ELEMENTARY THEORY OF REACTORS WITH CHAIN NUCLEAR REACTIONS*

E. Fermi.

The basic results and methods for calculating reactors with nuclear chain reactions occurring in a mixture of natural uranium and graphite were obtained by a large number of investigators, working partly independently of one another and partly in collaboration. An important contribution to the ideas underlying reactor theory was made by Szilard and Wigner. Substantial experimental results, the obtaining of which made it possible to advance further in understanding the laws of nuclear chain reactions, belong to many investigators. Among them one should first of all mention H. Anderson (Columbia University) and W. Zinn (Metallurgical Laboratory of the University of Chicago), as well as R. Wilson and E. Greitz of Princeton, Allison, Whitaker, and W. Wilson of the University of Chicago.

The first nuclear chain reaction was achieved in the Metallurgical Laboratory directed by A. Compton.

PRODUCTION AND ABSORPTION OF NEUTRONS IN THE REACTOR

Let us consider a certain mass of uranium distributed among suitably arranged blocks of graphite. We shall call such a system a reactor. If any one of the uranium nuclei placed in the reactor undergoes fission, then in the act of fission an average of $\nu$ neutrons will be produced. These neutrons have a continuous energy spectrum with an average energy of about $10^6$ eV.

A fast neutron produced in fission will be slowed down, losing its energy in elastic collisions with carbon and uranium nuclei, and also in inelastic collisions with uranium nuclei. As a result, the majority of neutrons will be slowed down to thermal energies. The process of slowing down from an energy of the order of $10^6$ eV to thermal energy requires about 100 collisions with carbon nuclei. After the neutron has been slowed down to thermal energy, it continues to diffuse in the reactor until it is absorbed by some nucleus. In some cases, however, it may happen that the neu-

* Translated from Science 5, 2715, 105 (1947).

tron will be absorbed even before it has completely slowed down. A neutron may be captured either by a carbon nucleus or by a uranium nucleus. The effective capture cross section of carbon nuclei is very small. For thermal neutrons it is about \(5\cdot 10^{-27}\,\text{cm}^2\).

For graphite of density 1.6, such a capture cross section corresponds to a mean free path before absorption of about \(25\,\text{m}\). If it is assumed that the effective cross section for capture by carbon follows the law \(\frac{1}{v}\), then absorption of high-energy neutrons in graphite may be completely neglected. Therefore, to a sufficiently good approximation one may assume that, in the process of slowing down, no appreciable absorption of neutrons occurs.

Capture of neutrons by uranium nuclei may lead either to fission or to radiative capture (an \((n,\gamma)\)-process). The latter process takes place in resonant capture of neutrons by uranium nuclei. The relative probability of fission and resonant neutron capture proves to be different depending on the energy of the neutron.

Roughly speaking, three energy intervals may be considered:

1) Neutrons with energy exceeding the fission threshold of U-238. Such neutrons we shall call fast neutrons. In the capture of a fast neutron the principal process is fission, which usually takes place in the principal isotope U-238. Resonant absorption is less probable than fission, but nevertheless occurs with appreciable probability.

2) Neutrons with energy lying in the interval between the fission threshold and thermal energy. We shall call such neutrons neutrons of intermediate energies. For neutrons of intermediate energies the principal process is resonant capture. The effective cross section of resonant capture is a rapidly varying function of the neutron energy. There is a series of resonance maxima which can be described rather well by means of the Breit–Wigner theory. In practice, resonant absorption becomes significant for neutrons with energy less than \(10\,000\ \text{eV}\). It increases as the neutron energy decreases.

3) Neutrons with thermal energy (thermal neutrons). For thermal neutrons, absorption followed by fission and resonant capture are equally important. The effective cross sections for both processes follow approximately the law \(\frac{1}{v}\), and therefore their relative probability does not depend on the energy.

Let \(\sigma_f\) and \(\sigma_r\) denote the effective cross sections for fission and resonant capture of neutrons with energy of the order of \(kT\), and let \(\nu\) be the mean number of neutrons arising when one neutron is captured by a uranium nucleus. Then \(\eta\) is related to \(\nu\) by the relation

\[ \eta=\frac{\nu\sigma_f}{\sigma_f+\sigma_r}, \tag{1} \]

since only the fraction \(\dfrac{\sigma_f}{\sigma_f+\sigma_r}\) of thermal neutrons is captured by uranium with subsequent fission.

The preceding considerations show that only a part of the initially present fast neutrons will be captured by uranium nuclei and will produce a fission reaction.

For systems having finite dimensions, it is necessary to take into account the additional loss of neutrons due to their leakage beyond the boundaries of the system.

For the time being we shall confine ourselves to systems having practically infinite dimensions. Let \(P\) denote the probability that an initially present fast neutron will, in one way or another, be captured by a uranium nucleus with subsequent fission. Then the average number of neutrons of the second generation will be equal to

\[ k=P\nu. \tag{2} \]

The quantity \(k\) is usually called the multiplication coefficient of the system. If \(k>1\), a self-developing chain reaction can proceed in the system; but if \(k<1\), such a reaction is impossible.

It should be emphasized that even if \(k>1\), a self-developing reaction can proceed only when the leakage of neutrons through the external boundaries of the system is sufficiently small. This can evidently be achieved by increasing the dimensions of the boiler.

LIFE HISTORY OF A NEUTRON IN THE BOILER

A fast neutron that has initially appeared in the boiler may take part in the following basic processes:

1) The neutron may be captured by a uranium nucleus before its energy has appreciably decreased because of collisions with nuclei. In this case, the absorbed neutron may cause fission of the U-238 nucleus.

However, the probability of fission by fast neutrons is small and usually amounts to about \(1\%\). Indeed, if the system contains much graphite and little uranium, elastic collisions of the neutron with carbon nuclei will lead to a rapid decrease of its energy to values lying below the fission threshold of U-238 nuclei. If, on the contrary, there is much uranium in the system, then inelastic collisions become very probable, as a result of which the neutron energy will decrease still more rapidly below the fission threshold of U-238.

2) Therefore, in the majority of cases the neutron will be slowed down without capture, losing its energy mainly because of collisions with carbon nuclei. It can be shown simply that, to reduce the neutron energy on the average by a factor \(e\), 6.3 collisions with carbon nuclei are required. Consequently, to reduce the neutron energy by a factor of 10, 14.6 collisions are required; and to slow a neutron from an energy of \(10^6\) eV down to thermal energy \(\dfrac{1}{40}\) eV, 110 collisions are required.

In the process of slowing down, a neutron may undergo resonance capture by a uranium nucleus. Let \(p\) denote the probability that the neutron will not be captured during its slowing down to thermal energy. One of the most important considerations in designing boilers is the possible reduction of the probability of resonance capture of the neutron during the slowing-down process.

3) If the neutron is not captured during the slowing-down process, it will reach thermal energy and subsequently be captured by a uranium or carbon nucleus already as a thermal neutron.

In the case where uranium and graphite are uniformly distributed in the boiler, the probabilities of these two competing processes are related as the corresponding effective absorption cross sections of uranium and carbon multiplied by the atomic concentrations of both elements. Since, however, uranium is in fact distributed nonuniformly in the boiler, the latter statement can have only an approximate character.

Let \(f\) denote the probability that a thermal neutron will be captured by a uranium nucleus. To carry out a chain nuclear reaction in a boiler, one usually tries to increase the probabilities \(p\) and \(f\) as much as possible. Unfortunately, however, increasing one of these quantities leads to a decrease of the other: to increase \(f\), it is necessary to create a system with a large uranium content, in order to reduce the probability of neutron capture in carbon. On the other hand, however, the slowing-down process in a system containing a relatively smaller amount of carbon will proceed more slowly. Therefore, in such a system the probability of resonance absorption of a neutron during the slowing-down process will be relatively large.

It is therefore clear that it is necessary to find the optimum ratio of the amounts of uranium and carbon in the system.

In a homogeneous mixture of uranium and graphite, the values of \(p\) and \(f\) depend only on the relative concentration of these elements. If, however, one does not restrict oneself to homogeneous mixtures, then more favorable conditions can be achieved by an appropriate geometrical arrangement of the two components.

In fact this indeed proves possible owing to the following circumstance: the effective cross section of resonance absorption, which removes neutrons from play during slowing down, is a very rapidly varying function of energy. It is expressed by the Breit–Wigner formula. Therefore, if we distribute the uranium nonuniformly, in the form of separate large blocks, it may be expected that the uranium located inside the block will be shielded by a thin surface layer from the action of neutrons whose energy lies close to the resonance energy. Resonance absorption of neutrons by uranium nuclei lying inside the block turns out to be considerably smaller than the resonance absorption by an isolated atom.

It is clear, of course, that along with the decrease in resonance absorption, the capture of thermal neutrons in uranium also decreases. However

theoretical calculations and experiment show that, for certain block dimensions, the gain obtained from reducing neutron losses to resonance capture outweighs the corresponding loss from a decrease in the capture of thermal neutrons.

A typical structure of a boiler is a lattice of uranium blocks inserted into a mass of graphite. The lattice may, for example, be a lattice of uranium rods or a cubic lattice of pieces of uranium. The first method of arrangement is somewhat less advantageous from the point of view of neutron absorption, but often has a number of practical advantages, since it makes it easier to solve the problem of removing the heat liberated in the boiler. In what follows, however, we shall consider only the cubic lattice.

As an example, one may give some typical values for the probabilities of the various competing processes of neutron capture. These probabilities are, of course, not unchangeable, but depend on the structure of the lattice. Below we give average values of the probabilities, taken for a good lattice.

Consider a neutron initially formed as a result of the fission of a nucleus in a uranium block. With a probability of the order of 3%, before the neutron has time to lose a noticeable fraction of its energy, it will be captured by another uranium nucleus and will cause in it a fission reaction.

In 97% of cases it will be slowed down and will either undergo capture in the resonance region, or will reach thermal energy. The probability of resonance capture in the process of slowing down is about 10%. In the remaining 87% of cases the neutron will be slowed down to thermal energy. In approximately 10% of cases a thermal neutron will be absorbed by a carbon nucleus, and in the remaining 77% of cases it will be absorbed in uranium. If, for example, the value of $\eta$ is taken equal to two, then the life history of one generation of neutrons may be represented schematically in the form of Table I.

Table I

For the multiplication factor in this case one may write the following expression:

\[ k = 0.06 + 0.77\,\eta . \tag{3} \]

It follows from (3) that if \(\eta\) is greater than \(1.22\), the multiplication factor in the lattice under consideration is greater than unity.

To compute the multiplication factor it is necessary to calculate the probabilities of the various processes listed. Below we shall briefly give some starting points for the practical calculation of these quantities.

Probability of fission by fast neutrons

The probability of fission by fast neutrons can easily be calculated for the case of very small blocks.

In this case, obviously,

\[ P_F = \sigma_F n d, \tag{4} \]

where \(\sigma_F\) is the mean value of the effective cross section for fission by fast neutrons, \(n\) is the concentration of uranium nuclei in the block, and \(d\) is the mean value of the distance that a neutron formed in the block must travel before it reaches the surface of the block.

In the case of large blocks the calculation of \(P_F\) becomes more complicated, since in a large block a neutron undergoes a considerable number of collisions with nuclei, and elastic and inelastic collisions become equally important.

In particular, the process of inelastic collisions in blocks of large dimensions leads to a rapid decrease of neutron energies below the fission threshold of U-238 and transfers them into the region of resonance energies.

Resonance absorption

If an isolated uranium atom is located in a graphite moderator, then the probability per unit time of resonance capture of a neutron with energy greater than thermal has the form:

\[ \frac{q\lambda}{0.158}\int \sigma(E)\frac{dE}{E}, \tag{5} \]

where \(q\) is the number of fast neutrons entering \(1\ \mathrm{cm}^3\) of the system per unit time, \(\lambda\) is the mean free path, and \(\sigma(E)\) is the value of the effective cross section for resonance absorption at energy \(E\).

The integral is taken over all energy values from thermal energy to the mean energy of fission neutrons. It can be shown that the main contribution to the value of integral (5) is made by the resonance peaks in the cross section \(\sigma(E)\), which is expressed by the well-known Breit-Wigner formula.

Application of formula (5) to resonance absorption in a lattice of uranium and graphite may lead to very noticeable errors. Po-

The latter is connected with the above-mentioned effect of self-shielding, which substantially reduces the density of neutrons with energy close to the resonance energy inside uranium blocks. Therefore the best way to solve the problem of resonance absorption is to measure directly the number of neutrons with resonance energy absorbed in uranium blocks of various sizes.

Such measurements were first carried out at Princeton University. The results of the measurements were reduced to empirical formulas convenient for calculations.

Probability of absorption of thermal neutrons

The probability of absorption of a thermal neutron in uranium in a system with a homogeneous distribution of uranium and graphite is expressed by the formula

\[ \frac{N_u\sigma_u}{N_c\sigma_c+N_u\sigma_u}. \tag{6} \]

Here \(N_u\) and \(N_c\) are the numbers of uranium and carbon atoms per unit volume, and \(\sigma_u\) and \(\sigma_c\) are the effective capture cross sections for thermal neutrons. A more complicated case is that in which the uranium is distributed in the graphite in the form of a lattice of blocks. In this case the density of thermal neutrons is not constant throughout the system. Namely, since the absorption of thermal neutrons in uranium is considerably greater than in graphite, it is large far from the uranium blocks and small near and inside them.

Let \(\overline{n}_c\) and \(\overline{n}_u\) be the mean densities of thermal neutrons in the graphite and in the uranium blocks. The number of thermal neutrons absorbed in uranium and in graphite will be proportional, respectively, to \(N_u\sigma_u\overline{n}_u\) and \(N_c\sigma_c\overline{n}_c\). Therefore, instead of equation (6), one may write:

\[ f=\frac{N_u\sigma_u\overline{n}_u}{N_u\sigma_u\overline{n}_u+N_c\sigma_c\overline{n}_c}. \tag{7} \]

The mean densities \(\overline{n}_u\) and \(\overline{n}_c\) can be calculated with a degree of accuracy sufficient for practice by using the diffusion equation. For simplicity of calculation, instead of the lattice cell one may consider a spherical cell of the same volume. The boundary condition on the surface of the sphere is the condition that the radial derivative of the neutron density be equal to zero.

One may also assume that the number of neutrons slowed down to thermal energy per unit volume per unit time is constant throughout the part of the cell filled with graphite. These simplifications do not lead to significant errors, provided only that the dimensions of the cell are not too large. Using the simplifying assumptions indicated, one can obtain the following expression for the probability of absorption of a thermal neutron in uranium:

\[ f=\frac{3\alpha^2}{\alpha^3-\beta^3}\, \frac{(1-\alpha)(1+\beta)e^{-\beta+\alpha}-(1+\alpha)(1-\beta)e^{\beta-\alpha}} {(\alpha+s-\alpha s)(1+\beta)e^{-\beta+\alpha}-(\alpha+s+\alpha s)e^{\beta-\alpha}}, \tag{8} \]

where \(a\) and \(\beta\) are the radii of the block and the cell, expressed in conventional units

\[ l=\sqrt{\frac{\Lambda}{3}}, \]

where \(\Lambda\) is the diffusion length in graphite; the quantity \(s\) denotes

\[ s=\frac{\lambda}{\sqrt{3}}\,\frac{1+\gamma}{1-\gamma}, \tag{9} \]

where \(\gamma\) is the reflection coefficient of the block for thermal neutrons.

A LATTICE CONSISTING OF A LARGE NUMBER OF CELLS

The density of neutrons of a given energy in a lattice consisting of a large number of cells varies from point to point along the lattice. A simple mathematical description of the behavior of such a system can be obtained if, in the first approximation, one neglects the local variation of the neutron density due to the periodic structure of the lattice and replaces the inhomogeneous system by a certain equivalent homogeneous system.

In this paragraph we shall simplify the problem in precisely this way, replacing all true neutron densities by their mean values obtained by averaging over the entire volume of a cell. In this case the average neutron density will be a smooth function of position, as it would be if we were dealing with a homogeneous uranium-graphite system.

Let \(Q(x,y,z)\) be the number of fast neutrons arising per unit volume per second at a certain point \((x,y,z)\) inside the boiler. These neutrons diffuse and are slowed down in the boiler. In the course of slowing down, some of them undergo resonance absorption. Let \(q(x,y,z)\) be the number of neutrons transformed per unit volume per unit time into thermal neutrons. We shall call \(q\) the density of the thermal neutrons being produced.

We shall assume that if initially a fast neutron appeared at the point \(O\), then the probability that it will slow down to thermal energy at some specified point will be expressed by a Gaussian function of the distance to the point \(O\). This assumption may be justified if the slowing-down process takes place over a distance considerably exceeding the mean free path.

It was found experimentally that the distribution of thermal neutrons being produced relative to a point source of fast neutrons can be expressed by a Gaussian function only approximately. To describe the results of measurements, formulas were used that represented a superposition of two or three Gaussian curves with different mean paths. We shall restrict ourselves, however, to the simpler case of a simple Gaussian distribution.

The \(p\)-th part of the initially produced fast neutrons reaches thermal energy. The distribution of the thermal neutrons being produced from a source of strength equal to unity, placed at na-

in the coordinate origin, has the form:

\[ q_1=-\frac{p}{\pi^{3/2}r_0^3}e^{-\frac{r^2}{r_0^2}}. \tag{10} \]

For graphite with density equal to 1.6, the mean free path \(r_0\) is approximately \(35\ \mathrm{cm}\).

The density of the thermal neutrons being born at the point \(P\) can be expressed through the density of fast neutrons \(Q\) by summing all infinitely small sources \(Q(P')\,d\tau'\) (\(d\tau'\) represents the volume element surrounding the point \(P'\)).

We thus obtain:

\[ q(P)=\frac{p}{\pi^{3/2}r_0^3}\int Q(P')e^{-\frac{(p'-p)^2}{r_0^2}}\,d\tau'. \tag{11} \]

The density of thermal neutrons \(n(x,y,z)\) is connected with \(q\) by the differential equation:

\[ \frac{\lambda v}{3}\Delta n-\frac{vn}{\Lambda}+q=0, \tag{12} \]

where \(\lambda\) is the mean free path (between two collisions) of thermal neutrons, \(v\) is their mean velocity, and \(\Lambda\) is the mean path length of thermal neutrons before capture. Equation (12) represents the balance of all processes leading to the appearance or disappearance of thermal neutrons at a given point. The first term represents the increase in the number of neutrons due to diffusion (the quantity \(\frac{\lambda v}{3}\) is equal to the diffusion coefficient of thermal neutrons). The second term is their loss due to absorption. Finally, the third term represents the number of thermal neutrons being born.

It should be noted that the mean path length of a thermal neutron before absorption, \(\Lambda\), in equation (12) is considerably shorter than the corresponding quantity \(\Lambda_0\) in pure graphite. Indeed, thermal neutrons in the lattice are captured predominantly by uranium nuclei. In a first approximation, \(\Lambda\) is expressed through \(\Lambda_0\) as follows:

\[ \Lambda=\Lambda_0(1-f). \tag{13} \]

For values encountered in practice, the quantity \(\Lambda\) is about \(300\ \mathrm{cm}\), whereas \(\Lambda_0\) in pure graphite is about \(2500\ \mathrm{cm}\). When one thermal neutron is absorbed in uranium, as a result of fission \(\eta\) new neutrons arise. This quantity must also be increased by several percent in order to take into account the effect of fission by fast neutrons. Let \(\varepsilon\eta\) represent the thus corrected number of fast neutrons arising as a result of fission. The number of thermal neutrons absorbed per unit volume per \(1\ \mathrm{sec}\) is equal to \(\frac{vn}{\Lambda}\). Of this number, the fraction \(f\) is captured by uranium nuclei.

We therefore find

\[ Q=f\varepsilon\eta\,\frac{vn}{\Lambda}+Q_0, \tag{14} \]

where \(f\varepsilon\eta\,\dfrac{v}{\Lambda}\) represents the number of fast neutrons arising in the course of the chain reaction, and \(Q_0\) is the number of fast neutrons from an external source (if there is one). In most cases, however, \(Q_0\) is equal to zero.

From equations (11), (12), and (14) one can eliminate all unknown quantities except \(n\). Then we have:

\[ \frac{3}{\lambda\Lambda}n-\Delta n = \frac{3r\varepsilon\eta f}{\pi^{3/2}r_0^3\lambda\Lambda} \int n(P')\,e^{-\frac{(p'-p)^2}{r_0^2}}\,d\tau' + \frac{3p}{\pi^{3/2}r_0^3\lambda\Lambda} \int Q_0(P')\,e^{-\frac{(p'-p)^2}{r_0^2}}\,d\tau'. \tag{15} \]

The solution of equation (15) can be obtained by expanding \(Q_0\) and \(n\) in Fourier series.

If \(Q_0\) can be represented in the form \(Q_0\sin\omega_1x\cdot\sin\omega_2y\cdot\sin\omega_3z\), then the general expression for \(n\) has the form:

\[ n= \frac{\lambda pQ_0}{v}\, \frac{\sin\omega_1x\,\sin\omega_2y\,\sin\omega_3z} {\left(1+\frac{\lambda\Lambda}{3}\omega^2\right)e^{-\frac{\omega^2 r_0^2}{4}}-\varepsilon pf\eta} \tag{16} \]

where

\[ \omega^2=\omega_1^2+\omega_2^2+\omega_3^2. \]

If the dimensions of the boiler are finite, but very large in comparison with the mean free path, then the boundary conditions of the problem may be formulated as the requirement that the neutron density vanish at the outer surface of the boiler.

Thus, for example, if the boiler is a cube with side \(a\) and the origin of coordinates is placed at one of the corners of the cube, the quantities \(\omega_1\), \(\omega_2\), and \(\omega_3\) are equal to:

\[ \omega_1=\frac{\pi n_1}{a},\qquad \omega_2=\frac{\pi n_2}{a},\qquad \omega_3=\frac{\pi n_3}{a}, \tag{17} \]

where \(n_1\), \(n_2\), and \(n_3\) are positive integers (the numbers of the corresponding Fourier components).

The critical size of the system can be found from the condition that the denominator in expression (16), taken for the first harmonic \((1,1,1)\), vanish, since in this case the neutron density increases without bound.

Therefore the critical conditions are determined by the equation

\[ \left(1+\frac{3\pi^2}{a^2}\frac{\lambda\Lambda}{3}\right) e^{-\frac{3\pi^2}{a^2}\frac{r_0^2}{4}} = \varepsilon pf\eta. \tag{18} \]

E. FERMI

On the right-hand side of equation (18) there is nothing other than the multiplication coefficient \(k\). Therefore (18) can be rewritten in the form

\[ \left(1+\frac{\pi^2\lambda\Lambda}{a^2}\right)e^{\frac{3\pi^2 r_0^2}{4a^2}}=k. \tag{19} \]

In most cases the quantity \(\dfrac{3\pi^2 r_0^2}{4a^2}\), standing in the exponent, and the second term in the parentheses are small in comparison with unity. Therefore expression (19) can be simplified by rewriting it in the form

\[ k=1+\frac{3\pi^2}{a^2}\left(\frac{\lambda\Lambda}{3}+\frac{r_0^2}{4}\right). \tag{20} \]

With the aid of the last formula one can determine the critical size of a cubical boiler. If, for example, the following numerical values are adopted: \(\lambda=2.6\ \mathrm{cm}\), \(\Lambda=350\ \mathrm{cm}\), \(r_0^2=1200\ \mathrm{cm}^2\), and \(k=1.06\), then from (20) we find for the critical size of the cubical boiler \(a=584\ \mathrm{cm}\).

Naturally, the values of the constants given above are hypothetical in character. The actual values of the constants depend substantially on the structure of the lattice, although, of course, they can vary only within a certain interval.

It is useful to derive an approximate relation between the power of the boiler and the density of thermal neutrons at the center of the boiler.

Approximately \(50\%\) of the thermal neutrons absorbed in the boiler produce fission. In each act of fission an energy of the order of \(200\ \mathrm{MeV}\) is released. This corresponds to the liberation of approximately \(1.6\cdot 10^{-4}\) ergs for each captured thermal neutron.

Since, in a unit volume, in 1 sec. \(\dfrac{vn}{\Lambda}\) thermal neutrons are absorbed, the energy released is

\[ \frac{vn}{\Lambda}\,1.6\cdot 10^{-4} = 4.6\cdot 10^{-7}vn\ \frac{\mathrm{erg}}{\mathrm{cm}^3\,\mathrm{sec}}. \tag{21} \]

Since the density of thermal neutrons \(n\) has its maximum value at the center of the boiler and decreases to zero at its surface, it is clear that the liberation of energy inside the boiler occurs nonuniformly.

For a cubical boiler \(n\) can be approximately represented in the form

\[ n=n_0\sin\frac{\pi x}{a}\sin\frac{\pi y}{a}\sin\frac{\pi z}{a}, \tag{22} \]

where \(n_0\) is the density of thermal neutrons at the center of the boiler. Integrating expression (21) over the entire volume of the boiler, one can obtain the following formula for the power:

\[ W=\frac{8}{\pi^3}\,4.6\cdot 10^{-7}n_0va^3 = 1.2\cdot 10^{-7}n_0va^3. \tag{23} \]

If \(a\) is set equal to \(584\ \mathrm{cm}\), then \(W=24n_0v\ \dfrac{\mathrm{erg}}{\mathrm{sec}}\). When the reactor operates at a power of \(1\ \mathrm{kW}\), the flux of thermal neutrons at the center of the reactor is

\[ n_0v = 3\cdot 10^8\ \frac{\text{neutrons}}{\mathrm{cm}^2\,\mathrm{sec}}. \]

DESCRIPTION OF THE ARGONNE LABORATORY URANIUM–GRAPHITE REACTOR

The first reactor was started up in 1942 on the grounds of the University of Chicago. After the reactor had operated for several months, it was moved to the Argonne Laboratory near Chicago. It operates there to the present day and serves for a wide variety of research work.

The lattice of the Argonne reactor has a nonuniform structure. Since at the time of its start-up there was an insufficient amount of metallic uranium, metallic uranium was placed only in the central part of the reactor. In the peripheral parts of the reactor the metallic uranium was replaced by uranium oxide.

The operating regime of the reactor is recorded by means of ionization chambers filled with \(\mathrm{BF}_3\) and connected to amplifiers or galvanometers. Since the reactor has no cooling system, its power is limited by the necessity of maintaining a sufficiently low temperature. The reactor can operate indefinitely at a power of \(2\ \mathrm{kW}\).

However, it is often run for a short time, on the order of an hour or two, at a power of about \(100\ \mathrm{kW}\). For research work with neutrons, a graphite column measuring \(5\times 5\) feet is often used; it is built on top of the reactor and shielded on all sides. Neutrons diffuse from the reactor into the column, where they are rapidly slowed down to thermal energy. Practically all neutrons in the column, at a distance of only a few feet above the top of the reactor, are already purely thermal neutrons.

The reactor is also equipped with a number of openings in the shield and with retractable graphite rods, which make it possible to study processes inside the reactor or to introduce samples into it for irradiation with neutrons.

When the reactor operates at a power of \(100\ \mathrm{kW}\), the neutron flux at the center is about

\[ 4\cdot 10^{10}\ \frac{\text{neutrons}}{\mathrm{cm}^2\,\mathrm{sec}}. \]