Physics 2D Lecture Slides Lecture 14: Feb 1 st 2005 Vivek Sharma UCSD Physics Compton Effect: what should Happen Classically? • Plane wave [f, λ ] incident on a surface with loosely bound electrons � interaction of E field of EM wave with electron: F = e E • Electron oscillates with f = f incident • Eventually radiates spherical waves with f radiated = f incident – At all scattering angles, Δ f & Δλ must be zero • Time delay while the electron gets a “tan” : soaks in radiation 1
Compton Scattering : Summary of Observations Δ λ = λ λ ∝ − θ ' ( - ) (1 cos ) ! Not isotropy in distribution of scatte red radiati n o How does one explain this startling anisotropy? Compton Effect : Quantum (Relativistic) Pool 2
Compton Scattering: Quantum Picture Energy Conservation: φ = − θ p cos p p 'cos e = + 2 E+m c E ' E φ = θ e e p sin p 'sin e Momentum Conserv : ⇒ Square and add θ φ p = p'cos +p cos e = − θ + 2 2 2 p p 2 pp 'cos p ' = θ φ 0 p'sin -p sin e e Eliminate p & using E Use these to e liminate e e = + 2 2 2 2 4 E p c m c & electron deflection e e e = − + 2 E ( E E ') m c angle (n ot measured ) e e Compton Scattering: The Quantum Picture Energy Conservation: = + 2 E+m c E ' E e e Momentum Conserv : θ φ p = p'cos +p cos e = θ φ 0 p'sin -p sin e Use these to e liminate ( ) electron deflection 2 − + = ⎡ − θ + ⎤ + 2 2 2 2 2 ( E E ') m c p 2 pp 'cos p ' ( m c ) ⎣ ⎦ e e angle (n ot measured ) E ⇒ For light p= c ⎡ ⎤ 2 2 ' ' E E E E + − + − = − θ + 2 2 2 2 E E ' 2 EE ' 2( E E ') mc ⎢ 2 co s ⎥ c 2 2 2 ⎣ ⎦ c c c ⇒ − + − = − θ 2 EE ' ( E E ') mc E E 'cos E-E' 1 h ⇒ = − − θ ⇒ λ − λ = − θ (1 cos ) ( ' ) ( )(1 co s ) 2 EE' m c m c e e 3
Rules of Quantum Pool between Photon and Electron h λ − λ = − θ ( ' ) ( )(1 cos ) m c e Checking for h in Compton Scattering Plot scattered photon data, calculate slope and measure “h” It’s the same value for h again !! c m e / h = λ C Δλ h h t λ − λ = − θ g ( ' ) ( )(1 cos ) n e m c l e e v a w n o t p m o Energy Quantization is a C UNIVERSAL characteristic of energy transactions ! 1-cos ϑ 4
Saw what light does, Now examine nature of matter • Fundamental Characteristics of different forms of matter – Rest Mass (m) Reading Assignment, one problem – Electric Charge ( q ) from here may be on the quiz • Measurable – using some combination of E & B fields interacting with the particle � � � � = + × F q E ( v B ) – Or E/B or some other macroscopic force e.g. Drag Force The “magic” is that one is measuring tiny tiny numbers using Macroscopic devices Thomson’s Determination of e/m of the Electron • In E Field alone, electron lands at D • In B field alone, electron lands at E • When E and B field adjusted to cancel each other’s force � electron lands at F � e/m = 1.7588 x 10 11 C/Kg 5
Millikan’s Measurement of Electron Charge Find charge on oil drop is always in integral multiple of some Q q e = 1.688 x 10 -19 Coulombs � m e = 9.1093 x 10 -31 Kg � Fundamental properties (finger print) of electron (similarly can measure proton properties etc) Bragg Scattering photographic film 6
Summary : From X Ray (EM Wave) Scattering data, Size of the Atom was known to be about 10 -10 m Where are the electrons inside the atom? Early Thought: “Plum pudding” model � Atom has a homogenous distribution of Positive charge with electrons embedded in them (atom is neutral) Positively charged e - matter e - + Core e - e - e - e - e - + e - or e - e - e - e - e - e - e - e - e - ? e - e - • How to test these hypotheses? � Shoot “bullets” at the atom and watch their trajectory. What Kind of bullets ? •Indestructible charged bullets � Ionized He ++ atom = α ++ particles •Q = +2e , Mass M α =4amu >> m e , V α = 2 x 10 7 m/s (non-relavistic) [charged to probe charge & mass distribution inside atom] 7
Plum Pudding Model of Atom • Non-relativistic mechanics (V α /c = 0.1) • In Plum-pudding model, α -rays hardly scatter because – Positive charge distributed over size of atom (10 -10 m) – M α >> M e (like moving truck hits a bicycle) – � predict α -rays will pass thru array of atoms with little scatter (~1 o ) Need to test this hypothesis � Ernest Rutherford Probing Within an Atom with α Particles Most α particles pass thru gold foil with nary a deflection • SOME ( ≅ 10 -4 ) scatter at LARGE angles Φ • Even fewer scatter almost backwards � Why • 8
“ Rutherford Scattering” discovered by his PhD Student (Marsden) Rutherford Discovers Nucleus (Nobel Prize) 9
Force on α -particle due to heavy Nucleus α particle trajectory is hyperbolic Scattering angle is related to impact par. •Outside radius r =R, F ∝ Q/r 2 ⎛ ⎞⎛ θ ⎞ kq Q = ⎜ α ⎟⎜ ⎟ •Inside radius r < R, F ∝ q/r 2 = Qr/R 2 Impact Parameter b cot 2 ⎝ ⎠ 2 ⎝ ⎠ m v α α •Maximum force at radius r = R Rutherford Scattering: Prediction and Experimental Result 2 2 4 k Z e NnA Δ = n 2 ⎛ ⎞ 1 ϕ 2 2 4 ⎜ ⎟ 4 R m v Sin ( / 2) α α ⎝ ⎠ 2 •# scattered Vs φ depends on : •n = # of incident alpha particles •N = # of nuclei/area of foil •Ze = Nuclear charge • K α of incident alpha beam •A= detector area 10
Rutherford Scattering & Size of Nucleus ∝ distance of closest appoach r size of nucleus 1 α 2 Kinetic energy of = K = 2 m v α α β α particle will penetrate thru a radius r nucleus until all its kinetic energy is used up to do work AGAINST the Coulomb potent ial of the Nucleus: ( )( ) 1 Ze 2 e = = 2 K = m v 8 MeV k α α β 2 r 2 2 kZe ⇒ = r K α = For K =7.7.MeV, Z 13 α Al 2 2 kZ e − ⇒ = = × 15 r 4.9 10 m K α nucleus - 15 Size of Nucleus = 10 m -10 Siz e of Ato m = 1 0 m Dimension Matters ! -15 Size of Nucleus = 10 m -10 Size of Atom = 10 m •how are the electrons located inside an atom •How are they held in a stable fashion •necessary condition for us to exist ! •All these discoveries will require new experiments and observations 11
Rutherford Atom & Classical Physics ? Continuous & Discrete spectra of Elements 12
Visible Spectrum of Sun Through a Prism Emission & Absorption Line Spectra of Elements 13
Kirchhoff’ Experiment : “D” Lines in Na D lines darken noticeably when Sodium vapor introduced Between slit and prism Emission & Absorption Line Spectrum of Elements •Emission line appear dark because of photographic exposure Absorption spectrum of Na While light passed thru Na vapor is absorbed at specific λ 14
Spectral Observations : series of lines with a pattern • Empirical observation (by trial & error) • All these series can be summarized in a simple formula ⎛ ⎞ 1 1 1 = − > = ⎜ ⎟ R , n n n , 1,2,3,4.. ⎜ ⎟ λ f i i 2 2 n n ⎝ ⎠ f i Fitting to spectral line serie s data − × 7 1 R= 1.09737 10 m How does one explain this ? The Rapidly Vanishing Atom: A Classical Disaster ! Not too hard to draw analogy with dynamics under another Central Force Think of the Gravitational Force between two objects and their circular orbits. Perhaps the electron rotates around the Nucleus and is bound by their electrical charge M M Q Q ⇒ 1 2 1 2 F= G k 2 2 r r Laws of E&M destroy this equivalent picture : Why ? 15
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