committee of the Contemporary Physics Education Project (CPEP)

Editorial assistance: G. Samuel Lightner, Westminster College, New Wilmington, PA and Vice-President of Plasma/Fusion Division of CPEP

Advice and assistance: T. P. Zaleskiewicz, University of Pittsburgh at Greensburg, Greensburg, PA and President of CPEP

Prepared with support from the Department of Energy, Office of Fusion Energy Sciences, Contract #DE-AC02-76CH03073.

This activity is intended for use in high school and introductory college courses to supplement the topics on the Teaching Chart, Fusion: Physics of a Fundamental Energy Source, produced by the Contemporary Physics Education Project (CPEP). CPEP is a non-profit organization of teachers, educators, and physicists which develops materials related to the current understanding of the nature of matter and energy, incorporating the major findings of the past three decades. CPEP also sponsors many workshops for teachers. See the homepage www.CPEPweb.org for more information on CPEP, its projects and the teaching materials available.

The activity packet consists of the student activity and these notes for the teacher. The Teacher’s Notes include background information, equipment information, expected results, and answers to the questions that are asked in the student activity. The student activity is self-contained so that it can be copied and distributed to students. Teachers may reproduce parts of the activity for their classroom use as long as they include the title and copyright statement. Page and figure numbers in the Teacher’s Notes are labeled with a T prefix, while there are no prefixes in the student activity.

Developed in conjunction with the Princeton Plasma Physics Laboratory and funded through the Office of Fusion Energy Sciences, U.S. Department of Energy, this activity has been field tested at workshops with high school and college teachers.

We would like feedback on this activity. Please send any comments to:

Robert Reiland Shady Side Academy 423 Fox Chapel Road Pittsburgh, PA 15238 e-mail: [email protected] voice: 412-968-3049 Simulating Fusion

Teacher’s Notes

Part of a Series of Activities in Plasma/Fusion Physics to Accompany the chart Fusion: Physics of a Fundamental Energy Source

Introduction:

In the section titled “Achieving Fusion Conditions” on the lower right hand corner of the Chart Fusion: Physics of a Fundamental Energy Source, a graph shows a history of fusion research and the region in which fusion reactions could be sustained. The three variables involved in the graph are the plasma confinement time,  (the Greek letter tau), the nuclear particle density, n and the absolute temperature, T. Putting these three variables together to understand the conditions for sustained fusion must seem like an abstract exercise to most high school students, and it is a good idea to supplement the Chart graph and text with more background and hands-on activities.

The activity, Simulating Fusion, is a hands-on activity to help students visualize the kinds of reactions that take place in fusion and to make sense of how the three variables are related. The activity uses plastic bottle tops, Velcro with adhesive backing, masking or Scotch tape, and a closable box or a bag. In doing this activity students produce a physical model of a fusion reactor by confining the bottle tops (with Velcro attached), which simulate nuclei, in the box or bag. Heating is simulated by shaking the system (nothing should actually get hot from this) and fusion is simulated by tops that stick together. The number of bottle tops and the time of shaking can be varied.

The students may generate mathematical models of how the variables combine to achieve sustained fusion in a reactor. Some thoughts on developing both kinds of models along with the limitations of such models will be presented later in this background section. It is important that students understand which aspects of the models are valid and which are questionable.

Materials:

Large shoe box or box of similar size 200 small plastic bottle tops (from bottles of two liters or less) 100 large plastic bottle tops (from bottles of three liters or more) Velcro masking tape stop watch

Prepare for this activity by saving about 200 of one size of plastic bottle tops and 100 of a different size for every group of students that will be doing the activity. This can be done as part of a recycling project, or just by asking students to bring in the plastic tops from used soda Simulating Fusion – Page T2

bottles, water bottles, etc. More of one size is needed because the number of one size will be varied in the model to investigate how variation in particle density of nuclei produces variations in fusion rates and in order to simulate nuclei of different masses and sizes.

With both sizes of tops, use the masking or Scotch tape to attach two bottle tops of the same size together - hole to hole. This part is mostly for aesthetics in that the combined tops look a lot more symmetrical than does a single top. If you’re in a hurry, you could use 100 uncombined tops of one size and 50 of the other size. Affix small pieces of the Velcro with hooks (the ones with a hard feeling to them are “hooks”) to the smaller tops and affix small pieces of the soft Velcro (“loops”) to the larger tops (see Figure T1). It is important in one part of the physical model that the hooked Velcro is on the smaller bottle tops and the looped Velcro is on the larger ones. The smaller bottle tops simulate deuterons and the larger ones simulate tritons. (A deuteron is the nucleus of a deuterium atom which is sometimes called heavy hydrogen made up of a proton and a neutron. A triton is the nucleus of a tritium atom -- the most massive of the three hydrogen isotopes and which consist of a proton and two neutrons. The symbols typically used for the deuteron and triton are “D” and “T” respectively, for example the D+T reaction on the chart. To avoid confusion with the symbol for temperature in this activity, T will be used to represent triton only as part of the combination D + T. This is the deuteron-triton fusion reaction.) You are now ready for students to put the model nuclei into bags or boxes for their simulated fusion experiments.

tape holding bottle tops tape holding bottle tops together soft velcro hooked together velcro

(b) A large bottle top nucleus (simulated (a) A small bottle top nucleus (simulated triton) deuteron)Figure T1: Bottle top nuclei details Background and Suggestions:

You might want to introduce the activity by asking your students about how their simulated nuclei are like real nuclei, such as deuterons and tritons, and how they are different. Obviously they differ in size, density and composition. Possibly the biggest difference that matters for the modeling of fusion is that the simulated nuclei do not repel at long range. Another is that, while the simulated nuclei can fuse, they will not emit a particle, such as a neutron, upon fusing.

(a) Bottle top nuclei on a collision path (b) Bottle top nuclei on a and likely to “fuse.” collision path with no chance Figure T2: Two of the many possible collision orientationsof “fusing”. of bottle top nuclei Simulating Fusion – Page T3

Begin the activity by showing students one of each kind of nucleus with the opposite Velcro strips, and ask them what would have to happen for the two to be able to stick together (see Figure T2). In particular, will they stick together the first time that the two collide?

A little thought or trial and error should convince students that for most collisions the Velcro will miss, and the nuclei will not stick together (see Figure T3). This corresponds to the fact that when real nuclei that can fuse collide, there is always a probability less than one that fusion will take place. In the bottle top model, the probabilities can be varied by using larger or smaller strips of Velcro. An alternative way to start the activity for those students who think better while playing with equipment or for classes that are very hands-on in orientation is to pass around bottle top nuclei for the brainstorming.

Figure T3: In any group of bottle top nuclei, the motions of bottle tops will be a variety of speeds and directions. Most collisions will not produce “fusion.”

The next question is the big one. How can the total number of collisions between nuclei be increased so that more fusions are likely to take place? Here’s where you can introduce the seemingly abstract variables from the chart in a natural way. You can ask the whole class to make suggestions, or you can ask them to work in smaller brainstorming groups and share their results. Three ways to get more collisions should emerge. (1) Increase the number of collisions by increasing the time during which the nuclei are confined, (2) increase the number of collisions by having more nuclei in a confined space, and (3) increase the number of collisions by increasing the speeds at which the nuclei move around (smaller average time between collisions).

The symbol for the confinement time is “” (tau), and it will be varied by shaking for different time intervals.

The number of nuclei in the confined space represents the particle density. Specifically, there is a particle density equal to the number of particles per volume for each type of nucleus. The symbol used for particle density is “n,” and it will be varied simply by changing the number of nuclei of one type in the bag or box. Simulating Fusion – Page T4

Since the speeds of real nuclei are not directly measurable and they are also distributed over a sizable range by the random nature of collisions, another variable must be used that depends on speed and is measurable. Temperature is measurable and depends on the average translational kinetic energy of the nuclei (rotational and vibrational kinetic energies of particles don’t affect system temperature), which in turn depends on the square of their speeds. The symbol used for temperature is “T.” In the simulated experiment T will be varied by how rapidly the bag or box is moved up and down or back and forth.

The experiment is then to control two of the variables while changing the third. The dependent variable in each case will be the number of bottle top connections formed (fusions), N. Since varying the shaking rate (the temperature) reproducibly and quantitatively is very difficult, this part is best done last or as an enrichment activity. Students should first get practice in getting fairly reproducible results from varying the confinement time and concentration, learning to shake the same each time, i.e. producing constant temperature. This increases the chance that students will be consistent in varying the temperature (shaking) later in a quantitative manner.

First, consider keeping the particle density and temperature constant and varying the time. Students might shake a certain collection of nuclei for 10 s. After each shaking, all joined bottle top nuclei should be counted and then separated (each connected pair is counted as one fusion). This shaking, counting, and separating should be repeated for several trials using the same time of 10 s. They should continue to do this until they get results that don’t change much from trial to trial. This is in order to demonstrate consistency in their simulation of constant temperature. Once the counts from repeated shakings have become fairly consistent, the last few numbers of fusions should be averaged as the number, N, of simulated fusions during 10 s. This average is then one data point in what will become a graph of fusions (N) versus time (). Additional data points should be found in the same way using the average number of fusions for 20 s, 30 s, 40 s, 50 s and 60 s of shaking. Then average number of fusions can be plotted against time or “”. The simplest outcome would be a linear trend, but this is not a sure thing. The smaller the number of particles used, the less likely the result will be linear.

Next, particle density at constant time and temperature can be varied by changing the number of one type of simulated nucleus (the type you prepared the most of). Since the size of the container is the same, increasing or decreasing the number of simulated nuclei increases or decreases the number per unit volume. The number of particles per unit volume is particle density. Again there should be several trials at each particle density averaged to give one data point of N versus n.

Here it is assumed that the effect of changing particle density doesn’t depend on which particle density is changed. This is intuitively reasonable, and it is also the way it works in experimental fusion reactors. Ideally, the average number of fusions depends linearly on the particle density and so a plot of N versus n would be a straight line.

The temperature variable is the hardest to vary in a quantitative way. Shaking more rapidly will cause the particles to have more average kinetic energy, but it’s not likely that students will be able to say that the energy, and therefore the temperature, is increased by a specific amount. Consequently, the main thing to be noted with confidence in this part, if you decide to do it at all, is whether or not the average number of fusions increases with temperature. Some students might Simulating Fusion – Page T5 want to see if they can find a degree of shaking below which no fusion occurs. This could lead to a discussion of a minimum temperature needed before collisions are energetic enough to overcome the electrical repulsive barrier to real fusion. In this physical model there is no such repulsion, but the Velcro might not stick unless the bottle tops have some minimum relative collision speed. It is also possible to show that fusion rates can actually drop at extremely high temperatures by shaking the bottle top system so vigorously that previously fused nuclei are roughly as likely to break up as new ones are to form.

Whether or not your students do an extensive investigation of the variables, at some point they should compare their results, observations, and thoughts with the plot, “Achieving Fusion Conditions,” in the lower right of the Fusion: Physics of a Fundamental Energy Source chart. Can they then understand what is being graphed? It is the bottom of the oval labeled as “Expected reactor regime” whose shape matters since this is the boundary between unsuccessful and successful operation. Can they propose more than one possible program for achieving successful operation on the basis of the variables examined? In other words, with temperature held constant, can they see that experimenters can get closer to achieving successful operation by increasing either confinement time or particle density or both? When they see cases in which magnetic and inertial confinement experiments result in points close together on the graph, can they comment on whether or not this means that the variables, confinement time and particle density, are really similar? Note that inertial confinement is limited to relatively short confinement times and high particle concentrations while magnetic confinement is characterized by relatively long confinement times and low particle densities (See “Creating the Conditions for Fusion” at the bottom center of the Fusion: Physics of a Fundamental Energy Source chart).

An additional outcome of the above investigation might be that now and then two of the lower mass (the ones with hooked Velcro) nuclei will stick together (whereas the higher mass ones with looped Velcro never stick together). If this happens, it presents an opportunity to talk about the fact that in a mixture of deuterons (lower mass nuclei) and tritons, most of the reactions will be between deuterons and tritons, but deuterons will fuse with deuterons at a much smaller rate as modeled by the relatively rare connections between two hooked Velcro tops.

A More Advanced Experiment:

If you have students who like a challenge and who are good at getting equipment to work, you might want them to attempt to use the bottle top model to find a relationship between fusion rate and simulated temperature. They won’t be able to model temperature very quantitatively, but to get close consider that the temperature of a gas is proportional to the average of the linear speed (as opposed to rotational or vibrational speed) squared of the particles. Since they will be modeling temperature by how vigorously they shake their fusion reactor, to double or triple the temperature, they should shake your reactor only slightly more vigorously each time.

Specifically, since T is proportional to the average of speed squared (v2), to double T, increase average speed by the square root of 2. To triple T, increase average speed by the square root of 3. To quadruple T, double the average speed. You can suggest to your students that they increase shaking speed from the initial rate by about 40% (square root of 2 - 1), 70% and 100%. To do this they could either rely on their sense of effort or build something to mechanically strike or shake a box at adjustable rates. Simulating Fusion – Page T6

They could then graph N versus T to get a third relationship. With a lot of luck they should find that N is directly proportional to T.

The three ideal proportions would be that N is directly proportional to each of the three variables, , n and T, and the product nT determines whether or not a reactor will be successful. (If you are not sure how this product follows from the individual proportionalities, look at “Producing a mathematical model from graphs of variables” at the end of this introduction.) In particular, to the extent that these three proportionalities hold, the bottom of the oval labeled “Expected reactor regime” is determined by nT = a constant. This means that if n is plotted against T the result would be a hyperbola.

Expected results and thoughts on model building:

The simplest results for fusions (N) versus confinement time () and (N) versus particle density (n) are that N is proportional to , when all other variables are held constant, and N is also proportional to n, when all other variables are held constant (N proportional to n means that as n doubles, so does N; as n triples, so does N, etc.). If this is true, then their graphs should be straight lines.

There are at least two ways in which the student’s physical model could fail to give these simple results. The first is that with only 50 of each type of bottle top interacting, basic uncertainties can be high in any given trial. That’s why it is often suggested that several trials be done to get an average for each measurement. In addition to getting better results from the averages, one can get a sense for the uncertainties. For example, if in three trials with the same variables, one gets N = 10, 8, & 11, the uncertainty is likely to be small since every number is within 2 units of the average. On the other hand if the results are N = 5, 15 & 9, the average is the same as it was in the first example, but the uncertainty is much larger, and if variations are this large from measurement to measurement, it’s hard to have confidence in a particular mathematical result. Probably the most one can be sure of is that the dependent variable either increases or decreases as an independent variable increases. Saying specifically that N is proportional to n would be hard to defend.

A second problem is one of saturation. In a sample of 50 of each type of bottle top, if 20 or more simulated fusions have taken place, the opportunities for fusions are then much less than if no fusions had yet taken place. This becomes really clear in the extreme case of all bottle tops getting connected. If this happens after 50 seconds of shaking, it’s a sure thing that N will not be greater than this after 60 seconds of shaking. It might even be less!

Saturation is a general problem when the number of fusions can be comparable to the number of nuclei in the system. This would not normally happen in a magnetic confinement fusion reactor, but it could happen (just barely) with inertial confinement, and it definitely can happen with the bottle top model when shaking is vigorous and goes on for a long time. This is a basic limitation of this physical model. This problem could be reduced by using a lot more bottle tops, but that would also greatly increase the preparation time. For this activity getting the basic sense of the trends is more important than getting exact mathematical results, and for most people it wouldn’t be worth the effort needed to greatly reduce the saturation problem. Simulating Fusion – Page T7

Answers to questions in the regular (non-optional) part:

From Procedure 6:

Question: Are your relationships between N and n and between N and  consistent with the graph of n versus T?

Answer: In the “Achieving Fusion Conditions” graph the oval labeled “Expected reactor regime” roughly corresponds to the attainment of a certain number of fusions, N, in a reactor. If N grows with n and with , N would increase as n increases at a certain temperature, and one expects that the “Expected reactor regime” should occur at large values of n, as it does in the graph.

Question: In the graph of n versus T, for a range of temperatures from less than 108 K to about 4 x 108 K, the product n needed to achieve successful reactor operation drops (the bottom of the oval gets lower). What does this suggest about the effect of increasing temperature in this range on the achievement of successful reactor operation?

Answer: At higher temperatures it is easier to sustain fusion. This is because, as can be seen in the Chart graph “Fusion Rate Coefficients” (lower left of the Chart), the probability of fusion taking place in a Deuteron + Triton (D + T) reaction goes up with increasing temperature in this temperature range.

Question: Beyond a certain temperature, about 4 x 108 K, the product n needed to achieve successful reactor operation increases (the bottom of the oval gets higher). What does this suggest about the effect of increasing temperature in this range on the achievement of successful reactor operation?

Answer: At temperatures higher than about 4 x 108 K it is actually harder to sustain fusion.

Question: Look at the Chart graph “Fusion Rate Coefficients” (lower left of the Chart). Does the plot of Rate Coefficient (indicates the probability of the reaction) versus Temperature for the D + T (Deuteron + Triton) reaction explain this effect?

Answer: At a temperature of roughly 4 x 108 K this graph shows that the probability of the D + T reaction is gently decreasing with temperature. This explains why the product n will have to be larger for successful reactor operation as the temperature goes beyond 4 x 108 K.

From the Questions Section:

Question 1: In what ways did your system model a real fusion reactor well?

Answer: This model shows that the number of fusions increases with particle density and confinement time. If any temperature variations were attempted, it probably shows that the number of fusions increases with temperature. Simulating Fusion – Page T8

Question 2: In what ways did your system model a real fusion reactor poorly?

Answer: The main things are that there are no long-range repulsion forces, and number of fusions (N) probably would not be directly proportional to time if saturation effects were important. The model also does not exhibit the detailed temperature dependence of reaction rates shown in the Chart graph “Fusion Rate Coefficients.”

Producing a mathematical model from graphs of variables:

Since it is often difficult for students to understand how a general relationship involving more than two variables can be produced from proportionalities found while only one independent and one dependent variable are allowed to change at a time, it is worth taking the time to show them how this works.

Many of them will be familiar with the Ideal Gas Law relationship, PV = nRT from chemistry. Start out by asking them what a graph of P versus T should look like if V and n are held constant (R is a constant). They will probably realize that it would be a straight line passing through the origin indicating that P is directly proportional to T. Do the same with P versus n while V and T are held constant.

At first the relationship between P and V may seem a little harder to handle, but if one thinks in terms of always getting a direct proportionality, it is a little easier. Clearly P is not directly proportional to V. What could be done to V to form a direct proportionality with P while n and T are held constant? For example, if V is squared, would P be proportional to V squared? If your students can’t see that this doesn’t work, show them that the entire right side of the equation is now being treated as a constant so that the product of P and V must be a constant. Then what happens to P if V is increased? At this point someone will likely note something to the effect that P and V are inversely related. That means that one should try relating the inverse of V (1/V) to P. It turns out that P is directly proportional to 1/V (another way to see this is to simply divide both sides of the original equation by V. Then P is being directly related to 1/V).

By breaking up the equation PV = nRT into three proportionalities, P proportional to T, P proportional to n, and P proportional to 1/V, you have come to the point that would be reached by experimentally determining the relationships. Now it can be seen that the way to produce a single proportionality from a group of proportionalities that have one variable in common (in this case it is P; for the fusion activity it is N) is to put the common variable to the left of the statement of proportionality and the product of all of the related variables (in this example n, T and 1/V) on the right. That is, P is proportional to n times T times 1/V or simply P is proportional to nT/V. So in this activity, if N is proportional to , N is proportional to n and N is proportional to T, then N is proportional to nT.

Going Further and Possible Projects:

The Chart graph Fusion Rate Coefficients (lower left of the Chart) suggests that at extremely high temperatures fusion reaction probabilities can actually go down as temperature increases. This may at first seem paradoxical, but students who really want to produce an example of this sort of thing happening, should be able to do so. Simulating Fusion – Page T9

This could be used as an independent project or as an entry in a science competition. With that in mind the following will not be detailed. Students wanting to go this route will find their own solutions to the problems of measurement and analysis.

The first problem to be solved is the quantitative control of simulated “temperature.” This probably requires the construction of a mechanical “shaker.” One possibility is the use of equipment used to randomize the selection of card positions in bingo and related games. Something like this could be constructed with an old bicycle pedal and chain that is used to rotate the container or lift and drop it. In the simplest arrangements the shaking rate would be proportional to the turning rate. That would make the temperature proportional to the square root of the turning rate (see the first two paragraphs of “A More Advanced Experiment”).

To enhance the probability of reaction rate coefficients going down as temperature goes up, students can attach relatively small strips of Velcro to the bottle tops. This will make it harder for simulated fusion to occur in the first place, but it will also make it easier for collisions to break fused “nuclei” apart.

With these preparations students can quantitatively investigate the rate at which nuclei form as a function of high simulated temperatures. From the information at the bottom left of the Chart they can calculate the “Fusion Rate Coefficients” as the total number of fusions per time (N/t) times the volume of the container divided by the product of the numbers of the two nuclei (bottle tops). These rates could then be plotted against simulated temperature and compared to the form of the “Fusion Rate Coefficients” graph for deuterium plus tritium (D + T).

A more direct analysis of the probability that collisions will result in fusions (typically called the reaction cross section) could be made by using a transparent container with a relatively small number of each type of nucleus. Collisions could then be filmed with a digital camcorder, and in frame-by-frame playback the number of fusions divided by the total number of collisions could be determined. This could be done as a function of simulated temperature. Simulating Fusion – Page T10

APPENDIX Alignment of the Activity Simulating Fusion with National Science Standards

An abridged set of the national standards is shown below. An “x” represents some level of alignment between the activity and the specific standard.

National Science Standards (abridged) Grades 9-12 A. Science as Inquiry Abilities necessary to do scientific inquiry X Understandings about scientific inquiry X B. Physical Science Content Standards Structures of atoms X Motions and forces Conservation of energy X Interactions of energy and matter x D. Earth and Space Origin and Evolution of the Universe X E. Science and Technology Understandings about science and technology G. History and Nature of Science Nature of scientific knowledge X Simulating Fusion – Page T11

Alignment of the Activity Simulating Fusion with AAAS Benchmarks

An abridged set of the benchmark is shown below. An “x” represents some level of alignment between the activity and the specific benchmark.

AAAS Benchmarks (abridged) Grades 9-12 1. THE NATURE OF SCIENCE B. Scientific Inquiry X 2. THE NATURE OF MATHEMATICS B. Mathematics, Science, and Technology X 3. THE NATURE OF TECHNOLOGY C. Issues in Technology 4. THE PHYSICAL SETTING A. The Universe X D. The Structure of Matter E. Energy Transformations F. Motion G. Forces of Nature X 11. COMMON THEMES A. Systems X B. Models X C. Constancy and Change X D. Scale X 12. HABITS OF MIND B. Computation and Estimation X