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 ...
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
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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 o Electronic transition 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)
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o Line spectra o Emission spectra
Bulb Sun
o Absorption spectra
Na
o Hydrogen spectrum Emission spectra
o Balmer Formula o Bohr’s Model
H Hg Cs
o Chapter 4, Eisberg & Resnick
Chlorophyll
Diethylthiacarbocyaniodid
Absorption spectra
Diethylthiadicarbocyaniodid 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:
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Gas cloud
3 1 2
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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.
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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. E! =h"
E2
E2
E1
E1
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. PY3P05
o
Radiative recombination can be either: a) Spontaneous emission: Electron minimizes its total energy by emitting photon and making transition from E2 to 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.
E! =h"
E2
E2
E!’ =h"’
E1
E1
E! =h"
Can only occur if E! = h" = E2 - E1 Also, h"’ = h" PY3P05
o In ~1850s, hydrogen was found to emit lines at 6563, 4861 and 4340 Å. H$
H#
6563
%& (Å)
4861
H!
4340
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 $ n2 ' " = 3646& 2 ) % n # 4( where n is an integer >2, and RH is the Rydberg constant. Can also be written: !
$1 1' 1/ " = R H & 2 # 2 ) %2 n ( PY3P05
!
$1 1' 1/ " = R H & 2 # 2 ) => " = 6563 Å %2 3 (
o If n =3, =>
o Called H# - first line of Balmer series. ! in Balmer series: o Other lines Name
Transitions
Wavelength (Å)
H#
3-2
6562.8
H$
4-2
4861.3
H!
5-2
4340.5
Highway 6563 to the US National Solar Observatory in New Mexico
o Balmer Series limit occurs when n '(. % 1 1( 1/ "# = R H ' 2 $ * => "# ~ 3646 Å & 2 #) PY3P05
!
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
o Rydberg showed that all series above could be reproduced using $1 1' 1/ " = R H && 2 # 2 )) %nf
Series Limits
ni (
o Series limit occurs when ni = !, nf = 1, 2, … !
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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, …
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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., L = mvr = n! 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.
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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. 1 Ze 2 v2 = m 4 "#0 r 2 r
o As Coulomb force is a centripetal, can write
(1)
mvr = n!
o As angular momentum is quantised (2nd postulate):
n = 1, 2, 3, …
n 2!2 (2) mZe 2 ! n! 1 Ze 2 => v = = mr 4 "#0 n!
o Solving for v and substituting into Eqn.!1 => r = 4"#0
o The total mechanical energy is:
E = 1/2mv 2 + V " En ! =#
mZ 2e 4 1 2 2 2 n 4 $% 2! ( 0)
n = 1, 2, 3, …
(3)
o Therefore, quantization of AM leads to quantisation of total energy. !
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o Substituting in for constants, Eqn. 3 can be written and Eqn. 2 can be written
r=
n 2 a0 Z
En =
"13.6Z 2 eV n2
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 PY3P05
o The wavelength of radiation emitted when an electron makes a transition, is (from 4th postulate): 4 & ) & )2 1/ " =
or
Ei # E f 1 me 1 1 =( Z 2 (( 2 # 2 ++ + 3 hc 4 $% 4 $ ! c n n ' 0* ' f i *
%1 1( 1/ " = R# Z 2 '' 2 $ 2 ** & n f ni ) !
(4)
% 1 ( 2 me 4 * 3 & 4 #$0 ) 4 #! c
where R" = ' !
o Theoretical derivation of Rydberg formula. !
o Essential predictions of Bohr model are contained in Eqns. 3 and 4.
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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?
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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): µ=
mM m+ M
o As m only appears in R!, must replace by: !
RM =
M µ R" = R" m+ M m
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.
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o Consider 1H (hydrogen) and 2H (deuterium): 1 = 109677.584 1+ m / M H 1 RD = R" = 109707.419 1+ m / M D R H = R"
cm-1 cm-1
o Using Eqn. 4, the wavelength difference is therefore: ! "# = # H $ #D = # H (1$ #D / # H ) = # H (1$ R H /RD )
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.
Balmer line of H and D
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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:
Z=1 H 0
$1 1' 1/ " = 4R He & 2 # 2 ) %3 n (
o See Fig. 8.7 in Haken & Wolf. !
13.6 eV
20
Z=2 He+
n 2
Z=3 Li2+
n 3 2
1
n 4 3 2
40
Energy (eV)
60
54.4 eV
1
80 100 1
120
122.5 eV
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Z=1 H
o 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
Z=2 He+
1
n 3 2
Z=3 Li2+
n 4 3 2
Energy (eV)
o From Bohr model, the ionization energy is:
40 60
54.4 eV
1
80 100
E1 = -13.59
Z2
eV
120
1 122.5 eV
o Ionization potential therefore increases rapidly with Z.
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o We also find that the orbital radius and velocity are quantised: rn =
n2 m a0 Z µ
and
vn = "
Z c n
o Bohr radius (a0) and fine structure constant (#) are fundamental constants: !
e2 4 "#0 ! 2 ! " = a0 = and 4 #$0 !c me 2 ! a0 = o Constants are related by mc" ! ! o With Rydberg constant, define gross atomic characteristics of the atom.
! energy Rydberg
RH
13.6 eV
Bohr radius
a0
5.26x10-11 m
Fine structure constant
!
1/137.04 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. o Have not been investigated spectroscopically … yet.
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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.
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o From the Bohr model, the linear momentum of the electron is $ Ze 2 ' n! p = mv = m& )= % 4 "#0 n! ( r
o However, know from Hiesenberg Uncertainty Principle, that ! ! ~ "x r o Comparing the two Eqns. above => p ~ n*p !
"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. 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.
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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
Hyperfine structure
10-6 - 10-5
Nuclear interactions PY3P05
