## Sunday, 3 January 2010

### Energy And Momentum In Nuclear Reactions

Neutron thermalization is important in the operation of nuclear reactors . A collision may be elastic or inelastic between an incident neutron and a target nucleus. An Inelastic collision results in the emission of gamma rays.

Q1) Explain why gamma rays would be emitted in an inelastic collision.

Inelastic collisions that do not involve capture are important only for relatively fast neutrons (E > 1 MeV) incident on fairly large nuclei. In lighter nuclei, and for less energetic neutrons in all substances, Elastic Scattering is the important process as the mode of energy loss.

Q2) Calculate the rest energy of a neutron . Hence determine the speed of a 1 MeV neutron using (a) A non - relativistic formula for kinetic energy.
(b) A  relativistic energy formula.
Is it necessary to use relativistic mechanics for analysing neutron scattering at this
energy ?

For the operation of nuclear reactors, slow moving neutrons in thermal equilibrium with their surroundings are required. This can be achieved by allowing neutrons to lose energy in elastic collisions. We now develop an analysis of this.

Q3) If a neutron of mass m1 and incident speed u1 collides with a stationary nucleus, mass m2, show that for an elastic , head - on collision:

(a) m1 ( u1 - v1 ) = m2 v2
.
(b) m1 ( u1 + v1 ) ( u1 - v1 ) = m2 v22

Q4) From these two equations , show that u1 + v1 = v2 and hence that

Q5) Hence show that the ratio of Kinetic Energy lost by object 1 in the collision to its initial kinetic energy is given by :

Q6) Plot a graph of the kinetic energy ratio (y - axis) against the mass ratio (m2 / m1) for mass ratios from 0 to 7.This should be a SMOOTH curve with no discontinuities.

Q7) Letting the ratio of masses be Rm and the ratio of energies as Re , and writing
Re = f ( Rm) , prove by differentiating with the product or quotient rule that Re is a Maximum when Rm equals 1 .Hence deduce that the most effective interchange of energy occurs when incident and target particle have equal mass.Now examine your graph again - is it correct ?

Q8) If a neutron collides with the following nuclei, what is the % loss of KE for the neutron in each case ?
(a) 1H (b) 9Be (c) 12C (d) 238U

It should now be clear that a neutron will lose energy most rapidly when scattered by lighter nuclei. 'Thermal' neutrons with an energy of just 0.025 eV are most effective in nuclear rectors as they are most readily absorbed by Uranium atoms , thus inducing fission.

Q9) Using your answer from 8c , calculate how many head-on collisions with a 12C nucleus are needed to 'thermalise' a 1 MeV neutron. You may find the result

If xy = z then y log10 x = log10 z useful here.

Q10) Explain carefully why the Moderator in a nuclear reactor is made from 'light' water , 'heavy' water (D20) or graphite .

Q11) If a moderator was NOT used , how many collisions would be needed per neutron to thermalise them if the target nuclei were 238U atoms ?

Q12) Explain qualitatively why your answers to Q8 are maximum values and your answer to Q9 and Q11 are minimum values.

Once thermal neutrons have been produced, then nuclear fission may begin to occur. It is NOT the impact of the neutron that has this effect. Instead , the low energy thermal neutrons are absorbed by target nuclei. The excited nuclei are unstable and explode immediately - producing high energy neutrons and gamma rays, as in the diagram below:

Q13) If the 'Q' value of a nuclear reaction is defined as the difference between the rest energies of the producs and the reactants, ( Q = Dmc2 ) , then

(i) Calculate the Q value of the reaction shown in the diagram.

(ii) Calculate the possible range of energies and frequencies of the gamma rays produced.

Actually, in the fission of 235U, about 83% of the energy appears as the kinetic energy of the fragments, about 2.5% as kinetic energy of the neutrons and about 3.5% in the form of instantly emitted gamma rays. 11% is given off in the subsequent decays of the daughter nuclei.

Q14) If the relativistic kinetic energyof a particle is given by the formula

then calculate the kinetic energy of ONE of the emitted neutrons, the speed at which it is travelling and the number of head on collisions with moderator atoms that would be needed to thermalise it.

Q15) A reactor is a very efficient source of energy.
Calculate the number of 235U atoms in a 1g sample.
Hence show that this mass of uranium per day would produce energy at the rate of
1 MW - 1 000 000 joules per second.
This is the equivalent of 2600 kg (2 600 000 g )of coal !

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