Lecture 1 2 Introduction to Atomic Spectroscopy o

Lecture 1 -2: Introduction to Atomic Spectroscopy o Line spectra Bulb o Emission spectra Sun o Absorption spectra Na o Hydrogen spectrum o Balmer Formula Emission spectra H Hg Cs o Bohr’s Model Absorption spectra Chlorophyll Diethylthiacarbocyaniodid Diethylthiadicarbocyaniodid PY 3 P 05

Types of Spectra o Continuous spectrum: Produced by solids, liquids & dense gases produce - no “gaps” in wavelength of light produced: o Emission spectrum: Produced by rarefied gases – emission only in narrow wavelength regions: o Absorption spectrum: Gas atoms absorb the same wavelengths as they usually emit and results in an absorption line spectrum: PY 3 P 05

Emission and Absorption Spectroscopy Gas cloud 3 1 2 PY 3 P 05

Line Spectra o Electron transition between energy levels result in emission or absorption lines. o Different elements produce different spectra due to differing atomic structure. H He C PY 3 P 05

Emission/Absorption of Radiation by Atoms o Emission/absorption lines are due to radiative transitions: 1. Radiative (or Stimulated) absorption: Photon with energy (E = h = E 2 - E 1) excites electron from lower energy level. E =h E 2 E 1 Can only occur if E = h = E 2 - E 1 o Radiative recombination/emission: Electron makes transition to lower energy level and emits photon with energy h ’ = E 2 - E 1. PY 3 P 05

Emission/Absorption of Radiation by Atoms o Radiative recombination can be either: a) Spontaneous emission: Electron minimizes its total energy by emitting photon and making transition from E 2 to E 1. E 2 E 1 E ’ =h ’ Emitted photon has energy E ’ = h ’ = E 2 - E 1 b) Stimulated emission: If photon is strongly coupled with electron, cause electron to decay to lower energy level, releasing a photon of the same energy. E =h E 2 E ’ =h ’ E 1 E =h Can only occur if E = h = E 2 - E 1 Also, h ’ = h PY 3 P 05

Simplest Atomic Spectrum: Hydrogen o In ~1850’s, optical spectrum of hydrogen was found to contain strong lines at 6563, 4861 and 4340 Å. H H 6563 (Å) 4861 H 4340 o Lines found to fall closer and closer as wavelength decreases. o Line separation converges at a particular wavelength, called the series limit. o Balmer found that the wavelength of lines could be written where n is an integer >2, and RH is the Rydberg constant. PY 3 P 05

Simplest Atomic Spectrum: Hydrogen o If n =3, => Å o Called H - first line of Balmer series. o Other lines in Balmer series: Name Transitions Wavelength (Å) H 3 -2 6562. 8 H 4 -2 4861. 3 H 5 -2 4340. 5 Highway 6563 in New Mexico o Balmer Series limit occurs when n . Å PY 3 P 05

Simplest Atomic Spectrum: Hydrogen o Other series of hydrogen: Lyman UV nf = 1, ni 2 Balmer Visible/UV nf = 2, ni 3 Paschen IR nf = 3, ni 4 Brackett IR nf = 4, ni 5 Pfund IR nf = 5, ni 6 o Rydberg showed that all series above could be reproduced using Series Limits o Series limit occurs when ni = ∞, nf = 1, 2, … PY 3 P 05

Simplest Atomic Spectrum: Hydrogen o Term or Grotrian diagram for hydrogen. o Spectral lines can be considered as transition between terms. o A consequence of atomic energy levels, is that transitions can only occur between certain terms. Called a selection rule. Selection rule for hydrogen: n = 1, 2, 3, … PY 3 P 05

Bohr Model for Hydrogen o Simplest atomic system, consisting of single electron-proton pair. o First model put forward by Bohr in 1913. He postulated that: 1. Electron moves in circular orbit about proton under Coulomb attraction. 2. Only possible for electron to orbits for which angular momentum is quantised, ie. , n = 1, 2, 3, … 3. Total energy (KE + V) of electron in orbit remains constant. 4. Quantized radiation is emitted/absorbed if an electron moves changes its orbit. PY 3 P 05

Bohr Model for Hydrogen o Consider atom consisting of a nucleus of charge +Ze and mass M, and an electron on charge -e and mass m. Assume M>>m so nucleus remains at fixed position in space. o As Coulomb force is a centripetal, can write (1) o As angular momentum is quantised (2 nd postulate): o Solving for v and substituting into Eqn. 1 => n = 1, 2, 3, … (2) o The total mechanical energy is: n = 1, 2, 3, … (3) o Therefore, quantization of AM leads to quantisation of total energy. PY 3 P 05

Bohr Model for Hydrogen o Substituting in for constants, Eqn. 3 can be written and Eqn. 2 can be written e. V where a 0 = 0. 529 Å = “Bohr radius”. o Eqn. 3 gives a theoretical energy level structure for hydrogen (Z=1): o For Z = 1 and n = 1, the ground state of hydrogen is: E 1 = -13. 6 e. V PY 3 P 05

Bohr Model for Hydrogen o The wavelength of radiation emitted when an electron makes a transition, is (from 4 th postulate): or (4) where o Theoretical derivation of Rydberg formula. o Essential predictions of Bohr model are contained in Eqns. 3 and 4. PY 3 P 05

Correction for Motion of the Nucleus o Spectroscopically measured RH does not agree exactly with theoretically derived R∞. o But, we assumed that M>>m => nucleus fixed. In reality, electron and proton move about common centre of mass. Must use electron’s reduced mass ( ): o As m only appears in R∞, must replace by: o It is found spectroscopically that RM = RH to within three parts in 100, 000. o Therefore, different isotopes of same element have slightly different spectral lines. PY 3 P 05

Correction for Motion of the Nucleus o Consider 1 H (hydrogen) and 2 H (deuterium): cm-1 o Using Eqn. 4, the wavelength difference is therefore: o Called an isotope shift. o H and D are separated by about 1Å. o Intensity of D line is proportional to fraction of D in the sample. Balmer line of H and D PY 3 P 05

Spectra of Hydrogen-like Atoms o Bohr model works well for H and H-like atoms (e. g. , 4 He+, 7 Li 2+, 7 Be 3+, etc). o Spectrum of 4 He+ is almost identical to H, but just offset by a factor of four (Z 2). o For He+, Fowler found the following in stellar spectra: Z=1 H 0 20 13. 6 e. V n 2 1 Z=2 He+ n 3 2 Z=3 Li 2+ n 4 3 2 o See Fig. 8. 7 in Haken & Wolf. Energy (e. V) 40 60 54. 4 e. V 1 80 100 120 1 122. 5 e. V PY 3 P 05

Spectra of Hydrogen-like Atoms o Hydrogenic or hydrogen-like ions: o o He+ (Z=2) Li 2+ (Z=3) Be 3+ (Z=4) … Z=1 H 0 Hydrogenic isoelectronic sequences 20 13. 6 e. V n 2 1 Z=2 He+ n Z=3 Li 2+ 3 2 n 4 3 2 Energy (e. V) o From Bohr model, the ionization energy is: 40 60 54. 4 e. V 1 80 100 E 1 = -13. 59 Z 2 e. V 120 1 122. 5 e. V o Ionization potential therefore increases rapidly with Z. PY 3 P 05

Implications of Bohr Model o We also find that the orbital radius and velocity are quantised: and o Bohr radius (a 0) and fine structure constant ( ) are fundamental constants: and o Constants are related by o With Rydberg constant, define gross atomic characteristics of the atom. Rydberg energy RH 13. 6 e. V Bohr radius a 0 5. 26 x 10 -11 m Fine structure constant 1/137. 04 PY 3 P 05

Exotic Atoms o Positronium o electron (e-) and positron (e+) enter a short-lived bound state, before they annihilate each other with the emission of two -rays (discovered in 1949). o Parapositronium (S=0) has a lifetime of ~1. 25 x 10 -10 s. Orthopositronium (S=1) has lifetime of ~1. 4 x 10 -7 s. o Energy levels proportional to reduced mass => energy levels half of hydrogen. o Muonium: o Replace proton in H atom with a meson (a “muon”). o Bound state has a lifetime of ~2. 2 x 10 -6 s. o According to Bohr’s theory (Eqn. 3), the binding energy is 13. 5 e. V. o From Eqn. 4, n = 1 to n = 2 transition produces a photon of 10. 15 e. V. o Antihydrogen: o Consists of a positron bound to an antiproton - first observed in 1996 at CERN! o Antimatter should behave like ordinary matter according to QM. o Have not been investigated spectroscopically … yet. PY 3 P 05

Failures of Bohr Model o Bohr model was a major step toward understanding the quantum theory of the atom - not in fact a correct description of the nature of electron orbits. o Some of the shortcomings of the model are: 1. Fails describe why certain spectral lines are brighter than others => no mechanism for calculating transition probabilities. 2. Violates the uncertainty principal which dictates that position and momentum cannot be simultaneously determined. o Bohr model gives a basic conceptual model of electrons orbits and energies. The precise details can only be solved using the Schrödinger equation. PY 3 P 05

Failures of Bohr Model o From the Bohr model, the linear momentum of the electron is o However, know from Hiesenberg Uncertainty Principle, that o Comparing the two Eqns. above => p ~ n p o This shows that the magnitude of p is undefined except when n is large. o Bohr model only valid when we approach the classical limit at large n. o Must therefore use full quantum mechanical treatment to model electron in H atom. PY 3 P 05

Hydrogen Spectrum o Transitions actually depend on more than a single quantum number (i. e. , more than n). o Quantum mechanics leads to introduction on four quntum numbers. o Principal quantum number: n o Azimuthal quantum number: l o Magnetic quantum number: ml o Spin quantum number: s o Selection rules must also be modified. PY 3 P 05

Atomic Energy Scales Energy scale Energy (e. V) Effects Gross structure 1 -10 electron-nuclear attraction Electron kinetic energy Electron-electron repulsion Fine structure 0. 001 - 0. 01 Spin-orbit interaction Relativistic corrections Hyperfine structure 10 -6 - 10 -5 Nuclear interactions PY 3 P 05