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Nucleogenesis! A Game with Natural Rules for Teaching Nuclear Synthesis and Decay Donald J. Olbris and Judith Herzfeld* Department of Chemistry MS #015 and Keck Institute for Cellular Visualization, Brandeis University, Waltham, MA 02454
Nucleogenesis! is a game designed to familiarize students with nuclear reactions and the general “geography” of the table of nuclear isotopes by simulating nucleogenic processes. It is intended as an inviting supplement to classroom instruction in a subject that is not amenable to hands-on experimentation or demonstration. In particular, the game is suitable as a lab exercise. Nucleogenesis! gives students extensive practice in manipulating nuclear reactions because the mechanics of the game require players to repeatedly calculate the outcome of nuclear reactions such as alpha and beta decays and fusions of protons, neutrons, or alpha particles with nuclei. The game board itself is a truncated table of nuclear isotopes. During the game, students will become familiar with the location of the “valley of stability” and the characteristic decay modes on each side of it. The game mechanics mimic some of the processes through which elements are synthesized in the universe. Each player starts with a nucleus of hydrogen (1H) and builds successively heavier elements by repeatedly fusing the nucleus with protons, neutrons, or helium nuclei. Occasional reactions with heavier nuclei are also included in simplified form (“bombardment” events). The first player to build a nucleus with at least 47 protons (silver) wins. In reality, synthesis of the elements proceeds through a series of stages, each with a different, dominant nuclear *Email:
[email protected].
Table 1. Decay Modes Included in Nucleogenesis! Decay Type
Example a
stable
12
beta emission
14
electron capture
15
positron emission
15
15
alpha emission
12
B → Li + α
proton emission
5
alpha absorption
8
C
C→
14
N + e + νe
{
{
O+e → O→
15
N + νe
N + e + νe +
8
Li → He + p 4
Be + α → 12C
aIn
the game, electron capture and positron emission have the same effect because we do not keep track of electrons (e{), positrons (e +), neutrinos (νe), or antineutrinos ( ¯νe).
Table 2. Fusion Reactions Included in Nucleogenesis! Fusion Type
Example
hydrogen fusion
12
C + 1H → 13 N
neutron (n) capture
60
Ni + n → 61Ni
helium fusion
16
O + 4 He → 20 Ne
process. Nucleogenesis! includes only the most basic processes and allows them to act over the whole range of atomic numbers rather than only in those regions where the processes naturally occur. Since the goal of the game is to learn the reactions and the geography of the table of isotopes, the kinetic details of nuclear synthesis are not as important. However, we provide a summary of the actual stages of nuclear synthesis in the discussion section. Nucleogenesis! comprises two 6-sided dice, an 81/2 × 11-in. paper game board, a 9 × 12-in. piece of cork board, multicolored map pins, the game rules, and a table of bombardment products. The game board, a color reproduction of part of the table of isotopes (Fig. 1), is pinned to the cork board, and map pins are used to mark each player’s position. Each square is color-coded to indicate the stability (black square) or preferred decay mode (colored squares) of the isotope. Although decay data are available for an astonishingly large number of isotopes (1, 2), we have omitted information for squares that are inaccessible within the rules of the game (white squares). Table 1 summarizes the decays included in the game. Playing the Game All players start with their markers on hydrogen (1H). Each player rolls the dice on his or her turn and follows the instructions in the chart on the lower right of the board. Rolling a 3–11 results in one of three fusion events. The player Table 3. Excerpt from the Bombardment Products Table Nucleons Initial No.
Product 1
Product 2
Nucleons Initial No.
Product 1 Product 2
2 p, 0 n
1 p, 0 n
1 p, 0 n
6 p, 8 n
0 p, 1 n
6 p, 7 n
2 p, 1 n
1 p, 0 n
1 p, 1 n
7 p, 6 n
1 p, 0 n
6 p, 6 n
2 p, 2 n
1 p, 0 n
1 p, 2 n
7 p, 7 n
1 p, 0 n
6 p, 7 n
3 p, 1 n
1 p, 0 n
2 p, 1 n
7 p, 8 n
1 p, 0 n
6 p, 8 n
3 p, 2 n
1 p, 0 n
2 p, 2 n
7 p, 9 n
0 p, 1 n
7 p, 8 n
3 p, 3 n
1 p, 1 n
2 p, 2 n
8 p, 7 n
1 p, 0 n
7 p, 7 n
4 p, 2 n
1 p, 0 n
3 p, 2 n
8 p, 8 n
2 p, 2 n
6 p, 6 n
4 p, 3 n
2 p, 1 n
2 p, 2 n
8 p, 9 n
0 p, 1 n
8 p, 8 n
4 p, 4 n
2 p, 2 n
2 p, 2 n
8 p, 10 n
2 p, 2 n
6 p, 8 n
4 p, 5 n
0 p, 1 n
4 p, 4 n
9 p, 8 n
1 p, 0 n
8 p, 8 n
5 p, 4 n
1 p, 0 n
4 p, 4 n
9 p, 9 n
2 p, 2 n
7 p, 7 n
5 p, 5 n
2 p, 2 n
3 p, 3 n
9 p, 10 n
2 p, 2 n
7 p, 8 n
5 p, 6 n
2 p, 2 n
3 p, 4 n
9 p, 11 n
0 p, 1 n
9 p, 10 n
6 p, 5 n
2 p, 2 n
4 p, 3 n
10 p, 9 n
2 p, 2 n
8 p, 7 n
6 p, 6 n
2 p, 2 n
4 p, 4 n
10 p, 10 n
2 p, 2 n
8 p, 8 n
6 p, 7 n
0 p, 1 n
6 p, 6 n
10 p, 11 n 0 p, 1 n
10 p, 10 n
N OTE: When one player bombards another, the number of protons (p) and neutrons (n) in their two nuclei are added together. Then this table is used to find the lowest-energy pair of product nuclei. Note the frequency with which the very stable isotope 4 He (2p, 2n) appears.
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No
In the Classroom
adds a proton (p or 1H), neutron (n), or alpha particle (α or 4 He) to his or her nucleus, calculates the product nucleus, and moves his or her marker to the appropriate space on the board. Table 2 summarizes the three types of fusion reactions. If the new nucleus is unstable (colored box), it decays according to the mode marked on the game board. The player calculates the product of the decay and moves his or her marker to the appropriate space. This process is repeated until the player reaches a stable state (black square). That player’s turn then ends. Two or more players are allowed to share the same stable square. The first player to build a nucleus with at least 47 protons (silver or higher) wins. On a roll of 2 or 12, the player has the option of “bombarding” another player’s nucleus. The protons and neutrons from the two nuclei are added together and then split into two new nuclei whose total energy as listed in tables of atomic masses (3) is the minimum for that total number of neutrons and protons. The bombardment chart (see Table 3) lists all such combinations accessible in the game. As before, unstable products decay until a stable isotope is reached. The player who rolls a 2 or 12 chooses which opponent to bombard (if there is more than one), whether to do the bombardment at all, and which of the two fragments to keep (and which to give to the other player) if the bombardment is done. The 65 0 64 63 62 61 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 n/p 0
1
2
3
4
5
6
7
8
9
bombardment rule has one major disadvantage: it takes some time to calculate the products. This situation is exacerbated when there is more than one opponent, because the player will want to calculate the results of each possible bombardment in order to choose the most advantageous. However, there are two balancing advantages: bombardment adds an element of choice for the players in an otherwise mechanistic game, and it allows a player who has fallen behind early to quickly reverse his or her situation. A few sample turns will help illustrate typical game play. For example, assume that Player 1 occupies 10B, Player 2 occupies 7Li, and it is Player 1’s turn. Player 1 rolls a 4, indicating fusion with a proton: 10B + p → 11C. 11C is unstable and decays by positron emission: 11C → 11 B + e+ + ν e, where e+ is a positron and ν e is a neutrino. Player 1’s pin is moved to 11B. Player 2 then rolls a 12—bombardment. The two players’ nuclei together have a total of 8 protons and 10 neutrons. The bombardment chart (see Table 3) indicates that the pair of nuclei with the lowest total binding energy for this number of nucleons is 4 He and 14C. Since Player 2 started on 7Li, she chooses to make the bombardment, moving her pin to 14C while moving her opponent’s to 4He. 4He is stable, but 14C is unstable and decays by beta emission: 14C → 14N + e{ + ν¯ e , where e{ is an electron and ν¯ e is an antineutrino. 14N is stable, so Player 2’s
10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45
Isotope Decay Modes stable beta emission electron capture or positron emission
55 54 53 52 alpha emission 51 50 simultaneous alpha and beta emission 49 48 proton emission 47 46 roll one die: 1-3 alpha absorption, 4-6 alpha emission 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 Dice Results 22 21 2 bombardment 20 19 3–5 hydrogen fusion 18 17 6–7 neutron absorption 16 15 8–11 helium fusion 14 13 12 bombardment 12 11 10 9 8 7 6 5 4 3 2 1 0 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 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 beta emission or electron capture (player's choice)
1
2
3
4
H He Li Be B
C
N
O
F
Ne Na Mg Al
Si
P
S
Cl Ar
K Ca Sc Ti
V
Cr Mn Fe Co Ni Cu Zn Ga Ge As Se Br Kr Rb Sr
Y
Zr Nb Mo Tc Ru Rh Pd Ag
Figure 1. The Nucleogenesis! game board. The game board is a truncated table of isotopes. The x-axis is the number of protons and the yaxis is the number of neutrons. Color coding indicates the decay mode of the isotope (see legend at upper left).
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In the Classroom
turn ends. Player 1 (now on 4He) then rolls a 6, neutron absorption: 4He + n → 5He. 5He is unstable and decays by alpha emission: 5He → n + α. Under the decay rules on the chart, this gives Player 1 a neutron (although he’d much rather have the alpha particle). The neutron is also unstable, so a second decay (a beta decay) brings Player 1 back to 1H. Although Player 1 seems far behind at this point, the existence of the bombardment rule makes quick turnarounds in the game possible (even likely). Discussion and Conclusions
Real-world Nuclear Synthesis The rules of Nucleogenesis! are only suggestive of the process by which the elements are formed in the universe. In this section we present a simple outline of how nuclear synthesis is believed to occur in nature. The information comes from Pagel (4). An astrophysicist’s interest in the elements often begins with hydrogen and ends with lithium, because measuring the relative abundances of those three elements is one way to probe conditions in the early universe. Shortly after the Big Bang, when the universe was hot, protons could freely convert to and from neutrons. As the universe cooled, n → p + e{ + ν¯ e continued, while the reverse reaction became impossible. Free neutrons then either decayed into protons or formed nuclei of hydrogen and helium by a number of reactions. Lithium was formed next by a variety of processes involving fusion of the nuclei of 1H, 2H, 3H, 3He, and 4He. Nuclear synthesis in the early universe halted at lithium because the universe had cooled enough that charged nuclei no longer had the kinetic energy necessary to overcome the electrostatic repulsion between them (the “Coulomb barrier”). The next steps in formation of the elements occurred (and still occur) in stars. Hydrogen in the cores of stars is compressed by gravity to very high densities and temperatures until hydrogen begins to fuse into helium. The energy released by fusion balances inward gravitational pressure. 3He is produced by p + p → d + e+ + ν e and then d + p → 3He, where d is a deuteron (2H). While the first reaction is a slow weak nuclear reaction, the second is very rapid, essentially depleting deuterons from stellar interiors. Most of the time (86% for the sun) the next step is 3He + 3He → 4He + p + p. The rest of the time, 3He + 4He → 7Be, and the 7Be decays again to 4He. In higher-temperature stars, hydrogen fusion is catalyzed by carbon, nitrogen, or oxygen. (Catalyzed hydrogen fusion did not occur in the first generation of stars formed after the Big Bang because the catalysts had yet to be created, as described below.) In the CNO (carbon-nitrogen-oxygen) cycle, a series of fusions and decays bring 12C to 15N, whereupon a fourth hydrogen fusion ends the cycle: 15N + p → 12C + 4He. At still higher temperatures, there are more catalytic hydrogenfusion cycles involving heavier elements such as neon, sodium, magnesium, and aluminum. Once the hydrogen in the core is exhausted, layers of previously uninvolved hydrogen surrounding the core collapse under gravity and also start to fuse into helium. Within the now-denser core, helium fusion begins: 4He + 4He → 8Be. 8Be either decays again into two alpha particles or fuses with another 4He to give an excited state of 12C, which then usually breaks into three alpha particles. Occasionally it decays into the stable 12C ground state. Other elements present in the
103
CNO cycle also fuse with helium to form heavier nuclei. Eventually helium, too, is depleted in the core, and the decrease in fusion-produced energy allows additional gravitational collapse to occur. Densities and temperatures again increase, and new shells of hydrogen and helium farther away from the core begin to burn. In the core itself, heavier elements begin to be consumed: first carbon (12C + 12C → 20Ne + 4He), then neon, oxygen, and successively heavier elements. Eventually iron is produced, from which no further energy can be extracted by fusion. The formation of elements heavier than iron occurs exclusively through neutron absorption via slow s-processes and rapid r-processes. While s-processes can occur in cool giant stars on the so-called “asymptotic giant branch” of stellar evolution, r-processes, which are the only mechanisms to produce elements heavier than 209 Bi, occur in supernovae and some high-energy events associated with neutron stars. Supernovae occur when fusion no longer provides enough energy to counter gravitational collapse. At this point, explosive collapse occurs, and the resulting shock waves and heating allow the r-processes that form heavier nuclei.
Simplifications Real-world nuclear synthesis has been greatly simplified for the game. The rules of Nucleogenesis! include a few of the more common processes (H and He fusion, neutron absorption, and heavy element reactions) and allow them to take place at all atomic numbers. The bombardment rules used here are a simplification that allows easy inclusion of the heavy element reactions. Although the bombardment rules ignore the kinetics of heavy ion collision and assume that the products always have the lowest energy possible, the rules do allow us to include reactions between heavy nuclei. Another change was made for improved playability. The original game board labeled 8Be as an alpha emitter, as it usually is. This had the effect that one could spend a dozen or so turns in the neighborhood of 4He because it was so difficult to bypass 8Be. It was recognized in the early ’50s that the instability of 8Be made it difficult to explain the observed abundance of 12C. However, the presence of a resonance in 12 C makes the reaction 8Be + α → 12C* (an excited state of 12C) possible. Occasionally 12C* decays via an electromagnetic transition to its ground state, 12C. We included the alternative that 8Be → 12C, instead of 8Be → 4He + 4He, by the roll of a die. Several of the reactions, particularly at the lower mass numbers, were simplified without altering the final end products. 12B → +e – + ν¯ e + 3α , so we made it an alpha emitter, which brings 12B to the correct end state (4He) in another step. In two places, our fusion rules allow players to form nuclei that do not form in nature: 1H + p → 2He, and 3He + p → 4Li. Both of the resulting nuclei were made into proton emitters so that the reaction is negated. Testing and Feedback Nucleogenesis! was tested by faculty, staff, and graduate students in the Department of Chemistry at Brandeis University. Additional testing was done by computer simulation of the game. The simulation allowed us to guard against some flaws in the game. In particular, it has been verified that it is impossible to “move off the board” and that the table of bombardment products includes all combinations possible within the game’s rules.
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The human testers first noted the problem with the instability of 8Be mentioned in the previous subsection (making 4He very hard to get away from). We tested our solution to the problem with the simulation. Our testers also noted problems with the length of the game, particularly due to the bombardment rule, which requires a more lengthy calculation than the other reactions. The problem is exacerbated with a larger number of players, as each player will want to calculate which of several possible bombardments will be most advantageous. Although in principle two to six players may play, in practice the game can become protracted with more than three. In general, we used the simulation to check how many turns the game would last on average, how many reversals would be brought on by the bombardment rule, and whether there were any other hidden problems with the game mechanics. This feedback led to some fine-tuning of the relative probabilities of bombardment and proton, neutron, and helium fusion.
Educational Use Nucleogenesis! was used in the introductory chemistry course at Brandeis as an optional exercise available in the library. As might be expected, however, few students found time for an optional exercise in a course already crowded with classwork, homework, and lab work. Nucleogenesis! has been more successfully used elsewhere as an in-class lab assignment (private communication with Dennis Flentge). The educational benefit of playing the game can be increased by asking the students some questions after they play the game. For example, the following questions address the basics of nuclear decay reactions: Which nuclear decay processes change the nucleus by 1 amu? Which nuclear decay processes increase number of the nucleus? Which nuclear decay processes decrease number of the nucleus?
mass of the mass of the the atomic the atomic
For these questions, students are expected to understand that electrons (beta particles) and positrons have a mass far less than 1 amu, so their emission and absorption does not significantly change the mass of the nucleus. They should also realize that the atomic number of the nucleus is the same as the number of protons. The atomic number only decreases when protons or alpha particles are emitted, or when protons are converted to neutrons during electron capture/positron emission. The atomic number only increases during beta decay. Other questions might address the “geography” of the table of isotopes: Where are the neutron-poor nuclei located on the isotope chart? What decay process(es) predominate there? Why? Where are the proton-poor nuclei located on the isotope chart? What decay process(es) predominate there? Why?
Students should see that the predominant decay processes in both the neutron-poor and proton-poor regions of the isotope chart tend to bring nuclei toward the central “valley of stability.” That is, in the proton-poor region of the isotope chart, nuclei tend to beta decay in order to convert a neutron to a proton. The opposite holds true for the neutron-poor isotopes. 352
One can also expect students to make observations about the relative stability of the isotopes while playing the game. Where in the Nucleogenesis! game is progress the slowest? Which stable isotopes are most frequently revisited? Can you relate this to nuclear binding energies?
Students may not immediately realize it, but the answer to the second of those questions can be read off the board by noting which squares have the most pinholes. One might expect the most stable elements in the game to be He (of the lighter elements) and Fe. Progress is slowest near He in the early part of the game. However, while 4He’s stability will be quite apparent, the stability of 56Fe will not be as obvious during any single game because nearby elements are almost as stable. It is possible that none of the heavier elements will appear more stable than the others after only one or two games. Students should appreciate that the more stable isotopes are the isotopes with the higher binding energies per nucleon. Finally, one can ask general questions about nuclear reactions. Why does fusion make stars hot? Why does fusion occur only in stars and certain nuclear devices (i.e., fusion reactors and fusion bombs)?
The latter question requires a student to understand that fusion can only occur when two nuclei overcome the Coulomb repulsion between them and approach closely enough that the strong nuclear force can bind them. The resulting release of energy fuels most stars, provides the destructive power of H-bombs, and may, in the future, drive electric generators in commercial power plants.
Availability We are making Nucleogenesis! freely available for classroom use via the World Wide Web at our “Games With Natural Rules” Web page: http://stanley.feldberg.brandeis.edu/ ~herzfeld/games.html. The game board, instructions, and bombardment products table can be downloaded in Adobe’s portable document format (PDF), which can then be viewed or printed from Macintosh, PC, and other computers with a freely downloadable reader (Adobe Acrobat Reader, available at http://www.adobe.com). Acknowledgments We would like to thank Diane M. Herrick and Kristin Kurz for helpful discussions on nuclear decay processes. We would also like to thank a reviewer for suggesting that we include sample discussion questions for use after playing the game. The development of Nucleogenesis! was supported by the National Science Foundation under grant HRD-9021929. Literature Cited 1. Firestone, R. B.; Shirley, V. S.; Baglin, C. M.; Chu, S. Y. F.; Zipkin, J. Table of Isotopes, 8th ed.; Wiley: New York, 1996. We used an online compilation of the data available at http://ie.lbl.gov/toipdf/ toi20.pdf and http://ie.lbl.gov/toipdf/chart1.pdf, …/chart2.pdf, and …/chart3.pdf (accessed Jan 1999). 2. CRC Handbook of Chemistry and Physics, 78th ed.; Lide, D. R., Ed.; CRC: Boca Raton, FL, 1998. 3. Audi, G.; Wapstra, A. H. Nucl. Phys. 1995, A595, 409–480. We used an online compilation of the data available at http://ie.lbl.gov/ txt/awm95.txt (accessed Jan 1999). 4. Pagel, B. E. J. Nucleosynthesis and Chemical Evolution of Galaxies; Cambridge University Press: New York, 1997.
Journal of Chemical Education • Vol. 76 No. 3 March 1999 • JChemEd.chem.wisc.edu
