Mechanics Physics 151

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Lee McGee
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1 Mechanic Phyic 151 Lecture 7 Scattering Problem (Chapter 3) What We Did Lat Time Dicued Central Force Problem l Problem i reduced to one equation mr = + f () r 3 mr Analyzed qualitative behavior Unbounded, bounded, and circular orbit l V () r = V() r + Condition for table circular orbit mr Defined orbit equation and olved it for the Kepler problem 1 mk El = co( θ θ ) Conic orbit depending on E r l mk Goal for Today Introduce the cattering problem What happen when a particle collide Define cattering cro ection How often a particle get cattered in a given direction How to calculate it from the potential Example 1/r force Rutherford cattering Rainbow cattering 1

2 Scattering Problem Conider unbound motion under central force Particle come from infinity goe to infinity Aume f(r) a r Orbit approache traight line at large r Straight ection A Interaction How are ection A and B related? Straight ection B Why Scattering Problem? Phyical obervation are cattering phenomena Photon cattered by an object Seeing Electron cattered by an object Electron microcope Experiment on microcopic object Electron-nucleu cattering to probe nuclear tructure Neutrino-electron cattering to meaure neutrino energy Claical decription fail with uch target Still a good approximation in many cae Claical framework of decribing cattering ued in QM a well and it more intuitive to undertand Louy Shooter Model Imagine hooting bullet at a mall target Suppoe you have very poor aim Bullet pread uniformly Number of bullet / area / time = intenity I Number of hit will be proportional to the target ize Hit frequency (bullet/ec) Nhit = I σ Target cro ection (m ) Intenity (bullet/m /ec)

Spherical Target Imagine the target i a olid phere We want to know which direction the bullet ricochet Number of bullet ricocheting into olid angle dω around a direction Ω i N = Iσ ( Ω) dω

3 Spherical Target Imagine the target i a olid phere We want to know which direction the bullet ricochet Number of bullet ricocheting into olid angle dω around a direction Ω i N = Iσ ( Ω) dω Differential cro ection (m /tr) Concentrate on the cattering angle Target i round = ha rotational ymmetry σ ( Ω) dω= σ( )inddφ Number of bullet between and + d i π N = dφσ I ( )in d= Iσ( )πind Differential Cro Section Claical mechanic i determinitic Scattering angle i determined by the impact parameter Probability of cattering between d and + d i proportional to the area of thi ring () π d = σ( )π ind σ ( ) = in Abolute value taken becaue d/d might be negative d d d Total Cro Section σ ( ) = in d d Let check if thi matche our idea of the total cro ection Integrating over the entire olid angle π σ = σ( Ω) dω= π in σ( ) d T 4π a = πd = πa Total area of the target 3

4 Central Force Scattering Now conider cattering by general central force How doe relate to? d σ ( ) = We need to know the hape of the orbit in d at large r du mdv(1/ u) Look at the orbit equation u dθ + + l du = Angular momentum l i related to by l = r p = rp inθ = p If we aume V(r) a r p E = T = l = p = me m p Central Force Scattering du 1 dv(1/ u) u dθ + + E du = One can in principle olve thi equation and get u = u( θ,, E) r at θ = u(,, E) = Solve = (, E) Then we can calculate d σ (, E) = in d Orbit equation in term of the impact parameter and the energy E Let look at the orbit we already know Invere-quare force Invere Square Force Conider a repulive 1/r force k k f() r = V() r r r = Ex: electrotatic force between two like-ign charged particle Equation and olution ame a Kepler problem Jut flip the ign of k 1 mk El = co( θ θ ) r l mk Radiu > ε = El 1+ > 1 mk Hyperbola Eccentricity 4

5 Hyperbolic Orbit 1 mk El = co( θ θ ) r l mk Solution i a hyperbola ε > 1 E > 1 1/r > co( θ θ ) < ε Scattering angle i = π Ψ co Ψ= 1/ ε El ε = 1+ > 1 mk Ψ l = me θ θ θ A bit of work E cot = ε 1 = k k = cot E We ve got what we need! Differential Cro Section k = cot E Differential cro ection i d 1 k d σ ( ) = = cot cot in d in E d 1 k 1 = 4 E in Scattering of particle with charge Ze and Z e 4 k = ZZe 1 ZZ e 1 σ ( ) = 4 E in 4 Rutherford cattering: α particle (Z = ) cattered by atomic nuclei with Z Exitence of nuclei in atom Rutherford Scattering Before Rutherford dicovery Electron wa known to exit in matter Poitive charge mut exit in atom, but the ditribution wa unknown 1 ZZ e 1 Meaurement of σ() howed σ ( ) = 4 4 E in Poitive charge of +Ze i in one particle Z Ze 1 e.g. Z particle of +e each would give 4 4 E in Dicovery of atomic nuclei 5

6 Total Cro Section Integrating Rutherford cro ection give π 1 ZZ e 1 σt = σ( Ω) dω= π in d 4 4 E in 4π ZZ e 1 d(in ) = π 3 E = in Becaue electrotatic force i long-range No matter how large i the impact parameter, the particle till get lightly deflected Reality: electrotatic field i hielded by the electron around the nucleu Finite cro ection Rainbow Scattering σ ( ) = in d d Equation for σ() aume that () i ingle-valued True for Rutherford cattering, but not alway If () i not monotonou i di σ ( ) = in d i Sum up for poible At maximum = m d = σ ( ) = d Called rainbow cattering 1 σ ( ) m Rainbow From Phyic 15c You ve probably heard of how rainbow are made But the cattering angle depend on where the light enter the drop If you add up all poible poition, rainbow will be wahed out They lied Real rainbow i made by θ 1 the light that reflected internally θ θ Total deflection i θ = θ 4θ + π 1 θ θ 1 6

7 Rainbow From Phyic 15c = θ 1 4θ + π inθ 1 = inθ1 = ninθ R π ha a minimum around Illuminate a water droplet with uniform light What i the ditribution of light intenity in? A bit difficult problem Covered in Phyic 143a and 151 The anwer: I( ) in d d min =.4 n =1.33 Thi goe to infinity at the turning point there R Rainbow From Phyic 15c Minimum of Sharp peak of intenity I() I ( ) min min R Reflection oberved only at min Thi depend on n, which depend lightly on λ Thi i really how rainbow i created min r < r min r > r min min Attractive Force Repulive force can only catter by Attractive force can do more << π If the potential and the energy are jut right, particle can make multiple turn before emerging Called piraling or orbiting E V () r Orbiting region: E V i mall r varie lowly r 7

8 Summary Dicued cattering problem Foundation for all experimental particle phyic Defined and calculated cro ection Differential cro ection σ ( ) = Rutherford cattering Done with central force problem Next: Rigid Bodie Nhit in = I σ d d 8

Lecture 7 Grain boundary grooving

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