PY3P05 o Lecturer: o Dr. Peter Gallagher o Email: o [email protected] o Web: o www.physics.tcd.ie/people/peter.gallagher/ o Office: o 3.17A in SNIAM PY3P05
o Section I: Single-electron Atoms
o Review of basic spectroscopy
o Hydrogen energy levels
o Fine structure
o Spin-orbit coupling
o Nuclear moments and hyperfine structure
o Section II: Multi-electron Atoms
o Central-field and Hartree approximations
o Angular momentum, LS and jj coupling
o Alkali spectra
o Helium atom
o Complex atoms
o Section III: Molecules
o Quantum mechanics of molecules
o Vibrational transitions
o Rotational transitions
PY3P05 o Lecture notes will be posted at
http://www.tcd.ie/physics/people/peter.gallagher/ o Recommended Books:
o The Physics of Atoms and Quanta: Introduction to experiement and theory Haken & Wolf (Springer)
o Quantum Physics of Atoms, Molecules, Solids, Nuclei and Particles Eisberg & Resnick (Wiley)
o Quantum Mechanics McMurry o Physical Chemistry Atkins (OUP) PY3P05 o Line spectra o Emission spectra o Absorption spectra o Hydrogen spectrum o Balmer Formula o Bohr’s Model
o Chapter 4, Eisberg & Resnick
Bulb Sun Na H Hg Cs Chlorophyll Diethylthiacarbocyaniodid Diethylthiadicarbocyaniodid Absorption spectra Emission spectra
PY3P05 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:
PY3P05
Gas cloud
1
2
PY3P05 o Electron transition between
energy levels result in emission or absorption lines.
o Different elements produce different spectra due to differing atomic structure.
o Complexity of spectrum increases rapidly with Z.
PY3P05 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 makes transition to lower energy level and emits photon with energy
h"’ = E2 - E1. E2 E1 E2 E1 E! =h"
PY3P05 o Radiative recombination can be either:
a) Spontaneous emission: Electron minimizes its total energy by emitting photon and
making transition from E2to E1.
Emitted photon has energy E!’ = h"’ = E 2 - 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 Also, h"’ = h" E2 E1 E2 E1 E2 E1 E2 E1 E!’ =h"’ E! =h" E!’ =h"’ E! =h" PY3P05 o In ~1850s, hydrogen was found to emit lines at 6563, 4861 and 4340 Å.
o Lines fall closer and closer as wavelength decreases.
o Line separation converges at a particular wavelength, called the series limit.
o Balmer (1885) found that the wavelength of lines could be written
where n is an integer >2, and RH is the Rydberg constant. Can also be written:
! 1/"=RH 1 22# 1 n2 $ % & ' ( ) 6563 4861 4340 H# H$ H! %&(Å) ! "=3646 n 2 n2 #4 $ % & ' ( )
PY3P05 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 '(. ! 1/"=RH 1 22# 1 32 $ % & ' ( ) =>"=6563 Å Highway 6563 to the US National Solar Observatory
in New Mexico ! 1/"#=RH 1 22$ 1 # % & ' ( ) * =>"# ~ 3646Å
Name Transitions Wavelength (Å)
H# 3 - 2 6562.8 H$ 4 - 2 4861.3 H! 5 - 2 4340.5 PY3P05 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 Series Limits
o Other series of hydrogen:
o Rydberg showed that all series above could be reproduced using
o Series limit occurs when ni = !, nf = 1, 2, …
! 1/"=RH 1 n2f # 1 ni2 $ % & & ' ( ) )
PY3P05 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, …
PY3P05 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.
!
PY3P05 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 (2nd postulate): n = 1, 2, 3, …
o Solving for v and substituting into Eqn. 1 => (2)
o The total mechanical energy is:
o Therefore, quantization of AM leads to quantisation of total energy. ! 1 4"#0 Ze2 r2 =m v2 r
!
mvr
=
n
!
! E=1/2mv2+V "En = # mZ2e4 4$%0(
)
22!2 1 n2 n = 1, 2, 3, … ! r=4"#0 n 2!2 mZe2 =>v= n! mr= 1 4"#0 Ze2 n! (3) PY3P05 o Substituting in for constants, Eqn. 3 can be written eVand Eqn. 2 can be written where a0 = 0.529 Å = “Bohr radius”.
o Eqn. 3 gives a theoretical energy level structure for hydrogen (Z=1):
o For Z = 1 and n = 1, the ground state of hydrogen is: E1 = -13.6 eV
! En="13.6Z 2 n2 ! r=n 2a 0 Z
PY3P05 o The wavelength of radiation emitted when an electron makes a transition, is (from
4th postulate):
or (4)
where
o Theoretical derivation of Rydberg formula.
o Essential predictions of Bohr model are contained in Eqns. 3 and 4.
! 1/"=Ei#Ef hc = 1 4$%0 & ' ( ) * + 2 me4 4$!3cZ 2 1 nf 2# 1 ni 2 & ' ( ( ) * + + ! 1/"=R#Z 2 1 nf 2$ 1 ni 2 % & ' ' ( ) * * ! R"= 1 4#$0 % & ' ( ) * 2 me4 4#!3 c
1. Calculate the bind energy in eV of the Hydrogen atom using Eqn 3?
o How does this compare with experiment?
2. Calculate the velocity of the electron in the ground state of Hydrogen.
o How does this compare with the speed of light? o Is a non-relativistic model justified?
3. What is a Rydberg atom?
o Calculate the radius of a electron in an n = 300 shell. o How does this compare with the ground state of Hydrogen?
PY3P05 o Spectroscopically measured RH does not agree exactly with theoretically derived
R!.
o But, we assumed that M>>m => nucleus fixed. In reality, electron and proton move about common centre of mass. Must use electron’s reduced mass (µ):
o As m only appears in R!, must replace by:
o It is found spectroscopically that RM = RH to within three parts in 100,000.
o Therefore, different isotopes of same element have slightly different spectral lines.
! µ = mM m+M ! RM = M m+MR"= µ mR" PY3P05 o Consider 1H (hydrogen) and 2H (deuterium):
o Using Eqn. 4, the wavelength difference is therefore:
o Called an isotope shift.
o H$ and D$ are separated by about 1Å.
o Intensity of D line is proportional to fraction of D in the sample. ! RH=R" 1 1+m/MH =109677.584 RD=R" 1 1+m/MD =109707.419 cm-1 cm-1
Balmer line of H and D
!
"#=#H$#D=#H(1$#D/#H) =#H(1$RH/RD)
PY3P05 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 n 2 1 2 3 Z=1 H Z=2 He+ Z=3 Li2+ n n 1 2 3 4 13.6 eV 54.4 eV 122.5 eV PY3P05 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 Energy (eV) 1 n 2 1 2 3 Z=1 H Z=2 He+ Z=3 Li2+ n n 1 2 3 4 13.6 eV 54.4 eV 122.5 eV 0 20 40 60 80 100 120
PY3P05 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.
! rn =n 2 Z m µa0 ! a0= 4"#0!2 me2 Rydberg energy RH 13.6 eV Bohr radius a0 5.26x10-11 m
Fine structure constant ! 1/137.04
! vn ="Z nc ! "= e 2 4#$0!c ! a0= ! mc" PY3P05 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 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.
PY3P05 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.
PY3P05 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.
! p=mv=m Ze 2 4"#0n! $ % & ' ( ) =n! r ! "p~ ! "x ~ ! r
PY3P05 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.
PY3P05
Energy scale Energy (eV) 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
