Upload
harry-de-los-rios
View
241
Download
1
Embed Size (px)
Citation preview
8/20/2019 Atomic Molecular Physics
1/31
E C
O L E P O L Y
T E C H N I Q
U E
Recueil
Programme d’approfondissement
Atomic and molecular
physicsTextes de contrôles
des connaissances proposésles années antérieures
Département de Physique
8/20/2019 Atomic Molecular Physics
2/31
8/20/2019 Atomic Molecular Physics
3/31
Édition 2014
Atomic and Molecular Physics
P ’
Textes de contrôles des connaissances
proposés les années antérieures
8/20/2019 Atomic Molecular Physics
4/31
8/20/2019 Atomic Molecular Physics
5/31
8/20/2019 Atomic Molecular Physics
6/31
2
Zinc (Zn) Cadmium (Cd) Mercury (Hg)
N° ! (cm-1
) J P N° ! (cm-1
) J P N° ! (cm-1
) J P
Zn 1
Zn 2
Zn 3
Zn 4
Zn 5Zn 6
Zn 7
Zn 8
Zn 9
Zn 10
Zn 11
Zn 12
Zn13
Zn 14
Zn 15
0
32311
32501
32890
4675453672
55789
61248
61274
61331
62459
62769
62772
62777
62910
0
0
1
2
11
0
0
1
2
2
1
2
3
1
+
-
-
-
-+
+
-
-
-
+
+
+
+
-
Cd 1
Cd 2
Cd 3
Cd 4
Cd 5Cd 6
Cd 7
Cd 8
Cd 9
Cd 10
Cd 11
Cd 12
Cd 13
Cd 14
Cd 15
0
30114
30656
31827
4369251484
53310
58391
58462
58636
59220
59486
59498
59516
59724
0
0
1
2
11
0
0
1
2
2
1
2
3
1
+
-
-
-
-+
+
-
-
-
+
+
+
+
-
Hg 1
Hg 2
Hg 3
Hg 4
Hg 5Hg 6
Hg 7
Hg 8
Hg 9
Hg 10
Hg 11
Hg 12
Hg13
Hg 14
Hg 15
0
37645
39412
44043
5406962350
63928
69517
69662
71208
71295
71333
71336
71396
71431
0
0
1
2
11
0
0
1
2
1
2
1
2
3
+
-
-
-
-+
+
-
-
-
-
+
+
+
+
4)
ATOMIC LEVELS POPULATION : Using Boltzmann equilibrium equation, evaluate (you don’t have to calculateeverything) the relative initial population (without any laser shining) of the levels corresponding to the ground state
and the excited states, when the atom vapour of Hg is heated at 100 °C. What assumption can we make ?
5) ATOMIC SPECTRUM OF Hg ATOMIC VAPOUR :
a. Give and explain shortly the selection rule for E1 dipolar electric transitions. Explain why, in the case of Hg,
the selection rule involving the spin, can be violated.
b.
Using these rules, represent in the diagram given with this problem (you should join it to your copy), the 4
absorption lines in Hg atomic spectra involving the levels given by the table (we will use the assumption
made in question 4, about the level populations).
c.
In the same diagram, represent the 14 emission lines, resulting from sequential desexcitation from levels
occupied by absorption. Show, the existence of 2 metastable levels i.e. levels populated by sequential
emission but that don’t possess dipolar electric transition E1 toward the ground state.
6) ATOMIC SPECTRA OF MERCURY LAMP (Hg-Cd-Zn) : The figure below shows the “experimental” emission
spectrum of Mercury lamp, obtained from data sheet. The intensities are given as function of the wavelength in nm in
the visible and near UV spectral range (i.e 250 nm-800 nm).
8/20/2019 Atomic Molecular Physics
7/31
3
a.
Using the transition diagrams and the levels table, show that the emission lines in the figure at the following
wavelength (254 nm, 405 nm, 408 nm, 436 nm, 546 nm) are Hg dipolar electric transition. Give for each of
those wavelength, the LS terms and J levels for the initial and final state involved in the transition.
b.
Show that the transitions in Cd and Zn, analogous to the Hg 546 nm lines, are present in the experimental
emission spectra given by the figure above.
c. Explain, how we can, using this spectrum, determine the relative quantities of Hg, Cd and Zn in the metallic
vapour in mercury lamp.
7) ATOMIC SPECTRA OF MERCURY LAMP (Hg-Cd-Zn) : The doublet in Hg (577 – 579 nm) does not correspond to the
transitions considered in the previous question. In fact the atoms in the metallic vapour are ionized when they are
submitted to a potential created between the electrodes of the lamp. The collisions between ionic flow attracted by one
of the electrode and the electronic flow attracted by the other, can populate higher energy levels.
a.
What is a doublet in emission spectra ? Show that the Hg doublet (577-579 nm), that can be seen in the
experimental data, result from the following transitions :3D2!
1P1 and
1D2 !
1P1.
b. Using qualitative arguments, can you guess what are the equivalent doublets, shown in the experimental
spectrum, for Cd and Zn ? Verify your assumption by calculation.
8) ZEEMAN EFFECT ON THE1D2
+ !
1P1
- TRANSITION IN Hg : We will focus, on this part, on the effect of a weak
external magnetic field on one of the components of the Hg doublet. The magnetic field is given by z o
u B B !
!
!=
a.
To what wavelength does this transition correspond ?
b. Give the contribution to the Hamiltonian, and the correction on energy (1st order perturbation theory) due to
this magnetic field interaction.
c. Determine the Landé factors for1D2 and
1P1 levels, and draw the energy levels obtained before and after
magnetic field levels splitting.
d. Draw Grotrian diagram and established transition probabilities relation using Fermi rules (symmetry rule,
sum rule and global non polarization rule) that you will explicit.
8/20/2019 Atomic Molecular Physics
8/31
8/20/2019 Atomic Molecular Physics
9/31
4
PROBLEM 2 – R OVIBRONIC SPECTRA IN O2 MOLECULE
The aim of this problem is to study the Shumann-Runge in O2 molecule i.e. an electronic transition corresponding to an
electrical dipolar transition between the B excited state and the X ground state.
1) MOLECULAR TERMS USING CORRELATION RELATION : We are going first to determine some of the molecular terms
for the homonuclear diatomic O2 molecule, using the correlation rules in the framework of the unified atom model.
a. Determine the atom that has the same number of electrons that O 2 molecule. Give for its ground state the
electronic configuration Ca, parity P, total degeneracy ga and associated Russell-Saunders LS terms (in
spectroscopic notation) for this atom.
b. Using the correlation rules between the LS terms in the equivalent unified atom and the molecular terms in
the molecule, determine the molecular terms that can be predicted from C a configuration. Give the
degeneracy of each molecular term obtained and verify the consistency of your result.
c. Using Hund rule for the atom, give the energy order expected for the molecular terms.
2) GROUND STATE ELECTRONIC CONFIGURATION : Give the ground state electronic configuration and its degeneracy
for O2 molecule (Z=8). For diatomic homonuclear molecule, the filling order for molecular orbitals is given by:
!g 1s < !"u 1s < !g 2s < !"u 2s < !g 2p < #u 2p
8/20/2019 Atomic Molecular Physics
10/31
5
We focused now on the electronic transition X-B, called « Schumann-Runge absorption band », of the O 2 molecule
which part of the spectra is given below (in cm-1 ). This X-B transition corresponds to an electric dipolar transition.
5) SCHUMANN-R ANGE ABSORPTION BAND: The O2 molecule presents in its absorption spectra a serie of narrow pics
called Schumann-Runge absorption band, that correspond to radiative dipolar electric transition between X3$
g
!
molecular term correlated to the atomic states O(3P) + O(
3P) to the excited B
3$
u
!
molecular term correlated to the
atomic states O(3P) + O(1D).
a. Using the selection rules in E1 molecular transitions explain why X-B transition is the first allowed
transition. Infer the excited electronic configuration to which it is associated.
b. The table below gives the vibrational levels for B state in O2 molecule
What is the minimal wavelength that can be transmitted in air, without being absorbed by O2 molecule ?
c.
Draw the difference between E(v+1)-E(v) as a function of v for B states. Can you infer by extrapolation, the
number of vibrational states trapped in the potential curve, and the dissociation energy ?d. Energy difference between
1D and
3P atomic states is 15867,7 cm-1, find the dissociation energy for O2
molecule in its ground state.
8/20/2019 Atomic Molecular Physics
11/31
1
Masters Physics for Optics and Nanophysic and HEP
Ecole Polytechnique X-2011 Phys 551B
Year 2013-2014
Final Exam – December 13th
2013
Atomic and Molecular Physics
PRELIMINARY REMARKS
•
Duration: 3 hours.• Allowed documentation: Lecture Notes – Periodic Table
•
Pocket calculator and dictionaries are allowed
• Each of your copy should contain your name. Please number your copies
• The 2 problems are completely independent
• You may answer in English or in French
PROBLEM 1 – NUCLEAR EFFECTS ON THE ATOMIC SPECTRA OF FRANCIUM
Nuclear structure has an effect on the structure of atom, in particular, effects linked to the nuclear spin (hyperfine structure,
magnetic dipole), nuclear mass (isotopic shift), nuclear volume (electrical quadripolar resonance) or weak interaction (parity
violation) are observed. These effects are generally rather weak and their highlighting is rather delicate and needs the use of
atoms cooling and atomic traps. This problem will try to illustrate some of those effects on the atom of Francium Fr (Z=87).
Constants c=3.108 m/s h = 6,6.10
-34 J.s k B=1,38.10
-23 J/K e=1,6.10
-19 C m p=1,6.10
-27 kg me=9,1.10
-31 kg
Energy equivalency 1 eV = 8065,73 cm-
= 11604,9 K = 1,6.10-
J = 2,42.10 Hz = 1/27,2 Hartree
1) GROUND ETATE ELECTRONIC CONFIGURATION : What are the electronic configuration C0, parity P, total degeneracy
gO and associated Russell-Saunders LS terms (spectroscopic notation) with their corresponding degeneracy, for
Francium atom (Z=87) in its ground state? (The atomic orbital’s filling order follows Klechkowsky-Madelung rule).
Justify the fact that this atom belongs to alkali atoms group like Na (Z=11) and K (Z=19).
2) EXCITED ELECTRONIC CONFIGURATIONS : The first excited electronic configurations correspond to the promotion of
the electron from the last occupied sub-layer to unoccupied sub-layer, without respecting necessarily the filling rule
given by Madelung-Klechkowsky. Determine the first excited electronic configurations C1 (i=1,2,3), parity, total
degeneracy gO and associated Russell-Saunders LS terms (spectroscopic notation) with their corresponding
degeneracy and associated J levels.
The table below gives for the first levels of Francium atom223 Fr (Z=87, A=223) : the experimental values of the
wave numbers expressed in cm-1
relatively to the ground state level, J quantum number and parity. Use this table to
discriminate between possible excited electronic configurations.
N° J Parity Energy (cm-1
)
1 1/2 + 02 1/2 - 12237
3 3/2 - 13924
4 3/2 + 16230
5 5/2 + 16430
6 1/2 + 19740
3) FINE STRUCTURE OF FRANCIUM ATOM : What physical phenomena are at the origin of the splitting of the electronic
configurations in Russell-Saunders LS terms, then in J levels ? (give the definitions of L, S and J). Give the
expression of the spin-orbit interaction operator and infer the Landé intervals rule. Determine when it is possible, the
value, in cm-1
and in Hz, of the spin-orbit constants.
4) ATOMIC LEVELS POPULATION : Using Boltzmann equilibrium equation, evaluate (you don’t have to calculate
everything) the relative initial population (without any laser shining) of the levels corresponding to the ground state
and the excited states, when the Fr vapour is heated to 300 °C. What assumption can we make ?
8/20/2019 Atomic Molecular Physics
12/31
2
5) ATOMIC SPECTRA OF FRANCIUM ATOM : Show that the atomic spectra of Francium, involving only the J levels given
in the table, is reduced, within the framework of the dipolar electric approximation (E1 transitions), to a doublet
(spectral lines close in energy and intensity). Calculate the corresponding wavelengths !1 and !2 for this doublet
(given in energy increasing order and in nm). To what spectral range does this doublet correspond ?
Francium possesses no stable natural isotopes. Its most stable isotope, Francium 223, has a lifetime lower than 22
minutes and is very rare. Francium can be however synthesized by the nuclear reaction:
197 Au + 18O! 210 Fr + 5 neutrons.
This synthesis process, developed at the New York State University, allows obtaining isotopes with atomic masse
equal to 210. This isotope is radioactive with a lifetime equal to 3 min, but can be isolated in magneto-optic atomic
traps. In the following we will focus on this isotope210 Fr (Z=87, A=210)
6) ISOTOPIC SHIFT IN STOMIC SPECTRA : How the atomic spectra, described previously, is modified, if we only take
into account the fine structure of the atom, by the isotopic change from to223
Fr to210
Fr ?
7) HYPERFINE STRUCTURE OF FRANCIUM ATOM : We shall now focus on the hyperfine structure of the atom i.e. the
coupling between the total kinetic momentum!
J and the nuclear spin!
I . Let’s call the resultant kinetic momentum :!
F=!
I +
!
J .
a.
Atoms possessing a nuclear magnetic moment present a hyperfine structure of their energy levels because of
the interaction of the total electronic magnetic momentum with the magnetic field created by nuclear
magnetic momentum. By analogy with the spin-orbit interaction, justify the expression of the perturbation
term in the hamiltonian induced by this coupling and given by : J I A H hf hf !!
!!=
Using first order perturbation theory, determine the gap in energy (hyperfine splitting ) induced by this term
on J levels function of I, J, F and Ahf
b. For the considered isotope,210Fr , the nuclear spin is I=6. Determine the possible values of F as well as the
gaps in energy, for the J levels associated with the ground state and the levels involved by the transitionsconcerned by the doublet determined at question 5).
Draw the obtained energy levels with their quantum numbers. The hyperfine structure constants are given by
Ahf = 7195 MHz for the ground state
A’hf = 946 MHz for the excited state with the same value of J as the ground state
A’’hf = 78 MHz for the excited state with the different value of J
c. Draw the allowed E1 (electrical dipolar) transitions using selection rules on F i.e. "F=0, +/-1. How theinitial atomic spectra will be modified ?
8) ZEEMAN EFFECT ON THE HYPERFINE STRUCTURE : We will focus, on this part, on the effect of a weak external
magnetic field on the hyperfine structure of Fr atoms. The magnetic field is given by z o
u B B !
!
!=
a.
Show that the interaction between the external magnetic field and electronic and nuclear magnetic momenta
leads to a contribution to the hamiltonian that can be written (clarify all the terms)
z N z z e z I S L H ! ! ++= )2(
b. Justify the fact that the ground state of Fr atom, is a pure spin state. In this case and in the case of a weak
external magnetic field, demonstrate that the total correction on energy is given by (clarify also all the terms)
I S S B mm Am B g E !"+!#=$ 0µ
8/20/2019 Atomic Molecular Physics
13/31
8/20/2019 Atomic Molecular Physics
14/31
4
PROBLEM 2 – MULLIKEN SYSTEM IN C2 MOLECULE
The aim of this problem is to study the called Mulliken system in C2 molecule i.e. an electronic transition corresponding to an
electrical dipolar transition between the D excited state and the X ground state.
1) GROUND STATE ELECTRONIC CONFIGURATION : Give the ground state electronic configuration and its degeneracy
for C2 molecule (Carbon : Z=6, A=12). For diatomic homonuclear molecule, the filling order for molecular orbitals isgiven by:
#g 1s < #$u 1s < #g 2s < #$u 2s < %u 2p
8/20/2019 Atomic Molecular Physics
15/31
8/20/2019 Atomic Molecular Physics
16/31
1
Master Physics for Optics and Nanophysics
Ecole Polytechnique X-2010 Phys 551B
Year 2012-2013
Exam – December 14th
2012
Atomic and Molecular Physics
PRELIMINARY REMARKS
• Duration: 3 hours. You may answer in English or in French• Allowed documentation: NONE. Pocket calculator and dictionaries are allowed • Each of your copy should contain your name. Please number your copies • !"#$ &'()* "+ ),$ &("-.$# / 0'1 -$ *".2$3 45),"6) *".2517 ),$ $'(.5$( &'()*8 9,$ $:$(05*$* 51
&("-.$# ; '($ 513$&$13$1)8
PROBLEM 1 - TOWARD OPTICAL CLOCK USING COOLED ATOMS OF STRONTIUM
Clocks used in the metrological applications get a ceaselessly improved accuracy. The atomic fountains with cold atoms of
Cs, which define the international time unit from the frequency of a microwave atomic transition, will soon achieve theirultimate performances. The optical clocks, using atomic transitions in the visible domain, are good candidates to exceed in
the future the best fountains and supply even more exact standards. An optical clock using cooled atoms of strontium was
developed at the beginning of 2000s in the SYRTE laboratory at Paris Observatory. This problem describes some physical
concept used for this experiment.
PART 1 : FINE STRUCTURE OF SR ATOM
We will focus, on this part, on Sr atom (atomic number Z=38)
1) What are the electronic configuration C0, parity P, degeneracy g and associated Russell-Saunders LSterms (in spectroscopic notation), of the Sr atom in its ground state ? (The atomic orbital’s filling order
follows Klechkowsky-Madelung rule). Justify the fact that this atom belongs to the earth alkali group ofthe periodic table.
2)
Determine the 3 first excited configurations Ci (i=1,2,3), for this atom. The electronic excitationconsists on the promotion of one of the external electron to an empty sub-shell with the same fillingorder rule. Give for this excited configurations, the parity Pi, degeneracy gi, associated LS terms usingspectroscopic notation and corresponding J levels. In stating a rule you will specify, give the energyorder for all terms LS of these configurations.
3) The table below gives, the values of the wave numbers observed for the deepest levels corresponding tofundamental and lowest excited configuration. These experimental values are expressed in cm -1 relatively to the ground state level. Identify then the configurations C i (i=0,1,2,3), LS terms andcorresponding J levels.
N° ! (cm-1) J P
123456789
1011
014318145041489918159182191831920150216992903930592
00121232110
+---++++-++
4) In many-electron atoms, the spin-orbit interaction Hamiltonian is given by S LS L A H SO
!!
!!= ),,(" ,where A(" , L,S ) is the spin-orbit constant, constant for all the quantum states associated to a given LSterm. Calculate the value of this constant in cm-1 and in Hz for all the multiplet in Sr where it is
possible to do it.Explain Lande intervals rule. Is this rule perfectly, moderately or not at all verified ? What can youconcluded concerning the validity of the LS coupling for Sr atoms ?
8/20/2019 Atomic Molecular Physics
17/31
2
PART 2 : SR ATOMS OPTICAL SPECTRA
5) Using the Boltzmann equilibrium equation, estimate (not especially calculate) initial relative populations of the levels corresponding to the ground state and the first excited states at 500 °C (Srvapour temperature in the experiment). What assumptions can be done ?
Constants:
123
10.38,1
!!
= JK k ,
1810.3
!
= msc
, Jsh
34
10.63,6
!
=
.
6) On a diagram, draw and count absorption line from populated levels at experiment temperature andsequential emission lines from levels populated by absorption. Selection rules in the case of electricdipolar transitions are given below :
• Level’s parity should change• "J = 0, ±1, except for J=0! J’=0 transitions
7) Give the wavelength #i in nm of the different absorption transition for Sr atoms. To what range of theelectromagnetic spectra do they belong to ?Define what is a metastable level. Are there, other than thermally populated, metastable levels amongthe deepest levels of the considered atom ?
PART 3 : ZEEMAN EFFECT AND OPTICAL PUMPING IN SR ATOM
We will focus, on this part, on the transition associated with #1=461 nm, between ground state level J andexcited level with different J’ value.
Placed in an external magnetic field!
B, the spectral lines split into different Zeeman components JM"J’M’. Westudy here the case where the effects of the magnetic field are much smaller than the ones induced by spin-orbitinteraction. In this case (weak field), we consider the interaction with the magnetic field as a perturbation to the
principal Hamiltonian and the spin-orbit interaction Hamiltonian. We have shown in this case that the Zeemancontribution can be written :
H z= µ
B B
!
J "!
L + 2!
J "!
S( ) J z!
J2
8) Show that the energy difference induced by H z is "W z = µ B Bg J M J , g J is the Lande factor you
should explicit.
9) Calculate this Lande factor for the concerned LS. Draw the energy levels obtained after that splittingand the allowed transitions.
10) Deduced from that the aspect of the spectra obtained when the vapour is placed in an external magneticfield B.
PART 4 : HYPERFINE STRUCTURE IN SR ATOM
87Sr is the only isotope of Sr that has a nuclear spin different from zero, actually I=9/2.We shall now focus on the hyperfine structure of the considered LS terms (ground state level and first excited
states levels associated to the first excited LS term) i.e. the coupling between the total kinetic momentum!
J and
the nuclear spin!
I =9
2. Let’s call
!
F the resultant kinetic momentum :!
F =
!
J +
!
I
11) Justify the expression of the perturbation contribution in the Hamiltonian induced by the hyperfine
structure coupling given by : J I A H HFS HFS
!!
!!=
12) Show that new transitions appears, induced by this coupling with the nuclear spin, the selection rule isnow : "F=0, +/- 1.
Calculate the wavelength # in nm associated to the transition of lowest energy.
8/20/2019 Atomic Molecular Physics
18/31
3
PROBLEM 2 – MOLECULAR PHYSIC
EXERCISE 1 : ELECTRONIC STRUCTURE OF CO MOLECULE
8/20/2019 Atomic Molecular Physics
19/31
4
EXERCISE 2 : HOMONUCLEAR DIATOMIC HALOGEN MOLECULES
The aim of this exercise is to study homonuclear diatomic halogen molecules and molecular anions (i.e. negativeanions). The electronic molecular orbital filling order is given by :
!g 1s < !$ u 1s < ! g 2s < !
$ u 2s < ! g 2p < % u 2p < % g $
2p < ! u $
2p1) Using fluorine as an example, show that both F2 and F2
- are stable molecules.
2) Without explicitly working out the solution, argue that the answer you have just found for fluorineapplies to all homonuclear diatomic halogen molecules.
3) The internuclear distance R e, the vibrational energy h&, and the dissociation energy De for fluorine andfluorine anion are given by the table below : (1 eV = 8066 cm -1)
Species R e (in Å) h& (in cm-1) De (in eV)
F2 1.411 916.6 1.60F2
- 1.900 450.0 1.31
4) Explain these data in terms of molecular orbital configurations. Compare the force constants k of thetwo species and explain the difference.
8/20/2019 Atomic Molecular Physics
20/31
5
5) Calculate the numbers of bound vibrational levels in F2 and F2-, when using the harmonic oscillator as a
model, and with taking into account the anharmonic correction.
6) Compare the rotational constants B (using rigid rotator model) of the two species and explain thedifference. Calculate the energy (in units of cm-1) of the principal absorption line in the rovibrationalspectrum starting from the ground state (v = 0, J = 0) of the fluorine molecule and molecular anions.
7) Explain how molecular vibrations and rotations are interconnected. In particular, do vibrationalexcitations affect a pure rotational spectrum ? And what about rotational excitation in a pure vibrationalspectrum ?
EXERCISE 3 : BONUS QUESTION
The Earth’s greenhouse effect is based on the principle that the energy from the Sun comes mostly in the form ofvisible and UV light, while the Earth radiates back in the infrared part of the electromagnetic spectrum. Since theatmosphere is mostly transparent in the Vis-UV range, but not in the IR, some of the outgoing energy is trapped
by the atmosphere and the planet is warmer than it would be otherwise.Explain why some of the most important greenhouse gases are molecules such as water (H 2O), carbon dioxide(CO2), and methane (CH4), while the most abundant molecules in the atmosphere, nitrogen (N2) and oxygen(O2), play no role.
8/20/2019 Atomic Molecular Physics
21/31
1
Master Physics for Optics and Nanophysic
Ecole Polytechnique X-2009 Phys 551B
Year 2011-2012
Final Exam – December 9th
2011
Atomic and Molecular Physics
PRELIMINARY REMARKS
Duration: 3 hours. Allowed documentation: NONE. Pocket calculator and dictionaries are allowed. Each of your copy should contain your name. Please number your copies. The 2 problems are completely independent The different parts on the problems are relatively independent too. You may answer in English or in French.
PROBLEM 1 – SPINS AND ORBITS COUPLING
This problem concerns the study of the fine structure for the atoms belonging to the same column of the periodic
table than carbon atom. The table below gives the list of those elements
Elements Carbon Silicon Germanium Tin Lead
Symbol M C Si Ge Sn Pb
Atomic Number Z 6 14 32 50 82
PART 1: ELECTRONIC CONFIGURATIONS – GROUND STATE AND FIRST EXCITED IN LS COUPLING
1) What are the electronic configuration C0(M), parity P, total degeneracy gO and associated Russell-Saunders LS terms (spectroscopic notation) with their corresponding degeneracy, of those atoms intheir respective ground state? (The atomic orbital’s filling order follows Klechkowsky-Madelung rule).In stating a rule you will specify, give the energy order for all terms LS of these configurations.
The first excited electronic configuration corresponds for the 5 atoms to the promotion of the electron from thelast occupied sub-layer to the first unoccupied sub-layer following the energy order given by Madelung-
Klechlovsky rule.
2) Determine the first excited electronic configuration C1(M), parity, total degeneracy gO and associatedRussell-Saunders LS terms (spectroscopic notation) with their corresponding degeneracy and associatedJ levels.
The table below gives, for the 5 elements of the series, the values of the wave numbers observed for the 4 levelscorresponding to the first excited configuration. These experimental values are expressed in cm
-1 relatively to the
ground state level of each element.
C Si Ge Sn Pb
60333 39683 37451 34641 3496060353 39760 37702 34914 3528760393 39955 39117 38629 48188
61982 40992 40020 39257 49439
3)
Explain Landé intervals rule. To what (s) atom (s) of the series this rule is perfectly, moderately or notat all verified. Infer the constant spin-orbit for the atom to which this rule is fully verified. What can
you already concluded concerning the validity of the LS coupling for this series of atoms?
8/20/2019 Atomic Molecular Physics
22/31
2
PART 2: FINE STRUCTURE INDUCED BY L-S AND J-J COUPLING
Lets now focuse on the fine structure hamiltonian H1 = W + HSO, taking into account the electrostatic
interaction between electrons (W) and spin-orbit interaction (HSO)
4)
Justify the physical origin of these contributions to the Hamiltonian and write the expression of W andHSO in the general case. Explain what physically happens in the atom in terms of orbital and spinmomenta coupling in the assumption of L-S coupling.To what hypothesis would j-j coupling be associated to ?
CASE 1: L-S COUPLING
Without calculating in details the electrostatic integrals, we will admit that it is possible, in the case of the series
of 5 atoms studied previously, to write W in the form of an effective Hamiltonian :
2121 llassaW ls
The quantities as and al are related to the electrostatic interaction integrals and don’t contain any spin-dependent term.
5) Express the energy correction S LS L SM LM W SM LM W induced by the hamiltonian W tothe energy of the first excited electronic configuration for the considered carbon-like atoms.
W will be expressed using :
the orbital and spin total kinetic momenta L and S that characterize the spectral term 2S 1 L the non-coupled orbital and spin kinetic momenta l1 and l2 and s1 and s2 of 2 electrons that
characterize these excited configurations,
and the interaction integrals as and al .
6) Calculate for each of the terms LS associated to the excited configuration C1, the correction W to thehamiltonian induced by the electrostatic interaction.
Show that only the value of as is involved in this case. What must be the sign of as to fulfilled Hundrule (this rule should be clearly defined)?
7) Give the expression of the effective Hamiltonian of spin-orbit interaction LS
SO H in the case of L-S
coupling and the one of the perturbation correction J LS
SO J
LS
SO LSJM H LSJM E induced by
spin-orbit interaction. LS
SO E will be expressed using the coupled quantum numbers J, L and S and thespin-orbit constant ALS.
8) Then calculate for each J levels, associated with the excited configuration C1, the correction LS
SO E as afunction of the spin-orbit constant ALS involved.
9) Represent in a diagram the relative positions of the levels of energy from the initial energy of the
excited configuration C1, taking into account successively the corrections W and then LS
SO E in thehypothesis of the L-S coupling.
The diagram should let appear the electrostatic interaction integral as as well as the spin-orbit constant
ALS involved. Give the numerical value of asfor the atom of the series for which the rule of Landéintervals is perfectly verified (see question 4).
8/20/2019 Atomic Molecular Physics
23/31
3
CASE 2: J-J COUPLING
For heavier atoms of the series, the spin-orbit interactions become more important than the electrostatic
interactions. The effective Hamiltonian of fine structure H1 is written then in this case :
W H H
jj
SO 1 with 222111 slasla H jj
SO
and W A j1 j2
'
j1
j2 .The electrostatic repulsion between the two electrons is expressed using the scalar product of their total kinetic
moments 111 sl j
and 222 sl j
.
10) Express the eigenvectors base, in the case where the j - j coupling replaces the L-S coupling, aftertaking into account the spin-orbit interactions, and then electrostatic interactions.
We note J
j j 21 , the corresponding energy levels.
The determination of energy gaps and relative positions, may be made in a way very similar to the case of the L-S coupling.
11)
Give the expression of the energy gap due to the spin-orbit interaction
2121 2121 j j
jj
SO j j
jj
SO m jm j H m jm j E using a1 and a2 quantities and the quantum numbers j1, l1and s1 for electron 1 and the quantum numbers j2, l2 and s2 for electron 2. Then calculate for the excited
configuration C1, the correction jj
SO E induced by the spin-orbit interaction for each of the levels
21, j j . Show that only the value of 1a is involved in this case.
12) Determine the expression of the gap energy due to the electrostatic repulsion
J J JM j jW JM j jW 2121 using the constants A j1 j2' appearing in W, and quantum numbers J,
j1 and j2. Then calculate for each of the levels 21, j j previously determined, the correction W induced by electrostatic repulsion, using the constants A j
1 j
2
'.
13) Represent on a diagram comparable to that of the previous case, the case of the j - j coupling, showing
the constants a1 and A j1 j2'
.
14) What type of coupling appears to be the best model for each case in the C, Si, Ge, Sn, Pb series ?Calculate, in the case of the atom for which the j - j coupling seems best adapted, the values of the
constant a1 and constants A j1 j2'
. Discuss the validity of the assumptions.
PART 3: ABSORPTION AND EMISSION SPECTRA IN CARBON AND LEAD
The table below gives the energies of the deepest levels for carbon and lead, corresponding to the ground stateand excited electronic configurations considered in part 1 of the problem.
Carbon C (Z = 6) Lead Pb (Z = 82)
N° J P (cm-1) N° J P (cm-1)12345678
9
01220012
1
+++++---
-
01643
1019321648603336035360393
61982
12345678
9
01220012
1
+++++---
-
07819106502145829467349603528748188
49439
8/20/2019 Atomic Molecular Physics
24/31
4
To determine the levels j1, j2 J in j-j coupling corresponding to a configuration with two equivalent electrons,we can use the rule stipulating that when j1 and j2 are equal then m j1 and m j2 must be different. This rule is
equivalent to the L+S even rule in the case of L-S coupling, for Pauli exclusion principle.
15) Using the previous rule, find the levels j1
, j2
J
corresponding to the considered ground state
configuration. To do this, you can draw a table for each possible pair 21, j j listing the values m j1 ,m j2 and MJ. You will then determine the possible values of J from MJ valuesJ. thus found. Due to
electron indistinguishability. 21, j j and 12 , j j are equivalent.
16) Identify the set of levels2S 1 L
J or j1, j2 J corresponding to the table above.
17) Using the Boltzmann equilibrium equation, estimate (not especially calculate) initial relative populations of different groups of levels at room temperature. What assumptions can be done for the
carbon and lead?Constants:
12310.38,1 JK k , 1810.3 msc , Jsh 3410.63,6 .
18) On two different diagrams (one for the carbon assuming a pure L-S coupling and one for the leadassuming a pure j-j coupling) draw and count absorption line from populated levels at room temperatureand sequential emission lines from levels populated by absorption.
Selection rules in the case of electric dipolar transitions is given below :
L-S coupling j-j coupling
levels parity should change
J = 0, ±1, except for J=0↔
J’=0 transitions
S = 0, L = 0, ±1(pure LS coupling)
j1 0 and j2 0,1
or j2 0 and j1 0,1
19) Give the wavelength in nm of the different transitions, absorption and sequential emissions for these 2atoms.
20) Define what a metastable level is. Are there other than thermally populated metastable levels among the
deepest levels considered atoms?
8/20/2019 Atomic Molecular Physics
25/31
5
PROBLEM 2 –HETERONUCLEAR MOLECULES
PART 1: ELECTRONIC STRUCTURE OF CO MOLECULE
Is given below (in eV) energy of the Atomic Orbitals (AO) involved in the formation of Molecular Orbitals(MO) for the CO molecule. These energies correspond to the binding energies for an electron placed in asubshell (nl) in carbon and oxygen atoms:
1s 2s 2p
C -270.4 eV -13.6 eV -5.44eV
O -510.3 eV -23.7 eV - 5eV
These energy values have been postponed on the diagram of figure 1, left axis corresponding to the carbon atomand right axis to the oxygen atom. These 2-axis scales are identical, from 0 to-30 eV linear and logarithmic from-30 to-1000 eV. This double scale was used to improve the clarity of the diagram and has no effect on thedelineation of the different molecular orbitals.
-30
-20
-10
0
-30
-20
-10
0
1000
100
-
-
-
-
1000
100
2p C
2p O
2s C
2s O
1s C
1s O
E n e r g i e e n e V
Figure 1 : Construction diagram of the molecular orbitals for the CO molecule
8/20/2019 Atomic Molecular Physics
26/31
6
1) Plot approximately on this diagram, the energies of the molecular orbitals that it is possible to construct
from the atomic orbitals of C and O knowing that in this case the interaction between the MO * 2 s and 2 p pushes this last just above MO 2 p. For each MO specify its symmetry and its usual name.
2) Explicit on this chart the filling leading to the neutral molecule in its electronic ground state. An electron
with a projection of the spin ms= + 1/2 will be represented by an arrow pointing to the top, an electron withspin projection ms=-1/2 will be represented by an arrow pointing down. Infer the electronic configurationC0 of this ground state. What is its degeneracy g0? What are the molecular terms associated ?
3) The first excited configuration C1 is obtained from the ground state configuration by the promotion of anelectron from the HOMO towards the LUMO. Clarify these appellations (HOMO, JUMO) and infer this
configuration and give its degeneracy g1. Write all determinantal states corresponding to this lastconfiguration and establish the molecular terms. Check that you conserve the same degeneracy.
4) The second excited configuration C2 corresponds to the excitation from the ground state configuration of anelectron from the penultimate occupied orbital (the one before the HOMO) to the first unoccupied orbital.Give this configuration and give its degeneracy g2. Using the same method that above, write all molecularterms from this last configuration. Verify that the number of states is the one given by the value of g2.
Comment on the results.
5) Table 2 below lists the positions in energy of the first 9 electronic states (some being degenerated) of the
CO molecule from the electronic ground state called X. Indicate, by filling the empty elements of this table,
the degeneracy of each of these states and the configuration with which it is associated.
label termenergy(cm-1)
degeneracy configuration
X 0
a 3 48686a’ 3 55825d 3 61120e 3 64230A 1 65075B 1 65084D 1 65928D’ 1 70000
PART 2: STUDY OF OH RADICAL
The molecule OH is known as a radical, which means that it has one unpaired electron, while still being neutral.
This is an important aspect for chemistry, since the presence of a half-filled orbital makes such a moleculehighly reactive.
1) Show that OH is indeed a radical and calculate its bond order.
2) Give the ground state electronic configuration and derive the term of its ground electronic state.
3) The OH molecule has a vibrational frequency of 3737.76 cm−1. If we were to replace the hydrogen atom bydeuterium (2H), what would be the expected vibrational frequency of the molecule considering only themass effect?
4) The actual vibrational frequency of OD is 2720.24 cm−1. How does this compare to your previous answer?
Discuss.
8/20/2019 Atomic Molecular Physics
27/31
7
PART 3: STUDY OF NITRIC OXIDE
This part concerns the diatomic molecule nitric oxide, 14 N16O.
1) Knowing that the term of the electronic ground state is 2 can you draw the energy diagram of themolecule without ambiguity? What is the bond order of the molecule and its electronic ground stateconfiguration?
2) The value of the rotational constant Be is reported, in units of cm−1, as 167195. Unfortunately, the decimal
point is missing in this number. Where should it appear? What is the equilibrium bond length of14 N
16O ?
PART 4: STUDY OF DIPOLAR CHARACTER OF LIH
1) The LCAO method gives for the molecular orbital of lower energy :
(1 ) 0,323. 2s( Li) 0,231. 2 p( Li) 0,685. 1s( H ) (1)Justify the binding characteristic of this molecular orbital. Calculate the electronic density carried by theatomic orbitals of Li and H atom.
2) (1 ) can also be written as a linear combination of atomic orbital 1s(H) and an hybrid orbital Li centered on Li atom : (1 )C a. 1s( H )C b. Li (2)
Using normalization properties and both expression (1) and (2) for (1 ) find Ca and C b coefficientsvalues (the overlap between 1s(H) and Li can be neglected). Deduce the expression of Li using 2s(Li)and 2pz(Li).
3) Estimate the partial charge carried by each atom and justify the dipolar character of LiH molecule. The
dipolar momentum of the molecule is given by : qi.
rii
CM
where qi correspond to the charge carried
by atom i and r i the distance between atom i and the mass center. Calculate for LiH molecule andcompare to the experimental value.
Experimental values (in atomic unit) : dipolar momentum LiH = 2.31, inter nuclei distance r Li-H =3.02,
masses M Li= 6,94 and M H =1,00
8/20/2019 Atomic Molecular Physics
28/31
Master Physics for Optics and Nanophysic
Ecole Polytechnique X-2008 Phys 551B
Year 2010-2011
Final Exam – December 10th
2010
Atomic and Molecular Physics
PRELIMINARY REMARKS
• Duration: 3 hours.
•
Allowed documentation: NONE. Pocket calculator and dictionaries are allowed • Each of your copy should contain your name. Please number your copies
• The 2 problems are completely independent
• You may answer in English or in French
PROBLEM 1 - FINE AND HYPERFINE STRUCTURE OF CS ATOM
An atomic clock is a clock that uses an electronic transition frequency in the microwave region of the electromagnetic
spectrum of atoms as a frequency standard for its time-keeping element. Atomic clocks are among the most accurate time and
frequency standards known, and are used as primary standards for international time distribution services, to control thefrequency of television broadcasts, and in global navigation satellite systems such as GPS. Standards agencies maintain an
accuracy of 10!9
seconds per day. Since 1967, the International System of Units (SI) has defined the second as the duration of
9 192 631 770 cycles of radiation corresponding to the transition between two energy levels of the Cs-133 atom. This
definition makes the Cs oscillator the primary standard for time and frequency measurements, called the Cs standard. Other
physical quantities, e.g., the volt and the meter, rely on the definition of the second in their own definitions.
This problem will illustrate this by the study of the fine and hyperfine structure for the ground state of the133
Cs atom, the only
stable isotope for Cs with : atomic number Z=55, atomic mass A=133 and nuclear spin I=7/2.
PART 1 : CLASSICAL EXPRESSION OF SPIN-ORBIT INTERACTION IN AN ALKALI METAL
The spin-orbit interaction is an interaction of a particle's spin with its motion. It causes shifts in the atomic spectra and
splitting of spectral lines, due to electromagnetic interaction between the electron's spin and the nucleus's magnetic field. This
part will focus on establishing the expression of spin-orbit interaction Hamiltonian HSO and energy ESO for the valence electronof an alkali metal atom, using classical electrodynamics and non-relativistic quantum mechanics.
The expression of the magnetic field created by an electron in motion in its own referential is given by : ,
where is the electron velocity, the electric field produced by the nucleus and c the velocity of light. The magnetic field
created interacts with the spin magnetic momentum of the valence electron.
1) Determine the expression of the energy depending on , , and characteristic constants.
2) Give the expression of the electric field , created by a nucleus of effective charge Z* on the electron
3)
Deduce from above that the expression for can be written . Express .
PART 2 : FINE STRUCTURE OF CS ATOM
We will focus, on this part, on Cs atom (atomic number Z=55, atomic mass A=133 and nuclear spin I=7/2)
4) What are the electronic configuration C O
, parity P, degeneracy g and associated Russell-Saunders LS terms
(spectroscopic notation), of the Cs atom in its ground state ? (The atomic orbital’s filling order follows Klechkowsky-
Madelung rule)
5)
Determine the first excited configurations C 1, for this atom. The electronic excitation consists on the promotion of
the external electron to an empty sub-shell with the same filling order rule. Give for this excited configuration, the
parity P1, degeneracy g1, associated LS terms using spectroscopic notation and corresponding J levels.
6)
The absorption spectra of the Cs from the ground state level exhibits two spectral lines with respective wavelength
!1=852,1 nm and !2=894,3 nm. Can you give an interpretation to these doublet spectral lines ? To what spectral
range does it correspond ? Evaluate in cm-1
, the energy difference between the two levels involved in this doublet.
8/20/2019 Atomic Molecular Physics
29/31
2
7) In many-electron atoms, the spin-orbit interaction Hamiltonian is given by , where
is the spin-orbit constant, constant for all the quantum states associated to a given LS term. Calculate the
value of this constant in cm-1
and in Hz for the multiplet in Cs involved in these two transitions.
8) Draw an energy diagram showing the LS terms, the energy splitting induced by the spin-orbit interaction, the
degeneracy, the energy of the levels in cm-1
, and the allowed transitions, associated to !1 and !2, between these levels.
We will label 1, 2, 3, and N1, N2, and N3, respectively the levels concerned and their populations using the growing
energy order.
9) Using Boltzmann equilibrium equation, evaluate the relative initial population (without any laser shining) of the
levels corresponding to the ground state and the excited states, when the Cs vapour is heated to 500 °C. What
assumption can we make ?
10)
We now, illuminate the Cs vapour with a laser to produce all the dipolar electric transitions allowed between the
concerned states. Draw Grotrian diagrams (symbolic diagrams showing the transitions JM"J’M’ between levels 1
and 2 and 1 and 3) associated to those transitions. What happen when the laser is polarized left hand circularly " +
?
PART 3 : ZEEMAN EFFECT IN CS ATOM
We will focus, on this part, on the transition associated with !1=852,1 nm, between ground state level J and excited level with
different J’ value.
Placed in an external magnetic field , the spectral lines split into different Zeeman components JM"J’M’. We study here
the case where the effects of the magnetic field are much smaller than the ones induced by spin-orbit interaction. In this case
(weak field), we consider the interaction with the magnetic field as a perturbation to the principal Hamiltonian and the spin-
orbit interaction Hamiltonian. We have shown in this case that the Zeeman contribution can be written :
H z= µ
B B
!
J "!
L + 2!
J "!
S( ) J
z!
J2
11) Show that the energy difference induced by H z is "W
z = µ B Bg J M J , g J is the Lande factor you should explicit.
12) Calculate this Lande factor for the concerned LS. Draw the energy levels obtained after that splitting and the allowed
transitions.
13) Deduced from that the aspect of the spectra obtained when the vapour is placed in an external magnetic field B.
PART 4 : HYPERFINE STRUCTURE IN CS ATOM
We shall now focus on the hyperfine structure of the considered LS terms (ground state level and excited states levels) i.e. the
coupling between the total kinetic momentum and the nuclear spin!
I =7
2. Let’s call the resultant kinetic moment :
!
F =!
J +!
I
14)
Justify the expression of the perturbation contribution in the Hamiltonian induced by the hyperfine structure coupling
given by :
15) Draw and label the obtained energy levels, give the degeneracy for each level and the energy splitting. The
approximate values for hyperfine structure constants are given by:
AHFS = 2500 MHz for ground state
A’HFS= 300 MHz for excited states
Determine the allowed transitions using the selection rule: "F=0, +/- 1
16) The actual time-reference of an atomic clock consists of an electronic oscillator operating at microwave frequency.
The oscillator is arranged so that its frequency-determining components include an element that can be controlled by a
feedback signal. The feedback signal keeps the oscillator tuned in resonance (maximum microwave amplitude) with
the frequency of the electronic transition between the two levels in the hyperfine structure of the ground state. The
second is defined as the duration of 9 192 631 770 cycles of radiation corresponding to this transition.
Using this definition find a better accurate value for the considered AHFS constant.
8/20/2019 Atomic Molecular Physics
30/31
3
PROBLEM 2 – STUDY OF LIH MOLECULE
The aim of this problem is to determine the Molecular Orbitals (MO) for LiH molecule from the Atomic Orbitals (AO) of its
constituents Li (Z=3) and H (Z=1).
PART 1 : ENERGY LEVELS AND ATOMIC ORBITALS FOR H AND LI ATOMS
We will first focus on the determination of the first energy levels for the constituent atoms.
1)
Give the expression of the Hamiltonian H 0, in atomic unit, for hydrogen-like atom with charge Z. Remind the
general expression of the wave functions and the expression of the energies.
2) What are the electronic configurations for H and Li in their ground states ? In the case of Li, indentify the core
electrons and the valence electron. For the determination of the energy level occupied by the core electrons in Li, we
make the assumption that this energy remain equal to the energy level for Li+. Justify this assumption.
3)
Li+ ion is isoelectronic to helium atom. Justify the
fact that it can be considered as a hydrogen-like
atom with effective charge Z# = (Z ! $), where $
is a screening constant.Give the expression of the Hamiltonian for Li
+
ion and show that the energy for its ground state
can be written using expression A, B and C.
Do not try to calculate these values. C = "
1
r12
" =5
8 # Z
4)
Determine using variational method that the value $0 of $ for which Li+ ground state energy E1 is minimum is equal to
5/16. Calculate this energy E1.
5)
We will now, determine the energy for the valence electron in Li atom, assuming that Li is an hydrogen-like atom
with effective charge Z*, Z*=(Z - #) where # is a screening constant, that can be estimate using Slater rules.
Slater rules are empirical rules that allow evaluating an effective charge seen by an electron. This effective charge
results from screening effect produced by the other electrons in the atom. The table below gives the different
contributions to take into account in the screening value depending on the considered electron.
Table 1 : Different contribution of the electrons to the screening effect
Calculate the screening constant # for the valence electron in Li and deduce its energy E2.
6) Table 2 gives the experimental values (in atomic
unit) for the energy levels for each electron for H
and Li atoms.
Discuss the differences between your theoretical
values and the experimental ones. What
assumptions are correctly confirmed ?
1s 2s 2p
H -0,5
Li -3,64 -0,20 -0,14
Table 2 : Energies (in atomic unit) for H and Li electron
Slater Rules
sub-shell of the electron
with quantum number n
considered
Contribution of the others electrons
Others electrons shell n shells n-2, n-3 shell n-1
s and p d f superior shells
s and p 1.00 0.85 0.35 0.00 0.00 0.00
d 1.00 1.00 1.00 0.35 0.00 0.00
f 1.00 1.00 1.00 1.00 0.35 0.00
8/20/2019 Atomic Molecular Physics
31/31
PART 2 : MOLECULAR ORBITALS FOR HETERONUCLEAR DIATOMIC MOLECULE LIH
The figure 1 below first gives a qualitative diagram interaction between valence Atomic Orbitals (AO) of H and
Li atoms giving the formation of the Molecular Orbitals (MO) for LiH molecule.
Figure 1 : Interaction diagram between Li and H atomic orbitals forming LiH molecular orbitals
1)
Indicate the symmetry (! or ") for each MO, and put the valence electrons on the different levels. Labelthe MO taking into account their symmetries e.g. 1!, 2!… 1", 2", …
2) What are the quantum operators that can be used to describe a diatomic molecule ? Give for each
operator its action on the molecular orbital .
3) We first consider all the molecular orbitals that give for an atomic orbital limit that can be
written . From what atomic orbitals are they issued ?
Express these molecular orbitals as a linear combination of the atomic orbitals concerned. Let us write
these molecular wave functions et . Justify the name non-binding for these MO. To which MO
as labelled in figure 1 do they correspond ?
4)
Find the relevant molecular quantum numbers associated to the wave and . Show that they have
the same energy E" = E2p(Li).
5)
We consider now, the MO with symmetries !. Show that these OM can be obtained by a linear
combination of the AO #1s(H), #2s(Li), et #2pz(Li). Justify the number of MO so-obtained.
6) The LCAO method gives for the molecular orbital of lower energy :
(1)
Justify the binding characteristic of this molecular orbital. Calculate the electronic density carried by the
atomic orbitals of Li and H atom.
7) can also be written as a linear combination of atomic orbital #1s(H) and an hybrid orbital $Li
centered on Li atom : (2)
Using normalization properties and both expression (1) and (2) for find Ca and C b coefficients
values (the overlap between #1s(H) and $Li can be neglected). Deduce the expression of $Li using #2s(Li)
and #2pz(Li).
8)
Estimate the partial charge carried by each atom and justify the dipolar character of LiH molecule. The
dipolar momentum of the molecule is given by : where qi correspond to the charge carried
by atom i and r i the distance between atom i and the mass center. Calculate µ for LiH molecule and
compare to the experimental value.
Experimental values (in atomic unit) : dipolar momentum µ LiH = 2.31, inter nuclei distance r Li-H =3.02,
masses M Li= 6,94 and M H =1,00