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- Release of the Binding Energy - If a massive nucleus like uranium-235 breaks apart (nuclear fission), then there will be a net yield of energy because the sum of the masses of the fragments will be less than the mass of the uranium nucleus. If the mass of each fragment is equal to or greater than that of iron at the peak of the binding energy curve (see Figure 14-01), then the products of the decay will be bound more tightly than they were in the uranium nucleus, and that decrease in mass comes off in the form of energy according to the Einstein's equation E = mc
^{2}, where m is the loss of mass in the reaction. For elements lighter than iron it is the process of fusion that releases the binding energy. - Theory of Fission - A stable nucleus is in a state of minimum energy in a potential well with a barrier further out as shown in Figure 14-11a. Classical theory dictates that the system can break apart only when it is excited with energy beyond the barrier height. In quantum theory there is a certain probability to tunnel out under the barrier as shown in Figure 14-11b. The transmission coefficient T for a one dimensional rectangular barrier is given by the formula:

T = 16(p_{1}/p_{2})^{2}e^{-2p2a/}

for p_{2}a/ >> 1, where p_{1}= (2mE)^{1/2}, and p_{2}= [2m(V-E)]^{1/2}.This formula provides a crude approximation in estimating the probability of alpha decay, which depends inversely on the height "V" and width "a" of the barrier. Alpha decay is a process of asymmetric fission because it usually involves a larger nuclear fragment and the much smaller alpha particle. #### Figure 14-11a Nuclear Potential [view large image]

#### Figure 14-11b Quantum Tunneling [view large image]

- Neutron Induced Fission - When an uranium-235 nucleus captures a neutron, the newly formed isotope uranium-236 turns into an excited state with excitation energy about 6 Mev, which is a bit more than the barrier height and splits into two fragments as shown by the equation below (also see illustration on top of Figure 14-11a, and schematic in Figure 14-11c). Actually, the products in the fission process can be any two in each of the humps shown by Figure 14-11d as long as charge conservation is observed (thus called symmetric fission). There are about 90 different isotopes that can be produced by the fission. The most probable and radioactive product is iodine-131 (half life 8 days), that's why the run on iodine pills and iodized salts during the Fukushima incident. They are supposed to prevent further intake of radioactive iodine by the thyroid. Another dangerous by product is cesium-137, which has a half life of 30 years, but the biological half-life (elimination from the body) is much shorter at about 70 days. Anyway, the following equation is an example of the neutron induced reaction :

_{92}U^{235}+ n_{92}U^{236}_{56}Ba^{144}+_{36}Kr^{89}+ 3n + 166 MevFor uranium-238, the excitation energy is about 1 Mev less, so fission is not possible with slow neutrons; it can take place only for neutrons with 1 Mev energy or more. An isotope like uranium-235 that can be split by both slow and fast neutrons, is called "fissile", while uranium-238 which can be split only by fast neutrons, is called "fissionable". #### Figure 14-11c Neutron Induced Fission and Chain reaction [view large image]

#### Figure 14-11d Symmetric Fission [view large image]

- Cross Section - The neutron released immediately (about 10
^{-14}sec) in the fission process is called prompt neutron. There are three possibilities for further reaction for such neutron (in descending possibility):

Thus, "hydrogenic materials" are used in nuclear reactor to slow down the neutrons for producing more power. However, slow neutrons are not suitable for nuclear weapons because the explosion is over in microseconds before they have a chance to participate in the fission process.- It bounces off the nucleus by elastic scattering, which occurs 5 times more often than the next process.
- It induces fission of another nucleus.
- The least probable is for the neutron to be captured with the emission of radiation or slower neutron (inelastic scattering).

^{-24}cm^{2}). The cross section for the various process is the fraction that leads to such process times the geometrical cross section. Figure 14-11e show the fission cross section as a function of neutron energy. The astonish U^{235}#### Figure 14-11e Fission Cross Section [view large image]

fission cross section of 640x10 ^{-24}cm^{2}for slow neutron (at 0.025 ev) is due to quantum effect. While the threshold of 1 Mev for U^{238}has been explained already above. - Chain Reaction and Critical Mass - According to Figure 14-11c one fission produces 3 neutrons to initiate more fissions. Since not all the neutrons in such reactions are available for fission, let's take 2 neutrons as example (actually a good choice). This means that in the next generation 4 neutrons are produced and in the next, 8, and so on and on - a chain reaction. After 80 generations the number of neutrons has grown to 2
^{80}~ 10^{24}~ the number of U^{235}nuclei in 1 kg with each fission releasing about 166 Mev energy for a total of 1.66x10^{26}Mev = 2.65x10^{13}joules, which corresponds to the explosive energy of 10,000 ton of TNT (in comparison, 2.5 tons of high explosives is used to blow up the Murrah Building in Oklahoma City). However, this amount of energy from 1 kg U^{235}would not be released because it has a radius of only 2.3 cm (assuming a spherical shape), all the induced neutrons would have escaped the assembly before initiating more fissions. That is because the mean free path of neutron in U^{235}is equal to 16.5 cm.

Table 14-02 displays the critical (bare) mass, half-life, number of neutrons generated in spontaneous fission, and the rate of heat generation by radioactive decay for various fissionable isotopes. They all undergo transmutation via alpha decay. For comparison, the table also includes the two major fertile materials, Thorium-232 and Uranium-238, which in the presence of neutrons can produce the fissionable isotopes Uranium-233 and Plutonium-239, respectively.Therefore, chain reaction can be maintained only when the diameter of the sphere is at least equal to that length. From the known density of 18.9 gm/cm ^{3}for U^{235}, the critical mass is 44 kg. The arithmetic to produce all these numbers is very simple via the formula:

M = [(4/3)R^{3}]

where is the density, R the radius, and M the mass of the isotope. The time scale can be estimated from: (# of mean-free-path)/(average velocity of fast neutrons ~ 2x10^{9}cm/sec). See Figure 14-11f for the neutron spectrum from the fission process. Prompt neutron is the one released together with the fission.#### Figure 14-11f Neutron Spectrum [view large image]

Fissionable Isotope Crtiical Mass (kg) Half Life (years) Neutron Generation

(# / sec-kg)Power Generation

(Watts / kg)Protactinium-231 162 3.28x10 ^{4}nil 1.3 Thorium-232 Infinite 1.41x10 ^{10}nil nil Uranium-233 ~ 8 1.59x10 ^{5}1.23 0.281 Uranium-235 47.9 7.0x10 ^{8}0.364 6x10 ^{-5}Uranium-238 Infinite 4.5x10 ^{9}0.11 8x10 ^{-6}Neptunium-237 59 2.14x10 ^{6}0.139 0.021 Plutonium-238 10 88 2.67x10 ^{6}560 Plutonium-239 10.2 2.41x10 ^{4}21.8 2.0 Plutonium-240 36.8 6.54x10 ^{3}1.03x10 ^{6}7.0 Plutonium-241 12.9 14.7 49.3 6.4 Plutonium-242 89 3.76x10 ^{5}1.73x10 ^{6}0.12 Americium-241 57 433 1540 115 Americium-242 9 - 18 - - - Americium-243 155 7.38x10 ^{3}900 6.4 Curium-244 28 18.1 1.1x10 ^{10}2.8x10 ^{3}Curium-245 13 8.5x10 ^{3}1.47x10 ^{5}5.7 Curium-246 84 4.7x10 ^{3}9x10 ^{9}10 Curium-247 7 1.55x10 ^{7}- - Berkelium-247 10 1.4x10 ^{3}nil 36 Californium-251 9 898 nil 56 #### Table 14-02 Critical Mass for Fissionable Isotopes

A nuclear device with mass close to the critical limit would most probably produce a "fizzle" with the chain reaction terminated prematurely in less than 1 microsecond by an explosion, which is sizable by most standards but not in nuclear scale. A workable nuclear bomb requires special design such as adding more fissile material to make it super-critical, increasing the density (i.e., to decrease the mean free path) through implosion, or using tamper to retard the expansion. These considerations lead to the topic of nuclear bomb construction in the next section.

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