Coulomb elastic scattering“Rutherford scattering”Elastic Scattering † dE t( )dt= 0 dr p t( )dt= 0 dr L t( )dt= 0 r p i= r p fV (r) =qQ4peor Vi= Vf= V •( )= 0Assumptions --elastic scatteringb † r p i † r p fq+q+Q † r p † r r † r r min† b † r p f † r p i † Dr p † Dp = 2 psinq2( )Elastic ScatteringEnergy conservation for the central force problem † E =p22m+qQ4peordE t( )dt=pm˙ p -qQ4peor2˙ r ˙ r p =qQ4peor2ˆ r Newton’s II Law for central force† dE t( )dt=pmqQ4peor2-qQ4peor2˙ r ; p = m˙ r dE t( )dt= 0Total energy in central force fieldrecall…Æ E is a constant of the motionElastic ScatteringAngular momentum conservation for the central force problem † r L =r r ¥r p dr L dt=r ˙ r ¥r p +r r ¥r ˙ p dr L dt= 0…because of the central force field= 0 = 0Æ L is a constant of the motionElastic Scatteringb † r p i † r p fq+q+Q † r p † r r † r r min† b † r p f † r p i † Dr p † Dp = 2 psinq2( ) † Dr p =r F dt0•Ú† Dp =qQ4peo1r2cosbdt0•Ú† a† dr = rdb† ds =drsina ; dt =dsv=rdbv sina† a† r† r† dr† ds † r v † Dp =qQ4peo1r2cosbrdbv sinaÚProjection along† Dˆ pElastic Scatteringb † r p i † r p fq+q+Q † r p † r r † r r min† b † r p f † r p i † Dr p † Dp = 2 psinq2( )† a† Dp =qQ4peo1r2cosbrdbv sina-p2-q2( )p2-q2( )ÚTotal deflection is p-qTherefore,† -p2-q2Ê Ë Á ˆ ¯ ˜ £b£p2-q2Ê Ë Á ˆ ¯ ˜ Note: r, b, v, aare all functionsof timeElastic Scatteringb † r p i † r p fq+q+Q † r p † r r † r r min† b † r p f † r p i † Dr p † Dp = 2 psinq2( )† a† Dp =qQ4peo1r2cosbrdbv sina-p2-q2( )p2-q2( )Ú † dr L dt= 0 Æ pib = pr sinamvob = mvrsina† Dp =qQ4peovobcosbdb-p2-q2( )p2-q2( )ÚElastic Scatteringb † r p i † r p fq+q+Q † r p † r r † r r min† b † r p f † r p i † Dr p † Dp = 2 psinq2( )† a† Dp =qQ4peovobcosbdb-p2-q2( )p2-q2( )Ú† Dp =2qQ4peovobcosq2Ê Ë Á ˆ ¯ ˜ † 2qQ4peovobcosq2Ê Ë Á ˆ ¯ ˜ = 2mvosinq2Ê Ë Á ˆ ¯ ˜ b =qQ4peomvo2cotq2Ê Ë Á ˆ ¯ ˜Elastic Scatteringb † r p i † r p fq+q+Q † r p † r r † r r min† b† a† b =qQ4peomvo2cotq2Ê Ë Á ˆ ¯ ˜ If b = 0, head-on collision -† b =qQ4peo2Tocotq2Ê Ë Á ˆ ¯ ˜ † 12mvo2=qQ4peod qQ4peomvo2=d2Æ b =d2cotq2Ê Ë Á ˆ ¯ ˜ and thereforeElastic Scattering† b =qQ4peo2Tocotq2Ê Ë Á ˆ ¯ ˜ Æ db =qQ4peo4Tocsc2q2Ê Ë Á ˆ ¯ ˜ dqAll particles entering between b and b + db willscatter into q and q + dq. c.f. Krane Fig. 11.8.b † r p i † r p fq+q+Q † r p † r r † r r min† b† adbdqq† ds= 2pbdbElastic Scattering† b =qQ4peo2Tocotq2Ê Ë Á ˆ ¯ ˜ Æ db =qQ4peo4Tocsc2q2Ê Ë Á ˆ ¯ ˜ dq† ds= 2pbdb† ds= 4pqQ4peo4ToÊ Ë Á ˆ ¯ ˜ 2cotq2Ê Ë Á ˆ ¯ ˜ csc2q2Ê Ë Á ˆ ¯ ˜ dq† ds=s q( )dW Æs q( )=ds q( )dWds q( )dW=4pqQ4peo4ToÊ Ë Á ˆ ¯ ˜ 2cotq2Ê Ë Á ˆ ¯ ˜ csc2q2Ê Ë Á ˆ ¯ ˜ dq2psinqdqElastic Scattering† sinq= 2sinq2Ê Ë Á ˆ ¯ ˜ cosq2Ê Ë Á ˆ ¯ ˜ † ds q( )dW=4pqQ4peo4ToÊ Ë Á ˆ ¯ ˜ 2cotq2Ê Ë Á ˆ ¯ ˜ csc2q2Ê Ë Á ˆ ¯ ˜ dq2psinqdq† ds q( )dW=qQ4peo4ToÊ Ë Á ˆ ¯ ˜ 2sin4q2† qQ4peoTo= d† ds q( )dW=d216sin4q2† dYq( )dW= I 0( )Ndxds q(