atomic molecular physics

8/20/2019 Atomic Molecular Physics

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Atomic and Molecular Physics

P ’

Textes de contrôles des connaissances

proposés les années antérieures


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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).


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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.


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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


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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.


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Masters Physics for Optics and Nanophysic and HEP

Ecole Polytechnique X-2011 Phys 551B

Year 2013-2014

Final Exam – December 13th

2013


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 ?


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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


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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


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1

Master Physics for Optics and Nanophysics


Year 2012-2013

Exam – December 14th

2012


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 ?


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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.


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PROBLEM 2 – MOLECULAR PHYSIC

EXERCISE 1 : ELECTRONIC STRUCTURE OF CO MOLECULE


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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.


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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.


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Master Physics for Optics and Nanophysic


Year 2011-2012


2011


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?


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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).


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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


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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?


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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


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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.


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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


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Master Physics for Optics and Nanophysic


Year 2010-2011


2010


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.


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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.


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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


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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

atomic molecular physics

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