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.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

**m**and incident speed

_{1}**u**collides with a stationary nucleus, mass

_{1}**m**, show that for an elastic , head - on collision:

_{2}**(a)**

**m**

_{1}( u_{1}- v_{1}) = m_{2}v_{2}**.**

**(b)**

_{ }m_{1}( u_{1}+ v_{1}) ( u_{1}- v_{1}) = m_{2}v_{2}^{2}**Q4)**From these two equations , show that

^{ }**u**and hence that

_{1}+ v_{1}= v_{2}**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 (m

_{2}/ m

_{1}) for mass ratios from 0 to 7.This should be a

**SMOOTH**curve with no discontinuities.

**Q7)**Letting the ratio of masses be

**R**and the ratio of energies as

_{m}**R**, and writing

_{e}**R**,

_{e}= f ( R_{m})**by differentiating with the product or quotient rule that**

__prove__**R**is a

_{e}**Maximum**when

**R**equals

_{m}**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 ?

**(b)**

^{1}H**(c)**

^{9}Be**(d)**

^{12}C

^{238}UIt 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

**nucleus are needed to 'thermalise' a**

^{12}C**1 MeV**neutron.

**You may find the result**

^{ }**x**then

^{y}= z**y log**useful here.

_{10}x = log_{10}z**Q10)**

__Explain carefully__why the

**Moderator**in a nuclear reactor is made from 'light' water , 'heavy' water (D

_{2}0) 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

**atoms ?**

^{238}U**Q12)**Explain q

__ualitatively__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 =**

**Dmc**, then

^{2})(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

**, about**

^{235}U**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 energy**of 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

**atoms in a**^{235}U**1g**sample.Hence show that this mass of uranium

1 MW - 1 000 000 joules per second.

This is the

**per day**would__produce energy__at the rate of1 MW - 1 000 000 joules per second.

This is the

__equivalent__of 2600 kg (2 600 000 g )of coal !
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