Rutherford Scattering & Size of Nucleus distance of closest - PDF document

Brian Wecht, the TA is back! Pl. give all regrade requests to him Quiz 4 is This Friday : Chapter 3 Emphasis will be On the Rutherford Scattering & The Bohr Atom 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 = 8 m v 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 4.9 10 r m K α nucleus - 15 Size of Nucleus = 10 m -10 Siz e of Ato m = 1 0 m

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 = − > = ⎜ ⎟ , , 1,2,3,4.. R n n n ⎜ ⎟ λ f i i 2 2 n n ⎝ ⎠ f i Fitting to spectral line series da ta − × 7 1 Rydberg Constant R=1.09737 10 m How does one explain this ? Bohr’s Bold Model of Atom: Semi Quantum/Classical -e 1. Electron in circular orbit m e around proton with vel=v 2. Only stationary orbits allowed . Electron does not F V radiate when in these stable +e (stationary) orbits 3. Orbits quantized: r M e v r = n h/2 π (n=1,2,3…) – 4. Radiation emitted when electron “jumps” from a stable orbit of higher energy 2 e � stable orbit of lower = − ( ) U r k r energy E f -E i = hf =hc/ λ 5. Energy change quantized 1 = 2 KE m v • f = frequency of radiation e 2

Reduced Mass of 2-body system -e m e General Two body Motion under a central force F V +e reduces to r m e Both Nucleus & e - revolve around their common center of mass (CM) • Such a system is equivalent to single particle of “reduced mass” µ that • revolves around position of Nucleus at a distance of (e - -N) separation ฀ µ= ( m e M)/(m e +M), when M>>m, µ= m (Hydrogen atom) ฀ Ν ot so when calculating Muon (m µ = 207 m e ) or equal mass charges rotating around each other (similar to what you saw in gravitation) Allowed Energy Levels & Orbit Radii in Bohr Model 2 Radius of Electron Orbit : 1 e − 2 E=KE+U = m v k r = e � 2 mvr n Force Equality for Stable Orbit � n ⇒ = , v ⇒ Coulomb attraction = CP Force mr 2 2 2 1 ke e m v = = 2 substitute in KE= 2 m v e k r e 2 2 r r 2 � 2 n 2 2 m v e ⇒ = = ∞ ⇒ = = , 1 ,2 ,.... r n e K E k n 2 mk e 2 2 r = ⇒ 1 B ohr Radius n a 2 e 0 Total En erg y E = KE+U= - 2 k r 2 � 2 1 − = = × 10 0.529 10 a m 0 2 ⇒ mk e Negative E Bound sy stem = = ∞ 2 In ge neral ; 1 ,2,... . r n a n Thi s much energy must be added to n 0 Quantized orbits of rotat io n the system to break up the bound atom

Energy Level Diagram and Atomic Transitions − 2 ke = + = E K U n 2 r = 2 since , n =quantum number r a n n 0 − 2 13.6 ke = = − = ∞ , 1, 2, 3.. E eV n n 2 2 2 a n n 0 → Interstate transition: n n i f ∆ = = − E h f E E i f ⎛ ⎞ − 2 1 1 ke = − ⎜ ⎟ ⎜ ⎟ 2 2 2 a n n ⎝ ⎠ 0 i f ⎛ ⎞ 2 1 1 ke = − ⎜ ⎟ f ⎜ ⎟ 2 2 2 ha ⎝ n n ⎠ 0 f i ⎛ ⎞ 2 1 f ke 1 1 = = − ⎜ ⎟ ⎜ ⎟ λ 2 2 2 c hca n n ⎝ ⎠ 0 f i ⎛ ⎞ 1 1 = R − ⎜ ⎟ ⎜ ⎟ 2 2 n n ⎝ ⎠ f i Hydrogen Spectrum: as explained by Bohr ⎛ ⎞ 2 2 ke Z = −⎜ ⎟ E n 2 2 ⎝ ⎠ a n 0 Bohr’s “R” same as the Rydberg Constant R derived emperically from photographs of the spectral series

Another Look at the Energy levels ⎛ ⎞ 2 2 ke Z = −⎜ ⎟ E n 2 ⎝ 2 ⎠ a n 0 Rydberg Constant Bohr’s Atom: Emission & Absorption Spectra photon photon

Some Notes About Bohr Like Atoms • Ground state of Hydrogen atom (n=1) E 0 = -13.6 eV • Method for calculating energy levels etc applies to all Hydrogen- like atoms � -1e around +Ze – Examples : He + , Li ++ • Energy levels would be different if replace electron with Muons • Bohr’s method can be applied in general to all systems under a central force (e.g. gravitational instead of Coulombic) Q Q M M = → 1 2 1 2 If change ( ) U r k G r r Changes every thing: E, r , f etc "Importance of constants in your life" Bohr’s Correspondence Principle • It now appears that there are two different worlds with different laws of physics governing them – The macroscopic world – The microscopic world • How does one transcend from one world to the other ? – Bohr’s correspondence Principle • predictions of quantum theory must correspond to predictions of the classical physics in the regime of sizes where classical physics is known to hold. when n � ∞ [Quantum Physics] = [Classical Physics]

Atomic Excitation by Electrons: Franck-Hertz Expt Other ways of Energy exchange are also quantized ! Example: • Transfer energy to atom by colliding electrons on it • Accelerate electrons, collide with Hg atoms, measure energy transfer in inelastic collision (retarding voltage) Atomic Excitation by Electrons: Franck-Hertz Expt Plot # of electrons/time (current) overcoming the retarding potential (V) Equally spaced Maxima and minima in I-V curve ∆ E ∆ E Atoms accept only discrete amount of Energy, no matter the fashion in which energy is transffered

Bohr’s Explanation of Hydrogen like atoms • Bohr’s Semiclassical theory explained some spectroscopic data � Nobel Prize : 1922 • The “hotch-potch” of clasical & quantum attributes left many (Einstein) unconvinced – “appeared to me to be a miracle – and appears to me to be a miracle today ...... One ought to be ashamed of the successes of the theory” • Problems with Bohr’s theory: – Failed to predict INTENSITY of spectral lines – Limited success in predicting spectra of Multi-electron atoms (He) – Failed to provide “time evolution ” of system from some initial state – Overemphasized Particle nature of matter-could not explain the wave- particle duality of light – No general scheme applicable to non-periodic motion in subatomic systems • “Condemned” as a one trick pony ! Without fundamental insight …raised the question : Why was Bohr successful? Prince Louise de Broglie • Key to Bohr atom was Angular momentum quantization • Why Quantization mvr = |L| = nh/2 π ? • Invoking symmetry in nature the Prince deBroglie postulated – Because photons have wave and particle like nature � particles must have wave like properties – Electrons have accompanying “pilot” wave (not EM) which guide particles thru spacetime. • Matter Wave : – “Pilot wave” of Wavelength λ = h / p = h / ( γ mv) – frequency f = E / h • If matter has wave like properties then there would be interference (destructive & constructive) • Use analogy of standing waves on a plucked string to explain the quantization condition of Bohr orbits