Application of High-Temperature Plasma Physics to the Implementation of Controlled Atomic Nucleus Fusion Reactions*)

R. Post

I. Introduction

The utilization of the energy released in large quantities in reactions of fusion of atomic nuclei will make it possible once and for all to solve the problem of the ever-growing demand for energy sources. Of the several light elements that can be used as fuel in a fusion reactor, the reserves of deuterium alone in the seas and oceans of the world are sufficient to produce, for a billion years, a thousand times more energy than is now supplied by all the electric power stations of the world. However, the technical difficulties arising in solving this problem are truly enormous. Some physicists, having familiarized themselves with this problem, have expressed doubts as to the possibility of solving it.

Fragmentary information on the problem of the energy of atomic-nucleus fusion appeared in the literature many years ago. It is known that several countries are already working on it, and at the Geneva Conference in 1955 the Indian physicist H. J. Bhabha ventured to predict that this problem would be solved within the next twenty years. At the end of 1955 the chairman of the American Atomic Energy Commission officially announced that the Commission was supporting a long-term research program under the code name Project Sherwood, aimed at carrying out controlled nuclear fusion reactions intended for peaceful use. It was published that the main part of the experimental work is being carried out in three places: at Princeton University and in two laboratories of the Atomic Energy Commission at the University of California (the Los Alamos Scientific Laboratory and the Radiation Laboratory of the University of California at Livermore). In addition, the Commission supports research being carried out at Oak Ridge and New York University. Although it was stated that the beginning of the implementation of Project Sherwood was the year 1951, in reality the problem had been studied in the Commission’s laboratories even before the end of the Second World War. The basic theoretical concepts were formulated at Los Alamos by Edward Teller, Enrico Fermi, James Tuck, and others. Some results of the theory were later used in the experimental program of Project Sherwood at Los Alamos, whose execution was directed by J. Tuck.

) Rev. Mod. Phys. 28, 338 (1956). Translated by I. M. Podgorny. The article appeared two months after I. V. Kurchatov’s report on the work of Soviet physicists in the field of the search for a controlled thermonuclear reaction (UFN, LIX, no. 4, 1956). These works were published in the journal Atomic Energy*, no. 3 for 1956. In the appropriate places in the proposed article references have been made to these, as well as to certain other works devoted to the question under consideration.

At the beginning of 1951, the Princeton University astrophysicist Lyman Spitzer, who knew nothing of the secret work being conducted at Los Alamos, became interested in a different approach to the problem of the fusion reaction, one other than that which was being considered in the United States and had been supported by the Atomic Energy Commission. To develop Spitzer’s work, the Commission created a new project, “Matterhorn,” at Princeton. Soon afterward, Herbert York of the Radiation Laboratory of the University of California at Berkeley, having studied the work of the Los Alamos and Princeton groups, proposed several new approaches to the problem. In parallel with the work of organizing the new Livermore Laboratory, York created a small experimental group, whose leader became the author of the present article. The creation of this group was prompted by the need to broaden the search and to develop new methods for analyzing the problem.

In June 1952, Johnson, director of the research division of the Atomic Energy Commission, convened a conference on thermonuclear reactions in Denver, Colorado, chaired by Edward Teller. At the conference the principal directions of research were formulated, in order to concentrate on the most promising of them. Attention was paid equally both to the great prospects and to the great technical difficulties arising in solving the problem of controlled fusion of nuclei. As a result, work on controlled fusion reactions was expanded; the coordination of the work was handled by the Commission’s research division.

Subsequently, a number of further conferences were held to study the experimental and theoretical aspects of the problem. At present the Sherwood scientific program is directed by the National Steering Committee, whose members are Edward Teller of the University of California, James L. Tuck of the Los Alamos Scientific Laboratory, Lyman Spitzer of Princeton University, and William Brobeck of the Radioengineering Laboratory of the University of California. Coordination and administrative direction of the entire program are under the management of Amos Bishop, chief of the Sherwood section of the research sector.

In many of its parts, the present article is a review of the work of many physicists who have made a definite contribution to the Sherwood project. The full understanding now achieved of the basic physical principles of a controlled nuclear fusion reaction is, to a considerable extent, the product of their individual and collective creativity. The work was based on the results of earlier published investigations by astrophysicists, who can now judge the possibility of the practical, peaceful application of processes occurring at ultrahigh temperatures, which previously had been observed only in stars or in the heart of an atomic bomb.

II. JUSTIFICATION

As has already been said, the realization of a controlled nuclear fusion reaction is one of the most important problems of our century. This circumstance provides sufficient incentive for physicists to take up its solution vigorously. But a more careful consideration of the problem shows that the reasons for the interest lie much deeper. The study of trends in the growth of population and in the standard of living of the majority of the world’s population points to the necessity of the most rapid possible increase in the capacity for producing energy throughout the world[^1].

If energy consumption can be extrapolated, it turns out that fossil fuels, such as coal and oil, will be exhausted in meeting the needs of the population in less than a century. Even the use of solar energy falling on the entire accessible area of the earth would hardly satisfy the assumed demand for energy a hundred years from now. Continuously,

growing needs can be met only by using the fission of heavy nuclei or the fusion of light nuclei as a source of large quantities of energy. As the results of the corresponding estimates show, the energy resources contained in the reserves of fissile materials whose development is economically justified exceed by roughly 25 times the energy stored in fossil fuel. Despite the considerable size of these reserves, they can satisfy energy demand only for several decades of the next century. For comparison, let us recall that the energy needs of the future era can be met for billions of years solely through fusion reactions of deuterium nuclei found in the seas and oceans of the Earth. The fuel for fusion reactions of atomic nuclei is essentially inexhaustible.

Interesting conclusions can be drawn from a comparison of the cost and the trends in the change of the cost of nuclear and fossil fuel. Even with the extraction methods now in use, the cost of deuterium fuel amounts to only a few percent of the cost of the quantity of coal that yields the same amount of energy. As the reserves of one or another type of fuel, including fissile materials, are depleted, its cost will become higher and higher. An exception in this respect is deuterium extracted from seawater, the cost of which will fall or, at least, remain approximately constant for an indefinitely long time.

On the other hand, the prospect of supplying hundreds of billions of kilowatts of energy from fission reactors in a hundred years’ time is unpleasant from the standpoint of harmful radioactivity and the difficulties associated with the disposal of radioactive wastes. At such a level of energy production, approximately \(10^{13}\) curies of long-lived radioactive solid and gaseous fission products would be produced per year, the safe storage of which would have to be ensured. The influence of this circumstance will, of course, not be fully felt by our generation, so that extrapolations into the future will not affect present-day efforts to create power installations based on fission reactions.

The danger associated with the possibility of an explosion, and the necessity of disposing of radioactive wastes, may in principle also arise in the construction of a fusion reactor. Here, however, there is apparently no appreciable danger that an explosion of a fusion reactor could occur; and the properties of fusion reactions are such that a noticeable amount of radioactive contamination will probably not be produced.

Looking ahead, we shall point out that the basic problem arising in the attempt to create a controlled fusion reaction consists in heating a suitable nuclear fuel to a kinetic temperature of \(100\,000\,000^\circ\mathrm{C}\) or higher and in the controlled confinement of the reacting substance at such temperatures for a sufficiently long time. During this time the fuel nuclei must undergo fusion reactions accompanied by the release of energy, despite the presence of losses from the reaction region. The corresponding energy flux can be used in the form of useful power. The production of fusion energy differs greatly from the production of fission energy. When fissile materials are used in a self-sustaining process, there is no need for initial heating or thermal insulation, whereas heating and confinement of the fuel are the initial problems of a controlled fusion reaction. At present it is not difficult to estimate more or less reasonably the cost of an operating fission power plant; however, for a fusion reactor no estimates can be made with comparable accuracy. Nevertheless, apparently there are no grounds—especially in view of the increasing cost of all other kinds of fuel—for believing that fusion energy cannot compete with energy from other sources that are already sufficiently developed at the present time.

In considering possible methods of obtaining energy from fusion reactions, it may reasonably be assumed that a significant part of the fusion energy can be converted directly into electrical energy. This possibility may lead to a sharp reduction in the cost of energy and to a simplification of the design of electric power installations through the elimination of the thermal cycle.

In summary, one may say that the continuous growth of industry requires the practical application of the fusion reaction within the coming decades. The primary fuel for fusion energy—deuterium—is inexhaustible; its use, even on large scales, will not lead to the problem of contamination by radioactive waste. The problem of radioactive contamination, requiring the overcoming of enormous scientific and technical difficulties, disappears with the creation of a practically usable controlled reactor for the fusion of atomic nuclei.

III. WHAT IS A CONTROLLED FUSION REACTOR?

At present, the question “what is a controlled fusion reactor?” is best answered by the following phrase: “It is a device in which the corresponding isotopes of light elements can undergo nuclear fusion.” As a result of the operation of the reactor, controlled production of useful energy is carried out in an amount exceeding that consumed in the operation of the installation. In discussing possible forms of the design of a future reactor, various ingenious proposals have been put forward. The study of such projects and of the corresponding physical phenomena, as well as the creation of the apparatus necessary for carrying out experiments, is the goal of the Sherwood project. Some physical problems are common to any attempts at research in the field that is the subject of this article. In this paragraph some basic aspects will be discussed which relate primarily to the nuclear reactions themselves.

Nuclear fusion reactions

Among the nuclear reactions that appear promising for use in a controlled fusion reactor are those in which various isotopes of hydrogen, helium, and lithium participate. Some of these reactions are listed in Table I.

Table I

Fusion reactions

\[ \mathrm{D}+\mathrm{D}\to \mathrm{He}^{3}+n+3.25\ \mathrm{MeV}, \tag{1} \]

\[ \mathrm{D}+\mathrm{D}\to \mathrm{T}+p+4\ \mathrm{MeV}, \tag{2} \]

\[ \mathrm{T}+\mathrm{D}\to \mathrm{He}^{4}+n+17.6\ \mathrm{MeV}, \tag{3} \]

\[ \mathrm{He}^{3}+\mathrm{D}\to \mathrm{He}^{4}+p+18.3\ \mathrm{MeV}, \tag{4} \]

\[ \mathrm{Li}^{6}+\mathrm{D}\to 2\mathrm{He}^{4}+22.4\ \mathrm{MeV}, \tag{5} \]

\[ \mathrm{Li}^{7}+p\to 2\mathrm{He}^{4}+17.3\ \mathrm{MeV}. \tag{6} \]

Reactions (1) and (2) occur with approximately equal probability. Reactions (3) and (4) are of interest because of their large energy yield, and also because they occur on the products of reactions (1) and (2).

The cross sections of reactions (1), (2), (3), and (4) have been measured down to low energies. Fig. 1 summarizes recently published data on these reactions. It should be noted that the Coulomb barrier has a strong influence on the magnitude of the cross sections at low energies. This effect becomes, of course, more pronounced for reactions (5) and (6), which makes their practical use more difficult.

Consideration of the Energy Balance

Since the cross sections attain values close to their maxima, in particular for reaction (3) in the energy range between 10 and 100 kev, whereas the energy yield of the reaction is several Mev, there is a real possibility of obtaining a net energy gain. One may say that even if only a few percent of the group of fast deuterons and tritons undergo fusion, the reaction energy will exceed the total kinetic energy of all the accelerated particles.

Fig. 1. Dependence of the cross sections for fusion reactions on the relative energy of the particles.

As is well known, the energy balance becomes unfavorable when a target, say a deuterium target, is bombarded by a beam of deuterons. In this case the greater part of the kinetic energy of the incident beam is uselessly expended on ionization, radiation, and the transfer of energy to the atomic electrons of the target, and as a result the yield is only \(10^{-5}\), or about \(10^{-4}\), of that required to obtain a favorable energy balance.

A possible solution of the problem that arises is known from astrophysics: the entire fuel charge must be heated to a kinetic temperature sufficient for the corresponding number of reactions to occur in mutual collisions of free nuclei. These kinetic temperatures correspond to mean particle energies measured in tens or, perhaps, even hundreds of kev*). At such temperatures all matter is completely ionized, and the ionization losses that predominate in the example of a deuteron beam incident on a target become negligible. The possibility of obtaining a useful energy balance under the indicated conditions is connected with the outcome of the competition between the nuclear energy produced in the volume of the reacting “fuel” and the energy lost through the outer surface. Here astrophysicists will say that, since the ratio of volume to surface increases with radius, a sufficiently large region in which the reaction proceeds can always ensure a positive energy balance, as occurs in the Sun and in the stars. Under terrestrial conditions this method of solving the problem is not especially attractive, and therefore other means must be sought. The search should lead to the selection of the most favorable fusion reactions, in order to create effective conditions that ensure a reduction of surface energy losses not only from the standpoint of achieving a favorable energy balance, but also for preventing evaporation of the material of the wall surrounding the region in which the reaction proceeds.

*) The mean energy of a particle in a Maxwellian distribution is equal to \(\frac{3}{2} kT\). In considering fusion reactions it is convenient to express the kinetic temperature in kiloelectronvolts. A temperature of \(1\) kev corresponds to \(1.16 \cdot 10^7\,^\circ\mathrm{K}\). The mean particle energy at this temperature is equal to \(\frac{3}{2}\) kev.

The material of the walls is heated by a gas at ultrahigh kinetic temperatures in a state of complete ionization, i.e., by a gas consisting of equal numbers of free electrons and ions. In this article, fuel in such a state is called plasma, in accordance with Langmuir’s definition. The study of the dynamics of fully ionized gases, or plasmas, is a new and important field of activity for physicists; it is also important in connection with problems of astrophysics. This is not surprising, since practically all matter in the world, with the exception of a negligible fraction of it, exists in the plasma state. Research on controlled fusion reactions represents an attempt to apply the dynamics of astrophysical plasma on terrestrial scales.

Theory of Binary Reactions

Fusion reactions are binary, since they occur upon the collision of two particles. In the case of fusion reactions taking place in a heated plasma, collision processes between all particles of the plasma occur continuously. It is therefore useful to consider the theory of such processes.

Suppose that mutual collisions occur between ions of the reacting substance, which is at kinetic temperature \(T\). The probability that two ions will react with one another during a close passage relative to each other is determined by their interaction cross section \(\sigma\), which is a function of their relative velocity \(v_{12}\). The probability that, per unit time, an ion of type 1 will react with some other ion of type 2 is given by the product of the reaction cross section \(\sigma\), the relative velocity \(v_{12}\), and the number of particles \(n_2\) of type 2 per unit volume. Since there is a certain distribution of relative velocities, rather than a fixed value of the relative velocity, the product \(\sigma v_{12}\) must be averaged over the distribution corresponding to the kinetic temperature of the plasma. Thus, the reaction probability per particle of type 1 is

\[ R_1 = n_2 \langle \sigma v_{12} \rangle . \tag{1} \]

The rate of occurrence of reactions per unit volume is found by multiplying \(R_1\) by the density of ions of type 1

\[ R_{12} = n_1 R_1 = n_1 n_2 \langle \sigma v_{12} \rangle \quad \text{reactions}/\text{cm}^3/\text{sec}. \tag{2} \]

If ions of types 1 and 2 are identical (as in the DD reaction), then expression (2) takes the form

\[ R_{11} = \frac{1}{2} n^2 \langle \sigma v \rangle \quad \text{reactions}/\text{cm}^3/\text{sec}. \tag{3} \]

The power density of the reaction is expressed as the product of the number of reactions occurring per unit time and the energy of each reaction \(W_{12}\) or \(W\)

\[ p = n_1 n_2 \langle \sigma v_{12} \rangle W_{12} \tag{4} \]

or

\[ p = \frac{1}{2} n^2 \langle \sigma v \rangle W . \tag{5} \]

If the velocity distribution of the plasma particles is known, then the calculation of \(\langle \sigma v \rangle\) is carried out directly, although it is rather cumbersome. If the Maxwellian distribution of ion velocities is assumed, then a convenient analytical expression is obtained, giving \(\langle \sigma v \rangle\) at small and medium energies.

In Ref. \(^{2}\) it is shown that the experimental values of the reaction cross section at low energies can be accurately brought into agreement with Gamow’s formula for

passing through the barrier. For the DD reaction this expression has the following form (energy in keV, cross section in barns, \(1\ \text{barn}=10^{-24}\ \text{cm}^2\)):

\[ \sigma_{DD}=\frac{288}{W}\exp\left[-45.8W^{-\frac12}\right]. \tag{6} \]

The quantity \(\langle\sigma v\rangle\) can be obtained by integrating the product of expression (6) by the particle-velocity distribution function over all relative velocities. For a Maxwellian distribution with temperature \(T\) keV, the expression obtained has the form*):

\[ \langle\sigma v_{DD}\rangle=260\cdot10^{-16}T^{-\frac23} \exp\left[-18.76T^{-\frac12}\right], \qquad T<50\ \text{keV}. \tag{7} \]

An analogous, but somewhat more complicated, expression can be obtained for the DT reaction. Graphs of the functions \(\langle\sigma v_{DD}\rangle\) and \(\langle\sigma v_{DT}\rangle\) for a Maxwellian particle distribution in the energy interval from 1 to 100 keV are shown in Fig. 2.

It is interesting to note that at very low temperatures, say below 5 keV, the reactions proceed mainly on particles whose energy is several times greater than the average. This fact is a consequence of the extremely small value of the effective cross section at low energies.

A rough estimate of the energy of the particles making the principal contribution in the DD reaction can be obtained simply by determining the maximum of the function representing the product of the exponential factor in the expression for the cross section and the expression for the Maxwellian velocity distribution:

\[ \exp\left[-45.8W^{-\frac12}-WT^{-1}\right]. \tag{8} \]

By differentiating (8), one can find the quantity \(W_m\)—the energy of the ions making the maximum contribution to the course of the reaction. The quantity \(T\) in keV, expressed as the ratio of \(W_m\) to the temperature, has the form**):

\[ \frac{W_m}{T}=\frac{8.1}{T^{1/3}}. \tag{9} \]

Fig. 2. Values of \(\langle\sigma v_{DD}\rangle\) and \(\langle\sigma v_{DT}\rangle\) for the case of a Maxwellian particle distribution.

Thus, for \(T=1\ \text{keV}\), this expression shows that the majority of reactions are produced by particles possessing an energy approximately eight times greater than the mean energy corresponding to this temperature.

Power released per unit volume

Of considerable interest are calculations, for certain possible cases, of the typical reaction power, the mean reaction time, and the mean path length of ions before reaction. Let us consider DD and DT reactions proceeding at

*) See the works of Gamow and Teller (Phys. Rev. 53, 608, 1938). Numerical values were obtained by Leith (private communication).

**) With a more accurate calculation, the coefficient obtained is not 8.1 but 6.2 (see, for example, W. B. Thompson, Pr. Phys. Soc. 70B, 1 (1957)). Translator’s note.

temperature of 100 keV. (This temperature was chosen simply because the cross sections for the DD and DT reactions are well known and their values change little near 100 keV, i.e., in this region the figures given are practically insensitive to temperature. It will be shown later that the maximum working temperature should be chosen an order of magnitude lower.) From Fig. 2 it is seen that the quantity \((\sigma v)_{DD}\) is approximately \(3 \cdot 10^{-17}\ \text{cm}^3/\text{sec}\). The mean energy of the DD reaction, as follows from Table I, is: \((3.25 + 4):2 = 3.6\) MeV, or about \(6 \cdot 10^{-13}\) joules per reaction. Thus the mean total power of the reaction per unit volume will be:

\[ P_t = \frac{1}{2}\, n_D^2 (3 \cdot 10^{-17})(6 \cdot 10^{-13}) = 9 \cdot 10^{-30} n_D^2\ \text{W}/\text{cm}^3 \tag{10} \]

\[ (T = 100\ \text{keV}). \]

It should be noted that this expression is proportional to the square of the deuteron density. In Fig. 3 the quantity \(P_t\) is plotted as a function of the particle density. The horizontal line corresponding to the value \(100\ \text{W}/\text{cm}^3\) gives a typical energy density that can be achieved in powerful fission reactors. The dashed vertical line \((n_D = 2.7 \cdot 10^{19})\) marks the particle density in a gas under normal conditions.

Fig. 3. Dependence of the power density of DD- and DT-reactions on the particle density of deuterium at a temperature of 100 keV.

In the same figure the power released per unit volume of a mixture consisting of 50% D and 50% T at a temperature of 100 keV, calculated from equation (2) using the data of Fig. 2, is represented graphically.

The curves for the DD- and DT-reactions show that at a particle density equal to \(10^{-4}\)—\(10^{-5}\) of the atmospheric density, the power of a fusion reactor turns out to be of the same order as for fission reactors. These densities seem paradoxically small; they are approximately the same as the densities encountered in many laboratory vacuum systems (\(10^{-3}\) mm Hg \(\sim 10^{-6}\) atm). It is clear that an exact parallel between the problems of pressure and heat removal, which limit the working specific power in fission reactors, and the corresponding problems in a fusion reactor cannot be drawn. Nevertheless, the problem of heat transfer appears in any continuously operating fusion reactor. This follows from the fact that prolonged operation at a temperature of 100 keV and at densities reaching atmospheric densities seems impossible. It is not difficult to be convinced of this if one recalls that the power of a large steam electric power station is 500,000 kW. In a fusion reactor with a deuteron density equivalent to atmospheric pressure and at a temperature of 100 keV, this power, as follows from Fig. 3, would have to be concentrated in a volume of only \(0.03\ \text{cm}^3\). In this case the gas-kinetic pressure reaches approximately \(10^7\) atm.

Another limiting case corresponds to operation at too low a fuel density. Many projects for the production of thermonuclear energy, which seem reasonable on qualitative examination, fail after quantitative calculations simply because the necessary

densities. From the curves it is seen that when the density falls, for example, to \(10^{12}\) particles/\(\mathrm{cm}^3\), the power from the reaction is only about \(10^{-5}\) W/\(\mathrm{cm}^3\). This value is too small to be economically advantageous. A density of \(10^{12}\) particles/\(\mathrm{cm}^3\) is typical for plasma generated in ordinary gas discharges.

Mean free path before reaction; distribution of energy among the reaction products

In principle, one can imagine the operation of a fusion reactor at temperatures both greater and less than 100 keV. However, in the following section it will be shown that operation below a certain minimum temperature is impossible.

Fig. 4. Dependence of the mean reaction time on the concentration of deuterium particles, calculated for various temperatures.

Fig. 5. Dependence of the mean free path before reaction as a function of particle concentration.

Starting from the value of \(\langle \sigma v\rangle\), one can find the mean lifetime of the fuel ions. Equation (1) gives the number of collisions per particle: \(R_1 = n\langle \sigma v\rangle\). Thus, the mean lifetime before reaction is

\[ \tau = \frac{1}{R_1} = \frac{1}{n\langle \sigma v\rangle}; \]

the numerical value of this quantity is given in Fig. 4 as a function of the density of deuterium particles \(n_D\) for the DD and DT reactions at 10 keV and 100 keV. The point on the graph corresponds to the pressure at a specific power of 100 W/\(\mathrm{cm}^3\) (the value corresponding to the deuterium reactor). From the graph it is seen that in these typical cases the mean lifetime of a particle before reaction may reach many seconds. Consequently, in order to maintain the energy balance, the mean time for an ion to leave the reactor region must not be too small in comparison with the indicated value.

The magnitude of the necessary reaction time can be estimated more completely from the curve for the mean free path before reaction \(\lambda = 1/n\sigma\). This time as a function of \(n_D\) is given in Fig. 5 for the DD and DT reactions, also at 10 and 100 keV. It should be noted that for a specific power of 100 W/\(\mathrm{cm}^3\)

the mean free path for DD reactions at 100 keV is \(5 \cdot 10^9\) cm, i.e., a distance approximately equal to the circumference of the Earth!

So far nothing has been said about the fate of the energy produced in the fusion reactor. The kinetic energy of these reactions is, of course, distributed among the reaction products, with the lion’s share going to the lighter particles. If the lightest particle is a neutron, then, because of the low density of the substance, it will almost certainly leave the reaction region and carry its kinetic energy outside. Whether charged reaction products will leave the reaction region depends on various circumstances. In any case, the total energy of the charged reaction products is all that remains in the system both to compensate losses and to sustain the reaction. The distribution of energy between charged particles and neutrons for the DD and DT reactions is given below (we neglect the initial kinetic energy).

\[ \begin{aligned} \mathrm{DD} &\to (\mathrm{T} + 1.0\,\mathrm{MeV}) + (\mathrm{p} + 3.0\,\mathrm{MeV}) \\ \mathrm{DD} &\to (\mathrm{He}^3 + 0.8\,\mathrm{MeV}) + (\mathrm{n} + 2.45\,\mathrm{MeV}) \end{aligned} \qquad \left\{ \begin{array}{l} \text{The probabilities of occurrence}\\ \text{of the reactions are approximately equal.} \end{array} \right. \]

\[ \mathrm{DT} \to (\mathrm{He}^4 + 3.6\,\mathrm{MeV}) + (\mathrm{n} + 14.1\,\mathrm{MeV}). \]

Only in the case of the DD reaction, which is the most favorable in this respect, does an average of 66% of the reaction energy pass to the charged reaction products and 34% to neutrons. Thus the internal energy generated during a DD reaction amounts to 0.66 of the value determined by equation (10) and by the DD curve in Fig. 3. In the case of the DT reaction only 20% of the energy is transferred to charged products, while 80% is imparted to neutrons.

To simplify the discussion, here and below, when considering DD reactions, we neglect the possible contribution of secondary fusion reactions caused by the charged products themselves (i.e., T and \(\mathrm{He}^3\)).

IV. COMPETING PROCESSES

Energy losses in a fusion reactor and the intensity of nuclear reactions are the determining processes. Some losses are inherent in any system. Among them one may name the radiation of the electron cloud which, together with the ions of the reacting substance, forms a plasma. If radiative equilibrium is assumed, then the flux of radiation from the region in which the reaction proceeds will be determined by the black-body relation \(I=\sigma T^4\) erg/cm\(^2\) sec. At 10 keV (\(10^8\,^\circ\mathrm{K}\)) the radiation flux will be of the order of \(10^{21}\) W/cm\(^2\)! We may therefore conclude that no controllable device based on nuclear fusion can operate under conditions of radiative equilibrium between the particles and the radiation field. However, radiative equilibrium can be attained only on the condition that the mean free path for self-absorption of radiation is significantly smaller than the dimensions of the system. Fortunately, these conditions are by no means satisfied in the plasma of a fusion reactor; this is precisely why, in describing the energy state of the plasma, we use the term “kinetic temperature” and not simply “temperature.”

The situation is analogous to that which obtains in the rarefied outer layers of the solar corona. Although it is known that the kinetic temperature of the electron gas in the corona is approximately \(10^6\,^\circ\mathrm{K}\), in the actual radiation of the Sun the principal role is played by the deep-lying layers, and the total radiation corresponds to a black-body temperature of only about \(5000^\circ\). In the outer parts of the corona the mean free path of the most important components of the intrinsic radiation is significantly greater than the thickness of the layer, so that radiative equilibrium is not attained.

Effective Radiation Intensity and the Minimum Temperature Required for a Self-Sustaining Fusion Reaction to Occur

For systems whose dimensions are small in comparison with the mean free path of photons before absorption, it may be roughly assumed that the energy of the effective radiation from the medium is, in order of magnitude, equal to the energy of black-body radiation multiplied by the ratio of the dimensions of the system to the mean free path of photons before absorption; it is assumed that the photon energy corresponds to the kinetic temperature. As an example, let us indicate that at \(10^8\,^\circ\mathrm{K}\), for a plasma with density \(10^{15}\) particles/\(\mathrm{cm}^3\), the mean free path before absorption is approximately \(10^{20}\,\mathrm{cm}\). From this it is easy to estimate the order of the decrease in radiation intensity for systems of any given dimensions.

The principal type of plasma radiation is ordinary bremsstrahlung, or x-ray radiation, arising from the deflection of rapidly moving plasma electrons in the Coulomb field of ions. The theory of this radiation was considered by Heitler\(^{3}\) and others. The value of the radiation power per unit volume in the form of x rays for the case of a completely ionized gas can be obtained by averaging over the Maxwellian distribution of electrons (\(T_e\) is the electron temperature in keV):

\[ p_r = 0.54 \cdot 10^{-30} Z^2 n_e^2 T_e^{1/2}\ \mathrm{W}/\mathrm{cm}^3. \tag{11} \]

For a hydrogen plasma \(Z = 1\) and \(n_e = n_i\) (\(n_e\) is the number density of electrons; \(n_i\) is the number density of ions); therefore, for the DD reaction one obtains:

\[ p_r = 0.54 \cdot 10^{-30} n_D^2 T_e^{1/2}\ \mathrm{W}/\mathrm{cm}^3. \tag{12} \]

Comparison of formulas (5) and (12) shows that both quantities \(p\) and \(p_r\) vary as \(n_D^2\). If equality of the electron and ion temperatures is assumed, then it is not difficult to see that, even in the absence of other losses (except those due to the escape of neutrons), a self-sustaining controlled fusion reaction cannot proceed at a temperature below some definite minimum temperature independent of density. Since \(\langle \sigma v\rangle\) increases exponentially with temperature, whereas the radiation energy increases only as \(T^{1/2}\), the generated power reaches the value of the radiation power at a critical temperature, whose value is determined by the relation \(p_r = p_i\) (we take into account only the charged products of the DD reaction, since neutrons freely leave the system). From equations (6), (7), and (12) we find the ratio of the energy released in nuclear fusion to the energy carried away by radiation:

\[ \frac{p_i}{p_r} = 1.92 \cdot 10^4 (T)^{-7/6}\exp[-18.76 T^{-1/3}]. \tag{13} \]

Taking \(p_i/p_r = 1\), we obtain \(T_c\) equal to \(35\ \mathrm{keV}\). The value of \(T_c\) for DT is considerably smaller and is approximately \(4\ \mathrm{keV}\).

Since \(p_r\) is proportional to \(Z^2\), the presence even of a small quantity of ionized elements with large \(Z\) can considerably increase energy losses. It follows from this that high fuel purity is a most important requirement imposed on the plasma in a fusion reactor. For this and other reasons, special attention must be paid to consideration of the role of the material walls of the reactor in plasma contamination during bombardment of the walls by fast particles and heating, although the influence of the walls in other respects is insignificant.

The necessity of confinement; possible methods of operation

The radiation of plasma electrons thus determines the minimum operating temperature of a fusion reactor, but it imposes no specific restrictions on the size of the region in which the reaction takes place, or on the value of the particle number density in the plasma (so long as radiation equilibrium has not been reached), since the specific powers of radiation and of the fusion reaction depend on the particle density in the same way. As was indicated, in a continuously operating reactor there will probably arise a limitation on the specific power, which will lead to a limitation on the particle number density.

It will be shown below that in a plasma at thermonuclear temperatures the mean free path for ordinary collisions between particles is very large. Until special measures are taken, collisions between particles are the only cause preventing particles from leaving the region in which the reaction proceeds. Thus, at a density of the order of \(10^{15}\) particles/\(\text{cm}^3\), there are a priori no causes preventing the instantaneous escape of a fuel ion from the reactor.

Two principal directions for overcoming the difficulties may be noted. The first possibility consists in abandoning the conception of a continuously reacting plasma and considering a pulsed process. In this case sufficiently high densities are necessary, at which, for at least a short interval of time, the rate of diffusion due to frequent collisions becomes low enough to prevent rapid escape of particles from the reaction region. Otherwise, the intense escape of fast particles would lead to cessation of the reaction. The logical limit of this process is the hydrogen bomb, which in the ordinary sense of the word is not fully controllable. The second path consists in introducing, between the material wall of the reactor and the plasma, some force field capable of exerting pressure so as to exclude direct contact between the particles participating in the reaction and the low-temperature region. It is clear that even at reduced density the high temperature of the fuel leads to pressures of considerable magnitude. For example, at a kinetic temperature of 100 keV and a total density of \(10^{15}\) particles/\(\text{cm}^3\), the gas-kinetic pressure reaches approximately 1000 atm. The gravitational field is evidently too weak, except in cases belonging to stellar scales. The remaining possibility is to use an electromagnetic field capable of transmitting momentum and, consequently, exerting a force action. Since the fuel in a fusion reactor is practically completely ionized, the charged particles of which it consists are capable of interacting directly with this field. The range of possibilities here is very large, and their discussion lies beyond the scope of this article; however, a brief mention of one example will be made below.

In summary, one may say that a controlled fusion reaction must proceed far from radiative equilibrium under such conditions that radiative losses are limited only by ordinary bremsstrahlung radiation. At a sufficiently high operating temperature the power released in the reaction will exceed the radiation power, making it possible to realize a self-sustaining fusion reaction, provided, of course, that other significant losses are absent. At the same time, to overcome excessive losses of particles (and, consequently, of energy) caused by direct escape from the reaction region, it is necessary either to choose an operating regime of the generator with a high particle density (and hence a very short-lived one), or to seek means of confining the particles for a sufficiently long time by means of an electromagnetic field of one kind or another.

V. EXAMPLE OF A POSSIBLE METHOD OF MAGNETIC CONFINEMENT—THE PINCH EFFECT

We encounter a classical example of an electromagnetic field capable of confining a group of charged particles when considering the so-called “pinch effect”—the self-contraction of a group of charged particles moving in such a way that they create a current in one direction. The pinch effect is, in essence, one of the cases of mutual attraction of parallel currents. The theory of the pinch effect was first advanced by Bennett^4 and subsequently developed by Tonks and Allis^5,6. Some experimental work on this question has already been reported^7,8, and at the present time there are references to a more direct connection between the phenomena of electromagnetic compression and research on controlled thermonuclear reactions^9,10.

The theoretical work of Bennett and others showed that, also in the ordinary pinch effect, at large currents the conducting region is concentrated near the axis of the discharge chamber, forming a plasma cord. The configuration of the magnetic field near the plasma cord is shown in Fig. 6. The fact that the current is compressed is direct proof of the confinement of the plasma in the radial direction under the action of the magnetic field.

To illustrate the principle of magnetic confinement of this type, let us consider a simplified picture of the pinch and derive from it the conditions for equality between the kinetic pressure of the plasma and the compression effect of the magnetic field. We shall assume, as shown in Fig. 6, that the compressing current fills a thin cylindrical layer or shell of outer radius \(a\) and thickness \(\varepsilon\). The plasma filling the region inside the layer will be characterized by a uniform density of singly charged ions \(n_i\), electrons \(n_e\), and kinetic temperature \(T\). We shall take the density \(n_i\) to be equal to \(n_e\) (the necessity of this will be shown below). Since the plasma is a gas, its kinetic pressure can be represented in the form \(P = (n_i + n_e) kT\). Outside the layer the density, and consequently also the kinetic pressure, is equal to zero. The pressure difference must be balanced by the gradient of the magnetic field.

Fig. 6. Schematic representation of the pinch effect in plasma.

In a macroscopic consideration, the balancing force is the force \(\mathbf{j} \times \mathbf{H}\), acting on each unit volume in which a discharge current of density \(j\) flows. In this case the force, of course, acts only within the limits of the thin layer carrying the current. Inside the conducting shell itself the magnetic-field strength falls from the value \(\dfrac{2I_0}{a}\) outside to zero at the inner surface. At the same time it is assumed that the particle density increases from zero outside to the value corresponding to the density at the center on the inner surface. If we assume that the current is distributed uniformly over the layer thickness \(\varepsilon\), then at any point within the layer the field will have the value

\[ H = \frac{2I}{r}, \tag{14} \]

where \(I\) is the total current inside the radius \(r\). Thus, if the shell thickness

if there is \(\varepsilon\), then

\[ I(r)=I_0\left(\frac{r+\varepsilon-a}{\varepsilon}\right), \qquad (a-\varepsilon)<r<a. \]

Let \(r=a-x,\quad 0<x<\varepsilon\); then

\[ I(x)=I_0\left(1-\frac{x}{\varepsilon}\right), \tag{15} \]

\[ H(x)=\frac{2I_0}{a-x}\left(1-\frac{x}{\varepsilon}\right)\simeq \frac{2I_0}{a}\left(1-\frac{x}{\varepsilon}\right), \quad \text{since } \varepsilon \ll a. \tag{16} \]

The current density \(\mathbf{j}\) is equal to the total current divided by the cross-sectional area of the current-conducting shell,

\[ j=\frac{I_0}{2\pi a\varepsilon}, \qquad \varepsilon \ll a. \tag{17} \]

The total force acting on a unit area of the inner surface of the shell is obtained by integrating the product \(\mathbf{j}\times\mathbf{H}\) over the area of the transverse section of the shell. Here \(\mathbf{j}\) is perpendicular to \(\mathbf{H}\), so that

\[ \int_0^\varepsilon (jH)\,dx = \left(\frac{2I_0}{a}\right) \left(\frac{I_0}{2\pi a\varepsilon}\right) \int_0^\varepsilon \left(1-\frac{x}{\varepsilon}\right)\,dx = \frac{I_0^2}{2\pi a^2}. \tag{18} \]

This force must balance the kinetic pressure of the plasma,

\[ P=(n_i+n_e)kT, \]

\[ \frac{I_0^2}{2\pi a^2}=P=(n_i+n_e)kT. \tag{19} \]

In other words,

\[ I_0^2=2NkT, \tag{20} \]

where \(N\) is the total number of plasma particles per centimeter of length of a column of radius \(a\). This relation is useful in determining the magnitude of the current required for magnetic confinement. Note that the expression for \(I_0\) does not depend on the radius of the plasma column.

Magnetic Pressure

Since \(H_0=2I_0/a\), formula (19) can be written in the form

\[ \left[\frac{H_0^2}{8\pi}\right]_{\text{outside}} = P_{\text{inside}}. \tag{21} \]

In this equation the role of the magnetic field is clearly visible. The magnetic field in the outer region confines the kinetic pressure in the inner region by means of magnetic pressure. The magnitude of the magnetic pressure is equal to

\[ \frac{H_0^2}{8\pi} \]

\[ \left(\frac{H_0^2}{8\pi} \text{ is also the energy density of the magnetic field}\right). \]

An analogous relation for any point inside the current-conducting layer has the form:

\[ \nabla\left(\frac{H^2}{8\pi}+P\right)=0, \quad \text{that is} \quad \left(\frac{H^2}{8\pi}\right)+P=\text{const}=\frac{H_0^2}{8\pi}. \tag{22} \]

This relation is valid if the curvature of the magnetic-field lines is negligibly small. It is often encountered in the plasma-physics literature \(^{11,12}\), and is also realized in the example considered here. It is important to note that the resulting magnetic pressure is a direct consequence of the interaction of the plas-

of currents and the magnetic field. Magnetic pressure acts only where the plasma currents tend to reduce the external field, i.e., where the medium possesses a property that may be called diamagnetism. In the example under consideration, the magnetic field produced by the currents is pushed out of the conducting region; such a phenomenon can occur only in ideal conductors, but an analogous picture can be observed for a short time also in a plasma having finite electrical conductivity. To illustrate the magnitude of the magnetic-field intensity required for magnetic confinement, let us return to the previous example with a DD reaction proceeding at a temperature of \(10\ \mathrm{keV}\) and a deuteron number density \(n_i = 3 \cdot 10^{15}\ \mathrm{cm}^{-3}\).

Suppose that the electron and ion temperatures are approximately equal. As in the previous example, the plasma pressure is

\[ P=(n_i+n_e)kT=6\cdot 10^{15}\cdot (1.6\cdot 10^{-7})=10^9\ \mathrm{dyn}/\mathrm{cm}^2 \]

or about \(1000\ \mathrm{atm}\), whence

\[ \frac{H_0^2}{8\pi}=10^9,\quad \text{i.e. } H_0=1.6\cdot 10^5\ \mathrm{gauss}. \]

If the radius of the plasma cylinder is, for example, \(10\ \mathrm{cm}\), then equation (19) shows that the required current is about \(8\cdot 10^5\) CGSM units, or \(8\cdot 10^6\ \mathrm{A}\).

Up to now there has been no need to discuss the specific microscopic mechanism by which the magnetic field confines the plasma. It is clear that in the example given above one may qualitatively suppose that ions and electrons, in trying to leave the shell in the radial direction, will be reflected from the region of strong magnetic field. The infinitely small currents that arise when charged particles are reflected from the boundary must add up to a self-consistent system of macroscopic currents creating the given configuration of the magnetic field. This conception may serve as the basis for the very important problem of the stability of the confined plasma.

Instability of the plasma cord; longitudinal confinement

Although it was shown above that pressure equilibrium can occur for a plasma cord confined by a magnetic field, the possibility of stable equilibrium in this case has not been established. In fact, stability is absent. In the now classic work of Kruskal and Schwarzschild\({}^{13}\) it was shown that the “self-sustaining” current flowing in the plasma is unstable with respect to bending perturbations. If a compressed cord undergoes an infinitely small local lateral displacement, and if the length of the displacement is greater than the diameter of the cord column, then the perturbation will grow exponentially with time until the column is destroyed. This state is to a certain extent analogous to the instability of a rotating long thin rod fixed only at its ends. The presence of the instability is physically due to the fact that the magnetic field (and, consequently, the magnetic pressure) is greater on the concave side of the bend of the current-carrying conductor than on its convex side. This picture is clearly visible in Fig. 7; the magnetic field lines become dense on the concave side of the cord and sparse near its convex side. The instability predicted by Kruskal and Schwarzschild is a special case of a more general type of hydromagnetic instability. The property of an instability of this type is that the time for the growth of unstable perturbations by a factor \(e\) is approximately equal to the time it takes an ion to traverse a distance equal to the “wavelength” of the perturbation. Therefore

Fig. 7. Instability of the plasma cord.

perturbations possessing “short wavelengths” grow more rapidly. The growth time predicted at thermonuclear temperatures is disastrously short (100 keV deuterons have a speed of about 300 cm/μsec). To transform the simple pinch effect into a device for producing thermonuclear energy, it is necessary to find ways of eliminating this fundamental instability. One possible way consists in creating the plasma cord in a time shorter than the growth time of the instability. If a sufficiently large instantaneous power of the reaction is achieved, then in this case a net energy gain may be obtained.

To complete the description of the picture of the elementary (quasi-stationary) pinch effect as a possible device for obtaining thermonuclear energy, it should be noted that the field configuration in this example (Fig. 6) does not provide effective methods of longitudinal confinement of the plasma. In the experiments of Cousins and Ware^9, in some of Tuck’s work at Los Alamos and in work by Taft’s group at the University of Southern California^10, a toroidal induction discharge was used, which in principle indicates possible ways of reducing losses to the electrodes. The plasma cord in this case is closed.

VI. PHENOMENA IN A FULLY IONIZED GAS

In recent years, problems in the physics of fully ionized gases have begun to be studied intensively. Despite the obvious importance of this subject for controlled fusion reactions, a detailed treatment of it lies outside the scope of this article; the reader is therefore referred to the study of the corresponding literature. However, some simple and at the same time fundamental features of this “fourth state of matter” will be considered here.

The behavior of a plasma is determined in its main features by three processes: 1) long-range, collective electrostatic effects (space-charge effects) generated by the plasma itself; 2) short-range effects due to collisions between individual plasma particles; 3) interaction between individual particles and externally applied electromagnetic fields. Strictly speaking, processes 1) and 3) are interconnected, since the plasma can completely change the character of external fields through collective effects. However, separate consideration of the interaction of the two types may be a reasonable approximation to reality.

Charge equality

Process 1) acts most effectively, establishing a state of charge equilibrium of the plasma. One may say that any tendency of space charges to deviate from neutrality causes the appearance of a large electrostatic field opposing this deviation. To illustrate this phenomenon, it is enough to calculate the electric field that exists around a sphere filled with plasma whose electron and ion densities are equal to \(10^{15}\) particles/cm\(^3\), after the plasma electrons have in some manner suddenly disappeared. If the radius of the sphere is 1 cm, then Gauss’s theorem gives for the field the value

\[ E=\frac{Q}{r^2}=\frac{4\pi}{3}\cdot 10^{15}\cdot 4.8\cdot 10^{-10}=2\cdot 10^6 \ \text{abs. el. st. units} \]

or 600 million V/cm.

Already for a relative deviation from charge equality equal to \(10^{-5}\),

the field strength will be 600 V/cm near a plasma sphere of radius 1 cm, or 600,000 V/cm near a sphere of radius 1 m. Effective equality of the electron and ion charge densities must be taken as a necessary condition in any operating fusion reactor.

Electrical Conductivity of Plasma

Another obvious property possessed by a cloud of freely moving negative and positive charges is its electrical conductivity. As in any conducting medium, the numerical value of the conductivity is inversely proportional to the number of collisions between the current carriers and other, neighboring particles. In a hot plasma, in contrast to ordinary conductors, as the temperature increases the number of collisions becomes smaller and, consequently, the electrical resistance of the plasma decreases with increasing temperature. The theory of the resistance of a fully ionized plasma shows that it varies inversely as the temperature to the three-halves power. For a hydrogen plasma the numerical value of the specific resistance is approximately determined by the expression ($T$ in keV)

\[ \rho_0=\frac{3\cdot 10^{-6}}{T^{3/2}}\ \text{ohm}\cdot\text{cm}. \tag{23} \]

Thus, at $T = 100$ keV the theoretical value of the resistance is approximately $3\cdot 10^{-9}$ ohm cm—less than 1% of the resistance of copper at room temperature. This expression is strictly applicable only in those cases in which the effective dimensions of the plasma are considerably greater than the mean free path of the electrons. At low densities and high temperatures this condition may not be satisfied. Nevertheless, for other reasons the qualitative result of low resistances remains valid in many practically interesting cases. It follows from this that in many cases a hot plasma behaves in some respects like a superconductor; in other words, it excludes or weakens a magnetic field of external origin (as in the case of the “pinch”) and can twist accumulated charges or the lines of force of an electric field. These qualitative results are not especially sensitive to the density of the plasma.

In the presence of a magnetic field, the theoretical value of the plasma resistance in the stationary state changes somewhat in the direction perpendicular to the direction of the magnetic field lines, and practically does not change along the magnetic field lines. Short-time effects, however, may be substantially different depending on the time scale.

Plasma Oscillations

Another property of a fully ionized gas, which manifests itself owing to the presence of space charge and the high effective conductivity of the plasma, is the occurrence of plasma electromagnetic oscillations. The simplest type of these oscillations, which do not depend on the presence of a magnetic field, are oscillations first predicted and observed by Tonks and Langmuir. These (longitudinal) plasma oscillations are connected with the fact that a small instantaneous displacement of charges of one sign with respect to charges of the other sign leads to the appearance of an electric field directed so as to counteract the displacement. The inertial forces arising during the displacement of charges provide a mechanism for periodic oscillations about the point of zero displacement of the average charge. The frequency of these

...oscillations is determined by the mass of the displaced particles and by the coefficient of elasticity of the electric field caused by the displacement. Plasma oscillations of this kind, by their nature, may be either electronic or ionic, i.e., their frequencies are determined by the masses of the electrons or ions. Ionic oscillations have a lower frequency than electronic ones, and they are not so easily identified. In recent years a considerable number of theoretical works on plasma oscillations have been carried out, many of them by Soviet researchers[^14]. This question is treated in detail in the literature[^11][^12][^15][^16].

It is important to note that the frequency of electronic plasma oscillations is measured by the characteristic time of reaction of the electron gas of the plasma to changes in the electric field, arising either as a result of charge separation or in the presence of an external alternating field. The plasma frequency is very often encountered in the description of many phenomena occurring in plasma. Neglecting corrections for the effect of thermal velocities, we find the electronic plasma frequency:

\[ \omega_p=\sqrt{4\pi n_e\frac{e^2}{m}}\ \text{radian/sec} \tag{24} \]

or

\[ f_p=9\cdot 10^3 n_e^{1/2}\ \text{cps}; \]

thus, for \(n_e=10^{15}\) we obtain:

\[ f_e=2.7\cdot 10^{11}\ \text{cps}. \]

If pulsed electric fields are applied for a time interval much greater than \(1/\omega_p\), then the previously considered effects of electrical conductivity play the principal role. If, however, they are applied for a time interval less than \(1/\omega_p\), then the inertial effects of the plasma predominate and the influence of conductivity becomes small. Thus, for example, ordinary light passes freely through plasma, whereas radio waves may be strongly absorbed or reflected by it. The effective dielectric constant of a plasma (in the case where collisions of particles and the action of the magnetic field may be neglected) can be written in the form

\[ K=1-\left(\frac{\omega_p}{\omega}\right)^2 =1-\frac{4\pi n_e e^2}{m\omega^2}. \tag{25} \]

This expression is familiar to all who have studied the classical theory of dispersion. Here \(\omega\) is the angular frequency of electromagnetic waves propagating in the plasma. The refractive index is determined by the relation

\[ \gamma=\sqrt{K}=\left[1-\left(\frac{\omega_p}{\omega}\right)^2\right]^{1/2}. \tag{26} \]

The corresponding phase velocity turns out to be greater than the speed of light. Propagation of an electromagnetic disturbance without attenuation can occur only in the case \(\omega>\omega_p\); the situation is similar to that which occurs in waveguides.

Hydromagnetic waves

If \(\omega\) and \(\omega_p\) are greater than \(\omega_c\), the cyclotron frequency of the electrons, then the presence of the magnetic field has little effect on the propagation of electromagnetic waves in the plasma. If the electric vector of the electromagnetic wave is parallel to \(H\), then the conditions for propagation of the wave, as follows from equation (26), depend only weakly on the frequency. In the case where \(\omega\) and \(\omega_p\) are comparable with or less than \(\omega_c\), the situation becomes more complicated and complex dispersion effects arise.

In the presence of a magnetic field, the appearance of new types of waves, of fundamental importance, becomes possible. These are the so-called magnetohydrodynamic or hydromagnetic waves, described by Alfvén^12. In the simplest case these waves are transverse and propagate along magnetic lines of force. They are similar to the waves that arise in transverse oscillations of material elastic strings (the lines of force of the field). The mass of the “string,” of course, is determined by the mass of the ions (or plasma electrons) and is related to a very important qualitative concept connected with the motion of plasma particles in the presence of a magnetic field. This concept consists in the fact that, for motions that occur sufficiently rapidly in comparison with the frequency of collisions between particles (but not too rapidly in comparison with the period of gyration of the particles), charged particles behave as though they were bound to magnetic lines of force (more precisely, to certain magnetic surfaces).*)

As has already been said, magnetic lines of force behave like mutually repelling elastic strings. Transverse oscillations, or torsional waves, can propagate along the lines of force with a velocity that is a function of the mass per unit length (i.e., the ion density) and of the force constant determined by the magnitude of the magnetic field. The propagation velocity of these waves is determined by the expression obtained by Alfvén:

\[ v_A=\left(\frac{H^2}{4\pi \rho}\right)^{1/2}. \tag{27} \]

Here \(\rho\) is equal to \((n_i m_i+n_e m_e)\simeq n_i m_i\). Squaring both sides of equality (27) and dividing by the square of the velocity of light, we obtain:

\[ \left(\frac{v_A}{c}\right)^2 = \frac{2\left(\dfrac{H^2}{8\pi}\right)}{\rho c^2}. \tag{28} \]

The right-hand side of this expression is the ratio of twice the magnetic-energy density to the rest-mass energy of the particles contained in \(1\ \mathrm{cm}^3\). In plasmas of practical interest for controlled nuclear reactions, this quantity is much less than unity and, consequently, the waves propagate with a velocity considerably smaller than the velocity of light.**)

One of the reasons why the description of hydromagnetic oscillations has been given here is that they illustrate an important property of a “hot” plasma in a magnetic field, namely the “freezing-in” in the plasma of magnetic lines of force (or, at any rate, of magnetic surfaces).

Unstable types of some such oscillations may also be important in considering the important problem of the stability of a plasma confined by a magnetic field. For example, one can see that the left-hand side of equation (28) is nothing other than the square of the refractive index of the plasma for Alfvén waves, so that the quantity inverse to \((v_A/c)^2\) is, in itself, analogous to the dielectric constant of a medium for certain types of hydromagnetic disturbances. This dielectric constant of the plasma, in cases of practical interest, may be very large.

*) W. A. Newcomb created an exhaustive theory of the “motion of magnetic lines,” which is presented in the technical reports of Princeton University, No. 1 (1955).

**) F. de Hoffmann and E. Teller, Phys. Rev. 80, 692 (1950), showed that at extremely high values of the magnetic field or low density, Alfvén waves transform into light waves.

Collision Processes

It has already been shown that the plasma in a thermonuclear reactor cannot exist in equilibrium with the field of its own radiation; moreover, it obviously must not be in thermal equilibrium with the material walls surrounding it. Thus, a confined plasma cannot even approach thermodynamic equilibrium in the usual sense. The existence of a plasma confined by a magnetic field must be regarded as an example of a nonequilibrium state, since the confining electromagnetic forces can manifest themselves only if electric currents flow through the plasma. Since the plasma is not an ideal conductor, the confining currents in an isolated plasma system will decrease with time. They can be maintained in a stationary state only by the continuous supply of kinetic or electromagnetic energy. As in the case of an ordinary conductor, collisions in a plasma provide the mechanism of energy dissipation and determine the rate at which equilibrium is established. However, in contrast to ordinary conductors, the number of collisions in a plasma decreases as the temperature increases, and consequently confinement by a magnetic field is facilitated.

In completely ionized gases the processes of scattering or collision are almost entirely determined by the Coulomb fields of the bare nuclei and free electrons of the plasma. The infinite range of the Coulomb forces leads to the fact that the Coulomb interaction of particles is usually divided, as was already pointed out, into two basic types. The combined action of all particles located at distances greater than a certain distance of “cutoff,” or shielding, merges into a collective effect, similar to the effect of a space charge, such that only the action of a large charge is substantial. On the other hand, within the shielding distance it is physically acceptable to regard collisions as discrete independent events, even if within the unshielded region there are many particles. The shielding length (“Debye length”) is determined by the smallest distance at which the plasma electrons, by their collective motions, can still shield the Coulomb field of a charged particle from another moving near it. Thus, the Debye length is directly connected with the minimum time of reaction of the plasma to a local electric disturbance. This time is determined by the frequency of plasma oscillations, i.e. it is approximately equal to \(1/\omega_p = (m/4\pi n_e e^2)^{1/2}\). The distance \(\lambda\) that a plasma electron, possessing the mean energy, can traverse during this time in attempting to shield the field is approximately equal to the mean electron velocity multiplied by the indicated time, i.e.

\[ \lambda = \left(\frac{\frac{3}{2} kT_e}{m}\right)^{1/2} \left(\frac{m}{4\pi n_e e^2}\right)^{1/2} = \left(\frac{kT_e}{\frac{8}{3}\pi n_e e^2}\right)^{1/2}; \tag{29} \]

\(\lambda\), therefore, is a roughly estimated shielding distance. Usually one uses the Debye theoretical value of this quantity (introduced in the theory of electrolytes), which is somewhat smaller than the value given above:

\[ \lambda_D = \left(\frac{kT_e}{4\pi n_e e^2}\right)^{1/2}. \tag{30} \]

In the calculation of all physically interesting cases this quantity enters under the logarithm sign, and therefore its exact numerical value is immaterial. In Fig. 8, \(\lambda_D\) is plotted as a function of \(n_e\) for various electron temperatures. There also is plotted the mean number of charges located

inside the Debye sphere (a sphere with a radius equal to the Debye length). As is seen from Fig. 8, any individual particle can interact with many others located at distances smaller than the Debye length. For this reason we shall turn to the effect of distant collisions (occurring at distances greater than nuclear dimensions but smaller than the Debye length), which are of greater importance than close collisions in the scattering of any charged particle passing through a plasma.

First of all, let us consider close collisions. The scattering cross section of a charged particle is determined by the classical Rutherford formula[^17]. For large deflections, corresponding to individual close collisions, it is not difficult to find the order of magnitude of the effective cross section. To do this it is necessary to determine the minimum distance between two interacting charged particles. In this state the mutual Coulomb potential energy is equal to the initial kinetic energy of the colliding particles. Particles that pass close to one another will evidently be scattered through large angles, and the effective cross section of such processes will be equal to the area of a disk with radius equal to the distance corresponding to closest approach. Thus, we set the initial kinetic energy of the particle \(W\) equal to the mutual electrostatic potential energy at the moment of closest approach. For two singly charged particles \(W_c = e^2/r_c\), where \(r_c\) is the minimum distance between the particles. Therefore,

\[ \sigma_c \simeq \pi r_c^2 = \frac{\pi e^4}{W^2}. \tag{31} \]

Expressing \(W\) in kev, we obtain

\[ \sigma_c \simeq \frac{6\cdot 10^{-20}}{W^2}\ \text{cm}^2. \tag{32} \]

It is evident from the formula that the effective cross section \(\sigma_c\) varies inversely as the square of the relative energy. At a relative energy of 100 kev the cross section is equal to 6 barns, i.e., 160 times greater than the cross section of the DD reaction at the same relative energy. For considerably higher energies, the relation written above gives a value of the effective cross section that is too low, because it does not take account of the direct nuclear interaction.

Fig. 8. Dependence of the Debye length and of the number of particles inside the Debye sphere on particle concentration.

To calculate the effect of distant collisions, let us assume that the deflecting effect in each such collision is small, and that the magnitude and direction of the deflection relative to the motion of the scattered particle are random. The increment of momentum acquired by the particle in such a collision is equal to the average force multiplied by the average collision time,

\[ F(\Delta t) = \Delta(Mv) = \Delta p . \tag{33} \]

(\(\Delta p\) is the increment of momentum; \(p\) is the momentum).

The force is precisely that which arises from the interaction of the moving charge with the electric field of the particles of the “background” with which this charge collides. Thus, \(F = Ee\), and \(\Delta t\) is approximately equal to the time,

which the particle traverses while at a distance equal to the distance of closest approach, as shown in Fig. 9. Thus, \(\Delta t \simeq r/v\) and \(E \simeq e/r^2\) (\(v\) is the particle velocity)

\[ \Delta p \simeq \frac{Eer}{v}=\frac{e^2}{rv}. \tag{34} \]

Fig. 9. Geometry of distant collisions.

Since in distant collisions there occur transfers of momenta whose directions are completely random, the mean square change in the momentum of the charged particle will be proportional to the number of such collisions, multiplied by the square of the momentum change in a single collision. During the time in which the scattered particle traverses a distance \(L\), it “collides” with a large number of particles of identical mass lying in a layer at distance \(r\) from its trajectory. The number of such particles is equal to the particle density multiplied by the volume of the layer:

\[ dN_L=n(2\pi rL\,dr). \]

Thus, for random collisions,

\[ \left\langle(\Delta p)^2\right\rangle=(nL2\pi r\,dr)\left(\frac{e^2}{rv}\right)^2. \tag{35} \]

The total change in momentum can be found by summing over all collisions along the path length \(L\), i.e., by integrating over all values of \(r\) in the interval from the smallest to the largest distance of closest approach,

\[ \left\langle(\Delta p)^2\right\rangle=\frac{2\pi e^4}{v^2}\,nL\ln(q), \tag{36} \]

where

\[ q=\frac{r_{\max}}{r_{\min}}. \]

Divide both sides of the equality by the square of the initial momentum of the scattered particle, \(p^2=(Mv)^2\):

\[ \frac{\left\langle(\Delta p)^2\right\rangle}{p^2} = \frac{\pi e^4}{2\left(\frac{1}{2}Mv^2\right)^2}\,nL\ln(q). \tag{37} \]

When \(\left\langle(\Delta p)^2\right\rangle\), increasing, becomes comparable with \(p^2\), we may assume that the particle has been scattered through a large angle, close to \(90^\circ\) (i.e., its energy has changed by a substantial amount). In other words, if \(L\) is taken as the mean free path before scattering of the charged particle through large angles in distant collisions, then \(L\) is found from the relation

\[ \frac{\left\langle(\Delta p)^2\right\rangle}{p^2}=1, \]

\[ 1\simeq \frac{\pi}{2}\, \frac{e^4}{\left(\frac{1}{2}Mv^2\right)^2}\, nL\ln(q) = \frac{\pi}{2}\, \frac{e^4}{W^2}\, nL\ln(q). \]

But in the sense of the effective cross section, \(L=\frac{1}{n\sigma}\). Thus, we find that the effective cross section for scattering through large angles, or for the transfer of a substantial part of the energy in distant collisions, is expressed as follows:

\[ \sigma_d \simeq \frac{\pi e^4}{W^2}\,\frac{\ln(q)}{2}. \tag{38} \]

It is not difficult to see that \(\sigma_d\) is equal to \(\sigma_c\) multiplied by \(\dfrac{\ln(q)}{2}\); the factor \(q\) is very large, but finite. It is approximately equal to the ratio of the greatest distance at which the interaction between colliding particles still takes place to the distance of their closest approach. The first of these distances is evidently determined by the Debye length, while the second is approximately equal to \(r_c\), calculated above, if the quantum effect is negligibly small. Thus, \(q \approx \dfrac{\lambda_D}{r_c}\), and for typical plasma densities and temperatures it is of order \(10^9\). The quantity \(\ln(q) \approx 20\) is comparatively insensitive to changes in the density and temperature of the plasma. If we neglect corrections for the relative energies of the background particles, then we obtain \(\sigma_d \approx 10\sigma_c\). Thus, the contribution of the effect of distant collisions in the scattering of charged particles is approximately an order of magnitude larger than the contribution of close collisions. If \(W\) is expressed in \(kev\), then

\[ \sigma_d \approx \frac{6\cdot 10^{-19}}{W^2}\ \text{cm}^2 . \tag{39} \]

It may be expected that this relation is approximately valid also for collisions between particles of equal masses, as well as for the case of scattering of a particle lighter than the scattering particles. Consequently, using equation (39), one can roughly estimate the values of the cross sections of ion-ion, electron-electron, and electron-ion collisions. Detailed and more accurate calculations of this type for the case of collisions of stars were made by the astrophysicist Chandrasekhar\(^{18}\). Spitzer\(^{11}\) extended these calculations to the case of a completely ionized gas.

One of the important consequences of the work of Chandrasekhar and Spitzer is the introduction of the concept of relaxation time for collisions in a plasma. It is clear that the mean time of scattering through large angles, or of momentum change in the interaction of a particle with the background created by other particles, is a measure of the rate at which a nonequilibrium plasma distribution approaches an equilibrium one.

Two cases are of special interest. In the first of them one considers the time of elastic or inelastic scattering of a certain specified ion or electron possessing high energy (“test particle”) as a result of interaction with particles similar to it.

The effective cross section \(\sigma_d\) can be used to estimate the relaxation time*). These times are respectively equal to \(t_{ii}\)—the mean collision time for scattering of ions in distant collisions with ions, \(t_{ee}\)—the analogous time of scattering of electrons by electrons, and \(t_{ei}\)—the time of scattering of electrons in distant collisions with ions. Evidently, \(t_{ei}\approx t_{ee}\), if the recoil effect is neglected. The symbols \(i\) and \(e\) refer respectively to ions and electrons. Thus,

\[ t_{ii}=\frac{1}{n_i\sigma_d v_i}, \tag{40} \]

\[ t_{ee}=\frac{1}{n_e\sigma_d v_e}\approx t_{ei}. \tag{41} \]

Since an electron has a velocity approximately 60 times greater than a deuteron of the same energy, \(t_{ii}\approx 60t_{ee}\), if the ion and electron energies are comparable. Using the expression given above for the quantity \(\sigma_d\), the time of energy change can be represented in the form of a function of the density of the background particles and of the energy of the scattered particles in \(kev\). Using the value of the mass

\[ \text{*) More exact values of } \sigma_d \text{ are given in the works of Chandrasekhar and Spitzer.} \]

of the deuton \(M\), we have \(v_i=3\cdot 10^7\sqrt{W}\ \text{cm/sec}\), and, consequently

\[ t_{ii}=6\cdot 10^{10}\frac{W^{3/2}}{n_i}\ \text{sec}, \tag{42} \]

\[ t_{ee}=10^9\frac{W_e^{3/2}}{n_e}. \tag{43} \]

Let us note that the relaxation time increases with increasing energy. These equations are suitable only for estimates of the relaxation time, since an error in the numerical factor of \(\sim 2\) or more is possible in comparison with the exactly derived value.

From equation (42) one can see, for example, that for \(n_i=3\cdot 10^{13}\) and \(W=150\ \text{keV}\), \(t_{ii}=0.04\ \text{sec}\) (corresponding mean free path about \(10^7\ \text{cm}\)). From Fig. 4 it follows that the mean lifetime of a deuton before the DD reaction at a kinetic temperature of \(100\ \text{keV}\) is \(10\ \text{sec}\), or approximately 250 times greater than the relaxation time of a deuton at the mean particle energy corresponding to this temperature. This means that at a plasma temperature equal to \(100\ \text{keV}\), a deuton, in order to undergo a reaction, must on average experience a considerable, though not excessively large, number of effective collisions. At lower temperatures this number becomes much larger.

A second interesting case concerns the effect of collision of an ion with electrons forming the background. Here one may consider two limiting cases: a) the ion energy is considerably smaller than the mean energy of the electrons; b) the ion energy exceeds the mean energy of the electrons. In case a) the ion will gain energy in collisions with the more rapidly moving electrons in an amount determined by the number of electron-ion collisions and by the mean energy transferred in each collision. In case b) the ion will lose energy, transferring it to the electrons, despite the fact that their mean velocity is greater than the ion velocity.

In case a), the energy transfer can be estimated by a method analogous to that used in estimating \(\sigma_d\). Since the ions move very slowly in comparison with the mean velocity of the electrons, this problem is very similar to the classical problem of Brownian motion. When fast electrons pass near an ion, they transfer momentum to it through interactions described in the calculation of \(\sigma_d\), with the difference that here the mean value of the electron velocity must be used

\[ \Delta p \approx \frac{e^2}{r\bar v_e}, \tag{44} \]

where \(\bar v_e\) is the mean velocity of the electron.

The number of collisions during a time \(\Delta t\) with electrons passing through a ring of area \(dA=2\pi r\,dr\) is proportional to the mean number of electrons passing per second through such a ring. From elementary kinetic theory it can be established that this number is approximately equal to \(3n_e\bar v_e\,dA\). Assuming that the collisions are random in character, and integrating over \(r\), we obtain:

\[ \langle(\Delta p)^2\rangle=\frac{6\pi n_e e^4}{\bar v_e}\ln(q)\Delta t. \tag{45} \]

Substituting \(\frac{1}{2}m\bar v_e^2\approx \frac{3}{2}kT_e\) and \(\langle\langle(\Delta p)^2\rangle=2M\langle\Delta W\rangle\) into equation (45), we obtain the value of the rate of energy acquisition by an ion under bombardment by fast electrons \(\left(W\ll \frac{3}{2}kT_e,\ \text{all expressed in CGSE units}\right)\):

\[ \frac{dW}{dt}\approx 3\pi\ln(q)\frac{n_e e^4}{(3m_e kT_e)^{1/2}}\cdot\left(\frac{m}{M}\right). \tag{46} \]

This expression shows that the rate at which an ion acquires energy does not depend on the magnitude of its energy as long as the latter is small in comparison with the mean energy of the electrons, and that it varies inversely as the square root of the electron temperature. This qualitative fact is of interest in connection with consideration of the possibility of heating “cold” ions by means of collisions with “hot” electrons. Since it is often considerably easier to transfer energy directly to the plasma electrons than to the ions, such a heating cycle is possible in which the electrons are heated first, and only afterward do they transfer part of their energy to the ions through collisions. This is probably possible at energies of several ev, but at thermonuclear temperatures the rate of energy transfer becomes small and the effectiveness of heating decreases.

In considering electron-ion collisions, another limiting case is also of interest, when the ion energy is large in comparison with the mean electron energy. In this case approximate methods do not permit the solution of the problem to be obtained immediately. Calculations show that even if the mean velocity of the electrons exceeds the velocity of the ion, the energy transfer is determined mainly by those few electrons whose velocities are less than the ion velocity. These collisions reduce the ion energy until its value reaches the mean electron energy. Spitzer and others obtained exact expressions for the rate of energy exchange between ions and electrons of a Maxwellian distribution. The formula given below includes both cases considered here. For hydrogen plasma this expression, in CGSE units, has the form:

\[ \frac{dW}{dt} = 4\pi \sqrt{2}\,\ln(q)\, \frac{n_e e^4}{(\pi m kT_e)^{1/2}} \left(\frac{m}{M}\right) \left( 1-\frac{W}{\frac{3}{2}kT_e} \right). \tag{47} \]

If the ion energy \(W\) is considerably less than \(\frac{3}{2}kT_e\), then this expression assumes the same form as the approximate formula (46), to within the numerical factor \((3\pi/32)^{1/2}=0.54\). Substituting numerical values in (47), expressing \(W\) and \(kT_e\) in kev and substituting the deuteron mass for \(M\), we obtain in this limiting case:

\[ \frac{dW}{dt} = 1.4\cdot 10^{-12}\, \frac{n_e}{T_e^{1/2}} \ \text{kev/sec}. \tag{48} \]

For \(n_e=3\cdot 10^{15}\) and \(T_e=0.1\) kev (such a temperature may be obtained in a gas discharge at large currents), an ion with small energy will acquire energy at a rate of \(1.3\cdot 10^4\) ev/sec; thus an ion possessing an energy of \(1\) ev can double its energy in a time equal to approximately \(0.1\) msec. However, at the same density and \(T_e=100\) kev this quantity is \(4.3\cdot 10^5\) ev/sec, i.e., an ion with an energy of \(10\) kev will double its energy only after \(0.023\) sec.

If \(W \gg kT_e\), then expression (47) assumes the form:

\[ \frac{1}{W}\frac{dW}{dt} = -0.9\cdot 10^{-12}\, \frac{n_e}{T^{3/2}}, \tag{49} \]

where the energy and temperature are expressed in kev. After integration one obtains:

\[ W=W_0 e^{-t/\tau_e}, \tag{50} \]

where \(\tau_e=1.1\cdot 10^{12}\,(T_e^{3/2}/n_e)\).

It follows from this formula that collisions with electrons of the “cold” distribution lead to an exponential decrease of the energy of fast ions with a time constant proportional to the electron temperature.

to the power \(3/2\); thus, under these conditions, the ion energy asymptotically approaches the value \({}^{3}/_{2}\,kT_e\), in accordance with the classical law of equipartition among degrees of freedom. At \(n_e=3\cdot10^{15}\) and an electron temperature of \(10\) keV (typical for an ordinary discharge), \(\tau_e\) is only \(0.3\) μsec, i.e., it is approximately \(3\cdot10^{-7}\) of the mean lifetime of an ion before the DD reaction at a temperature of \(100\) keV. This is essential for some of the problems discussed in Section VIII.

One of the interesting consequences of equation (47) is that at a high plasma temperature the equilibrium electron temperature may be somewhat lower than the ion temperature. If no appreciable amount of energy is supplied to the plasma from external sources or from reactions occurring within it, then the electron temperature will be determined by the balance between the intensity of bremsstrahlung radiation and the rate of energy transfer from ions to electrons in collisions; i.e., it can be found by combining equations (47) and (12). For deuterium plasma one obtains approximately the following temperature difference (temperature in keV):

\[ T_i - T_e = 4.4\cdot10^{-3}T_e^2 . \tag{51} \]

Thus, for \(T_i=100\) keV the value of \(T_e\) will asymptotically approach \(75\) keV, if, of course, other processes do not participate in the energy balance.

Motion of Particles

As has already been noted, at thermonuclear temperatures the mean free path before collisions of the ions and electrons of the plasma will probably be very large. In this case, the displacement of electrons and ions of the plasma over distances of the order of the dimensions of the discharge chamber will be determined primarily by electrodynamic forces, and each charged plasma particle will move practically in accordance with the usual equations of motion of a particle in an electromagnetic field. This field is the result of a self-consistent superposition of fields formed by external sources, space charge, and currents created by the plasma itself. A sufficiently rarefied plasma has a comparatively weak effect on the externally applied field, and its behavior can be understood by considering the motion of individual particles in the external field.

The equation of motion of a charged particle in an electromagnetic field in the Gaussian system of units has the form:

\[ m\frac{d\mathbf{v}}{dt} = e\left(\mathbf{E}+\frac{\mathbf{v}}{c}\times\mathbf{H}\right). \tag{52} \]

The component of the electric force directed parallel to the motion causes a change in the kinetic energy of the particle. The force acting on a particle in a magnetic field is directed perpendicular to the motion and bends the trajectory of the particle, but does not change its energy.

If \(\mathbf{E}=0\) and \(\mathbf{H}\) is constant in space and time, then the motion consists of displacement with an arbitrary constant velocity, directed along the magnetic lines of force, and rotation with the cyclotron angular frequency \(\omega_c=eH/mc\). In other words, the trajectories are helical lines whose axes are parallel to the field lines. Because of the different signs of the charges of ions and electrons, they rotate in opposite directions with frequencies differing by a factor of \((m/M)\). For equal rotational energy, the diameter of the ion orbit is \((M/m)^{1/2}\) times larger than the diameter of the electron orbit (approximately 60 times in the case of deuterons). The product of the magnetic-field strength by the radius of curvature of the orbit for deuterons is equal to:

\[ H\rho_c = 6.4\cdot10^3\sqrt{W_\perp}\ \text{gauss}\cdot\text{cm}, \tag{53} \]

The energy of the deuterons here is expressed in kev. The angular frequency of rotation of the deuterons is

\[ \omega_c = 4.8 \cdot 10^3\ \mathrm{H}\ \text{radians/sec}. \tag{54} \]

If the electric field has a component directed perpendicular to the magnetic field, then the trajectory of the particle is a superposition of motion along a helical line and drift with constant velocity in a direction perpendicular both to the direction of the magnetic field and to the transverse component of the electric field. If the drift velocity is small in comparison with the velocity of the particle’s motion in its orbit, then the actual motion may be represented as a transverse drift of the instantaneous center of rotation of the particle (the guiding center). The drift velocity in crossed fields is

\[ \mathbf{v}_0 = c\,\frac{\mathbf{E}\times\mathbf{H}}{H^2}, \qquad v_0 = c\,\frac{E}{H}, \tag{55} \]

if \(\mathbf{E}\) and \(\mathbf{H}\) are perpendicular. Let us note that the direction of this velocity and its magnitude do not depend on the magnitude or sign of the charge, nor on the mass of the particle.

If we recall that even in a static magnetic field a moving observer experiences an electric field, then the formula for the drift velocity \(\mathbf{v}_0\) can be easily derived from the equations of motion. Suppose that the actual motion of the particle is composed of a displacement with constant velocity \(\mathbf{v}'_0 \ll c\) and a motion with variable velocity \(\mathbf{v}_1\), so that \(\mathbf{v}=\mathbf{v}'_0+\mathbf{v}_1\); in other words, let the velocity of the particle be equal to \(\mathbf{v}_1\) with respect to an observer in a coordinate system moving with velocity \(\mathbf{v}'_0\). In such a moving coordinate system, in addition to the applied electric field, an additional field \(\mathbf{E}_m=(\mathbf{v}'_0\times\mathbf{H})/c\) appears. Therefore, in the moving system the equation of motion will have the form

\[ m\frac{d\mathbf{v}_1}{dt} = e\left( \mathbf{E}+\mathbf{E}_m+\frac{\mathbf{v}_1\times\mathbf{H}}{c} \right). \tag{56} \]

We choose the velocity \(\mathbf{v}'_0\) so that \(\mathbf{E}_m=-\mathbf{E}\), and hence \(\mathbf{E}+\mathbf{E}_m=0\). In this coordinate system the electric field is absent, and the trajectory of motion will again be a helical line. Choosing the velocity

\[ \mathbf{v}'_0=\mathbf{v}_0=c\,\frac{(\mathbf{E}\times\mathbf{H})}{H^2}, \]

we obtain:

\[ \mathbf{E}_m=\frac{(\mathbf{E}\times\mathbf{H})\times\mathbf{H}}{H^2}=-\mathbf{E}, \]

since \(\mathbf{H}\) and \(\mathbf{E}\) are perpendicular.

When the magnetic field varies with time, the drift velocity \(c(E/H)\) has yet another important meaning. Suppose that a homogeneous magnetic field, produced by a long solenoid, increases with time. In this case an electric field appears whose lines of force are circles with centers located on the axis of the solenoid. The intensity of this electric field in the laboratory system is found from the law of induction:

\[ \oint \mathbf{E}\cdot d\mathbf{l} = -\frac{1}{c}\int \left(\frac{d\mathbf{H}}{dt}\right)\cdot d\mathbf{A}, \tag{57} \]

or, for a circle of radius \(r\):

\[ 2\pi r E = -\frac{\pi r^2}{c}\frac{dH}{dt}. \tag{58} \]

At some radius \(r\) the drift velocity \(v_0=c(E/H)\) is equal to

\[ v_0=\frac{dr}{dt}=c,\quad \frac{E}{H}=-\frac{r}{2}\frac{dH}{dt}\frac{1}{H}, \tag{59} \]

or

\[ \frac{1}{r}\frac{dr}{dt}=-\frac{1}{2}\frac{1}{H}\frac{dH}{dt}. \tag{60} \]

After integration one obtains

\[ \frac{r}{r_0}=\left(\frac{H_0}{H}\right)^{1/2}, \tag{61} \]

or

\[ \pi r^2 H=\pi r_0^2 H_0. \tag{62} \]

The last equality shows that, when any element moves with the local drift velocity \(c(E/H)\) inward in an axially symmetric region of magnetic field, the magnetic flux inside a circle whose radius at each instant is equal to the distance between the moving point and the axis is conserved. In other words, the guiding centers of orbits moving with the local drift velocity remain on the surface of a certain contracting tube of magnetic field.

Fig. 10. Drift of charged particles in a magnetic field whose absolute-value gradient is perpendicular to the direction of the field.

In the presence of a magnetic-field gradient, a more complicated motion of charged particles occurs. If this gradient is small, then the motion of the particles is determined by simple relations. Of special interest is the case in which the magnetic-field gradient at every point is directed perpendicular to the lines of force of the field. The character of the trajectories obtained in this case is shown qualitatively in Fig. 10. Where the field is stronger, the curvature of the trajectories is greater than the average; where the field is weaker, it is smaller. Thus, as a result, a cycloidal drift is obtained in a direction perpendicular to the gradient, with the centers of rotation of negatively and positively charged particles moving in opposite directions; therefore tendencies toward charge separation appear in the plasma, which may in turn lead to the appearance of local electric fields. Ultimately the plasma may drift as a whole in the direction opposite to that of the transverse gradient. The details of the true drift motion of a plasma are best described by the macroscopic equation of motion of the plasma, for example in the form considered by Spitzer \(^{11}\). The drift velocity of individual particles (if the drift is not inhibited by collective effects) is equal to \(^{12}\)

\[ v_d=\frac{\rho_c v_\perp}{2}\frac{\nabla_\perp(H)}{H}, \tag{63} \]

where \(\nabla_\perp(H)\) is the component of the gradient of the absolute value \(H\) in the plane perpendicular to \(H\), \(\rho_c\) is the radius of curvature of the particle, and \(v_\perp\) is the rotational component of the velocity. This expression is valid only if \(v_d\ll v_\perp\), i.e. \((\rho_c/2)(\nabla(H_\perp)/H)\ll 1\). The latter condition simply means that the relative change in magnetic-field strength over the extent of the orbit must be small.

A drift which, by its origin, may be called “gyroscopic” arises in the presence of gravitational or centrifugal fields. These fields lead to drift motion perpendicular to the applied force and directed in opposite directions for ions and electrons. In a gravitational field whose component directed perpendicular to the mag-

nitic field, is equal to \(g_\perp\), the drift velocity is

\[ v_g=\frac{g_\perp}{\omega_c}, \tag{64} \]

where \(\omega_c\) is the cyclotron frequency of the particle in the magnetic field. For large magnetic fields this drift velocity is small.

If particles move along curved field lines, a centrifugal acceleration appears, which also leads to a drift. This drift can be calculated from (64) if the centrifugal acceleration is substituted for \(g_\perp\). Suppose that a particle moves along a helical line along a curved field line, and that the component of its velocity directed parallel to the field lines is \(v_\parallel\). Then, if the radius of curvature of the magnetic lines of force is \(R\), the centrifugal acceleration is \(v_\parallel^2/R\), so that

\[ v_c=-\frac{v_\parallel^2}{R\omega_c}. \tag{65} \]

In contrast to gravitational drift, this drift in some cases may be substantial. Let us note that the drift velocities of the centers of the orbits described by the particles, \(v_d\), \(v_g\), and \(v_c\), are directed oppositely for ions and electrons. The different direction of the velocities may lead to a separation of charges and, consequently, to the appearance of electric fields. As a result, in a number of cases the plasma begins to move in the direction in which the magnetic field weakens; in other words, there is a tendency for the plasma to be expelled from the region of a strong magnetic field.

This evident tendency of charged particles moving in a magnetic field to be repelled from regions of strong magnetic field, i.e., to exhibit diamagnetic properties, is also found in the motion of particles in a magnetic field with a gradient directed along the lines of force. A well-known effect of this type is the reflection of charged cosmic particles by the Earth’s magnetic dipole. Fermi used analogous considerations to explain the origin of cosmic rays\(^{19}\). The repulsive action of a positive gradient parallel to the magnetic lines of force is most easily explained with the aid of one of the so-called “adiabatic invariants” of the motion of a charged particle. Particles, moving in a constant magnetic field in a spiral around the lines of force, are acted upon by forces directed perpendicular to their motion; therefore the angular momentum corresponding to their rotational motion about the field lines will be an approximate integral of the motion. The magnetic moment arising from the rotational motion of a charge will likewise be an integral of the motion. This can be seen from elementary considerations.

Let us write the known equation expressing the equality of the forces acting on a particle in a magnetic field:

\[ \frac{mv_\perp^2}{r}=\frac{Hev_\perp}{c}. \tag{66} \]

It may be rewritten in the following form:

\[ \frac{\frac{1}{2}mv_\perp^2}{H}=\frac{e}{2mc}(mv_\perp r), \tag{67} \]

where \(mv_\perp r\) is the angular momentum of the rotating particle, which we shall subsequently denote by \(a\); \(\frac{1}{2}mv_\perp^2=W\) is the rotational energy of the particle, \(e/2mc=\text{const}\). Then

\[ \frac{\frac{1}{2}mv_\perp^2}{H}=\frac{W_\perp}{H}=\frac{e}{2mc}(a)=\mu, \tag{68} \]

where \(\mu\) is the magnetic moment in erg/gauss. The constancy of \(\mu\) signifies the constancy of \(\mu\), and, consequently, \(W_\perp/H\) is an adiabatic invariant of the motion. By an adiabatic invariant here is meant a quantity which remains constant under a small change in the magnitude of the magnetic field over the course of one revolution of the particle around the circle. Various authors have shown that \(W_\perp/H\) is also an invariant under a slow change of the magnetic field in time\(^{12,20}\).

For a charged particle moving freely in a static magnetic field, another invariant of the motion is its total kinetic energy. In other words, when the particle moves along magnetic lines of force, the sum of its rotational and translational energies must remain constant. If the magnetic-field intensity in two different regions (1) and (2), through which the particle passes, is respectively equal to \(H_1\) and \(H_2\), then, evidently, one may write:

\[ W_\parallel(1)+W_\perp(1)=W_\parallel(2)+W_\perp(2). \tag{69} \]

The invariance of \(\mu\) also means that

\[ \frac{W_\perp(1)}{H_1}=\frac{W_\perp(2)}{H_2}, \tag{70} \]

whence

\[ W_\parallel(2)=W_\parallel(1)-W_\perp(1)\left[\frac{H_2}{H_1}-1\right]. \tag{71} \]

If \(H_2/H_1>1\), then \(W_\parallel(2)<W_\parallel(1)\). Obviously, the component of the particle velocity directed along the field will decrease as the particle moves toward increasing field. If the initial value of \(W_\parallel\) is not too large, the particle may be reflected from the region of strong magnetic field, whose action in this case is similar to that of a retarding potential. Since the force acting on the particle is equal to the gradient of the potential taken with the opposite sign, differentiation of (71) gives the approximate equation for the axial motion of a particle winding around a magnetic line of force:

\[ F_z=-\mu\,\frac{\partial H_z}{\partial z}, \tag{72} \]

where the \(z\)-axis coincides with the direction of the magnetic line of force at the given point.

It follows from the equations written above that a particle whose energy of rotational motion is very small will not be reflected from a region of stronger magnetic field.

The constancy of \(\mu\) leads to one further interesting conclusion. Since the ratio \(W_\perp/H\) is constant and \(H\rho_c\sim W_\perp^{1/2}\) (according to equation (53)), it follows that \(\rho_c^2H=\text{const}\), i.e. the magnetic flux through the circular orbit is an approximate constant of the motion.

Diffusion Across the Magnetic Field

Although the use of the methods considered above has heuristic value, the representation of a plasma as a complex of independently moving charged particles, obeying only simple adiabatic laws, is an excessive simplification of the true picture. For the solution of many problems it is necessary to use the macroscopic equations of plasma\(^{11}\). An example of a phenomenon that is most simply analyzed with the aid of such macroscopic or hydrodynamic equations is the diffusion of charged particles across a strong magnetic field, i.e. the case in which the gas-kinetic pressure is negligibly small in comparison with the magnetic pressure.

In a microscopic approach, the diffusion of charged particles across a magnetic field should be regarded as a random walk of particles in two-dimensional space; the random-walk velocity in this case is determined by the frequency of mutual collisions of the particles, and the step is approximately equal to the cyclotron radius. Thus, qualitatively, the diffusion velocity decreases if 1) the magnetic-field strength increases, i.e., the orbit diameter becomes smaller, or if 2) the temperature increases or the particle density decreases—in other words, if the collision frequency decreases.

In a quiescent plasma consisting of electrons and identical ions, diffusion can arise both from ion–ion or ion–electron collisions and from collisions of electrons with one another. Recently Simon\(^{21}\) calculated the relative role of these processes and apparently resolved the existing paradox. The paradox consists in the fact that consideration of collision processes between identical particles in the first approximation leads to the absence of diffusion, since the position of the center of mass of the colliding particles does not change in such collisions.

Usually the effect of electron–ion collisions predominates in diffusion; for these the diffusion velocity is given by the expression\(^{11}\)

\[ v_p = -\frac{c}{\sigma}\frac{\nabla P}{H^2}\ \text{cm/sec}, \tag{73} \]

where \(\sigma\) is the electrical conductivity of the plasma in \(\text{cm}^{-1}\) (CGSM) and \(P\) is the plasma pressure. The diffusion velocity, expressed as a function of temperature in kev for a deuterium plasma, is approximately equal to

\[ v_p \simeq -\frac{6.3\cdot 10^3}{T^{3/2}}\frac{\nabla P}{H^2}. \tag{74} \]

If it is assumed that \(P\) is some small constant fraction \(\beta\) of the magnetic pressure \(H^2/8\pi\), and that the characteristic distance associated with \(\nabla P\) is equal to \(L\), then

\[ v_p \simeq -\frac{1.5\cdot 10^5}{T^{3/2}}\left(\frac{\beta}{L}\right)\ \text{cm/sec}. \tag{75} \]

For \(T = 100\) kev, for example, \(v_p \simeq 150(\beta/L)\ \text{cm/sec}\), which corresponds to a very small diffusion velocity and, consequently, to a considerable decrease in heat transport across the field.

Simon\(^{21}\) showed theoretically that under certain circumstances diffusion due to collisions between identical particles can play an important role. In this case, however, the diffusion velocity varies as \(H^{-4}\), and its direct proportionality to \(\nabla P\) is absent.

The expressions given above were derived under the assumption that particle transport across the magnetic field occurs only because of mutual collisions. However, the results of investigations of arc-discharge plasma carried out in the radiation laboratory of the University of California several years ago\(^{22}\) seem to indicate the presence of more rapid diffusion across the magnetic field than follows from the formula given above. Bohm and his collaborators postulated that random electric fields arising as a result of turbulent plasma oscillations are responsible for the increased drift velocity. Bohm proposed the following formula for the diffusion velocity:

\[ v_B = 6\cdot 10^9 T_e \frac{\nabla n}{n}\frac{1}{H}\ \text{cm/sec}. \tag{76} \]

The temperature here is expressed in kev.

As is seen from (76), the diffusion velocity is directly proportional to the electron temperature and inversely proportional to the magnetic-field strength. The diffusion losses predicted by the formula turn out to be much greater than those following from formula (74). Correspondingly, the heat flow also increases.

Some unpublished experimental work carried out recently by Simon and Neidigh at the National Laboratory in Oak Ridge apparently indicates that the Bohm diffusion mechanism, if it exists, can be connected only with a high asymmetry of the plasma system. Their work, performed in an arc discharge with axial symmetry, apparently indicates the presence of a dependence of the type \(1/H^2\), in agreement with (74). However, the assumption of nonturbulent plasma may be far from the truth for many cases of interest in the investigation of controlled thermonuclear reactions. An example may be the pinch-effect instability mentioned above.

Plasma Compression

To conclude the consideration of phenomena occurring in a fully ionized medium, one must consider phenomena connected with plasma compression. Depending on the nature of the compression and its time scales, the plasma may be regarded as a one-, two-, or three-dimensional gas. This unusual circumstance arises because processes that are slow in comparison with the motions of individual particles may still be fast in comparison with the relaxation time due to collisions of plasma particles. In this case the gas-kinetic degrees of freedom of the plasma become unconnected with one another. If the plasma is confined and is compressed adiabatically in only one dimension, then the energy belonging to the corresponding degree of freedom will increase in accordance with the adiabatic law for a one-dimensional gas. In compression across the magnetic field two degrees of freedom take part, and therefore the gas behaves as a two-dimensional one. In general, for a gas with number of degrees of freedom \(f\), the change of temperature \(T\) is given by the relation

\[ T_f \sim n^{\gamma-1}. \tag{77} \]

In general, \(T_f\) should be understood as a generalized kinetic temperature associated with the compression. In this expression

\[ \gamma = \frac{2+f}{f}. \tag{78} \]

and \(n\) is the number density of particles.

For \(f=1\), \(\gamma=3\), so that \(T_1 \sim n^2\). This fact was noted by Fermi \({}^{19}\) in his theory of the origin of cosmic rays, where he assumed that the acceleration mechanism consists in the multiple reflection of fast particles from two approaching magnetic clouds separated by a distance \(L\), so that \(T_1 \sim \frac{1}{L^2}\).

If the compression is two-dimensional (\(f=2\) and \(\gamma=2\)), then \(T_2 \sim W_\perp \sim n\). If the compression is adiabatic and occurs over a time interval greater than the relaxation time, then \(f=3\), \(\gamma=5/3\), and \(T \sim n^{2/3}\), as in an ordinary gas.

The existence of processes of adiabatic compression in a plasma confined by a magnetic field indicates their possible reversibility and, consequently, creates the possibility of using them to extract energy from the reacting plasma during expansion against the forces of the magnetic field. One of the results of such an expansion may be the generation of electrical energy in external circuits, and possibly with a high thermodynamic efficiency.

In a plasma there may also occur nonadiabatic, i.e., irreversible shock-hydrodynamic effects. In this case the physical conditions are complicated to a considerable degree by the presence of a magnetic field, and here phenomena of new types may arise. Gofman and Teller[^23] have considered some of them, and recently new papers devoted to this question have begun to appear. This branch of plasma physics is new and interesting; it is related to the search for controlled thermonuclear reactions, for example, to the attempt to use the short-lived pinch effect, which has already been discussed.

To summarize, one may say that a plasma at thermonuclear temperatures can be qualitatively described as a mixture of two gases consisting of charged particles—ions and electrons. These two gases interact weakly through collisions (in the microscopic treatment), but interact strongly through large Coulomb forces (in the macroscopic treatment). Depending on the scales, times, and the manner in which external electromagnetic forces are used, the gas may behave, in compression processes, as a one-, two-, or three-dimensional ideal gas. Owing to its high effective electrical conductivity, the plasma exhibits a tendency toward motion in which the electric fields existing in a coordinate system at rest relative to the plasma are reduced to a minimum. This latter property may be identified with the tendency of the plasma, in the course of its motion, to preserve a constant magnetic flux and thus to become locally “glued” to the magnetic lines of force. As a consequence of this, there is a tendency for instantaneous centers of rotation of individual particles in a sufficiently rarefied plasma to be retained on the surfaces of the tubes of force of the magnetic field when the field changes slowly (but not too slowly) with time. Diffusion arising from collisions tends to destroy this state, so that the diamagnetism of the plasma is necessarily a nonequilibrium phenomenon. In addition, the presence of induced particle drifts may lead to charge separation and to the appearance of electric fields. These drift effects may in turn be self-sustaining, so that in particular cases a hydromagnetic instability may manifest itself in the plasma.

VII. SIMILARITY LAWS; ENERGY BALANCE

Even a cursory examination shows that the phenomena of nuclear fusion and chemical synthesis are similar in many respects. Both methods of obtaining energy are realized through the close interaction of reacting particles as a result of their combination or rearrangement. Both processes can be self-sustaining only under favorable physical circumstances: the attainment of a definite temperature to excite the reaction, the presence of a minimum amount of fuel, and the provision of sufficiently low energy losses to the surrounding space are necessary. The latter is required in order to prevent the reaction from dying out. If too weak a flame is brought to a fireplace, the fire will not be able to flare up; it is difficult to ignite a single piece of coal by placing it on a stove. Such everyday observations on chemical synthesis have analogies in nuclear fusion reactions. In this part of the article some general similarity laws that are valid for a fusion reactor will be discussed. In their light it is evident that the first success in creating a thermonuclear reaction in a hot plasma will require considerably less skill than the subsequent creation of an energetically favorable self-sustaining reaction. The success of attempts to obtain energy from the fusion of nuclear particles will depend not only on the results of theoretical studies. It is also necessary to show that a self-sustaining reactor built on the basis of the proposed principles will not be too large or require unattainable physical conditions.

The achievement of a favorable energy balance in the operation of a fusion reactor will depend on the relation between the nuclear energy produced and the direct and indirect energy losses. Direct energy losses include the escape of radiation or fast particles from the reaction region. Indirect energy losses are associated with the inefficient operation of the apparatus intended for heating and confining the plasma. Excessively high direct losses lead to extinction of the reaction. Intensive indirect losses do not permit the realization of a closed cycle capable of producing useful energy, even if the reaction proceeds sufficiently efficiently. The latter is similar to what happens in a moving automobile when the circuit of the electric generator is broken. The process of fuel combustion in the automobile cylinder continues only until the ignition battery is discharged and the spark plug ceases to operate.

Similarity laws for energy output

To illustrate the similarity relations that may be derived for real systems, let us consider in simplified form one patently unrealizable scheme—a stationary pinch (Part V). For this purpose let us assume that the Kruskal–Schwarzschild instability does not exist and that the longitudinal dimensions of the gas-discharge cord are so large that energy losses to the electrodes may be neglected. In the case of a stationary process, a simple similarity law for the energy output can be found. If \(R\) is the radius of the system in centimeters, then the power released in thermonuclear DD reactions per unit length of the column is determined by the relation

\[ p=\frac{1}{2} n_D^2 \langle \sigma v\rangle W\pi R^2 . \tag{79} \]

Recall that, in accordance with equation (21), the pressure is proportional to the square of the magnetic-field strength, i.e. \(n_D \sim H_0^2/T\). Omitting numerical factors, we obtain the following dependence of \(p\) on the temperature, radius, and magnetic field:

\[ p\sim \frac{\langle \sigma v\rangle}{T^2} R^2 H_0^4 . \tag{80} \]

Several conclusions may be drawn from this expression. First, if the losses were small, then the most effective operating temperature would be determined by the position of the maximum of the function \(\langle \sigma v\rangle/T^2\). This function has a broad maximum for DD and DT reactions in the region \(10\)—\(15\) keV. Since these temperatures are above the critical “braking” temperature \(T_c\) for the DT reaction, but below the corresponding value for the DD reaction (see Part IV), in the indicated temperature range it is possible to work only on one of them. Second, the strong dependence of the energy output on the magnetic field indicates the advantage of strong confining fields and, consequently, the probable infeasibility of any controlled thermonuclear device operating at small fields. Since (80) was derived on the basis of the simple concept of magnetic pressure, it should be expected that analogous similarity laws characterizing the operation of thermonuclear reactors can be obtained in the analysis of more complicated examples of magnetic compression than the case of the simple pinch effect considered in this paper.

Without a detailed analysis it is impossible to write more complete expressions for the similarity laws for energy losses. However, it is possible to formulate certain regularities whose validity does not depend essentially on the choice of the particular example.

Direct losses

As has already been shown, the ratio of the energy carried away by bremsstrahlung to the energy released in the fusion reaction does not depend on the concentration of particles or on the dimensions of the reactor. Consequently, this part of the energy balance is affected neither by a change in the reactor dimensions nor by a change in the magnitude of the magnetic field (at constant temperature). Another mechanism of direct losses consists in the escape of particles from the system by diffusion. The simplest diffusion mechanism is characterized by a transport velocity proportional to the concentration gradient. Thus, for example, both expressions (74) and (76) for the diffusion of particles across the magnetic field are proportional to the density gradient, although they depend differently on the strength of the magnetic field.

As the simplest example, let us consider a system whose longitudinal dimensions are large compared with the transverse ones, and suppose that, when the scale is changed in the radial direction, the relative spatial distribution of the plasma is preserved. In this case the energy losses per unit length due to diffusion in the radial direction are proportional to: a) the area of the plasma boundary per unit length, b) the diffusion velocity of the plasma, c) the concentration of particles, and d) the mean energy of the particles, which in turn is proportional to the kinetic temperature:

\[ p_L \sim R v_D n T . \tag{81} \]

If radial diffusion occurs through ordinary collisions, then equation (74) is applicable, and consequently:

\[ p_L \sim \frac{n^2}{T^{1/2} H_0^2}. \tag{82} \]

Thus, \(p_L\) does not depend on the radius, since the gradient of \(n\) under a similar change of scale changes as \(n/R\). The quantity \(p_L\), like \(p\), is proportional to \(n^2\), because in calculating both \(p\) and \(p_L\) the theory of binary collisions is used. Eliminating the particle density and using (21), we obtain:

\[ p_L \sim \frac{H_0^2}{T^{5/2}} . \tag{83} \]

An order-of-magnitude estimate of the ratio of the energy released in nuclear reactions to the direct energy losses (without radiation) can be obtained by dividing \(p\) by \(p_L\):

\[ \frac{p}{p_L} \sim \left( \langle \sigma v \rangle T^{1/2} \right) R^2 H_0^2 . \tag{84} \]

It is clear from this estimate that, in the case where direct escape of particles due to diffusion in the radial direction predominates, in order to achieve a favorable energy balance it is necessary to work at high temperatures with systems of large dimensions and with strong magnetic fields.

Indirect losses

The calculation of indirect losses is a considerably more complicated problem. In carrying out the calculations it is necessary to estimate the energy loading of each element of a thermonuclear reactor, and also to estimate the magnitude of the immediate losses, such as, for example, Joule heating of conductors (both plasma and metallic). In the case of a closed cycle, when part of the electrical energy is spent both on initiating and on maintaining the thermonuclear reaction, an important factor may prove to be the efficiency of conversion of the thermal energy of the fusion reaction into electrical energy.

One of the scaling laws that must hold for any magnetic-confinement scheme concerns the magnitude of the power required to maintain the magnetic field. If a current of density \(j\) flows through a conductor (a plasma pinch or an external conductor, in other magnetic-confinement methods), then the power released per unit volume of the conductor is equal to \(\rho_0 j^2\), where \(\rho_0\) is the resistivity. The magnetic field created by any system of conductors is proportional to the current density multiplied by the linear dimensions of the system. For simplicity let us consider a cylindrical system whose length is large compared with its diameter. We write the scaling relation for the magnetic field created by the system of conductors:

\[ H_0 \sim j \cdot R_c, \tag{85} \]

where \(R_c\) is the radius of the current-carrying system. Then

\[ H_0^2 \sim j^2 R_c^2 . \tag{86} \]

But the total Joule heat released per unit length of the system is equal to the heat released per unit volume, \(\rho_0 j^2\), multiplied by the volume per unit length of the system, i.e.

\[ p_j \sim \rho_0 j^2 R_c^2 \sim \rho_0 H_0^2, \]

which does not depend on the diameter of the system. Thus, for the ratio \(p/p_j\) we obtain (taking \(R_c \sim R\)):

\[ \frac{p}{p_j} \sim \left(\frac{\langle \sigma v\rangle}{T^2}\right) \frac{1}{\rho_0} R^2 H_0^2 . \tag{87} \]

Here, as in the case of the competition between energy release due to nuclear reactions and direct losses due to particle diffusion, the energy balance becomes more favorable as the radius of the system is increased and when magnetic fields of high intensity are used.

It is interesting to note that in those cases where volumetric Joule losses are the limiting factor in designing the coils that create the magnetic field, equation (85) shows that large magnetic fields are easier to obtain in systems having a large diameter.

Other scaling factors

To conclude the discussion of scaling laws, it is necessary to mention the problems of radiation and the choice of materials. As has already been noted, heat removal and other problems must impose a limitation on the practically attainable power density in thermonuclear reactors. However attractive it may be to fantasize about the construction of a compact thermonuclear reactor the size of a desk telephone, the limitations imposed by the properties of materials on specific power are already, by themselves, a serious obstacle to its construction. Moreover, all types of nuclear-fusion fuel that are presently of interest lead to reactions accompanied by the emission of neutrons. Therefore, in constructing a thermonuclear reactor that releases an appreciable amount of energy, it is necessary to use shielding whose thickness must be measured in meters. This leads to the result that a fantastically miniature reactor proves to be of little economic attractiveness even if it were technically feasible.

Another obviously important factor determining the possibility of obtaining a favorable energy balance is the choice of reactor fuel. It makes sense to consider only DD and DT reactions. DT reactions, as is evident from (1), have a considerably larger cross section than DD. The total energy released in each individual reaction is also signifi-

considerably higher. However, among the various factors that must be taken into account when comparing the advantages of using DD and DT as primary fuel, it is necessary to bear in mind that the main share of the energy released in the DT reaction goes into the kinetic energy of the neutrons. At the same time, in the case of the DD reaction a considerable part of the released energy is imparted to charged particles and can be returned to the reactor by directly converting the particle energy into electrical energy through their interaction with confining fields.

VIII. FRUITLESS ATTEMPTS

Research on controlled thermonuclear reactions imposes extraordinarily contradictory demands on physicists. On the one hand, great inventiveness is required in formulating qualitative ideas for heating and confining plasma; on the other hand, the absence of exacting criticism of the quantitative side of the problem may lead to the most ingenious plans later turning out to be completely meaningless. For illustration we shall give several fruitless projects for carrying out a controlled thermonuclear reaction. It is often risky to try to prove general theorems on the impossibility of carrying out certain experiments. However, knowing of the existence of the first and second laws of thermodynamics, only a few physicists of our day dare to build a perpetual-motion machine. (Let us hope that the creation of a practically useful thermonuclear reactor does not belong to the category of such attempts!)

Earnshaw’s Theorem. (“Electrical Confinement”)

The fuel of a thermonuclear reactor consists of charged particles, and an electrostatic field acts on charged particles. It is therefore natural to suppose that a thermonuclear reactor should be created in the form of a system of electrodes producing an electrostatic trap for the plasma. The electrostatic field in this case should ensure the absence of contact between the plasma and the electrode material.

The proposal to use an electrostatic field is ruled out for two reasons of a qualitative nature and one quantitative reason. First, Earnshaw’s theorem, formulated in classical electrostatics, shows that there is no equilibrium state even for a single charged particle situated in the field of arbitrarily arranged charged conductors. Second, an electrostatic field that is a potential well for charges of one sign serves as a potential “hill” for particles of the other sign. The third reason is connected with the magnitude of the “pressure” that an electric field can exert. As in the case of a magnetic field, the attainable pressure is limited by the energy density, which for an electric field is equal to \(E^2/8\pi\) erg/cm\(^3\). Let us again use the example in which a plasma with density \(6 \cdot 10^{15}\) particles/cm\(^3\) and temperature 100 keV is considered. In this case the field \(E\) required to confine the particles is \(4.8 \cdot 10^7\) V/cm. This field value is gigantic, and since the electrostatic pressure is proportional to the square of the field, one should not place great hopes on the applicability of this method in any cases of practical interest, even if the first two objections did not exist.

Bombardment of a Target

As was already noted earlier, nuclear fusion reactions can easily be carried out under laboratory conditions by bombarding a deuterated target with a beam of deuterons possessing high energy. However, this method is not promising because of the small magnitude of the energy yield,

incident upon one incident deuteron. The major part of the energy of the incident particle is then uselessly spent on ionization of the atoms of the target and is then emitted when electrons are captured into atomic orbits, or in the course of other analogous processes. An improvement of the method of direct bombardment is often proposed, consisting in using an ordinary plasma as the bombarded target in order to eliminate ionization losses. This idea passes under a qualitative consideration, but fails under a quantitative one. First, the plasma of an ordinary gas discharge is not completely ionized, and the ionization losses, together with other losses, will exceed the energy released in the reaction. Second, the electron temperature in most such discharges is of the order of \(0.01\) keV; it is usually established at this level by processes occurring at the boundaries of the plasma. Expression (50) shows that the mean time during which the energy of an incident “hot” ion will be lost through collisions with “cold” electrons is approximately \(10^9/n\) sec. Thus, if, for example, \(n = 10^{15}\), then the “energy-loss time” of the ion is of the order of \(10^{-6}\) sec, which is considerably less than the mean free time of a deuteron before a collision leading, at the indicated densities, to a nuclear reaction. The idea under consideration becomes more attractive if one uses a plasma having a higher electron temperature. In this case, however, the same difficulties arise as in the attempt to create a hot plasma in which thermonuclear reactions occur. If the problem of creating such a plasma is solved, then the need immediately disappears to consider the possibilities of obtaining an energy yield from fusion reactions occurring when targets are bombarded by an ion beam.

Beam collision

Another variant belonging to the same circle of ideas is the proposal according to which reactions are to be carried out by means of two ion beams directed toward one another. The merit of this proposal is that the problem of attaining sufficiently high energies may be regarded as solved, and, at least for a time, the possibility of interaction of the focused beam with the walls is eliminated. This project proves untenable when considered quantitatively. A very intense ion beam has a current density of approximately \(0.1\ \mathrm{a/cm^2}\). The ion density in such beams is easily computed by dividing the number of charges passing in 1 sec through a cross section of \(1\ \mathrm{cm^2}\) by their velocity. For deuterons with an energy of \(100\) keV, at the indicated current density, the particle number density obtained is \(2 \cdot 10^9\ \mathrm{cm^{-3}}\). Equation (10) of Part III then gives, for the specific power released in this case, the value \(4 \cdot 10^{-11}\ \mathrm{W/cm^3}\)!

Many other analogous examples could be given. They all show that, in discussing any projects for carrying out controlled fusion reactions of light nuclei, an extraordinarily important role is played by the simultaneous fulfillment of two criteria: the qualitative possibility and the quantitative suitability of the project under consideration.

IX. PLASMA INVESTIGATION

An experimenter undertaking a systematic study of phenomena occurring inside a hot plasma encounters painful difficulties when attempting to make the necessary measurements. If the experimenter studies, say, the properties of solid quartz, then he first of all turns to a supplier of chemicals for quartz. He must then think through the appropriate experimental research procedure and make use of

be used in carrying out experiments with well-known experimental techniques. He may employ optical and electrical measurements, and perhaps acoustic methods as well. By transferring a piece of quartz from one experimental setup to another, he will collect the necessary data, on the basis of which it will then be possible to compose an accurate picture of the properties of the substance under study.

The work of a physicist in the field of controlled thermonuclear reactions is the complete opposite of this. First of all, he will encounter great difficulties here in attempting to obtain from a supplier some quantity of hot plasma. The substance that the experimenter studies he must himself prepare in the course of the experiment. Secondly, the experimenter becomes convinced that the closer he comes to the final result, the more limited becomes the number of experimental methods he can use in the course of the investigation. The best of the methods is the recording of neutrons arising in fusion processes. However, although this method is convenient, it may lead to false results, since fusion reactions leading to the appearance of neutrons may be not of thermonuclear origin but caused by entirely different secondary processes^25. This circumstance compels increased demands to be made on observations. Lack of space does not permit a detailed discussion here of all experimental methods. However, to illustrate possible paths of experimental approach, some of them will be briefly considered.

The following measurement methods are considered here: 1) optical, or spectroscopic, 2) methods based on electromagnetic interactions, 3) methods of experimental nuclear physics.

Spectroscopic Measurements^28

The problem of the controlled fusion reaction is perhaps the most direct descendant of astrophysical science. It is therefore natural that astrophysical measurement techniques are also applicable in the study of controlled fusion reactions. Before the advent of radio astronomy, optical measurements were one of the few methods suitable for obtaining astrophysical data. The plasma in a successfully operating fusion reactor is completely ionized. Therefore the light emitted in atomic processes must have negligible intensity. It may turn out, therefore, that spectroscopic measurements will become auxiliary at later stages of the study of thermonuclear reactions, but in the initial stages, when the temperatures are far from thermonuclear and complete ionization has not yet been achieved, spectroscopic measurements can yield very useful information. Data on the purity and composition of the plasma can be obtained by studying emission spectra.

The study of the broadening of individual spectral lines provides information about the density of the plasma and its temperature. The passage of ions in the plasma close to one another causes the appearance of local electric fields, which in turn lead to fluctuational Stark broadening^24, the magnitude of which is related to the plasma density. In addition, the spectral emission lines of an ion moving relative to the observer are shifted owing to the Doppler effect. If the light comes from a large number of such ions and the directions of their motion are random, then an additional broadening of the spectral lines arises. In this case the profile of the spectral line differs from the profile corresponding to Stark broadening. From analysis of the line profile the ion temperature can be determined. Ordered or strongly turbulent motion of the plasma can also cause

Doppler broadening. Thus, special caution must be exercised in interpreting spectroscopic data of this kind, especially in view of the fact that the radiating atoms may belong to an unrepresentative part of the plasma.

Electromagnetic measurements. Microwave technique

The study of electromagnetic phenomena arising in a plasma is probably the most important type of experimental investigation of plasma. Some of these investigations originated in the old field of gas-discharge physics, to which researchers in thermonuclear reactions are greatly indebted. Electromagnetic measurements used in studying gas-discharge phenomena have been distinguished, especially in recent years, by a high level of experimental technique. One should note the brilliant experimental and theoretical work of Allis and Brown and their collaborators at the Massachusetts Institute of Technology. In these works, in the analysis of discharges of interest for thermonuclear research, microwave technique was used. The properties of the plasma were determined from the detuning of hollow resonators containing the gas-discharge plasma. Such a detuning effect can be explained most simply in terms of the dielectric constant of the plasma, the concept of which was given earlier (see (25)):

\[ K = 1 - \frac{4\pi n e^2}{m\omega^2} = 1 - \frac{f_p^2}{f^2} = 1 - 8{,}1 \cdot 10^7 \frac{n}{f^2}. \tag{88} \]

Since the Massachusetts Institute of Technology group considered it most convenient to use cavities operating at a wavelength of \(10\ \text{cm}\) (\(f = 3000\ \text{MHz}\)), it follows, as is seen from formula (88), that in most of their measurements the electron concentration should not have exceeded \(10^{11}\ \text{particles}/\text{cm}^3\). Larger dimensions of the experimental chamber would require still lower working densities.

Because of the limitations imposed by size and density, microwave technique cannot always be used in research on controlled thermonuclear reactions. However, special microwave technique can be employed. The use of millimeter-wave technique leads to an increase in the values of the concentrations at which \(K\) becomes negative (and, consequently, the root of \(K\) imaginary). The creation of a microwave interferometer in which the extended path through the plasma is one of the [[unclear: arms]] of the interferometer may make it possible to measure electron density as a function of time or of other variables. The state of millimeter-wave technique does not yet permit the entire spectrum of plasma densities that may be of interest in these investigations to be covered. Nevertheless, even now its application makes it possible to obtain much useful information.

Electromagnetic measurements\(^ {27}\). Induction effects and probes

It has already been noted that plasma possesses a property that may be called diamagnetism, and that the high electrical conductivity of a plasma may lead to the “freezing-in” of magnetic lines of force into it. For this reason the presence, in a system containing plasma, of internal electric currents in it may substantially alter the pattern of the magnetic field near the plasma. The motion of the plasma may induce an electromotive force in the external circuit of the discharge circuit, the results of measuring which may be used to obtain information on the motion of the plasma, its possible density and temperature. For example, in the case of the pinch effect

a change in the radius of the cord should lead to the appearance of an electromotive force in the external circuit.

Let us consider the cord discharge shown in Fig. 6. The potential difference between its ends is measured by the sum of the inductive and ohmic voltage drops

\[ V = RI + \frac{d}{dt}(LI). \tag{89} \]

If we assume that the ohmic resistance is small, then

\[ V = \frac{d}{dt}(LI) = L\frac{dI}{dt} + I\frac{dL}{dt}. \tag{90} \]

\(V\), \(I\), and \(\frac{dI}{dt}\) can be measured by instruments located outside the plasma. From this equation the inductance \(L\) can be determined as a function of time. In this case the geometry is known; hence the dependence of \(L\) on the radius of the discharge cord is also known (the flow of current over the surface of the cord is meant). Thus, measurement of the current and voltage makes it possible to determine the behavior of the discharge cord in time. More detailed information on the compression can be obtained by suitable placement of measuring coils in the region of the field produced by the flow of current through the discharge cord.

Information on the electrostatic field surrounding the plasma can sometimes be based on the use of electric probes similar to those employed in the classical physics of gas discharges. Here, however, the situation is not so clear, and the need for at least a minimum of physical contact between the confined plasma and the probe makes this method of limited applicability.

Nuclear Measurements

Measurement of the intensity of the neutron radiation arising when fusion reactions occur in the plasma is undoubtedly a satisfactory way of clarifying the conditions of plasma behavior. Under proper circumstances such measurements can provide information on the plasma temperature, its absolute density, and its spatial distribution. If the plasma consists of ionized deuterium and its temperature is somewhat below \(10\) keV, then the dependence of the reaction yield on temperature can be used to measure the plasma temperature. A clearly expressed proportionality of the neutron yield to the square of the plasma density should correspond to a thermonuclear neutron source, whereas a linear dependence on density should probably indicate the production of neutrons as a result of bombardment of the target. The experimenter may fall into these and other traps when attempting a facile interpretation of the regularities of neutron radiation in investigations of controlled thermonuclear reactions. For example, in some cases the acceleration of ions may be caused by various kinds of electric fields\(^{25,30}\). The neutrons obtained in this way will not be of thermonuclear origin.

The study of the state of the plasma and the determination of its properties constitute one of the most important aspects, requiring special attention, in the realization of a controlled thermonuclear reaction. Unfortunately, the difficulties associated with such measurements are especially great at the initial stage of research.

X. CONCLUDING REMARKS

The present article is an attempt to present the physical picture and the practical problems that arise in the course of investigations whose development at present is still in its “infancy.” The aim of the article is not to astonish the reader with the complexity of the problem and

difficulties arising in carrying out research in the field of nuclear-fusion reactions, but rather to bring together in one place the most important of the facts obtained. It is necessary to attract many scientists to this problem, since success in this undertaking will be a very great achievement. Most physicists actively engaged in research on thermonuclear reactions in the USA are firmly convinced that all the scientific and technical problems of controlled fusion reactions will be solved, perhaps even within the next few years. In the USA several different new approaches to the problem are being studied, including both pulsed and steady-state processes. There is no doubt that analogous studies are also being carried out in other countries of the world.

In the search for a solution to this problem, new and fruitful areas of experimental and theoretical physics arise. One may hope that a complete understanding of the physics of high-temperature plasma and of the laws of its interaction with electromagnetic fields will lead not only to the creation of a controlled thermonuclear reactor, but will also raise the level of our knowledge of nature. New and important applications of this branch of knowledge to other fields of science and technology will arise.

Of course, it is unlikely that the first success in realizing a self-sustaining controlled reaction will lead in the near future to the creation of an economically advantageous reactor. However, thermonuclear reactions contain new possibilities: they allow the direct conversion of nuclear energy into electrical energy, using inexpensive, safe fuel whose reserves are practically inexhaustible. These possibilities will undoubtedly someday play a decisive role in shaping the world of the future.

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Some works by Soviet physicists carried out in connection with the development of controlled thermonuclear reactions

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3. L. A. Artsimovich, A. M. Andrianov, O. A. Bazilevskaya, Yu. G. Prokhorov, N. V. Filippov, Atomic Energy, No. 3, p. 76 (1956).
4. S. Yu. Lukyanov, V. I. Sinitsyn, Atomic Energy, No. 3, p. 88 (1956).
5. S. Yu. Lukyanov, I. M. Podgornyi, Atomic Energy, No. 3, p. 97 (1956).
6. A. L. Bezbatenko, I. N. Golovin, D. P. Ivanov, V. D. Kirillov, N. A. Yavlinskii, Atomic Energy, No. 5, p. 26 (1956).
7. V. D. Shafranov, Atomic Energy, No. 5, p. 38 (1956).

Reviews

1. I. V. Kurchatov, UFN LIX, 603 (1956).
2. E. Teller, Nucl. Scien. Engin. 1, 253 (1956).
3. L. A. Artsimovich and S. Yu. Lukyanov, Priroda, No. 1, p. 18 (1957).