9/5/2018. Physics colloquium today -- 9/05/2018 PHY 711 Fall Lecture /05/2018 PHY 711 Fall Lecture 4 3

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1 PHY 7 Classical Mechanics and Mathematical Methods 0-0:50 AM MWF Olin 03 Plan fo Lectue 4: Reading: Chapte F&W. Summay of pevious discussion of scatteing theoy; tansfomation etween la and cente of mass fames. Scatteing theoy in the cente of mass fame; Calculation of the scatteing coss section 3. Coss section fo Ruthefod scatteing PHY 7 Fall Lectue 4 Physics colloquium today PHY 7 Fall Lectue PHY 7 Fall Lectue 4 3

2 90508 PHY 7 Fall Lectue 4 4 Scatteing theoy: detecto PHY 7 Fall Lectue 4 5 Diffeential coss section Nume of detected paticles at pe taget paticle Nume of incident paticles pe unit aea Aea of incident eam that is scatteed into detecto at angle d sin d d sin sin Figue fom Maion & Thoton, Classical Dynamics PHY 7 Fall Lectue 4 6

3 Note: Notion of coss section is common to many aeas of physics including classical mechanics, quantum mechanics, optics, etc. Only in the classical mechanics can we calculate it using geometic consideations d Figue fom Maion & Thoton, Classical Dynamics d d sin sin Note: We ae assug that the pocess is isotopic in PHY 7 Fall Lectue 4 7 Tansfomation etween cente-of-mass and laoatoy efeence fames: (assug that enegy is conseved La ( vs ( v V V sin tan cos m m cos cos m m m m cos m m V V v PHY 7 Fall Lectue 4 8 Diffeential coss sections in diffeent efeence fames d cos d cos m m cos m m m m cos PHY 7 Fall Lectue 4 9 3

4 Diffeential coss sections in diffeent efeence fames whee : m m cos m m m m cos sin tan cos m m xample: suppose m = m sin In this case : tan cos note that PHY 7 Fall Lectue cos 3 Focusing on the cente of mass fame of efeence: Typical two-paticle inteactions Cental potential: V V V D Had sphee: V X 0 D K Coulom o gavitational: V A B Lennad-Jones: V PHY 7 Fall Lectue 4 Fo a continuous potential inteaction in cente of mass efeence fame: el V ( V V ( Need to elate these paametes to diffeential coss section d el mm m m =angula momentum PHY 7 Fall Lectue 4 4

5 el V Tajectoy of elative vecto in cente of mass fame Need to find an equation fo ( PHY 7 Fall Lectue 4 3 Consevation of enegy in the cente of mass fame: d el V( dt Tansfomation of tajectoy vaiales: t ( ( d d d dt dt Hee, constant angula momentum is: d V( dt PHY 7 Fall Lectue 4 4 Solving fo ( ( : d Fom: V ( 4 d V( d V( PHY 7 Fall Lectue 4 5 5

6 PHY 7 Fall Lectue 4 6 ( V d d Special values at lage sepaation ( : v v v v PHY 7 Fall Lectue 4 7 max When the dust cleas: ( ( (, d d V d d V PHY 7 Fall Lectue ( : whee ( 0 max V V d d

7 Relationship etween max and ( max max PHY 7 Fall Lectue 4 9 max V( 0 d d du V( V( u u PHY 7 Fall Lectue 4 0 xample: Diagam of Ruthefod scatteing expeiment PHY 7 Fall Lectue 4 7

8 Scatteing angle equation: whee: ( du V 0 0 V( u u Ruthefod scatteing example: V( =0 du sin 0 u u PHY 7 Fall Lectue 4 Ruthefod scatteing continued : sin cos sin sin d 6 sin PHY 7 Fall Lectue 4 3 sin d 6 sin 4 What happens as 0? Fom wepage: PHY 7 Fall Lectue 4 4 8

9 Oiginal expeiment pefomed with paticles on gold ZZAue ZZA ue 4 80v 60 el PHY 7 Fall Lectue 4 5 Recap of equations fo scatteing coss section in the cente of mass fame of efeence d sin d V( whee is found fom V( PHY 7 Fall Lectue 4 6 9