Lecture 32: Impulse and Cross Sections

Transcription

1 Lectue 3: Impulse and Coss Sections Even though we may not know what foces act duing a collision, we can detemine something about those foces fom Newton s Second Law Duing the collision: d( mv) F p dt Fdt d ( mv) t t 1 Fdt ( mv) m v if m is constant The quantity (mv) is called the impulse, and givcn the symbol P The left-hand side is elated to the time-aveage of the foce acting duing the collision, so we have: F avg t P

2 Scatteing Coss Sections We vc seen that if we know the initial velocities, and know the foces involved, we can exactly pedict the esults of the collision In paticle physics we want to evese the pocess i.e., we want to lean about the foces acting between paticles by obseving the esults of a collision Thee s a catch, though: we can t specify the initial state exactly thee s a limit to how well we can aim the incoming paticles We can typically diect a beam of paticles (say, electons o potons) into a block of mateial, o at each othe In each case, we don t know which two paticles ae going to collide, o the initial distance between them

3 In the CM fame, the collision looks like: T 1 θ b T The quantity b is called the impact paamete, and we usually don t know its value fo a collision We typically do contol the initial enegies accuately, and can measue the scatteing angle and final enegies We also know the intensity (numbe of paticles pe unit aea) of the incoming beams

4 So we define a quantity in tems of the things we know o can measue: σ ( θ) Collisions pe paticle that cause scatteing into a solid angle element dω at angle θ Beam intensity Note that the numeato has no dimension, while the denominato has dimensions of 1/aea So, σ(θ) has dimensions of aea, which is why we give it the name coss section As defined, σ(θ) elates to scatteing into a diffeential element of solid angle, so it s called the diffeential coss section

5 The equation fo diffeential coss section is σ ( θ) σ ( θ) dω dn / dω I dn If the foce between the colliding paticles is symmetic about the collision axis (as all cental foces ae), we can integate ove φ to find: dω πsinθdθ Fo paticles of a given enegy, the impact paamete detemines the scatteing angle If collisions with impact paamete between b and b + db esult in scatteing into d Ω, we have: dn IdA I π bdb I

6 Relating the two expessions fo dn gives: I πbdb Iσ ( θ) dω Iσ ( θ) πsinθdθ σ ( θ) b db sinθ dθ Actually, fo most collision foces db/dθ is negative the close the objects come, the geate the scatteing angle Since the coss section should be a positive quantity, we wite: σ ( θ) b sinθ db dθ

7 Coss-section fo a Given Foce We see that calculating the coss section hinges on detemining db/dθ fo a given foce To do this, we etun to what we leaned in ou study of cental foces: Fo the equivalent one-body poblem (i.e., using the educed mass) we have the following elationship between change in angle and change in adius: Θ 1 l/ d ( ) µ E U l /µ Since E is constant, and initally (when the objects ae vey fa apat) U is zeo, we have E T o (the initial kinetic enegy) Also, 1 l lo µ vob µ µ vo b µ Tob

8 So, Θ 1 b µ To / d ( ) ( ) µ To U b µ To /µ 1 b/ d ( ) ( ) 1 U / To b / We can elate Θ to the scatteing angle by letting be infinite and 1 be the distance of closest appoach: Θ Θ θ Θ min π Θ θ

9 Coss Section fo k/ Foces Fo gavity and the electic foce, So we have: U( ) This integal is exactly the same as the one we did in detemining the obits of planets unde gavity. Rathe than do it again, I ll just give the answe: k / / b d Θ ( ) min 1 / / b d k/ T b k T o b min o ( κ / b) k cos Θ ; κ 1 + ( κ / b) To

10 Solving fo b gives: 1 + ( κ / b) cos Θ ( κ / b) ( ) [ ] κ / b 1 cos Θ cos Θ cos Θ ( κ / b) cot Θ sin Θ b κ tan Θ Theefoe, db db d Θ 1 κ sec Θ dθ d Θ dθ κ κ cos Θ cos ( π / θ / ) κ sin ( θ /) Note that this is negative, as we expect

11 New we e set to find the diffeential coss section: b κ κ( κ tan Θ) σ ( θ) sinθ sin ( θ / ) sinθsin ( θ / ) κ cot ( θ / ) sin ( θ /)( sin ( θ /) cos ( θ /)) κ k 4 4 4sin ( θ / ) 16T ( ) o sin θ / This fomula was deived by Ruthefod in 1911 (classical mechanics in the 0 th centuy!) By coincidence, the esult also holds tue fo a quantum mechanical calculation (not tue fo othe types of foces) Note that esult does not depend on sign of k (attactive and epulsive foces have same coss section) Also, coss section deceases as enegy inceases Moe difficult to scatte a fast-moving paticle

12 Total Coss Section Sometimes we want to know the total pobability fo a collision to occu when we fie beams of paticles at a taget (o at each othe). The total coss section is detemined by integating ove the diffeential coss section: σ ( ) ( ) T σ θ dω π σ θ sinθdθ Fo k/ foces we have: π π πκ sinθ ( ) ( ) sin θ / cos θ / σt dθ πκ d 4 4 sin ( /) θ θ sin ( θ /) 0 0 π 1 ( ) cos θ / du πκ dθ πκ ; u sin ( θ /) 3 3 sin ( θ / ) u πκ u 0! But most of this infinite coss section esults in scatteing at vey small angles