lectures 1-2
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PY3P05
o Lecturer: o Dr. Peter Gallagher
o Email: o peter.gallagher@tcd.ie
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
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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
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
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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
1 2
3
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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.
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"
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o Radiative recombination can be either:
a) Spontaneous emission: Electron minimizes its total energy by emitting photon and making 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 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"
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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/" = RH122#1n2
$
% &
'
( )
6563 4861 4340
H# H$ H!
%& (Å)
!
" = 3646 n2
n2 # 4$
% &
'
( )
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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/" = RH122#132
$
% &
'
( ) => " = 6563 Å
Highway 6563 to the US National Solar Observatory
in New Mexico
!
1/"# = RH122$1#
%
& '
(
) * => "# ~ 3646Å
Name Transitions Wavelength (Å)
H# 3 - 2 6562.8
H$ 4 - 2 4861.3
H! 5 - 2 4340.5
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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/" = RH1n f2 #
1ni2
$
% & &
'
( ) )
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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., 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.
!
L = mvr = n!
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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.
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.
!
14"#0
Ze2
r2= m v 2
r
!
mvr = n!
!
E =1/2mv 2 +V
"En = #mZ 2e4
4$%0( )22!21n2 n = 1, 2, 3, …
!
r = 4"#0n2!2
mZe2
=> v =n!mr
=14"#0
Ze2
n!
(3)
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o Substituting in for constants, Eqn. 3 can be written eV
and Eqn. 2 can be written where a0 = 0.529 Å = “Bohr radius”.
o Eqn. 3 gives a theoretical energy level structure for hydrogen (Z=1):
o For Z = 1 and n = 1, the ground state of hydrogen is: E1 = -13.6 eV
!
En ="13.6Z 2
n2
!
r =n2a0Z
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o The wavelength of radiation emitted when an electron makes a transition, is (from 4th postulate):
or (4)
where
o Theoretical derivation of Rydberg formula.
o Essential predictions of Bohr model are contained in Eqns. 3 and 4.
!
1/" =Ei # E f
hc=
14$%0
&
' (
)
* +
2me4
4$!3cZ 2 1
n f2 #
1ni2
&
' ( (
)
* + +
!
1/" = R#Z2 1n f2 $
1ni2
%
& ' '
(
) * *
!
R" =14#$0
%
& '
(
) *
2me4
4#!3c
1. Calculate the bind energy in eV of the Hydrogen atom using Eqn 3?
o How does this compare with experiment?
2. Calculate the velocity of the electron in the ground state of Hydrogen.
o How does this compare with the speed of light? o Is a non-relativistic model justified?
3. What is a Rydberg atom?
o Calculate the radius of a electron in an n = 300 shell. o How does this compare with the ground state of Hydrogen?
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o Spectroscopically measured RH does not agree exactly with theoretically derived R!.
o But, we assumed that M>>m => nucleus fixed. In reality, electron and proton move about common centre of mass. Must use electron’s reduced mass (µ):
o As m only appears in R!, must replace by:
o It is found spectroscopically that RM = RH to within three parts in 100,000.
o Therefore, different isotopes of same element have slightly different spectral lines.
!
µ =mMm + M
!
RM =M
m + MR" =
µmR"
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o Consider 1H (hydrogen) and 2H (deuterium):
o Using Eqn. 4, the wavelength difference is therefore:
o Called an isotope shift.
o H$ 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 )
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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:
o See Fig. 8.7 in Haken & Wolf. !
1/" = 4RHe132#1n2
$
% &
'
( )
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
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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
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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 =n2
Zmµa0
!
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 ="Znc
!
" =e2
4#$0!c
!
a0 =!
mc"
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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
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 Ze2
4"#0n!$
% &
'
( ) =
n!r
!
"p ~ !"x
~ !r
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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
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