Physics 116 Session 32 Models of atoms Nov 22, 2011 Thomson Rutherford R. J. Wilkes Email: ph116@u.washington.edu
Announcements Exam 3 next week (Tuesday, 11/29) Usual format and procedures I ll post example questions on the website tomorrow afternoon, as usual We ll go over the examples in class Monday 11/28 Enjoy your holiday weekend!
Lecture Schedule (up to exam 3) Today 3
Let s back up a bit: Subatomic discoveries ~100 years ago J. J. Thomson (1897) identifies electron: very light, negative charge E. Rutherford (1911) bounces alpha rays off gold atoms We now know: α = nucleus of helium: 2 protons + 2 neutrons Scattering experiment = model for modern particle physics Size of atoms was approximately known from chemistry He finds: scattering is off a much smaller very dense core (nucleus ) Rutherford s nuclear model of atom: dense, positively charged nucleus surrounded by negatively charged lightweight electrons Niels Bohr (1913): applies Planck/Einstein quanta to atomic spectra Atoms have fixed energy states: they cannot soak up arbitrary energy Quanta are emitted when atom jumps from high to low E state Assumed photon s energy E=hf, as Planck and Einstein suggested Simple model of electrons orbiting nucleus, and classical physics (except for quantized E) gives predictions that match results well (at least, for hydrogen spectrum) Next topics: atoms, nuclei, radioactivity, subatomic particles 4
Back to the puzzles of 1900 Excite a low-pressure sample of noble gas (like neon) with an electric discharge: Pass this light through a slit and prism and you see sharp, separated lines, NOT a continuous rainbow: look closely at spectrum of sunlight and you see dark lines in it wavelength (in Angstroms = 10-10 m) "Holes" in the rainbow? What causes these sharp lines, in both emission and absorption spectra? Boltzmann s thermodynamics + Maxwell s electrodynamics explain only continuous spectra: physical quantities are described by real numbers (decimals) Electric charges in atoms can oscillate at any frequency...emit any wavelength of light 5
Atomic spectra Nice illustration of progress of a science: 1. Masses of data collected ( bug collections ) 2. Empirical rules discovered suggesting underlying regularities 3. Rules lead to models of atomic structure 4. Models lead to a refined theory that (eventually) can explain everything and make predictions of as yet unseen phenomena, to provide a test Theory has to be testable and refutable! (otherwise: speculation) Example of item 2: Hydrogen s line specrum (1885-) Heat hydrogen in a tube and run through a diffraction grating and you see lines with wavelengths that satisfy the rule (Balmer, 1885) 1 λ = R 1 2 1 2 n 2, n = 3, 4, 5K ( R = 1.097 10 7 m 1 ) Outside the visible range, similar series of lines are found, in different EM wavelength regions, named after the rule-finders: 1 λ = R 1 n 1 2 n 2, n = 1, 2, 3K n = ( n + 1), ( n + 2), ( n + 3)K R = Rydberg constant n Series name (range) 1 Lyman (UV) 2 Balmer (visible) 3 Paschen (IR) 6
Early ideas about atoms Atom = concept since Democritus; physical evidence circa 1900 Plum pudding model (J. J. Thomson): electrons are very small negative (q=-e) particles; atoms are larger, and neutral (q=0) perhaps positive charge occupies a blob the size of the atom, and the electrons are like plums in a pudding? Nuclear model (Rutherford, 1911) Alpha-rays (q=+2e) scatter off atoms as if there were a tiny hard core, like a billiard ball: large scattering angles, sometimes even knocked backwards Perhaps positive charge occupies only a small volume in the atom, and most of the mass is in this nucleus? Phosphorescent screen Radioactive mineral in a lead box with a pinhole Gold foil Thomson Rutherford Beam of alpha-rays Rutherford experiment
Bohr s model of the atom Semi-classical synthesis, combines Planck/Einstein quanta with Maxwell/Newton physics Assume (N. Bohr, 1911) 1. Electrons are negative particles, occupying circular orbits around a positively charged nucleus (Rutherford model + classical physics) 2. Only certain orbits are allowed: ones where electron s angular momentum L = integer multiple of hbar (quantized) L n = nh ( h= h /2π) 3. Electrons do not radiate while in stable circular orbits (contrary to Maxwell!) 4. Radiation occurs only when electrons move between allowed orbits, absorbing or releasing energy (quantum jumps) Bohr found this model explained the hydrogen series relationships Assumption 1 means electron speed/momentum depends on radius mv 2 r = ke2 r 2 v 2 = ke2 rm L = ( mv)r v n = L n mr = nh 2πr n m 8
Bohr s model of the atom Assumption 2 defines allowed radii: equate v from assumption 1 with v derived from quantization condition: 2 2 v 2 = ke2 rm = L n nh mr n = 2πr n m r n = n 2 h 2 4π 2 mke 2, n = 1, 2, 3K All the constants above were known fairly well in 1911: r 1 =5.3 x 10-11 m Assumption 4 means allowed radii correspond to energy levels (quantized) E = K + U = mv2 2 kze2 r = kze2 2r kze2 r = 1 2 kze 2 Put in the value of r from above: E n = 2π 2 mk 2 e 4 h 2 Z 2 h 2 = ( 2 13.6eV n )Z 2 n, n = 1, 2, 3K 2 Energy released when electron jumps from one n to another: r Z=number of + charges in nucleus (Z=1 for hydrogen) Negative means we must supply this much energy to extract the electron from the atom E( n i n f )= 2π 2 mk 2 e 4 1 h 2 2 n 1 2 f n i E = hf = hc λ 1 λ = E hc = 2π 2 mk 2 e 4 1 h 3 c n 1 Rydberg constant! 2 2 = 1.097 10 7 m 1 f n i ( ) 1 n f 2 1 n i 2 Bohr explains hydrogen spectra: Lyman series has n f =1, Balmer has n f =2, etc 9
Familiar misleading picture of an atom We ve all seen this Electrons like tiny planets orbiting popcorn-ball nucleus at center You know better Nucleus is tiny (would be invisible on this picture s scale) Particles (protons and electrons) are not really at any point in space probability distribution describes their location You can observe an electron s path, but to do so you must knock it out of the atom! Electron tracks in a cloud chamber (1937) sciencemuseum.org.uk 10
debroglie revisited (this time in context) Einstein says photons simultaneously have wave and particle character Bohr can explain hydrogen spectra with orbiting electrons that have quantized angular momentum and energy De Broglie (1923): if we Assume e s have a wave character on the same basis as photons have particle character: p = h λ for photons λ = h for electrons p Calculate the wavelengths corresponding to Bohr s allowed e orbits Bohr L n = r n mv = p = mv = h = for electrons λ DeBroglie found that Bohr s orbit rules corresponded to having circumference of orbit exactly fit m (integer number) wavelengths! Other radii not allowed because overlapping waves interfere destructively. Semi-classical picture: related quantum facts to well-known classical phenomena debroglie 11
Wave mechanics E. Schrödinger (1927): particles obey a wave equation which can be used to understand subatomic phenomena Wave equation defines behavior of a wave function Example: particle s motion can be described by giving its position and momentum at any time: wave function = Ψ(x, p, t) this means Ψ depends on x, p and time Mathematical form ensures proper wavelike behavior of particles Interference effects (constructive and destructive) are possible! Wave function contains all information about quantum system (particle, or atom, or nucleus, or whatever) Deep consequence: any question you may ask that cannot be answered by solving the wave equation for a completely-specified wave function has no physical meaning! 12
Analogy to E-M waves (this is a cultural supplement) E-M wave = moving, time-varying electric and magnetic fields We can measure E field amplitude (volts per meter) with special hardware More commonly, we measure intensity of light (energy/sec) Intensity = (amplitude) 2 (this gave Schrödinger a hint! ) For your cultural benefit: look at and compare some wave equations Here is the equation describing waves on a string: 2 2 f 1 f f ( x, t) = vertical displacement at position x along string = 2 2 2 x v t v = wave speed on strn i g These are called Here is the wave equation governing E-M waves: differential equations: they involve partial 2 2 E 1 E Ext (, ) = Eat position x = derivatives 2 2 2 x c t c = light speed (concept from calculus: Here is Schrödinger s wave equation derivative = rate of change) 2 Ψ h Ψ Ψ ( x, t) = Schrodinger wave function = x 2 m t h = Planck ' s consta nt /2π ("h - bar" ) Notice a difference: Schrödinger s is first order equation (no squares) 13
Interpreting Schrodinger s wave function What is the wave function made of? Wave function Ψ is not a physical quantity like momentum or E Has no units, cannot be directly measured or detected Wave function squared gives probability of finding particle at position x (or with momentum p) 0.5 Wavefunction Ψ(x) (has no units!) of a particle, vs position x Probability of finding particle described by Ψ(x) at position x: P(x) = Ψ 2 Psi 0.4 0.3 0.2 0.1 0-0.1-0.2-32 -22-12 -2 8 18 28 Psi 2 0.2 0.15 0.1 0.05 0-0.05-32 -22-12 -2 8 18 28 X X 14