20
1 Energy Release in Nuclear Reactions, Neutrons, Fission, and Characteristics …
1979). In an ingenious experiment carried out about the same time and reported in
the April 8 edition of Nature, Feather (1939) reported that the fission process must
take place within a time of no more than about 10
−13 s. To researchers in the field of
nuclear physics, it was apparent in the spring of 1939 that a rapid, extremely energetic uranium-based neutron-initiated-and-maintained chain reaction was at least a
theoretical possibility.
1.7 The Bohr-Wheeler Theory of Fission: The Z 2 /A Limit
Against Spontaneous Fission
Much of the material in this section is adopted from Reed (2003).
Within a few weeks of the discovery of slow-neutron-induced uranium fission,
Niels Bohr published a paper in which he argued that of the two then-known isotopes
of that element (
235 U and
238 U), it was likely to be nuclei of the lighter, much rarer
one that were undergoing fission, whereas nuclei of the heavier isotope would most
probably capture any bombarding neutrons and subsequently decay (Bohr 1939).
Experimental verification of this prediction came in early 1940 when Alfred Nier
separated a small sample of uranium into its constituent isotopes via mass spectroscopy (Nier et al. 1940). In the meantime, Bohr continued with his work on the
theory of nuclear fission in collaboration with John Wheeler of Princeton University,
efforts which culminated with the publication of a landmark paper in the September
1, 1939, edition of The Physical Review (Bohr and Wheeler 1939). In this seminal
work, they reported two important discoveries: (i) That there exists a natural limit
Z
2 /A ~ 48 beyond which nuclei are unstable against disintegration by spontaneous
fission, and (ii) That in order to induce a nucleus with Z
2 /A < 48 to fission, it must be
supplied with a necessary “activation energy,” a quantity also known as the “fission
barrier.” Uranium isotopes fall into this latter situation.
Before proceeding with any calculations, it is important to point out that the Z
2 /A
~ 48 limit is not a hard-and-fast one. Uranium is known to fission spontaneously,
and has Z
2 /A = (92
2 /238) ~ 36. That this can happen is a consequence of quantum
tunneling, a wave-mechanical effect discovered independently by George Gamow
and the team of Ronald Gurney and Edward Condon in (1928). In this effect, a
nucleus can decay by alpha-decay or spontaneous fission even though the process
would be energetically forbidden on the basis of classical mechanics. In such decays,
the characteristic half-lives can be extremely long; for example, the spontaneous
fission half-life for
238 U is nearly 10
16 years (see Sect. 4.2). The Bohr and Wheeler
theory was purely classical, and sets an upper limit beyond which any nucleus will
be expected to essentially instantaneously fission.
Bohr and Wheeler’s calculations are extremely challenging, even for advanced
physics students. We can, however, get some idea of what they did by invoking some
simplifying approximations and by taking some empirical numbers at face value. This
section is devoted to an analysis of the issue of the limiting value of Z
2 /A against
1 Energy Release in Nuclear Reactions, Neutrons, Fission, and Characteristics …
1979). In an ingenious experiment carried out about the same time and reported in
the April 8 edition of Nature, Feather (1939) reported that the fission process must
take place within a time of no more than about 10
−13 s. To researchers in the field of
nuclear physics, it was apparent in the spring of 1939 that a rapid, extremely energetic uranium-based neutron-initiated-and-maintained chain reaction was at least a
theoretical possibility.
1.7 The Bohr-Wheeler Theory of Fission: The Z 2 /A Limit
Against Spontaneous Fission
Much of the material in this section is adopted from Reed (2003).
Within a few weeks of the discovery of slow-neutron-induced uranium fission,
Niels Bohr published a paper in which he argued that of the two then-known isotopes
of that element (
235 U and
238 U), it was likely to be nuclei of the lighter, much rarer
one that were undergoing fission, whereas nuclei of the heavier isotope would most
probably capture any bombarding neutrons and subsequently decay (Bohr 1939).
Experimental verification of this prediction came in early 1940 when Alfred Nier
separated a small sample of uranium into its constituent isotopes via mass spectroscopy (Nier et al. 1940). In the meantime, Bohr continued with his work on the
theory of nuclear fission in collaboration with John Wheeler of Princeton University,
efforts which culminated with the publication of a landmark paper in the September
1, 1939, edition of The Physical Review (Bohr and Wheeler 1939). In this seminal
work, they reported two important discoveries: (i) That there exists a natural limit
Z
2 /A ~ 48 beyond which nuclei are unstable against disintegration by spontaneous
fission, and (ii) That in order to induce a nucleus with Z
2 /A < 48 to fission, it must be
supplied with a necessary “activation energy,” a quantity also known as the “fission
barrier.” Uranium isotopes fall into this latter situation.
Before proceeding with any calculations, it is important to point out that the Z
2 /A
~ 48 limit is not a hard-and-fast one. Uranium is known to fission spontaneously,
and has Z
2 /A = (92
2 /238) ~ 36. That this can happen is a consequence of quantum
tunneling, a wave-mechanical effect discovered independently by George Gamow
and the team of Ronald Gurney and Edward Condon in (1928). In this effect, a
nucleus can decay by alpha-decay or spontaneous fission even though the process
would be energetically forbidden on the basis of classical mechanics. In such decays,
the characteristic half-lives can be extremely long; for example, the spontaneous
fission half-life for
238 U is nearly 10
16 years (see Sect. 4.2). The Bohr and Wheeler
theory was purely classical, and sets an upper limit beyond which any nucleus will
be expected to essentially instantaneously fission.
Bohr and Wheeler’s calculations are extremely challenging, even for advanced
physics students. We can, however, get some idea of what they did by invoking some
simplifying approximations and by taking some empirical numbers at face value. This
section is devoted to an analysis of the issue of the limiting value of Z
2 /A against