TheoreticalTask3(T-3):Solutions
A2+1Ea2A1
G(α,θ)==+cosθ=[(1+α)+(1?α)cosθ].
Eb(A+1)2(A+1)22
(OR)
(iv)Fromde?nitionofcenterofmassframevm=
9of9
vb
.Afterthecollision,letvandVA+1
bemagnitudeofthevelocitiesofneutronandmoderatoratomrespectivelyintheCMframe.FromconservationlawsintheCMframe,
v=AV
and
11211(vb?vm)2+Avm=v2+AV2.2222
Avbb
Solvinggivesv=AandV=Av.UsinσandUcosσaretheperpendicularandparallel+1+1componentsofU,intheLF,resolvedalongtheinitialdirectionoftheneutronbeforecollision.TransformingthesetotheCOMFgivesUsinσand?Ucosσ+vmastheperpendicularand
2
?2Vvmcosθ.SinceV=vmparallelcomponentsofV.SowegetU2=V2sin2θ+V2cos2θ+vm
2
(1?cosθ).SubstitutingforUfromeq(4)andsimplifyinggiveswegetU2=2vm
2
vaEaA2+2Acosθ+1==.2vbEb(A+1)2
EaA2+12A1
G(α,θ)==+cosθ=[(1+α)+(1?α)cosθ].
Eb(A+1)2(A+1)22
√
A2+2Acosθ+1Note:Wehaveva=vb.Substitutingforva,v,vminvcosθ=vacosθL?vm
A+1
givestherelationbetweenθLandθ,
cosθL=√Acosθ+1
.
A2+2Acosθ+1Treatingtheaboveequationasquadraticincosθgives,
??
2
?sinθL±cosθLA2?sin2θL
cosθ=.
A
ForθL=0?therootwiththenegativesigngivesθ=180?whichisnotcorrectso,
??
cosθLA2?sin2θL?sin2θL
cosθ=.
A
2va
Substitutingtheaboveexpressionforcosθintheexpressionfor2givesanexpressioninterms
vb
ofcosθL??
22
vaEaA+2cosθLA2?sin2θL+cos2θL
==.22vbEb(A+1)