**Lecture 1-2: Introduction to Atomic Spectroscopy** • Line spectra • Emission spectra • Absorption spectra • Hydrogen spectrum • Balmer Formula • Bohr’s Model Bulb Sun Na H Emission spectra Hg Cs Chlorophyll Diethylthiacarbocyaniodid Absorption spectra Diethylthiadicarbocyaniodid PY3P05

**Types of Spectra** • Continuous spectrum: Produced by solids, liquids & dense gases produce - no “gaps” in wavelength of light produced: • Emission spectrum: Produced by rarefied gases – emission only in narrow wavelength regions: • Absorption spectrum: Gas atoms absorb the same wavelengths as they usually emit and results in an absorption line spectrum: PY3P05

**Emission and Absorption Spectroscopy** Gas cloud 3 1 2 PY3P05

**Line Spectra** • Electron transition between energy levels result in emission or absorption lines. • Different elements produce different spectra due to differing atomic structure. H He C PY3P05

**Emission/Absorption of Radiation by Atoms** • Emission/absorption lines are due to radiative transitions: • Radiative (or Stimulated) absorption: Photon with energy (E= h = E2 - E1) excites electron from lower energy level. Can only occur if E = h = E2 - E1 • Radiative recombination/emission: Electron makes transition to lower energy level and emits photon with energy h’ = E2 - E1. E2 E2 E =h E1 E1 PY3P05

**Emission/Absorption of Radiation by Atoms** • Radiative recombination can be either: • Spontaneous emission: Electron minimizes its total energy by emitting photon and making transition from E2to E1. Emitted photon has energy E’ = h’ = E2 - E1 • Stimulated emission: If photon is strongly coupled with electron, cause electron to decay to lower energy level, releasing a photon of the same energy. Can only occur if E = h = E2 - E1Also, h’ = h E2 E2 E’ =h’ E1 E1 E2 E’ =h’ E2 E =h E1 E =h E1 PY3P05

**Simplest Atomic Spectrum: Hydrogen** • In ~1850’s, optical spectrum of hydrogen was found to contain strong lines at 6563, 4861 and 4340 Å. • Lines found to fall closer and closer as wavelength decreases. • Line separation converges at a particular wavelength, called the series limit. • Balmer found that the wavelength of lines could be written where n is an integer >2, and RHis the Rydberg constant. H H H 6563 4861 4340 (Å) PY3P05

**Simplest Atomic Spectrum: Hydrogen** • If n =3, => • Called H - first line of Balmer series. • Other lines in Balmer series: • Balmer Series limit occurs when n . Å Highway 6563 in New Mexico Å PY3P05

**Simplest Atomic Spectrum: Hydrogen** • Other series ofhydrogen: • Rydberg showed that all series above could be reproduced using • Series limit occurs when ni = ∞, nf = 1, 2, … Series Limits PY3P05

**Simplest Atomic Spectrum: Hydrogen** • Term or Grotrian diagram for hydrogen. • Spectral lines can be considered as transition between terms. • 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, … PY3P05

**Bohr Model for Hydrogen** • Simplest atomic system, consisting of single electron-proton pair. • First model put forward by Bohr in 1913. He postulated that: • Electron moves in circular orbit about proton under Coulomb attraction. • Only possible for electron to orbits for which angular momentum is quantised, ie., n = 1, 2, 3, … • Total energy (KE + V) of electron in orbit remains constant. • Quantized radiation is emitted/absorbed if an electron moves changes its orbit. PY3P05

**Bohr Model for Hydrogen** • Consider atom consisting of a nucleus of charge +Ze and mass M, and an electron on charge -e andmass m. Assume M>>m so nucleus remains at fixed position in space. • As Coulomb force is a centripetal, can write (1) • As angular momentum is quantised (2nd postulate): n = 1, 2, 3, … • Solving for v and substituting into Eqn. 1 => (2) • The total mechanical energy is: • Therefore, quantization of AM leads to quantisation of total energy. (3) n = 1, 2, 3, … PY3P05

**Bohr Model for Hydrogen** • Substituting in for constants, Eqn. 3 can be written eV and Eqn. 2 can be written where a0 = 0.529 Å = “Bohr radius”. • Eqn. 3gives a theoretical energy level structure for hydrogen (Z=1): • For Z = 1and n = 1, the ground state of hydrogen is: E1 = -13.6eV PY3P05

**Bohr Model for Hydrogen** • The wavelength of radiation emitted when an electron makes a transition, is (from 4th postulate): or (4) where • Theoretical derivation of Rydberg formula. • Essential predictions of Bohr model are contained in Eqns. 3 and 4. PY3P05

**Correction for Motion of the Nucleus** • Spectroscopically measured RHdoes not agree exactly with theoretically derived R∞. • 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 (): • As m only appears in R∞, must replace by: • It is found spectroscopically that RM = RHtowithin three parts in 100,000. • Therefore, different isotopes of same element have slightly different spectral lines. PY3P05

**Correction for Motion of the Nucleus** • Consider 1H (hydrogen) and 2H (deuterium): • Using Eqn. 4, the wavelength difference is therefore: • Called an isotope shift. • Hand Dare separated by about 1Å. • Intensity of D line is proportional to fraction of D in the sample. cm-1 cm-1 Balmer line of H and D PY3P05

**Spectra of Hydrogen-like Atoms** Z=1 H Z=2 He+ Z=3 Li2+ n 2 n n 0 20 40 60 80 100 120 3 4 1 2 3 13.6 eV 2 Energy (eV) 1 54.4 eV 1 122.5 eV • Bohr model works well for H and H-like atoms (e.g., 4He+, 7Li2+, 7Be3+, etc). • Spectrum of 4He+is almost identical to H, but just offset by a factor of four (Z2). • For He+, Fowler found the following in stellar spectra: • See Fig. 8.7 in Haken & Wolf. PY3P05

**Spectra of Hydrogen-like Atoms** Z=1 H Z=2 He+ Z=3 Li2+ n 2 n n 0 20 40 60 80 100 120 3 4 1 2 3 13.6 eV 2 Energy (eV) 1 54.4 eV 1 122.5 eV • Hydrogenic or hydrogen-like ions: • He+ (Z=2) • Li2+ (Z=3) • Be3+ (Z=4) • … • From Bohr model, the ionization energy is: E1 = -13.59 Z2 eV • Ionization potential therefore increases rapidly with Z. Hydrogenic isoelectronic sequences PY3P05

**Implications of Bohr Model** • We also find that the orbital radius and velocity are quantised: and • Bohr radius (a0) and fine structure constant () are fundamental constants: and • Constants are related by • With Rydberg constant, define gross atomic characteristics of the atom. PY3P05

**Exotic Atoms** • Positronium • 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). • 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. • Energy levels proportional to reduced mass => energy levels half of hydrogen. • Muonium: • Replace proton in H atom with a meson (a “muon”). • Bound state has a lifetime of ~2.2 x 10-6 s. • According to Bohr’s theory (Eqn. 3), the binding energy is 13.5 eV. • From Eqn. 4, n = 1 to n = 2 transition produces a photon of 10.15 eV. • Antihydrogen: • Consists of a positron bound to an antiproton - first observed in 1996 at CERN! • Antimatter should behave like ordinary matter according to QM. • Have not been investigated spectroscopically … yet. PY3P05

**Failures of Bohr Model** • 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. • Some of the shortcomings of the model are: • Fails describe why certain spectral lines are brighter than others => no mechanism for calculating transition probabilities. • Violates the uncertainty principal which dictates that position and momentum cannot be simultaneously determined. • Bohr model gives a basic conceptual model of electrons orbits and energies. The precise details can only be solved using the Schrödinger equation. PY3P05

**Failures of Bohr Model** • From the Bohr model, the linear momentum of the electron is • However, know from Hiesenberg Uncertainty Principle, that • Comparing the two Eqns. above => p ~ np • This shows that the magnitude of p is undefined except when n is large. • Bohr model only valid when we approach the classical limit at large n. • Must therefore use full quantum mechanical treatment to model electron in H atom. PY3P05

**Hydrogen Spectrum** • Transitions actually depend on more than a single quantum number (i.e., more than n). • Quantum mechanics leads to introduction on four quntum numbers. • Principal quantum number: n • Azimuthal quantum number: l • Magnetic quantum number: ml • Spin quantum number: s • Selection rules must also be modified. PY3P05

**Atomic Energy Scales** PY3P05