* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download R - University of St Andrews
Density functional theory wikipedia , lookup
Bremsstrahlung wikipedia , lookup
Molecular Hamiltonian wikipedia , lookup
Particle in a box wikipedia , lookup
Elementary particle wikipedia , lookup
Chemical bond wikipedia , lookup
Scalar field theory wikipedia , lookup
Renormalization group wikipedia , lookup
James Franck wikipedia , lookup
History of quantum field theory wikipedia , lookup
Hidden variable theory wikipedia , lookup
X-ray fluorescence wikipedia , lookup
Matter wave wikipedia , lookup
X-ray photoelectron spectroscopy wikipedia , lookup
Quantum electrodynamics wikipedia , lookup
Renormalization wikipedia , lookup
Bohr–Einstein debates wikipedia , lookup
Theoretical and experimental justification for the Schrödinger equation wikipedia , lookup
Wave–particle duality wikipedia , lookup
Atomic orbital wikipedia , lookup
Electron scattering wikipedia , lookup
Tight binding wikipedia , lookup
Electron configuration wikipedia , lookup
Physics of Atoms Donatella Cassettari Email: dc43 Tel: 3109 Room 218 (office) 105 (lab) Course Plan •Atomic theory started around ~1900 with J.J. Thomson’s model of the atom •Succession of atomic models: Rutherford (1911), Bohr (1913), Bohr-Sommerfeld “Old” quantum theory or “semi-classical” theory •Schroedinger equation for the Hydrogen atom (1 electron, simplest case) •Many-electron atoms (Alkali atoms, Helium atom) •Spin-orbit interaction, spin-spin interaction • Atoms in external fields (electric, magnetic, e.m.) • Novel applications: Laser cooling of atoms, Bose-Einstein condensation (BEC) Why is atomic physics interesting? Richard Feynman: “The understanding of the structure of atoms represents the greatest success of the theory of quantum mechanics” •Modern physics is based on the understanding of atoms •Applications: solid state, chemistry, biophysics, astrophysics, lasers…. Material for this course: • Textbook: Haken and Wolf The Physics of Atoms and Quanta 7th ed (2005) (only selected chapters) • These notes! • Tutorial problems • Practice with past exams Models of the atom – a short history J J Thomson’s model ~ 1900 Known that an atom with atomic number Z contains Z electrons “Plum-pudding model”: these electrons are embedded in a continuous spherical distribution of positive charge amounting in total to +Ze. e Diameter of atom is ~0.1nm e e e e e e + This idea though was immediately proved invalid by Rutherford’s expts in 1911 on the scattering of α-particles (He nuclei) by atoms: Measure no. of particles scattered between θ and θ +d θ. In Thomson’s model, the spread out positive charge can’t produce a large deflection, and the electrons are too light to do so. However, a significant fraction (1/10,000) of the α particles is found to be deflected thru angles θ greater than 90 degrees, whereas the theory says this fraction should be negligible. “It was quite the most incredible event that has ever happened to me in my life. It was almost as if you fired a fifteen inch shell at a piece of tissue paper and it came back and hit you.” Rutherford Model 1911 Positive charge is concentrated in a very small nucleus. So αparticles can sometimes approach very close to the charge Ze in the nucleus and the Coulomb force 1 ( Ze)(2e) F= 4πε o r2 Can be large enough to cause large angle deflections. Nuclear model of the atom Detailed calculations now show that there is a non-negligible probability of large angle scattering Stability problem + e e Because the electrons are in orbit, they are continuously being accelerated and therefore, classically, should emit radiation. They should lose energy and spiral into the nucleus! Bohr’s postulates Bohr 1913: circumvented stability problem by making two postulates: 1. An electron in an atom moves in a circular orbit for which the angular momentum is an integral multiple of h 2. An electron in one of these orbits is stable. But if it discontinuously changes its orbit from one where energy is Ei to one where energy is Ef , energy is emitted or absorbed in photons satisfying: Ei − E f = hν Photon frequency Analysis based on Bohr postulates for hydrogen-like atom (1 electron) Force of attraction Ze 2 m0 v 2 F= = 2 4πε o r r Kinetic energy 2 1 1 1 Ze KE = m0 v 2 = 2 2 4πε o r Potential energy 1 r e v +Ze (m 0= electron mass) Ze 2 PE = − 4πε o r 1 Ze 2 Total E = − 4πε o 2r 1 E<0 because it’s a bound state 1st postulate eliminate v from system m0 vr = nh 1 Ze m0 v = 4πε o r 2 2 2 2 2 nh 1 Ze m0 2 2 = 4πε o r m0 r 2 2 nh r = 4πε o 2 m0 Ze 1 2 4 m0 Z e E=− 2 2 2 (4πε o ) 2n h Energy is quantised For convenience we define the “Bohr Radius” as: 2 4πε o h a0 = = 0.0529nm 2 m0 e And the energy unit 1 Rydberg (1Ry) as: 4 m0 e = 13.6eV 2 2 (4πε o ) 2h 2 n r = a0 Z and E= − Z 2 n2 Ry n=1 n=2 n=3 “excited states” “ground state” a 0 = radius of ground state (n=1) orbit for hydrogen (Z=1) 1 Ry=13.6eV is the ground state (n=1) binding energy for hydrogen (agrees with experiment). For larger Z-values, radius of orbit shrinks for given n. Binding energies get much bigger. Emission Spectra Electron makes transition from initial quantum state ni to final state nf . The frequency ν of the photon emitted satisfies: 2 4 1 m0 Z e hν = Ei − E f = 2 2 (4πε o ) 2h 1 1 − n2 n2 i f Express this in terms of wavelength 1 1 2 4 m0 Z e = 2 3 λ (4πε o ) 4πh c 1 1 − n2 n2 i f 1 1 2 = R∞ Z − n2 n2 λ i f 1 1 4 m0 e 7 −1 R∞ = = 1.0974 × 10 m 2 3 (4πε o ) 4πh c is the Rydberg constant There are infinite spectral lines…. Let’s organize them in families. Family = all transitions with the same final state Note that for each family there is a series limit for For the Lyman series, this is at 1 λ = R∞ = 91nm ni → ∞ well within the UV range The Balmer series lies in the near UV and visible region, and the others are all in the infrared. At time of Bohr’s proposal, only Balmer-Paschen series were known, and the remaining series were therefore predicted in advance of the discovery (triumph for Bohr theory). Absorption spectrum General formula above also applies to the case where an electron gains the energy of a suitable photon having energy hν exactly equal to the difference between initial and final states. Normally, electron will start off in ground state so only the Lyman series is observed in absorption spectrum. ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Next, we are going to consider two extensions of the Bohr model: • Finite nuclear mass • Elliptical orbits (Bohr-Sommerfeld model) Finite Nuclear Mass In the theory above, we assumed that the nucleus is so massive that it is effectively at rest. But the mass of the nucleus is finite and the classical model of the atom therefore envisages that the electron and the nucleus both revolve around their common centre of mass. Therefore, any theory should be modified by replacing the mass of the electron by the “reduced mass” of the system. m0 M m0 µ= = m0 + M 1 + m0 M where m0 = electron mass M = nuclear mass 1 1 2 = R.Z − n2 n2 λ i f 1 4 µe R = 2 3 (4πε o ) 4πh c 1 µ R M 1 = = = m0 R∞ m0 m0 + M 1 + M R∞ is the limit of R as M tends to infinity Difference between R and R∞ is small (~ 1 part in 1800) but shows up in precise measurements of spectral lines. For hydrogen 7 −1 7 −1 R = 1.0968 × 10 m Cf. R∞ = 1.0974 ×10 m Isotope shift of spectral lines: It occurs because different isotopes of the same atomic species have different reduced masses, hence different values of R. Example: Positron “atom” =system containing 1 positron (like an e- with a positive charge) and 1 electron. How does emission spectrum differ from hydrogen? Set M=m0 , so R∞ R∞ R= = m0 2 1+ M 1 1 R∞ 1 = − = × λ 2 n 2f ni2 2 1 previous formula All wavelengths are doubled Effect on energy and radii of quantum states? We replace m0 in the formulae above by m0 µ= 2 Therefore, all energies are halved in magnitude, radii doubled. Elliptical orbits (Bohr-Sommerfeld) When viewed at high resolution, transitions split. Transitions between any two Bohr energy states involve several spectral lines. This is known as fine structure. Explanation: each energy level actually consists of several distinct states with almost the same energy. The first theory that justified this was done by Wilson and Sommerfeld: they conjectured that electron orbits can be elliptical, of which a circular orbit is a special case. Each orbit is specified by 2 parameters instead of 1. Geometrically by semi-major and semi-minor axes a,b, no just radius r. Therefore we have now two quantum numbers. e nθ b = n a a +Ze b nθ = “azimuthal” quantum number Thus, energy levels turn out to be dependent on two quantum numbers, but only when one takes relativistic considerations into account. Without relativity, we get the same formula for E as before. Relativistic correction: electrons in very eccentric orbits have large velocities when they are near the nucleus, so v/c is NOT negligible. Each energy level n is split into several sub-levels corresponding to orbits of different eccentricity (i.e. different nθ ) fine structure Final remarks on the early atomic models 1) Bohr’s theory works! It generates the correct formula for the energy levels in a hydrogen atom. This is because the Bohr postulate: m0 vr = nh inadvertently makes use of the de Broglie wave associated with an electron. Supposing the classical orbit is circular, let’s look at that the associated de Broglie wave following the motion: in order for the wave to return to its initial value (i.e. we are requiring that the wave be single valued), we must have 2πr = circumference = nλ de Broglie wavelength p= h h λ= = p m0 v h λ 2πrm0 v = nh m0 vr = nh The Bohr condition 2) Inadequacies of the Bohr Theory •Does well to describe hydrogen, but can be extended only to 1-electron atoms, i.e. hydrogen-like, with higher Z values. •Theory does not explain rate at which transitions occur between states, i.e. the relative intensities of spectral lines. •The theory is ad hoc and lacks a satisfying basis. Superseded by Quantum Mechanics, initiated by de Broglie (1924) and Schroedinger (1926).