II. Depolarization cross-section of the (1s 3p) 3II u (N = 1) level H*2 due to electrostatic long range interactions

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II. Depolarization cross-section of the (1s 3p) 3II u (N = 1) level H*2 due to electrostatic long range interactions

M.-A. Mélières-Maréchal, M. Lombardi

To cite this version:

M.-A. Mélières-Maréchal, M. Lombardi. II. Depolarization cross-section of the (1s 3p) 3II u (N = 1) level H*2 due to electrostatic long range interactions. Journal de Physique, 1977, 38 (6), pp.547-552.

�10.1051/jphys:01977003806054700�. �jpa-00208615�

II. DEPOLARIZATION CROSS-SECTION OF THE (1s 3p) 3IIu (N

⁼

1)

LEVEL OF H*2 ^{DUE TO} ELECTROSTATIC LONG RANGE INTERACTIONS

M.-A.

MÉLIÈRES-MARÉCHAL

and M. LOMBARDI

Laboratoire de

Spectrométrie Physique,

Université

Scientifique

^{et Médicale de}

Grenoble,

^B.P.

53,

38041 Grenoble

Cedex,

^France

(Reçu

^{le 5 octobre}

1976, accepté

^le

10 fevrier 1977)

Résumé. ²⁰¹⁴ La section efficace de

dépolarisation

^{du niveau} (1s 3p)

3IIu

(v ⁼ 0, ^{N =} 1, I = 0)

de H*2 dans les collisions H*2-H2, ^{calculée en ne} considérant _que les interactions

électrostatiques

^à longue distance, ^{est de 160} Å2. L’influence du recouplage ^du spin

électronique

après la collision est faible (03C3 ^{= 175} Å2

lorsqu’on néglige

^le spin). ^Le résultat trouvé est en bon agrément ^avec l’expé- rience; ^{ceci montre} que l’interaction

quadrupôle-quadrupôle

peut ^être responsable de la collision.

Abstract. ²⁰¹⁴ The

depolarization

cross-section of the (Is

3p) 3IIu

(v ⁼ 0, ^{N =} 1, I = 0) ^{level of} H*2 ⁱⁿ H*2-H2 collision is found to be 160

Å2,

^when only ^the

long

range electrostatic interactions

are considered. The influence of electronic

spin recoupling

^{is weak} (03C3 ^{= 175} Å2 when

neglecting

spin).

Agreement

^with experimental ^{value is}

satisfactory

^and shows that the

quadrupole-quadrupole

interaction may be responsible ^{for the collision.}

Classification Physics ^Abstracts

5.480

1. Introduction. ^- In this _paper we calculate the

depolarization cross-section, Qk-2,

^{of the molecular}

triplet (S

⁼

1)

excited level

(Is 3p) 3nu (v

⁼

0,

^N

= 1,

I ⁼

0)

^of

H*

^{due to} collisions

H*-H2-

^{We use the}

theory developed

ⁱⁿ paper I

[1],

^{based on the}

predo-

minance of the

long

range electrostatic

forces,

^which

are

usually responsible

^for

depolarization.

^{We then}

compare the calculated value of

al=2

with

previously published experimental

^results

[2].

^In

appendix

^A

we

briefly

recall how the observed

polarization signal

is related to

ak=2

in this type ^of

magnetic depolari-

zation

experiment [2].

The interactions considered here ^are the

quadru- pole-quadrupole

interaction at first order of pertur-

bation,

^{and the}

dipole-dipole

interaction at second order of

perturbation.

^{We shall}

give

^{their exact forms}

in §

^{2. The collision} cross-section

ak (k

⁼

0,

^{1 or 2}

correspond

^{to the}

population, orientation,

^or

align-

ment,

destruction) resulting

from the relation

(1.48)

when transfers

(No - N)

^are

neglected

^is

given by

The

analytical expression

^{of the}

AF’ matrix,

^when

neglecting

the electronic

spin recoupling,

^is

given

ⁱⁿ

§

^{3. In}

§ 4,

^we present the calculated values of

a’

when

neglecting

^the

spin recoupling.

^In

§ 5,

^we

study

the influence of the

spin recoupling,

^and

in §

^{6 we}

compare the theoretical and

experimental

^{values of}

the

depolarization

cross-section.

2. Interactions. ^- The electrostatic

long

range interaction

potential, V,

^is

given by (I.22, 23).

^At

first order of

perturbation,

^{the first non zero term of}

interaction, CO’lKx corresponds

^{in the} present ^{case to} the interaction of the permanent

quadrupole

^moments

in the excited

(K

⁼

2)

and in the fundamental

(X -= 2)

states. This term which varies as

R-5 (R being

^the

intermolecular

distance)

^is

given by (I.24)

^where

The

following

term, which varies

as R -’,

^corres-

ponds

^to

(K

⁼

2,

x ⁼

4).

^{Note that resonant colli-}

sions do not exist as one state is a

triplet,

^{and the}

other one a

singlet.

At second order of

perturbation,

^{the first non zero}

term of

interaction, ’BJ 2K lK2X lX2 corresponds

^{to the}

interaction of the induced

dipole-dipole

^{moments in}

the excited

(Kl = K2 =1)

and fundamental

(Xl = X2 = 1)

states. This term, which varies ^as

R- 6,

^is

given by (I . 32, 33)

^where

K,

x and J are obtained

by coupling respectively Kl

^and

K2,

^{Xl and}

x2, K

^{and x.}

^K,

^X

and J can then have

only

^the

following

^sets of values :

i) K = 2, x=0

^{and J = 2.}

ii) K = 2, X = 2

^and

J = 0, 2, 4.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01977003806054700

548

Cases where K ⁼ 0 are not considered as

they correspond

^{to a zero} value of the AT matrix. The tensorial ’6

operators given by (I . 30)

^{and the}

y(KX, J)

coefficients

given by (I.34)

involved in the

present

calculation are :

where the summation concerns all the

charged

par- ticules of the molecule

(in fact,

^{in the}

present

^case of

H2,

homonuclear diatomic

molecule,

the contri- bution of the nuclei is zero and the summations

concern the electrons

only).

Here AE is an average excitation _energy between the initial state of the molecular ensemble

and the various intermediate states : e.g.

composed

^{of a}

triplet

^{state connected}

by

^a

dipolar

matrix element to

(ls 3p) 3 llu

^{and a}

singlet

^state

connected to

(1s2) 1 Eg .

^{Due to the}

^peculiar

^{form of}

the _energy level

diagram

of the molecule

(the

^funda-

mental state is well

separated (~ 100

⁰⁰⁰

cm-’)

^from

the bunch of the excited levels

( ±

^{20 000}

^{cm -1}

^around

the

mean)),

^{we have} taken AE = 100 000

cm -’

(i.e.

^0.5

Rydberg).

^{Since as we shall see} this interaction is not the dominant _one, this

rough approximation

is sufficient.

3. The Ar matrix. ^- The dTk matrix _expresses the evolution of the tensorial

density

^matrix compo- nent,

p’,

of the excited molecule

during

^collisions

(1.6, 9).

When coherences ^are

neglected (which

^is

the case in the

experiments

under consideration

[2]),

the

AF’

matrix is

given by (I.42)

^{which can be written}

in the present ^case in the form

(1.45)

with three different terms :

-

ð.rk(’tJ lKx’ U1Kx)

^{with K = x = 2 which will}

be written

ð.r:uad-qUad.

^{This term}

expressed by (1.43)

varies as

b-8 (b being

the collision

impact parameter)

and

depends

^on the cut-off function defined

by (I . 41),

4

^5,5-

which will be written as

4Td;p-a;p.

^{This term}

^given

explicitly by (1.44)

^{varies as}

^{b - 10}

^and

depends

^on

al,

^{with J =}

^{0, 2,}

^{4. The two}

corresponding

^different

cases

i)

^and

ii)

^mentioned

in §

¹ will be written ^as

Arb;p-d;p ^{(K = 2, X}

⁼

⁰⁾

^and

Arb;p- d;p ^{(K = 2, X = 2).}

Note that the last case must be considered

only

^when

a

permanent quadrupolar

^moment exits in the fun- damental state

(x

⁼

2),

and is therefore not considered in atomic collisions.

- 2

Ark(’BJ lKx, ’BJ2KIK2XIX:J

^or

2 Ar:uad-dip.

^This

term is

given by

^the

general expression (1.42)

^with

s = 1 KX (K = X = 2) and s’ = 2 K1 K2 X1 X2 (Kl

⁼

K2

^{= xi =} X2 ⁼

1)

^where the b functions TABLE I

Mean values

of

the electronic part

of

^the

UKQ

^operators

imply Ks’ = ki

⁼

2,

xs, = Ki ⁼

2, Js’

^{= 4. This term}

varies as

b-9

and

depends

^on

a 4

The

analytical expressions

of the cut-off functions

are

given

^{in the}

appendix

^B. The reduced matrix elements of the ’B1

operators

^are calculated

using expressions (1.67)

for electronic and

(1.68)

^for

nuclear contributions. In these calculations we have used the electronic wavefunctions of

Rothenberg

^and

Davidson

[3]

^{relative to the}

(Is 3p) 3llu,

^{v =}

0,

^state and the mean values of the

operators z2, r2,

ZI z2, XI X2 in the fundamental state

given by

^{Kolos and}

Wolniewicz

[4].

^The

corresponding

^{results are}

given

in table I.

The nuclear contribution has to be calculated to the first order of

perturbation only (see § 1).

^The

hK(P) (see § ^1, appendix D)

reduce then to

where _p is the intemuclear distance. The mean value of this function has been calculated

using

^{the results}

of Matcha

[5].

^{We found}

that p >2

⁼

4.147(a2)

ⁱⁿ

the excited state

and ( p >2

⁼

2.074(aõ)

in the fun- damental.

4. Results. ^- The cross-section

ak

relevant to the excited state

(Is 3p) 3llu (v

⁼

0,

^{N =}

1, I = 0)

^is

calculated

using (I.48).

^The

states involved in the calculation ^are related to N

by

the selection rules N’ ⁼ N ±

(2,0).

^{The same}

selection rules connect v and _vo in the fundamental

state. The level _energy values are

given by

^Dieke

[6]

and the first 11 fundamental rotational ^{states are} included. The

following approximations

^{are made :}

the relative

velocity

^{v is}

replaced by

^{its mean value}

v(T), ^Ilrk

^is calculated for b

varying

^from 5 ao ^to 25 _ao

(ao being

^{the Bohr}

^radius).

We have also

neglected

^the

spin recoupling.

^{In table}

^II,

^and

figure 1,

the results

correspond

^{to T = 600}

^K; figure

^{2 shows}

the variation of the

depolarization

cross-section

ak= 2

with

temperature.

FIG. 1.

- I1r2 total.

--- (1) ^{AF 2} dip.-dip. (K ⁼ 2, x ⁼ 2).

---

(2) ^{I1r 2} dip.-quad.

--- (3) ^{AF 2} dip.-dip. (K ⁼ 2, x ⁼ 0).

---

(4) ^{I1r 2} quad.-quad.

FIG. 2.

In order to show the

respective

contributions of the different interactions we have

presented

^{in the}

first

column,

of table

II,

the value of

a’,

^{when the}

quadrupole-quadrupole

^{and the}

dipole-dipole

^interac-

tions are

simultaneously

considered and in the second

column,

the value of

a’

when

only

part of

årk

is included. We indicate the value of the cut-off _para- meter,

bo,

^{in, each case.}

We have

plotted

ⁱⁿ

figure

¹ the function

årk=2(b)

and its different components.

TABLE II

Values

of ^ak corresponding

^{to T = 600 K}

550

5. Spin

^{influence on the}

^fIc=2.

^{- The}

recoupling

^of

the electronic

spin (S

⁼

1)

of the excited molecule with the orbital momentum N after collision reduces the

anisotropy

^{of this state and therefore} reduces the

polarization

of the emitted

light.

^The

depolarization

cross-section will therefore be smaller when the

spin recoupling

is included. Calculations

concerning

^the

spin recoupling

^are

given

ⁱⁿ

(I, § 4).

^We have calculated the

quantities NJ NJ NJoNJ°ð.rk using

^the

expression (I . 54) ;

^the

corresponding

^{values of ak}

(nvak

⁼

AF)

are

reported

^{in table III.}

TABLE III

Values

Of

^{NJ NJ NJo}

^NJOak

ⁱⁿ

(Á 2) for

^{T = 600 K}

In the present ^{case the fine structure} of the level considered is not

optically

resolved and we

experi- mentally [2]

observe the

superposition

of the three

optical

lines emitted

by

the three fine structure levels.

The cross-section

Uk = 2 experimentally

determined is the

slope

^{of the curve}

dH( p) where p

^{is the} gas- pressure and AH the width at

half-height

^{of the}

polarization, P(H),

^{as a function of the}

magnetic field,

^H

(Hanle

^or

depolarization

^effect

[7]).

The

polarization

^{P is} calculated

using

the D’Yako-

nov

[8] expressions

^{for the}

intensity

of the emitted

light,

^{and is}

proportional

^to

The

p’s

^are solutions of a system ^of

coupled equations :

The excitation

density

matrix has been

previously

calculated

[2]

with the Percival and Seaton

[9] hypo-

thesis of fixed

spin

excitation

(it

^was shown that all the ^NJ

NJ’pqxc. k ^depend

^on one

NNpq

^parameter

^only).

When k ⁼

2,

^this system ^{reduces to two}

coupled equations.

^{We have here}

supposed

^as usual that the radiation lifetime T is

independent

^of

J,

^{and have} taken

for 9 [2] :

The calculated

quantity AH(p)

^{shows a} linear varia- tion with pressure, and the

slope gives

^{the cross-}

section value of

Qk - 2

⁼ 160

A2.

6. Discussion of results.

Comparison

between expe- riment and

theory.

^{- 6.1. COMMENTS} ON RESULTS. ^-

From table II and

figure

¹ we can see that the _qua-

drupole-quadrupole

interaction is

mainly responsible

for the collision. We notice

that,

^{in the}

dipole-dipole interaction,

the main contribution comes from the term

Ar k=2 ip ^(K

⁼

^2,

^{x =}

^0),

the contribution of the other term,

Ar k=2 ip ^(K

⁼

^2,

^{x =}

^2),

^which

makes the calculation somewhat more difficult is almost

negligible.

^{In the same} way the contribution of the cross terms

Ar k=2 is quite negligible.

Figure

^{2 shows a}

relatively

small variation of

Uk=2

with temperature

(180 A

^{to 155}

A

when T vary from 500 to 1 000

K) (1).

^However this variation has to be considered

because,

^{in the} _present ^electron

impact

excitation type ^of

experiment [2],

^the temperature ^is

not very

homogeneous

in the observation zone and the mean value of T has not

actually

^been

precisely

determined

( ±

¹⁰⁰

K) (see § 6.2).

From the results obtained

in §

^5, we can see that the

spin recoupling

^{does not}

strongly

affect the cross-

section value

(minus

¹⁰

%).

6.2 COMPARISON BETWEEN EXPERIMENTAL AND THEORETICAL RESULTS. ^- The

published experimen-

tal

[2]

^{value of}

^ak=2

associated with the temperature 800 K was 230 ± 30

A2.

Later temperature ^measu-

rements have shown

that,

^{in that}

early experiment

^the temperature ^was

overestimated,

^{and the}

uncertainty

in T

underestimated;

^{the new} estimation is T ⁼ 600 ± 100 K. The

corresponding experimental

value of

Uk=2,

^{relative to the}

slope AH(p),

^{is after}

this correction

Uk=2

⁼ 200 +

40 A2. Independent

measurements have been

performed by Baltayan [10]

using

^{a different} apparatus ^{and have}

given

^similar

results

(167 ±

²⁵

A2

for T ⁼ 800 ± 100

K).

^The

theoretical value of 160

A2

_agrees with the

experimen-

tal value. This agreement suggests ^{that the}

quadru- pole-quadrupole

interaction is

responsible

^{for the}

collision.

7. Conclusion. ^- In this

article,

^we have calculated the

depolarization

cross-section

Qk-2

of the

(Is 3p) 3nu (v

⁼

0,

^{N =}

1,

^{1 =}

0)

^of

H2,

^{in a collision}

H2-H2.

This calculation was based on the

hypothesis

of

predominant long

range electrostatic

interactions,

which has been

theoretically

described in ^a

preceding

article

[1]. Agreement

^between

experiment

^and

theory

(1) ^When neglecting ^the spin recoupling.

allows us

only

^to conclude that if

long

range forces

are

predominant,

^the

quadrupole-quadrupole

^interac-

tion is

responsible

of the collision. It does not exclude the

possibility

of the existence of other mechanisms

(short

range

forces),

if these mechanisms

give

^the

same value for Uk. This could be verified

by

^an expe- rimental

study

^{of the}

dependence

of cross-section on

temperature.

Appendix

^{A. - In the}

magnetic depolarization experiment [2],

the excited molecular state is

aligned

^and

a static

magnetic field, Hz,

^is

applied perpendicularly

^{to this}

alignment.

^The

polarization

^of the emitted

light

is studied as a function of Hz at different pressures. This

polarization

is calculated as follow.

The components ^{of the}

intensity

of the emitted

light (and

therefore the

polarization)

^are

expressed

^{as a}

function of the tensorial

density

^matrix components,

NN’pk, ^using

the D’Yakonov

expression [8]

^{when no}

spin coupling

^exist

(N

⁼

J,

^{S = I =}

0). The NN’pq

^{are obtained}

^by resolving equation (I.5); neglecting

^coherences

and transfers from

No

^{to N and}

using (I.6)

^{one obtains}

where g

is the Lande factor

The type ^of excitation

imposes

The

dependence

^{of the}

polarization

^{on Hz is then}

given by

The full width at half maximum

of P(Hz)

^varies

linearly

^with pressure, the

slope being

^{related to}

^(J’k=2.

Appendix

^{B. -}

Analytical expression

^{of the} cut-off function. ^-

l. a 45,5

^{CAN BE} EXPRESSED AS FOLLOWS

where

J.

^is

given by

These

integrals

^{can be}

expressed

^{in terms of other}

integrals in

^{which are related to} Bessel functions of the second

kind, Kn(1

^a

1),

^{with a =}

wblv. If in

^is

given by

we

finally

^{obtain :}

- These functions can be

expressed using Jm integrals

^where

Jm

^is

given by

We obtain :

552

The

a. integrals

^{can be}

expressed

^{in terms of}

other jn integrals,

^these

being

evaluated in the

complex plane

and

given by

Thus we

finally

^{obtain :}

with

3.

a 45,6 IN THE SAME WAY WE HAVE :

^:

with

References [1] MÉLIÈRES-MARÉCHAL, ^{M. A. and} LOMBARDI, M., ^J. Physique

38 (1977) ^527.

[2] MARÉCHAL, ^M. A., JOST, R., LOMBARDI, M., Phys. ^{Rev. A 5} (1972) ^740.

[3] ROTHENBERG, ^{S. and} DAVIDSON, E. F., J. Chem. Phys. ⁴⁵ (1966)

2560.

[4] KOLOS, W., WOLNIEWICZ, L., J. Chem. Phys. ⁴³ (1965) ^2429.

[5] MATCHA, Private communication.

[6] DIEKE, ^G. H., Wavelength Tables of the Hydrogene ^Molecule (Interscience, ^New York) ^1971.

[7] HANLE, W., ^Z. Phys. ³⁰ (1924) ^93.

[8] D’YAKONOV, ^M. I., ^{J.E.T.P. 20} (1965) ^1484.

[9] PERCIVAL, J. C. and SEATON, ^M. J’, Phil. Trans. Roy. ^Soc.

251 (1958) ^113.

[10] BALTAYAN, P., Thesis, ^Grenoble (1973).