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Download Lectures 1-2: Introduction to Atomic Spectroscopy Types of Spectra
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Types of Spectra Lectures 1-2: Introduction to Atomic Spectroscopy o Continuous spectrum: Produced by solids, liquids & dense gases produce - no “gaps” in wavelength of light produced: o Line spectra o Emission spectra o Absorption spectra o Hydrogen spectrum o Balmer formula o Bohr’s model Bulb Sun Na Atomic emission spectra o Emission spectrum: Produced by rarefied gases – emission only in narrow wavelength regions: H Hg Cs o Absorption spectrum: Gas atoms absorb the same wavelengths as they usually emit and results in an absorption line spectrum: Chlorophyll Molecular absorption spectra Diethylthiacarbocyaniodid Diethylthiadicarbocyaniodid PY3004 PY3004 Emission and Absorption Spectroscopy Line Spectra Gas cloud o Electron transitions between energy levels result in emission or absorption lines. 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 o Radiative recombination can be either: Emission/absorption lines are due to radiative transitions: a) Spontaneous emission: Electron minimizes its total energy by emitting photon and making transition from E2 to E1. 1. Radiative (or Stimulated) absorption: Photon with energy (E# = h$ = E2 - E1) excites electron from lower energy level. E# =h$ E2 E2 E1 E1 E2 E2 E1 E1 E#’ =h$’ Emitted photon has energy E#’ = h$’ = E2 - 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 2. Radiative recombination/emission: Electron emits photon with energy (h$’ = E2 - E1) and makes transition to lower energy level. E# =h$ E2 E2 E#’ =h$’ E1 E1 E# =h$ Can only occur if E# = h$ = E2 - E1 Also, h$’ = h$ PY3004 PY3004 Simplest Atomic Spectrum: Hydrogen o Simplest Atomic Spectrum: Hydrogen In 1850s, the visible spectrum of hydrogen was found to contain strong lines at 6563, 4861 and 4340 Å. H% H& 6563 4861 !" (Å) H# o If n =3, => o Called H% - first line of Balmer series. o Other lines in Balmer series: 4340 o Lines found to fall more closely as wavelength decreases. o Line separation converges at a particular wavelength, called the series limit. o In 1885, Balmer found that the wavelength of lines could be written o Name Transitions Wavelength (Å) H% 3-2 6562.8 H& 4-2 4861.3 H# 5-2 4340.5 Road to National Solar Observatory in New Mexico: “Highway 6563” Balmer series limit occurs when n '(. where n is an integer >2, and RH is the Rydberg constant. PY3004 PY3004 Simplest Atomic Spectrum: Hydrogen o Simplest Atomic Spectrum: Hydrogen Rydberg showed that all series above could be reproduced using o Ritz Combination Principle: Transitions occur between terms: where n are principal quantum numbers. o Series limit occurs when ni = !, nf = 1, 2, … 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 where Tf = RH/nf2 etc. o Transitions can occur between certain terms => selection rule. Selection rule for hydrogen: *n = 1, 2, 3, … o Transitions between ground state and first excited state produce a resonance line. o Grotrian diagram shows terms and the transitions. Series limits Energy level or term diagram for hydrogen PY3004 PY3004 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: Bohr Model for Hydrogen 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., L = mvr = nh where n = 1, 2, 3, … 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: o And assuming angular momentum is quantised: o Solving for v and substituting into Eqn. 1 => (1) (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. PY3004 o Quantized AM has restricted the possible circular orbits of the electron. PY3004 Bohr Model for Hydrogen o The total mechanical energy is: o Using Eqn. 1, o The potential energy (V) can be calculated from o Total mechanical energy is therefore o Using Eqn. 2 for r, o Therefore, quantization of AM leads to quantisation of total energy. Bohr Model for Hydrogen o Substituting in for constants, Eqn. 3 can be written 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 For Z = 1 and n = 1, the ground state of hydrogen is: E1 = -13.6 eV (3) PY3004 PY3004 Bohr Model for Hydrogen o Correction for Motion of the Nucleus The wavelength of radiation emitted when an electron makes a transition, is (from 4th postulate): or 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 (µ): 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. (4) where o Theoretical derivation of Rydberg formula. o Essential predictions of Bohr model are contained in Eqns. 3 and 4. PY3004 PY3004 Correction for Motion of the Nucleus o Spectra of Hydrogen-like Atoms Consider 1H (hydrogen) and 2H (deuterium): cm-1 o o Bohr model works well for H and H-like atoms (e.g., 4He+, 7Li2+, 7Be3+, etc). o Spectrum of 4He+ is almost identical to H, but just offset by a factor of four (Z2). o For He+, Fowler found the following in stellar spectra: Using Eqn. 4, the wavelength difference is therefore: $1 1' 1/ " = 4R He & 2 # 2 ) %3 n ( 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. o See Fig. 8.7 in Haken & Wolf. Z=1 H 0 13.6 eV 20 Z=2 He+ n 2 1 n 3 2 Z=3 Li2+ 2 Energy (eV) 40 60 54.4 eV 1 80 ! 100 Balmer line of H and D 1 120 122.5 eV PY3004 PY3004 Spectra of Hydrogen-like Atoms o o Z=1 H Hydrogenic or hydrogen-like ions: o o o o He+ (Z=2) Li2+ (Z=3) Be3+ (Z=4) … 0 Hydrogenic isoelectronic sequences 20 13.6 eV n 2 Implications of Bohr Model Z=2 He+ 1 n 3 2 Z=3 Li2+ o 4 3 Energy (eV) o eV and o 40 60 54.4 eV Bohr radius (a0) and fine structure constant (%) are fundamental constants: 1 and 80 100 E1 = -13.59 We also find that the orbital radius and velocity are quantised: n 2 From Bohr model, the ionization energy is: Z2 n 4 3 120 1 o Constants are related by o With Rydberg constant, define gross atomic characteristics of the atom. 122.5 eV Ionization potential therefore increases rapidly with Z. PY3004 Rydberg energy RH 13.6 eV Bohr radius a0 5.26x10-11 m Fine structure constant ! 1/137.04 PY3004 Exotic Atoms o o o Failures of Bohr Model 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 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. Muonium: o Replace proton in H atom with a µ 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 eV. o From Eqn. 4, n = 1 to n = 2 transition produces a photon of 10.15 eV. 2. Violates the uncertainty principal which dictates that position and momentum cannot be simultaneously determined. 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. o Bohr model gives a basic conceptual model of electrons orbits and energies. The precise details can only be solved using the Schrödinger equation. PY3004 PY3004 Failures of Bohr Model o o Hydrogen Spectrum 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. However, know from Hiesenberg Uncertainty Principle, that o Principal quantum number: n o Comparing the two Eqns. above => p ~ n*p o Azimuthal quantum number: l o This shows that the magnitude of p is undefined except when n is large. o Magnetic quantum number: ml 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. o Spin quantum number: s PY3004 o Selection rules must also be modified. PY3004 Atomic Energy Scales Energy scale Energy (eV) Effects Gross structure 1-10 electron-nuclear attraction Electron-electron repulsion Electron kinetic energy Fine structure 0.001 - 0.01 Spin-orbit interaction Relativistic corrections Hyperfine structure 10-6 - 10-5 Nuclear interactions PY3004