Lectures 1-2: Introduction to Atomic Spectroscopy Types of Spectra o Continuous spectrum: Produced by solids, liquids & dense gases produce - no o Line spectra “gaps” in wavelength of light produced: o Emission spectra Bulb o Absorption spectra Sun o Hydrogen spectrum Na o Emission spectrum: Produced by rarefied gases – emission only in narrow o Balmer formula Atomic emission H wavelength regions: o Bohr’s model spectra Hg Cs Chlorophyll o Absorption spectrum: Gas atoms absorb the same wavelengths as they usually emit and results in an absorption line spectrum: Molecular absorption Diethylthiacarbocyaniodid spectra Diethylthiadicarbocyaniodid PY3004 PY3004 Emission and Absorption Spectroscopy Line Spectra o Electron transitions between energy levels result in emission or absorption lines. Gas cloud o Different elements produce different spectra due to differing atomic structure (discovered by Kirchhoff and Bunsen). H 3 1 2 He C ! " PY3004 PY3004 Emission/Absorption of Radiation by Atoms Emission/Absorption of Radiation by Atoms o Radiative recombination can be either: o Emission/absorption lines are due to radiative transitions: a) Spontaneous emission: Electron minimizes its total energy by emitting photon and making transition from E to E . 1. Radiative (or Stimulated) absorption: 2 1 Photon with energy (E# = h$ = E2 - E1) excites electron from lower energy E2 E2 level. ’ E# =h$’ E1 E1 E2 E2 E =h$ ’ ’ # Emitted photon has energy E# = h$ = E2 - E1 E1 E1 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. Can only occur if E# = h$ = E2 - E1 E E2 E ’ =h$’ 2. Radiative recombination/emission: 2 # E =h ’ # $ E Electron emits photon with energy (h$ = E2 - E1) and makes transition to E1 1 E# =h$ lower energy level. ’ Can only occur if E# = h$ = E2 - E1 Also, h$ = h$ PY3004 PY3004 Simplest Atomic Spectrum: Hydrogen Simplest Atomic Spectrum: Hydrogen o In 1850s, the visible spectrum of hydrogen was found to contain strong lines at o If n =3, => 6563, 4861 and 4340 Å. H% H& H# o Called H% - first line of Balmer series. o Other lines in Balmer series: 6563 4861 4340 !" (Å) o Lines found to fall more closely as wavelength decreases. Name Transitions Wavelength (Å) H 3 - 2 6562.8 % Road to National Solar o Line separation converges at a particular wavelength, called the series limit. 4 - 2 4861.3 H& Observatory in New Mexico: “Highway 6563” H 5 - 2 4340.5 o In 1885, Balmer found that the wavelength of lines could be written # o Balmer series limit occurs when n '(. where n is an integer >2, and RH is the Rydberg constant. PY3004 PY3004 Simplest Atomic Spectrum: Hydrogen Simplest Atomic Spectrum: Hydrogen o Rydberg showed that all series above could be o Ritz Combination Principle: Transitions reproduced using occur between terms: where n are principal quantum numbers. 2 where Tf = RH/nf etc. o Series limit occurs when ni = !, nf = 1, 2, … o Transitions can occur between certain terms o Other series of hydrogen: => selection rule. Selection rule for hydrogen: *n = 1, 2, 3, … Lyman UV nf = 1, ni)2 Balmer Visible/UV nf = 2, ni)3 o Transitions between ground state and first Series limits excited state produce a resonance line. Paschen IR nf = 3, ni)4 Brackett IR nf = 4, ni)5 o Grotrian diagram shows terms and the transitions. Pfund IR nf = 5, ni)6 Energy level or term diagram for hydrogen PY3004 PY3004 Bohr Model for Hydrogen Bohr Model for Hydrogen o Consider atom consisting of a nucleus of charge +Ze and mass M, and an electron o Simplest atomic system, consisting of single electron-proton pair. on charge -e and mass m. Assume M>>m so nucleus remains at fixed position in space. o First model put forward by Bohr in 1913. He postulated that: o As Coulomb force is a centripetal: (1) 1. Electron moves in circular orbit about proton under Coulomb attraction. o And assuming angular momentum is quantised: 2. Only possible for electron to orbits for which angular momentum is quantised, o Solving for v and substituting into Eqn. 1 => ie., L = mvr = nh where n = 1, 2, 3, … (2) 3. Total energy (K + V) of electron in orbit remains constant. ! and 4. Quantized radiation is emitted/absorbed if an electron changes orbit. The frequency emitted is $ = (Ei - Ef ) / h. o Quantized AM has restricted the possible circular orbits of the electron. PY3004 PY3004 Bohr Model for Hydrogen Bohr Model for Hydrogen o The total mechanical energy is: o Substituting in for constants, Eqn. 3 can be written o Using Eqn. 1, and Eqn. 2 can be written where a0 = 0.529 Å = “Bohr radius”. o Eqn. 3 gives a theoretical energy level structure for hydrogen (Z=1): o The potential energy (V) can be calculated from o Total mechanical energy is therefore o Using Eqn. 2 for r, (3) o Therefore, quantization of AM leads to quantisation of total energy. o For Z = 1 and n = 1, the ground state of hydrogen is: E1 = -13.6 eV PY3004 PY3004 Bohr Model for Hydrogen Correction for Motion of the Nucleus o The wavelength of radiation emitted when an electron makes a transition, is (from o Spectroscopically measured RH does not agree exactly with theoretically derived 4th postulate): R!. o But, we assumed that M>>m => nucleus fixed. In reality, electron and proton or (4) move about common centre of mass. Must use electron’s reduced mass (µ): where o As m only appears in R!, must replace by: o Theoretical derivation of Rydberg formula. o Essential predictions of Bohr model are contained in Eqns. 3 and 4. 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. PY3004 PY3004 Correction for Motion of the Nucleus Spectra of Hydrogen-like Atoms o Bohr model works well for H and H-like atoms (e.g., 4He+, 7Li2+, 7Be3+, etc). o Consider 1H (hydrogen) and 2H (deuterium): o Spectrum of 4He+ is almost identical to H, but just offset by a factor of four (Z2). cm-1 o For He+, Fowler found the following Z=1 Z=2 Z=3 H He+ Li2+ o Using Eqn. 4, the wavelength difference is in stellar spectra: n n n 0 therefore: 2 3 4 1 2 3 $ 1 1 ' 20 13.6 eV 1/" = 4RHe& 2 # 2 ) 2 % 3 n ( 40 Energy 1 o Called an isotope shift. (eV) 60 54.4 eV o See Fig. 8.7 in Haken & Wolf. ! 80 o H& and D& are separated by about 1Å. 100 Balmer line of H and D 1 o Intensity of D line is proportional to fraction of 120 122.5 eV D in the sample. PY3004 PY3004 Spectra of Hydrogen-like Atoms Implications of Bohr Model o Hydrogenic or hydrogen-like ions: Z=1 Z=2 Z=3 o We also find that the orbital radius and velocity are quantised: H He+ Li2+ n n n 0 2 + 3 4 o He (Z=2) 1 2 3 and 13.6 eV 2+ 20 o Li (Z=3) Hydrogenic 2 3+ isoelectronic o Be (Z=4) 40 o Bohr radius (a0) and fine structure constant (%) are fundamental constants: sequences Energy o … 1 (eV) 60 54.4 eV and 80 o From Bohr model, the ionization energy is: 100 o Constants are related by 1 2 E1 = -13.59 Z eV 120 122.5 eV o With Rydberg constant, define gross atomic characteristics of the atom. o Ionization potential therefore increases rapidly with Z. Rydberg energy RH 13.6 eV -11 Bohr radius a0 5.26x10 m Fine structure constant ! 1/137.04 PY3004 PY3004 Exotic Atoms Failures of Bohr Model o Positronium o Bohr model was a major step toward understanding the quantum theory of the atom o electron (e-) and positron (e+) enter a short-lived bound state, before they - not in fact a correct description of the nature of electron orbits. 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 Some of the shortcomings of the model are: o Energy levels proportional to reduced mass => energy levels half of hydrogen. 1. Fails describe why certain spectral lines are brighter than others => no o Muonium: mechanism for calculating transition probabilities. o Replace proton in H atom with a µ meson (a “muon”). -6 o Bound state has a lifetime of ~2.2 x 10 s. 2. Violates the uncertainty principal which dictates that position and momentum o According to Bohr’s theory (Eqn. 3), the binding energy is 13.5 eV. cannot be simultaneously determined. o From Eqn. 4, n = 1 to n = 2 transition produces a photon of 10.15 eV. o Bohr model gives a basic conceptual model of electrons orbits and energies. The o Antihydrogen: precise details can only be solved using the Schrödinger equation. 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. PY3004 PY3004 Failures of Bohr Model Hydrogen Spectrum o From the Bohr model, the linear momentum of the electron is 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 However, know from Hiesenberg Uncertainty Principle, that o Principal quantum number: n o Comparing the two Eqns.