The capability to generate power by fusing relatively abundant light nuclei, such as those from deuterium and tritium, promises to ease the world's problem of rapidly diminishing fossil fuel. Unfortunately, the technology to harness this almost unlimited source of fuel has not been fully developed yet. This is the reason why many countries have taken the technological challenges to improve the design of a practical fusion power reactor that uses water as its fuel source. One such device is called a tokamak. The word is an acronym for the Russian phrase, "toroid-kamera-magnit-katushka," which translates as the "toroidal chamber and magnetic coils." This implies that the reactor is shaped like a hollow donut with magnetic fields spiraling around to confine the hot plasma flowing inside.

Two conditions that must first be met to produce fusion power from plasma are: (1) The temperature must be high enough to allow the kinetic energy of the positively charged nuclei to overcome their natural repulsion and fuse when they collide. At this high temperature atoms are ionized into a collection of electrons and nuclei, commonly called plasma. (2) The density of the plasma ions must be high enough to ensure a high probability of collision. In addition, (3) the plasma confinement time, which is the time the interacting ions are maintained at a temperature equal to or greater than that required for the reaction to proceed successfully, must be large in order that more fusion energy will be released than is required. This extra energy is (4) the auxiliary heating power that heats the nuclei and makes them move at a faster velocity.

The ultimate goal of conducting fusion experiments is to find how well a tokamak reactor design would work under different conditions of temperatures, densities, magnetic fields, and auxiliary heating powers. Each experiment, called a shot, is given a number. To see how the output variables are affected by the changes made in the input variables, the raw data are plotted and statistically analyzed.

Raw data for plasma currents and the three output variables were taken from the PPPL* fusion energy experiments shot 41087 (magenta), shot 41152 (cyan), and shot 45599 (brown). *Reference: http://ippex.pppl.gov/ippex The data were tabulated and then plotted. Question: How does the change in plasma current affect the stored energy, energy confinement time, and fusion power output?

The GA Teacher-Scientist Team Giving a Workshop on Nuclear Fusion during the State Convention
of the California Science Teachers Association held in Long Beach, CA.

Energy Confinement Times
Time	Shot 41087	Shot 41152	Shot 45599
Seconds	Seconds	Seconds	Seconds
0.2	0.00192	0.0374	0.00298
0.6	0.0436	0.0919	0.114
1	0.105	0.182	0.286
1.4	0.127	0.209	0.352
1.8	0.133	0.153	0.425
2.2	0.138	0.153	0.228
2.6	0.144	0.155	0.183
3	0.158	0.153	0.2
3.4	0.197	0.183	0.193
3.8	0.0585	0.111	0.144
4.2	0.0505	0.0956	0.142
4.6	0.061	0.149	0.266
5	0.0184	0.193	0.21
5.4	0.0074	0.194	0.17
5.8	0.000157	0.129	0.085
6.2	0	0.104	0.0639
6.6	0	0.0545	0.011
7	0	0.00168	3.76E-05
7.4	0	0.000599	3.76E-05
7.8	0	0.00021	3.76E-05

Stored energy
Time	Shot 41087	Shot 41152	Shot 45599
Seconds	MegJoules	MegJoules	MegJoules
0.2	0.000402	0.0248	0.00254
0.6	0.0211	0.0848	0.0334
1	0.0632	0.141	0.0908
1.4	0.0786	0.161	0.179
1.8	0.0825	0.157	0.269
2.2	0.0843	0.152	0.341
2.6	0.0859	0.146	0.334
3	0.0867	0.142	0.336
3.4	0.114	0.188	0.446
3.8	0.663	1.22	1.73
4.2	0.66	1.27	1.56
4.6	0.299	0.745	0.533
5	0.0258	0.258	0.328
5.4	0.00982	0.191	0.193
5.8	7.13E-05	0.117	0.131
6.2	0	0.0773	0.167
6.6	0	0.0355	0.0182
7	0	0.00124	1.32E-12
7.4	0	0.000232	1.48E-12
7.8	0	0	1.40E-12

A close examination of the graphs of the input variable and the output variables, show that as the plasma current increases so do the stored energy, the energy confinement time, and the fusion power output. This may lead one to assume that to get the maximum power output from a tokamak, all it takes is to get the plasma current as high as may be feasible. The operation of a tokamak is more complex than this assumption, since past experiments have shown that there are limits to the values that can be assigned to these variables beyond which the tokamak will disrupt. This is another way of saying that the tokamak's power output will actually go down than up as predicted, when these limits are exceeded. For example:

• With the magnetic field and auxiliary power held constant, a point is reached beyond which increasing the plasma density further will cause the plasma current to disrupt. This is the upper limit that the plasma density can be raised.
• By changing the magnetic field and/or the auxiliary power to other values, and keeping them there, gradually increasing the plasma current will result in another density limit.

Tokamak experiments are carried out to understand how the density limits depend on the values of the other two variables. In fact, many other experiments are designed to find how different combinations of the other input variables affect the output variables. Another example is the limit imposed on the temperature, which if raised too high will make the plasma very hot for the magnetic field to confine it.

The following limits have been found from many experimental attempts to control fusion reactions:

• Kinetic energies of the order of 10 KeV or greater are needed to overcome the repulsion between ²H and ³H nuclei in order to get them close enough for the attractive nuclear forces to become effective and cause fusion.
• This means that the temperature should correspond to kT = 10 KeV, which is of the order of 10⁸ K. This is roughly the temperature in the interiors of stars, where fusion reactions normally take place.
• The energy required to heat a plasma long enough to produce power is proportional to the density n, of the ions.
• However, the collision rate is proportional to the square of the density n².
• If the confinement time is t, the above proportions can be combined into an inequality, which is applicable to fusion reactors: C₁n²t > C₂n.
• In 1957, the British physicist J. D. Lawson computed the constants C, and came up with what is now known as Lawson's Criterion: C₁n²t > 10¹⁴ s× particles/cm³.

The statistics formula, sampling by replacement (y = mⁿ), is used to calculate the number of experiments needed in a particular run, where m input variables are allowed to change n times. Given 5 input variables as listed above, how many experiments should be performed if each variable is given 3 different values while holding the other 4 variables constant? Solution: 125 (=5³) experiments are needed to give each variable an equal opportunity to change 3 times. This overly simplified calculation gives a reason why it is taking a long time to design an industrial prototype of the tokamak.