An interesting insight into the rate at which the energy is released in a fission explosion can be obtained by treating the fission chain as a series of “generations.” Suppose that a certain number of neutrons are present initially and that these are captured by fissionable nuclei; then, in the fission process other neutrons are released.
These neutrons, are, in turn, captured by fissionable nuclei and produce more neutrons, and so on. Each stage of the fission chain is regarded as a generation, and the “generation time” is the average time interval between successive generations. The time required for the actual fission of a nucleus is extremely short and most of the neutrons are emitted promptly. Consequently, the generation time is essentially equal to the average time elapsing between the release of a neutron and its subsequent capture by a fissionable nucleus.
This time depends, among other things, on the energy (or speed) of the neutron, and if most of the neutrons are of fairly high energy, usually referred to as “fast neutrons,” the generation time is about a one-hundred-millionth part (10-8) of a second, i.e., 0.01 microseconds.
It was mentioned earlier that not all the fission neutrons are available for maintaining the fission chain because some are lost by escape and by removal in nonfission reactions.
Suppose that when a nucleus captures a neutron and suffers fission f neutrons are released; let l be the average number of neutrons lost, in one way or another, for each fission. There will thus be f – l neutrons available to carry on the fission chain. If there are N neutrons present at any instant, then as a result of their capture by fissionable nuclei N(f – l) neutrons will be produced at the end of one generation; hence, the increase in the number of neutrons per generation is N(f – l) – N or N(f – l – 1). For convenience, the quantity f – l – 1 , that is, the increase in neutrons per fission, will be represented by x. If g is the generation time, then the rate at which the number of neutrons increases is given by:
Rate of neutron increase
dN/dt = Nx/g.
The solution of this equation is
N = N0ext/g,
where N0 is the number of neutrons present initially and N is the number at a time t later. The fraction t/g is the number of generations which have elapsed during the time t, and if this is represented by n, it follows that
N = N0exn.
If the value of x is known, equation can be used to calculate either the neutron population after any prescribed number of generations in the fission chain, or, alternatively, the generations required to attain a particular number of neutrons. For uranium235, f is about 2.5, l may be taken to be roughly 0.5, so that x, which is equal to f – l – 1, is close to unity; hence, equation may be written as
N ~= N0en or N ~= N010n/2.3. (1.56.1)
According to the data in Table 1.45, it would need 1.45 x 1022 fissions, and hence the same number of neutrons, to produce 0.1 kiloton equivalent of energy. If the fission chain is initiated by one neutron, so that N0 is 1, it follows from equation (1.56.1) that it would take approximately 51 generations to produce the necessary number of neutrons.
Similarly, to release 100 kilotons of energy would require 1.45 x 1025 neutrons and this number would be attained in about 58 generations. It is seen, therefore, that 99.9 percent of the energy of a 100-kiloton fission explosion is released during the last 7 generations, that is, in a period of roughly 0.07 microsecond. Clearly, most of the fission energy is released in an extremely short time period. The same conclusion is reached for any value of the fission explosion energy.
In 50 generations or so, i.e., roughly half microsecond, after the initiation of the fission chain, so much energy will have been released-about 1011 calories-that extremely high temperatures will be attained. Consequently, in spite of the restraining effect of the tamper and the weapon casing, the mass of fissionable material will begin to expand rapidly.
The time at which this expansion commences is called the “explosion time.” Since the expansion permits neutrons to escape more readily, the mass becomes subcritical and the self-sustaining chain reaction soon ends. An appreciable proportion of the fissionable material remains unchanged and some fissions will continue as a result of neutron capture, but the amount of energy released at this stage is relatively small.
To summarize the foregoing discussion, it may be stated that because the fission process is accompanied by the instantaneous liberation of neutrons, it is possible, in principle to produce a self-sustaining chain reaction accompanied by the rapid release of large amounts of energy. As a result, a few pounds of fissionable material can be made to liberate, within a very small fraction of a second, as much energy as the explosion of many thousands of tons of TNT. This is the basic principle of nuclear fission weapons.