Lectures 1-2: Introduction to Atomic Spectroscopy Types of ... · Lectures 1-2: Introduction to...
7
PY3004 Lectures 1-2: Introduction to Atomic Spectroscopy Lectures 1-2: Introduction to Atomic Spectroscopy o Line spectra o Emission spectra o Absorption spectra o Hydrogen spectrum o Balmer formula o Bohr’s model Bulb Sun Na H Hg Cs Chlorophyll Diethylthiacarbocyaniodid Diethylthiadicarbocyaniodid Molecular absorption spectra Atomic emission spectra PY3004 Types of Spectra 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: PY3004 Emission and Absorption Spectroscopy Emission and Absorption Spectroscopy Gas cloud 1 2 3 PY3004 Line Spectra Line Spectra 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 He C ! "
Transcript of Lectures 1-2: Introduction to Atomic Spectroscopy Types of ... · Lectures 1-2: Introduction to...
PY3004
Lectures 1-2: Introduction to Atomic SpectroscopyLectures 1-2: Introduction to Atomic Spectroscopy
o Line spectra
o Emission spectra
o Absorption spectra
o Hydrogen spectrum
o Balmer formula
o Bohr’s model
Bulb
Sun
Na
H
Hg
Cs
Chlorophyll
Diethylthiacarbocyaniodid
Diethylthiadicarbocyaniodid
Molecular absorption
spectra
Atomic emission
spectra
PY3004
Types of SpectraTypes 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 narrowwavelength regions:
o Absorption spectrum: Gas atoms absorb the same wavelengths as theyusually emit and results in an absorption line spectrum:
PY3004
Emission and Absorption SpectroscopyEmission and Absorption Spectroscopy
Gas cloud
12
3
PY3004
Line SpectraLine Spectra
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
He
C
! "
PY3004
Emission/Absorption of Radiation by AtomsEmission/Absorption of Radiation by Atoms
o Emission/absorption lines are due to radiative transitions:
1. 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
2. Radiative recombination/emission:
Electron emits photon with energy (h$’ = E2 - E1) and makes transition tolower energy level.
E2
E1
E2
E1
E# =h$
PY3004
Emission/Absorption of Radiation by AtomsEmission/Absorption of Radiation by Atoms
o Radiative recombination can be either:
a) Spontaneous emission: Electron minimizes its total energy by emitting photon andmaking transition from E2 to E1.
Emitted photon has energy E#’ = h$’ = E2 - E1
b) Stimulated emission: If photon is strongly coupled with electron, cause electron todecay to lower energy level, releasing a photon of the same energy.
Can only occur if E# = h$ = E2 - E1 Also, h$’ = h$
E2
E1
E2
E1
E2
E1
E2
E1
E#’ =h$’
E# =h$
E#’ =h$’
E# =h$
PY3004
Simplest Atomic Spectrum: HydrogenSimplest Atomic Spectrum: Hydrogen
o In 1850s, the visible spectrum of hydrogen was found to contain strong lines at6563, 4861 and 4340 Å.
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
where n is an integer >2, and RH is the Rydberg constant.
6563 4861 4340
H% H& H#
!" (Å)
PY3004
Simplest Atomic Spectrum: HydrogenSimplest Atomic Spectrum: Hydrogen
o If n =3, =>
o Called H% - first line of Balmer series.
o Other lines in Balmer series:
o Balmer series limit occurs when n '(.
Road to National SolarObservatory in New
Mexico: “Highway 6563”4340.55 - 2H#
4861.34 - 2H&
6562.83 - 2H%
Wavelength (Å)TransitionsName
PY3004
Simplest Atomic Spectrum: HydrogenSimplest Atomic Spectrum: Hydrogen
Series limits
o Rydberg showed that all series above could be
reproduced using
where n are principal quantum numbers.
o Series limit occurs when ni = !, nf = 1, 2, …
o Other series of hydrogen:
nf = 5, ni)6IRPfund
nf = 4, ni)5IRBrackett
nf = 3, ni)4IRPaschen
nf = 2, ni)3Visible/UVBalmer
nf = 1, ni)2UVLyman
PY3004
Simplest Atomic Spectrum: HydrogenSimplest Atomic Spectrum: Hydrogen
o Ritz Combination Principle: Transitionsoccur between terms:
where Tf = RH/nf2 etc.
o Transitions can occur between certain terms=> selection rule. Selection rule forhydrogen: *n = 1, 2, 3, …
o Transitions between ground state and first
excited state produce a resonance line.
o Grotrian diagram shows terms and thetransitions.
Energy level or term diagram for hydrogen
PY3004
Bohr Model for HydrogenBohr 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., where n = 1, 2, 3, …
3. Total energy (K + V) of electron in orbit remains constant.
4. Quantized radiation is emitted/absorbed if an electron changes orbit. Thefrequency emitted is $ = (Ei - Ef ) / h.
!
L = mvr = nh
PY3004
Bohr Model for HydrogenBohr Model for Hydrogen
o Consider atom consisting of a nucleus of charge +Ze and mass M, and an electronon charge -e and mass m. Assume M>>m so nucleus remains at fixed position inspace.
o As Coulomb force is a centripetal: (1)
o And assuming angular momentum is quantised:
o Solving for v and substituting into Eqn. 1 =>
o Quantized AM has restricted the possible circular orbits of the electron.
and
(2)
PY3004
Bohr Model for HydrogenBohr 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.
(3)
PY3004
Bohr Model for HydrogenBohr 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
PY3004
Bohr Model for HydrogenBohr Model for Hydrogen
o The wavelength of radiation emitted when an electron makes a transition, is (from4th postulate):
or (4)
where
o Theoretical derivation of Rydberg formula.
o Essential predictions of Bohr model
are contained in Eqns. 3 and 4.
PY3004
Correction for Motion of the NucleusCorrection for Motion of the Nucleus
o Spectroscopically measured RH does not agree exactly with theoretically derivedR!.
o But, we assumed that M>>m => nucleus fixed. In reality, electron and protonmove 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.
PY3004
Correction for Motion of the NucleusCorrection for Motion of the Nucleus
o Consider 1H (hydrogen) and 2H (deuterium):
o Using Eqn. 4, the wavelength difference istherefore:
o Called an isotope shift.
o H& and D& are separated by about 1Å.
o Intensity of D line is proportional to fraction ofD in the sample.
cm-1
Balmer line of H and D
PY3004
Spectra of Hydrogen-like AtomsSpectra of Hydrogen-like Atoms
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:
o See Fig. 8.7 in Haken & Wolf.!
1/" = 4RHe
1
32#1
n2
$
% &
'
( )
0
20
40
60
80
100
120
Energy(eV)
1
n2
1
23
Z=1H
Z=2He+
Z=3
Li2+
n n
1
2
34
13.6 eV
54.4 eV
122.5 eV
PY3004
Spectra of Hydrogen-like AtomsSpectra of Hydrogen-like Atoms
o Hydrogenic or hydrogen-like ions:
o He+ (Z=2)
o Li2+ (Z=3)
o Be3+ (Z=4)
o …
o From Bohr model, the ionization energy is:
E1 = -13.59 Z2 eV
o Ionization potential therefore increases rapidly with Z.
Hydrogenic isoelectronic
sequences
0
20
40
60
80
100
120
Energy(eV)
1
n2
1
23
Z=1H
Z=2He+
Z=3
Li2+
n n
1
2
34
13.6 eV
54.4 eV
122.5 eV
PY3004
Implications of Bohr ModelImplications of Bohr Model
o We also find that the orbital radius and velocity are quantised:
and
o Bohr radius (a0) and fine structure constant (%) are fundamental constants:
and
o Constants are related by
o With Rydberg constant, define gross atomic characteristics of the atom.
1/137.04!Fine structure constant
5.26x10-11 ma0Bohr radius
13.6 eVRHRydberg energy
PY3004
Exotic AtomsExotic Atoms
o Positronium
o electron (e-) and positron (e+) enter a short-lived bound state, before theyannihilate 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 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.
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.
PY3004
Failures of Bohr ModelFailures 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 => nomechanism for calculating transition probabilities.
2. Violates the uncertainty principal which dictates that position and momentumcannot be simultaneously determined.
o Bohr model gives a basic conceptual model of electrons orbits and energies. Theprecise details can only be solved using the Schrödinger equation.
PY3004
Failures of Bohr ModelFailures 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.
PY3004
Hydrogen SpectrumHydrogen Spectrum
o Transitions actually depend on more than asingle quantum number (i.e., more than n).
o Quantum mechanics leads to introduction onfour 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.
PY3004
Atomic Energy ScalesAtomic Energy Scales
Nuclear interactions10-6 - 10-5Hyperfine structure
Spin-orbit interaction
Relativistic corrections
0.001 - 0.01Fine structure
electron-nuclearattraction
Electron-electronrepulsion
Electron kinetic energy
1-10Gross structure
EffectsEnergy (eV)Energy scale
