第46届国际物理奥林匹克竞赛(IPhO2015)理论试题与解答(7)

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)

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