Nuclear Physics PHY303

6 Fission, fusion and the bomb

From the curve of binding energy per nucleon it can be seen that the most stable form of nuclear matter is as medium mass nuclei.

Thus energy will be released in the fission of a heavy nucleus into two medium mass nuclei and in the fusion of two light nuclei into a single medium mass nucleus.

6.1 Spontaneous and induced fission - We can quantify the conditions under which fission will proceed by examining the process

M(A,Z) M(A1,Z1) + M(A2,Z2)

The Q value (energy release) of this process will be given by

Q = B(A1,Z1) + B(A2,Z2) - B(A,Z)

The mass of constituents cancels out since the total number is the same before and after.

The binding energies can be estimated from the liquid drop model (neglecting pairing and asymmetry terms). As the total number of nucleons A = A1 + A2 remains the same, the difference in binding energy only involves the surface and coulomb terms.

In order to simplify the calculation we assume that in splitting up the nucleus the protons and neutrons divide in the same ratio. That is we assume< BR>
A1/A = Z1/Z = y1 say and A2/A = Z2/Z = y2 with y1 + y2 = 1

The energy released can then be expressed in terms of the surface energy Es and the coulomb energy Ec of the original nucleus (A,Z).

Q = Es(1 - (y1)2/3 - (y2)2/3) + Ec(1 - (y1)5/3 - (y2)5/3)

The maximum Q is found by setting dQ/dy1 = 0 and noting that dy2/dy1 = -1. Not surprisingly the maximum occurs when y1 = y2 = 1/2. Then

Q = 0.37Ec - 0.26Es

Thus on the basis of this simple model fission into two equal nuclei - so called symmetric fission - produces the largest energy output or Q value and the process is exothermic (Q > 0) if Ec/Es > 0.7. This limit is often specified in terms of the fission parameter, x defined as (note that ac and as are the coefficients in the liquid drop model expression for the binding energy)

x = Ec/(2Es) = (Z2/A)/(2as/ac) ~ (Z2/A)/50 > 0.35

Indicating that all nuclei with (Z2/A) > 18 (ie heavier than 90Zr) would release energy in undergoing symmetric fission. However even nuclei well beyond this limit do not fall apart but have very long lifetimes against spontaneous fission. A plot of half-lives against (Z2/A) is shown in the figure below.

The process of fission is suppressed by the presence of a potential barrier. The figure below shows schematically the shape of the barrier as a function of the deformation or fragment separation for different values of the atomic mass number, A.

At large separations the barrier shape is determined by the coulomb potential between the charges of the two nuclei.

At small distortions there is competition between the surface tension which pulls the nucleus back to a more spherical shape and the disruptive coulomb repulsion which is pushing the protons apart.

The liquid drop model can be used to determine the shape of the potential barrier and an example is given below

for quadrupole distortion of the nuclear shape. The figure shows the energy of deformation in units of the surface energy Es for a spherical nucleus plotted against the size of the quadrupole distortion for different values of the fission parameter x. When x > 1 the nuclei are completely unstable with respect to this distortion. At smaller values of x fission by barrier penetration can occur, however we should recall that the transmission factor discussed in alpha particle decay is given by

X = exp(-) where = 2(h/2)-1(2m)1/2 (V(r) - E)1/2dr

Note the dependence of on mass and that in this case the mass is very large. This means that the transmission factor is very small and so explains why the lifetimes are so long.

6.2 Energy released - The Liquid Drop model can be used to estimate the energy release in fission. For 235U it is about 200 MeV per nucleus and it mostly comes from the coulomb energy. The energy released is approximately divided up as follows -

Fragment kinetic energy 165 MeV
Prompt neutrons 5 MeV
Prompt gamma rays 7 MeV
Radioactive decay fragments 25 MeV

The radioactive decay occurs because the fragment nuclei produced have a neutron excess. Recall that heavy nuclei have more neutrons per proton than lighter nuclei and so when a heavy nucleus splits in two, the product nuclei will either have to shed neutrons or transform them into protons in order to achieve a distribution of nucleons which forms a stable nucleus.

The Liquid Drop model has been successfully employed in examining what type of distortion leads to fission. There are certain critical shapes at which a narrow neck between two proto fragments appears. This is known as the scission point. However the model's basic prediction of symmetric fission or equal masses for the fragments is not obeyed for either spontaneous fission or that induced by low energy excitation. It does however appear as the dominant mode for high energy excitation.

It is clear that for low energies some modification of the simple picture is required and this is thought to be due to the effect of individual particle states. At low excitation there is hardly enough energy to drive the two fragments of the nucleus apart and the process of division will only proceed if as much binding energy as possible is transformed into the motion separating them out. Thus the individual nucleons settle into the lowest energy configurations. There is a strong tendency to produce a heavy fragment of A ~ 140 with double magic numbers N = 82 and Z = 50. This trend is fairly clear in the fragment mass distributions shown below.

It is no longer a simple case of barrier penetration but one involving a complicated map of the potential with valleys allowing an easier transition to fission. One such surface for 236U is shown in the next figure.

6.3 Chain reaction and the fission reactor - As indicated above, neutrons are released in the fission process. The absorption of neutrons can lead to the formation of a compound nucleus in an excited state and thereby to fission of the absorbing nucleus. An example is

n + 235U 236U* 139Xe + 95Sr + 2n

and a schematic picture of such a process is shown below.

The induction of fission by neutron absorption and the susequent emission of neutrons in the process of fission leads to the possibility of a self sustaining or chain reaction. However such a reaction cannot occur in natural uranium (0.7% 235U, 99.3% 238U) and we will examine the reason for this by noting the following points from the neutron cross sections for 235U and 238U sketched below.

The particular reaction involved is indicated in brackets in each case.

  1. 235U will fission (n,f) at all energies of the absorbed neutron. It is a FISSILE material.

  2. 238U has a threshold for fission (n,f) at a neutron energy of 1MeV. The difference between these two isotopes of uranium is explained by the presence of the pairing term in the Semi-empirical mass formula.

  3. There is very strong reasonant capture of neutrons (n,) for energies in the range 10-100 eV - particularly in the case of 238U where the cross-section reaches very high values.

  4. The fission neutron energy spectrum (see figure below) peaks at around 1 MeV and at this energy the inelastic cross-section (n,n') in 238U exceeds the fission cross-section. This effectively prevents fission from occuring in 238U.

The fact that the fission cross-section is small at the energy of the fission neutrons and that neutrons are generally absorbed in other processes means that a lump of natural uranium cannot sustain a fission chain reaction.

We can make this more quantitative by examining what happens to the fission neutrons in a lump of natural uranium. Starting with 100 fission neutrons about 98 are captured in 238U and only 8 of these captures result in fission. The remaining 2 neutrons cause fission of 235U. Since each fission produces 2-3 neutrons there will be only about 25 neutrons in the second generation - clearly insufficient to sustain a chain reaction.

To enable the use of fission for power generation there are basically two ways round this limitation.

  1. By enriching the 235U content then sufficient fission of this isotope will occur. For example a 50-50 mix of the two isotopes will sustain a chain reaction with most of the fission events occuring in 235U at neutron energies in the range 0.3 - 2.0 keV. A reactor working in this way is called a FAST REACTOR.

  2. By mixing natural uranium with a material which slows down the neutrons without absorption, sufficient of them will get to low energies where the fission cross-section for 235U is extremely large. Such a slowing down medium is called a moderator. In this case most of the fissions are induced by neutrons with thermal energy (0.025 eV) - hence the name THERMAL REACTOR. We will conclude this section with a brief description of the principle features of each type of fission reactor.

Generally the fuel is packaged in the form of rods which are arranged in a regular lattice inside the moderator. The rods are typically about 2-3 cm in diameter and spaced by about 25 cm. The fuel is contained within a metal sheath or cladding - most commonly stainless steel or alloys of zirconium. This cladding serves to support the fuel mechanically, to prevent the release of radioactive fission products into the coolant stream and to provide extended surface contact with the coolant in order to promote effective heat transfer.

Periodically within the lattice there are control rods containing cadmium or boron. These elements are particularly strong neutron absorbers and the rods can be moved in and out of the core by remote operation in order to control the neutron flux.

The reactor core is usually surrounded by a reflector which consists of moderator without fuel elements. The reflector has the effect of keeping the neutron flux high throughout the active part of the core thereby ensuring efficient fuel consumption. The core is contained within a pressure vessel of welded steel which typically has to withstand a pressure of about 1.55 x107 Pa or 153 bar. Outside the pressure vessel there is a concrete biological shield.

The first example shown below is of a Pressurised Water (PWR) thermal reactor.

This is a very common type and makes use of the excellent properties of water both as a coolant and a moderator. A slight disadvantage of ordinary water is that it does absorb neutrons. This is through the conversion of hydrogen into deuterium. This means that for water moderated reactors the fuel has to be enriched with fissile material (eg 235U or 239Pu).

A limitation of using water to convey the heat from the core is that the highest temperature of liquid water is its so called critical temperature 374°C. This has an effect on the efficiency of energy conversion. Recall that for a Carnot engine the efficiency th is given by

th = 1 - T2/T1

where in this ideal case the heat is received isothermally by the working fluid at T1 and rejected isothermally at T2 - all processes being reversible. No practical power plant operates on a Carnot cycle but this expression serves to illustrate that the higher T1 is so the higher is the efficiency (T2 cannot be lower than the temperature of the earth's atmosphere or oceans). The first land based pressurised water reactor started operation at Shippingport USA in 1957.

An obvious step from the PWR is to allow the water to boil in the reactor core and use the steam thus generated to drive the turbines. The Boiling Water Reactor (BWR) is sketched in the second figure. As with the PWR the fuel has to be enriched but the operating pressure is lower at about 70 bar.

As mentioned above the Fast Reactor has no moderator and consquently has a much smaller core.

The very high power density means that liquid metals have to be used as coolants. Liquid sodium is the most common but it has the disadvantage of becoming radioactive through 23Na(n,)24Na. As well as generating power fast reactors are used for breeding fissile material through

238U(n,)239U (-) 239Np(-) 239Pu.

The content and information presented here is for the academic session 2013-2014.
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