Relative intensities and polarizations in H2 rotational lines excited by electron impact

HAL Id: jpa-00208236

https://hal.archives-ouvertes.fr/jpa-00208236

Submitted on 1 Jan 1975

HAL

is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire

HAL, est

destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Relative intensities and polarizations in H2 rotational lines excited by electron impact

P. Baltayan, O. Nedelec

To cite this version:

P. Baltayan, O. Nedelec. Relative intensities and polarizations in H2 rotational lines excited by electron impact. Journal de Physique, 1975, 36 (2), pp.125-133. �10.1051/jphys:01975003602012500�.

�jpa-00208236�

RELATIVE INTENSITIES AND POLARIZATIONS

IN H2 ROTATIONAL LINES EXCITED BY ELECTRON IMPACT

P. BALTAYAN and O. NEDELEC Laboratoire de

Spectrométrie Physique (*)

Université

Scientifique

^et Médicale de

Grenoble,

^BP

53,

³⁸⁰⁴¹

Grenoble-Cedex,

^France

(Reçu

^{le 22}

juillet 1974)

Résumé. ²⁰¹⁴ Les valeurs relatives des intensités et des polarisations ^sur les raies de rotation de H2

d’une même bande excitée _par impact électronique ^sont calculées dans

l’approximation

^{des atomes}

réunis par addition de ^vecteurs. Elles

correspondent

^{à une valeur} quelconque de la vitesse des élec- trons au-delà du seuil. Les expériences ^{faites sur} les bandes 3p

303A0u

^{~ 2s}

303A3+g

^{sont en} accord raison- nable avec la théorie. Les effets de cascade sont discutés.

Abstract. ²⁰¹⁴ The relative values of the intensities and

polarizations

ⁱⁿ H2 rotational lines in a

band excited by ^electron impact ^are calculated in the united atom

approximation

by ^{vector addition.}

They ^{are valid at} any electron energy beyond threshold.

Experiments

^{made on the} 3p

303A0u ~

^2s

303A3+g

bands show reasonable agreement ^{with the}

theory.

Cascade effects are discussed.

Classification

Physics ^Abstracts

5.482

1. Introduction. ^- If an atomic _{vapour is} excited

electronically,

^{it is well known that} the emitted

light

is

polarized.

^It has then been

possible

^{to extend to}

many excited states the

level-crossing

and double

resonance

experiments

^done

previously by optical

excitation. As such

experiments

^{were extended to}

molecules

by optical

and electronic excitation in many laboratories

[1],

^we

attempted

^{to calculate}

the

polarization

of molecular lines

by

^electron

impact

in a few

particular

^cases

[2].

^{In this} paper, the calcula- tions are

applied

^to

H2

^and

compared

^{to the measure-}

ments.

The

polarization

of the emitted

light

^{is due to the}

privileged

direction of the incident electrons. In atoms, and in the united-atom

approximation

^for

molecules,

the orbital

angular

^{momentum L} transferred

by

the electrons is

perpendicular

^to the electron beam at threshold

[3]

^{and has}

cylindrical

symmetry ^around the beam

beyond

threshold. Calculations of the

polarization

^at threshold have been made in atoms

[4]

using simple

^{vectôr addition}

involving

the various

angular

^{momenta in the}

ground

and the excited states and the

angular

^{momenta L and S}

given by

the electrons. We have extended these calculations to molecules in the

Born-Oppenheimer

and united- atom

approximations. Among

the lines in a

band,

the electronic and vibrational part ^{of the wave func-} tion of the involved levels are the same : the variations in the intensities and

polarizations

^{of the emitted}

light depend

^on the values of the rotational momen- tum and it is

possible

^{to obtain a} relation valid for any energy of the incident electrons.

In helium

°[5J,

absolute values of the

polarization

as a function of the electron

speed

have been measured and

compared

^{to the}

theory

^at threshold.

Agreement

has been found for a few lines excited

by

^{a mono-}

energetic

electron beam

(AE -

^0.02

eV).

^{No rela-}

tion is known between the

polarization

^{at threshold}

and at a

given

energy. In

H2,

^such

precise

^measure-

ments are made much more difficult

by

^{the weak}

intensities and the small

separations

of the molecular lines.

However,

^the

polarization

has been measured

as a function of the electron _energy

[6]

^{in a}

H2 triplet

line. The curve is _very similar to the

corresponding

curve obtained for atoms with an energy

spread

^of

about 1 eV : the

polarization

is maximum for electron

energies

^{about two times the} threshold value.

In our

experiments

^for

H2,

^we chose the _energy of the electrons which

yielded

^the maximum value of the

polarization (35 eV)

^{and the} measurements were restricted to the relative

polarizations

^{of the}

rotational lines in a few intense bands.

AÏ

the

speed homogeneity

^{and the}

directivity

of the beam have

no

explicit importance

^{we used a beam current and}

a gas pressure

high enough

^{to obtain a}

good signal-to-

noise

ratio,

^{but low}

enough

^to avoid collision

depola-

rization

[7, 8].

It is well known that

complications

ⁱⁿ the excita- tion _process occur near threshold due ^to the

temporary

attachment of the incident electron. This

phenome-

(*) ^{Associé au C.N.R.S.}

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01975003602012500

126

non is called core or Fechbach resonances

[9].

^At

thirty

^{eV in}

H2,

where the ionization energy is ^15.4

eV,

no excited state is obtained

through

resonances.

As in

He,

resonances may ^occur

only

^via

H2 triply

excited states which

give

^{rise to}

auto-ionizing

^levels.

The absence of such resonances

permits

^a

simple description

of the collision.

The conditions for the

validity

^{of the}

theory

^are

considered and numerical calculations are

perform-

ed

(§ 2).

^The

polarization

^and

intensity

^measure-

ments are described and the cascade effects are also estimated

(§ 3).

2.

Theory. -Z .1

CONDITIONS OF VALIDITY. - The calculations are

simply

^{vector additions if} the follow-

ing

conditions are fulfilled :

- The molecule is described in the

Born-Oppen-

heimer

approximation :

^the

electronic, vibrational,

and rotational parts ^{of the wave function are inde-}

pendent [10].

- The united-atom

approximation

is valid : the orbital

angular

^{momentum L is a}

good

quantum number.

- L ⁼ 0 in the

ground

^{state : the}

magnitude

^{of L}

to be added to excite a

given

electronic state is

uniquely

defined.

- The collision process is instantaneous : the nuclei of the molecule are fixed

(Franck-Condon principle),

^{and L} in the excited state is that

given by

the incident electron.

- The addition of orbital and

spin angular

momenta may be considered

independently

^{and the}

spins

of the incident electrons are

isotropic [4].

- The

diffusing potential

^has

spherical

symmetry : the electron

scattering

^is

independent

^{of the direc-}

tion of the molecular axis.

- The difference in _energy

required

^{to excite}

the rotational levels in a band is small

compared

^to

the

spread

ⁱⁿ energy of the incident electron beam : this _energy difference has no influence on the relative cross-sections.

The

validity

of these conditions are discussed in

our

experimental

^case in 2.2 and the

physical meaning

of the calculations is

explained

^{in 2.3.}

2.2 THE EXPERIMENTAL CONDITIONS. ^- We have chosen the more intense and

separable

lines in the

H2

spectra : the Fulcher

bands, 3p ’Hu --+

^2s

3Eg

^tran-

sition,

excited from the 1 s

’,E ^ground

^state

^{(Fig. 1).}

The

Born-Oppenheimer approximation

^{is not} gene-

rally

^{valid for}

H2

^{due to the}

strong

vibration-rotation interaction.

However,

the variation of the Franck Condon factors with N is

negligible

^{when v = 0 for}

one of the levels of the transition or when the two

potential

energy ^curves may be

superposed by

^a

translation in energy. If these conditions are

fulfilled,

the calculations for the excitation or the radiative

decay

^are

simplified

^as the relative transition

probabi-

lities between different rotational states may be _expres-

FIG. 1. ^-

Possible

transitions for electronic excitation

and radiative emission 3p 3IIu -> ^2s

3 E g+.

^{o and p mean respec-}

tively ortho and para.

sed

using Wigner’s 3 - j

^and

6 - j symbols.

^This

is the case for our

experiment :

^{in the}

ground

state,

only

^{v = 0 is}

populated;

ⁱⁿ the Fulcher

Q ^bands,

the calculated Franck Condon factors show a variation of less than 1

%,

^but their variation is greater ^{for the R} and P lines which arise from levels

perturbed by

3p 3Eû (11,12].

The united atom

approximation

is valid for the

highest

^{excited states of}

H2,

^{which tend} towards the

Rydberg

^series

[10, 13].

^{For n =}

^3,

the internuclear distance is small

(1 A) compared

^to the diameter of the classical orbit

(13 ^A according

^{to Slater}

[14]

and Messiah

[15]).

^{Thus we} take the orbital

angular

momentum L of the molecular excited state as a

good

quantum number with the

origin

^{at the center}

of the molecule.

The united-atom

approximation

^also supposes that the electrons are diffused

by

^a

spherical potential, independent

of the direction of the molecular axis.

Calculation of inelastic cross-sections for a few eV electrons in

H+ [16]

have shown that the

anisotropy

due to the molecular axis is

negligible compared

^to

the

polarization

^induced

by

the incident

électron.

In our

experimental

^{case, it is}

possible

^{to show}

by simple

considerations that this

hypothesis

may be reasonable. In the

H2 molecule,

^the

probability density

distribution of the electrons in the

ground

state has

spherical

symmetry

beyond

^0.8

^A

^{from the}

center

[10

p.

352].

^{For 35 eV incident} energy and 13 eV energy

loss,

^the

probability density

^distribu-

tions of incident and emergent ^waves

[15]

^{such as}

AI. ⁼ 1 are the most

important

^at the limit of the

spherical

part of the molecule.

The collision

time,

^about

^10-16

^s

( 17],

is the shortest of all the considered times. The

period

of vibration is greater ^{than 4 x}

lO-15

s which is the value for the

highest

considered v

(v

⁼

3).

^The

period

^{of rota-}

tion is greater ^{than 5 x}

10- 14

s which is also the value for the

highest

considered N

(N

⁼

4).

^The

period

^{of L due to} the nuclear electric field _may be

deduced from the _energy

separation

^between

3p 3HU

and

3p 3 Eu+ :

^{10- Il s.}

Although

^a few of these

approximations

appear to be

approximately valid,

^{it seems}

interesting

^to

apply

^the

simple theory

^{as a first}

approximation

^to

calculate the relative intensities and

polarizations

in

H2

^{lines. It} allows the calculations ^to be done

using only

^vector addition and not molecular wave-func- tions. Its

validity

^{will be}

justified a posteriori by

^the

experimental

^results.

2. 3 PHYSICAL MEANING. - We shall

emphasize

^the

fact that the calculations

give

the relative

polariza-

tions and intensities at any electron

velocity.

The different

angular

^momenta of the molecule

are denoted as usual

[10]

except that the notation for the rotational

angular

^{momentum is}

changed

from K to 1V to conform to more recent usage.

The orbital

angular

^{momentum L}

given

^{from the}

electron beam to the molecule _may be defined in the united-atom

approximation :

^{L = 1 for a s -} p

transition,

^{L = 2 for a s -} d transition. If the _quan- tization axis oz is taken

parallel

^to the electron

beam,

the excited state has

cylindrical

symmetry ^{around oz} and the

density

matrix is

diagonal.

^At

threshold,

L is

perpendicular

^to the beam :

ML

^{= 0. The}

density

matrix _p of the excited state is at threshold :

T ML

^{is an} irreducible tensor operator

[ 15] ;

^{N’and N"}

denote rotational levels in the excited and in the

ground

^state

respectively.

Beyond threshold,

^{for a}

given speed

^{for an incident}

electron,

^{we are unable to}

predict

^the

probability 1

aLM,,

l’

^{for an orbital}

angular momentum [ lez )

to be added to the molecule. The

density

^matrix

may be

expressed

^{as a} function of aLML :

with

We assume that the

ground

^{state is}

isotropic :

and we express the matrix elements

[18]

^{in the b}

coupling

^case.

The component ^{of the}

density

^{matrix on the}

tensorial

basis N’TÔ

^defined

^by

^Omont

^[19]

^are

^{[2, 7] :}

with

where Vi,

^and

F.

^are

respectively

the vibrational and the electronic wave functions.

Q = 0

^{since oz is}

taken

parallel

^{to the beam. We know that}

as

they correspond

^to

opposite

deviations of the incident

electron,

^{and the non-zero} _components

N’ Il correspond

^{to even values of x}

[20].

To describe the

linearly polarized light

^emitted

after electron

impact,

^{we have to consider} x ⁼ 0

(population)

^and x ⁼ 2

(alignment).

The relative values of the

populations

may be calculated from

(3).

The

alignments

ⁱⁿ the various rotational states are

maximum at threshold where

Lpô- 2

is maximum.

Regardless

of the electron _energy, their absolute value cannot be

calculated,

but their ratios are

given by (3) :

the relative values

of

^the

alignment

^{in the}

’ rotational levels in a band are

independent of

^the

speed of

the incident electrons.

If a level N’ is excited from _many levels N" in the

ground

^state, the relative

populations

^or

align-

ments are obtained from a sum over N"

weighted by

the relative

populations PN" :

The

population

^{of N’} excited from N" is

given by :

The

degree

^of

alignment

^{on a line} N’ ---> N is defined

as :

where

III

^and

^Il

^are the intensities of the

light

propa-

gating

^{in a direction}

perpendicular

^{to the electron}

beam,

^with

polarization parallel

^or

përpendicular

to the electron beam. _{It may} be obtained from the

weighted

^{sum :}

The

tN’->N(N")

may be found in

[2].

We now consider the

experimental

^{case of an inci-}

dent electron beam

having

^an

angular spread.

^Let

oz’ be the direction of an ensemble of incident elec-

128

trons

making

^an

angle

^{8 with} respect ^{to oz. The obtai-} ned

alignment along

^{oz is}

given by :

where

rôo-(B)

⁼

2(3 ^cos2

^{9 -}

1)

^{is a rotation matrix}

element

[15, 20]

^and

(N’p0x=2)oz’

^is

given by (3).

^The

resulting alignment

^will be obtained

by

^the

weighted

sum over the various 0

angles.

^{A lack of}

directivity

of the electron beam therefore decreases the

magni-

tude of the

alignment

^{but does not}

change

the relative values of the

alignments

in the rotational lines of a

band.

2.4 CALCULATIONS. ^- After this discussion of the

physical problems applied

^to

singlet

states, ^we

give

the formulae in a form suitable for numerical calcula- tions when electronic and nuclear

spins

^are present for the case of b

coupling.

We first consider the case where J and F are

good

quantum ^{numbers. We then} consider the

special

case of the ortho

3p 3 nu

^{levels of}

H2

^{where J is not a}

good

quantum ^number.

Finally,

^we

give

the calculated values of the

degree of alignment

^{of the}

^{P, Q,}

^{R Fulcher}

lines.

2.4.1 Natural width «

hyperfine structure «fine

structure. ^-

Using

^the

hypothesis

^of Percival and Seaton

[4],

^{it can} be assumed that there is an

equal probability

^for

populating

^{all the}

spin

^{states. When}

an

isotropic

electronic

spin

^is

added,

^{one obtains}

[19, 7, Ô 1

- -. -. -

The addition of an

isotropic

^nuclear

spin gives, assuming

^{that J’ is a}

good

quantum ^{number :}

The observed

intensity I(eu, À)

^{derived from the}

d’Yakonov

[21] theory

^and

expressed

^{in the nota-}

tion of Omont

[19],

when the fine and

hyperfine

structures are unresolved

by

^the

optical

system, ^{is :}

where

cP:(eu) ^[21]

^are determined

by

^the

polarization

^{e used to} observe the

light

^emitted

along

the direction u.

We have assumed that the

hyperfine

^{structure is much}

greater

than the natural width of the

emitting

^{level. The}

intensities

expressed

^{in that} way

give

^{the same results as} the Hônl-London formulae

[10].

The

degree of alignment t

^{measured on a line}

originating

^{from the} rotational level N’

(N’ -

^N

transition)

^{is :}

For lines

originating

^{from levels N" which are} excited from

only

^{one level N" of the}

ground

^state

(e.

^{g. the}

Q

branches of the Fulcher

bands) :

The case with zero

spin (S

^{= 0 or I =}

0)

^is

easily

deduced from

(11)

^and

(12).

(1) ^Formula (5) of reference [2], with S" ⁼ 0, gives ^{the same}

result as (8) using the relation :

where S, L, V, ^{V’ are} integers.

2.4.2 Natural

width « hyperfine

structure =

fine

structure. ^- The measurements made

by

^Freund

and Miller

[22]

have shown that for

orthohydrogen,

in the

3p 3 nu ^level,

^the

hyperfine splittings

^are compa- rable to the fine structure. That is to say, J’ is no

longer

^a

good

quantum

number,

and the wavefunc- tion for a state of total

angular

^{momentum F’}

is :

where the sum is over all

Jl

^{values for a}

single

^rotatio-

nal level N’. The

wavefunction (J’) F’MF >

^{is an}

eigenfunction

of the total hamiltonian

where

Jefs

^and

Jehfs

^denote

respectively

^{the fine}

and

hyperfine

^structure hamiltonians. The matrix

elements of

Jefs

^and

Jehfs

with the functions

1 yN’ SJ’ I ;

^F’

MF >

have been calculated

by

^Fon-

tana

[23],

^Chiu

[24]

^and

by

^Jette and Cahill

[25].

^We

have

diagonalized

the matrix of X, and calculated the coefficients

C(F’

^{, for N’ =}

^{1, 2, 3}

^{and 4}

^[7],

using

the fine and

hyperfine

structure constants measured

by

Freund and

Miller,

and calculations of Lombardi

[26].

(1)

^{The constants are}

expressed

ⁱⁿ

Mc/s.

(2) d

has been calculated

by

^Lombardi.

c is deduced from c - 3 d measured

by

Freund and Miller.

When J’ is not a

good

quantum

number, @)px

^and

^{I(e., Â)}

^are

^expressed

^{in terms of}

C(j,)j,

^and

⁽¹²⁾

^{becomes :}

The numerical values of t calculated with these formulae are

given

^{in table 1.}

z

TABLE 1

3.

Experiments.

^{- 3 .1} POLARIZATION MEASURE- MENTS. ^- We shall _compare the relative values of the

degree

^of

alignment

^{measured on} the Fulcher bands with the values obtained from our

theory

^and

given in §

^2.4.

In our

experiment,

the molecular

hydrogen

gas is excited in a triode

glass

^{cell. We}

^measûred

^the

degree

of

polarization

_p ^{of a line P =}

IL

and then calcu-

Il

^II

^+Il

1 - I.l

^{2 P}

lated the

degree

^{g of}

alignment

⁼

1"" + 2 11- = 3 - P . _/jj +211

^{i 3-P .}

A

simple

theoretical relation between the values of t at threshold and at a

given

energy is not

known,

but

in § ^{2. 3,}

^we have shown that the relative values of t for the lines of a band at threshold are

equal

^to

the relative values of t at any energy whatsoever.

3.1.1 The

experimental

set-up. - The cells were

continuously pumped

while molecular

hydrogen

^cir-

culated at the

working

pressure.

High purity (99,999

⁵

%) H2

gas ^was used. The pressure in the cell

was measured with a Pirani _gauge which was cali- brated

against

^{a MacLeod} gauge. Before _use, the cells were baked at 300 °C over a

period

^{of 24 hours.}

In the

triode,

^the electrodes were made of

molyb-

denium which absorbs

considerably

^less

H2

^than

does the tantalum

generally

^used.

The

grid-cathode voltage

^{was 35 V and the current}

was 20 mA. The _energy

spread

^{due to} the cathod emission is about 1 V which is

greater

than the diffe-

rences between the threshold

energies

^{used to excite}

the rotational levels in a

band,

^as

required

^{for the}

validity

^{of the}

theory.

The pressure ^was 4 x

10 - 2

torr.

130

It was low

enough

^{for the}

depolarization by

^collision

to be at the limit of the

experimental uncertainty

for all the studied lines

[7, 8],

^but

high enough

^to

obtain a

reasonably good signal-to-noise

ratio. From the inelastic collision cross sections

[30]

^{we estimate}

that the chance for an electron to

experience

^an energy loss is about 1

%. However,

^we recall that the

speed homogeneity

of the electron beam has no

explicit importance (§ 2.3).

The emitted

light

^was

analysed by

^{a Jobin-Yvon}

type ^{HRS 1} monochromator

having

^a

dispersion

of 12

Â/mm.

^{In order to} separate ^the

lines,

^{we used}

a resolution of 1

Á,

^as shown in the spectrum ^of

figure

^{2. The}

light

^{was detected}

by

^{an EMI 9558}

QB photomultiplier

^{cooled to - 50 OC.}

FIG. 2. ^- The Fulcher bands. Triode excitation.

3.1.2 Polarisation measurements. ^- The

polari-

zation measurements were made

by

^{a modulation}

technique.

The studied

light passed through

^{a linear}

polarizer rotating

^{at a}

frequency f,

^before

entering

the

grating

monochromator. The instrumental

pola-

rization

mainly

^{due to} the monochromator is elimi- nated with a set of

plane-parallel glass plates placed

before the entrance slit of the monochromator.

Then the

photomultiplier signal

^{is the sum of a}

time-independent

^current

io proportional

^to

Ili

⁺

^Il,

and of a component

i2 f

^{at a}

frequency 2 f,

propor- tional to

III

^-

^Il.

^{With a system}

regulating

^the

photomultiplier

^current

[27], io

^is

kept

^constant.

The relative values of the

polarization

^of the lines of

a band are then

given by

^{the relative}

amplitude

^of

i2 f

measured for each line with a lock-in detection.

To

improve

^the

signal-to-noise ratio,

^the

signal

^was

accumulated in a

computer (T 2000)

^{over more than}

two hours : one data was taken _every second and the

sum of 8000 such values was used for each line.

During

^this

time,

^{the current and the} pressure ^were

kept

^constant in the cell. For each line we have taken the mean value of about seven such

signal

^averages.

In our

experiments,

^{the earth’s}

magnetic

^field

was

compensated.

The width of the Hanle curves

observed on these lines

[7, 8]

show that the

depolari-

zation

by

this field would be a few per ^cent.

For the 2 -+ 2 Fulcher

band,

^the

wavelengths

^of

the considered

Q

branch lines

(Q(2), Q(2), Q(3), Q(4))

and R branch lines

(R(0), R(l), R(2), R(3))

^{are within}

25

A

and 50

A respectively.

^The

spectral

response of the

photomultiplier

^{can then} be assumed constant in each branch

[7].

3.1.3 Results. ^- Absolute

polarization

^measure-

ments have been made

by comparing

^the

signal

obtained with the studied line to the

signal given by

a

completely polarized light. They

^{were all found}

to be

positive

and less than 12

% :

The lines of the

Q

^branches

originate

from levels which are excited from

only

^one rotational level of the

ground

^state

(Fig. 1).

^The

degrees

^of

alignment t

of the

Q

^{branches are then}

independent

^{of the relative}

FiG. 3. ^- Comparison of observed and calculated values of the relative degrees ^of alignment of Fulcher lines.

Calculated : ^- without cascade correction. - - - with cascade correction from dl, dH, ^dA (from ^{left to} right respectively).

Observed : D v ⁼ 2 ; 0 ^{v =} 3. The vertical lines are the estimated

populations

^{in the}

ground

state, i.e. of the temperature.

We have measured P for the lines

Q 1, Q 2, Q 3, Q

⁴

of the 2 - 2

band,

and for the lines

Q 1, Q

^{2 of}

the 3 --> 3

band,

^{which are} well-resolved with our

monochromator. Our results are

given

ⁱⁿ

figure

^3a.

Agreement

^is

good

within the limits of

experimental

error. The effect of cascades is discussed

in §

^3.3.

The lines of the R and P branches

originate

^from

levels which are excited from two rotational levels of the

ground

^state

(Fig. 1).

^The

degree t

^of

alignment

of these lines

depends

^{then on} the relative

population

of the rotational levels of the

ground

state, i.e. ^on the

temperature.

^The calculations show that t is not very sensitive to the

temperature :

^for

T ⁼

(750

^±

200) K ,

At/t

is less than 4

%.

^The

change

of the relative values of t for the lines of a band is therefore much smaller than the

precision

^{of our} measurements. The tempe-

rature of the _gas between the

grid

^{and the}

plate

^{in a}

triode

glass

^cell

analogous

^{to ours,} has been estimated in our

laboratory [8]

^{to be} 800 K. Our

intensity

measurements

yield

^a temperature ^{of about 750 K}

(see Fig.

^{2 and}

Fig. 4).

FIG. 4. ^- Comparison of observed and calculated values of the relative intensities in Q ^branches.

Calculated : -. Observed : D, 0 , d, correspond ^to v = 2, 3, ⁴ respectively. (a) Calculated at various températures. (b) ^This work, ⁷⁷ K, 5 x 10-3 torr, HF discharge, Fulcher bands. (c) ^This work, ^700-800 K, 1.5 x 10-2 torr, triode, Fulcher bands. (d) ^Ref.

[29], ³¹¹ K, ^{5 x} 10-2 torr, 15 keV, Werner bands.

For the R

branches,

^we have measured P for the lines R

0,

^R

1,

^R

2,

^{R 3 of the 2 --. 2}

band,

^which

are well resolved

by

^our monochromator. Our results

are

given

ⁱⁿ

figure

^{3b. The} agreement is better than would be

expected

^{when one} considers the fact that the R branches

originate

^from levels of il + type which are

perturbed by

^{the 3s}

3 E g+

^level

^[10].

For the P

branches,

^the

polarizations

^{are small}

and the lines are not well resolved

by

^{the monochro-}

mator. Thus we believe that our

experimental

^results

are not

significant

^{and we do not} quote ^{them here.}

3.2 INTENSITY MEASUREMENTS. ^-

Intensity

^measu-

rements at very low temperature may be a sensitive test for our

theory

because the band head

only

^is

expected

^{to be} present.

Conversely,

relative intensities may be used to estimate the temperature ^{of the} gas.

We compare the intensities of

Q

^{lines of}

pn --+

^sE

transitions.

The

populations

^{in the}

ground

state must be cal- culated

separately

for the ortho and _para modifica-

tions,

^{as in our}

experiments,

^{even at 77}

K,

^{the ortho H}

para transformation does not have sufficient time to occur

[10].

^The

proportions

of ortho and _para remain 3 : 1. The

population

^{in the}

ground

^state

rotational level N of each ortho or para

species

^is

proportional

^to

(2

^{N +}

1) exp(- BN(N

⁺

1) hc/kT),

where the rotational constant B ⁼ 60.8 cm -1. To compare the intensities of all the lines in a

band,

^we

calculate their relative intensities for

equal

pressure of ortho and _para, and the

experimental

intensities of the ortho lines ^are divided

by

^three.

The

populations

ⁱⁿ the excited state may be calcula- ted from formula

(5).

^In

H2,

A Il = 0 in the

ground

state, ^{L = 1} and A’ ⁼ 1 for

3p 3ilu.

^{For the}

^Q

^bran-

ches of the Fulcher

bands,

^{N’ = N"}

(Fig. 1).

^{In this}

case, formula

(5)

shows that the

populations

^{of the}

levels N’ are

proportional

^{to the}

population

^of

N" = N’ in the

ground

^{state. As the N’} levels radiate

only

^{to N = N’}

(Fig. 1),

^the

intensity

^{of each}

Q(N)

line is

proportional

^{to the}

population

of the rotational level N" ⁼ N of the

ground

^state.

3 . 2 .1 HF

discharge (Fig. 4b).

^{- In order to main-}

tain the walls at a well defined low temperature, ^an electrodeless

discharge

^was

placed

ⁱⁿ

liquid nitrogen,

in a

glass

^{Dewar which was} fitted with an observa- tion window. The

applied

electric field was about 100

V/cm

^{at 100}

MHz,

^which

corresponds

^{to a}

maximum electron _energy of 40 eV.

When

decreasing

^the power and the _pressure, the relative intensities of

Q(l), Q(2), Q(3)

lines tend to

a limit. At 77

K, Q(2)

^and

Q(3)

^{are not}

predicted

^but

do occur. We can conclude that the united-atom

approximation

^{is not}

precise enough

^to

explain

^the

experimental

intensities. The

discrepancy

is however most

likely

^{due to cascades. It can be used to estimate}

the contribution of cascades in

populating

^the

3p 3 nu

level. This estimation is an upper value as the rota- tional temperature ^{in the}

discharge

^can

only

^be

higher

than 77 K. Its small

importance

agrees with the conclusion of reference

[8]

^{who detected no effects}

of cascades in

magnetic

^resonance

experiments.

3.2.2 Triode

(Fig. 4c). - We

report ^{the relative} intensities of the

Q

lines in the triode where the

polarization

measurements have been made. The best fit for the temperature

gives

⁷⁵⁰

K, although

the temperature ^is

probably

^not

homogeneous.

3. 2. 3

High energy electrons (Fig. 4d ).

^-

We report

the results of reference

[29],

^on

2p 1 il u --+ 1s1Fg+ .

The authors

verify

that the relative intensities of the

Q

^branches agree with the

optical

selection rules at

their measured temperature ^{311 K. The} agreement is better than in

figure

^{4b :} the cross-section of the

n ⁼ 2 level is much greater ^{than for the} upper levels which _may cascade onto it.

132

3.3 CASCADE EFFECTS. - We attempt ^{to estimate} the

importance

of cascades and their effects on the measured

degrees

^of

alignment.

We can use our

intensity

measurements at 77 K

(see Fig.

^{4 and}

§ 3.2)

^{to estimate the} contribution of thé cascades to

populate

^the

3p 3 llu level, assuming

that the

Q(2)

^and

Q(3)

^{lines are due}

only

^{to cascades.}

A p level can be

populated

from s and d

levels,

^and also

from

^{f levels}

^(4f

^-

^2p

transitions have been

observed).

The contribution of d levels is

probably

the most

important,

because the 4d -

2p

^{lines are}

much more intense that the s ^- p ^or the f -> p lines

[28].

^{We calculate}

(see § 2.4)

^{the relative}

popula-

tions of the first rotational levels of each d

3 EII d,

and the

resulting

^relative

population

after their

decay

on

3p 3II u.

These relative

populations

^are

nearly independent

^{of the}

higher

^level

(dl, dll,

^or

d4)

and are

proportional

^to

0.35, 0.40,

^{0.25 for}

Q(l), Q(2), Q(3) respectively. By comparing

the observed intensities of

Q(2)

^and

Q(3)

with that of

Q(l),

^{we find}

a cascade contribution of 17

%

^{to the total}

popula-

tion of

3p 3 nu.

The

degree

^of

alignment

of the line

corresponding

to the transition N’ - N, ^for

singlet

levels is :

For

singlet levels,

^the influence of cascades can

be calculated with formula

(16)

^which

gives

^the

ratio of the

degree

^of

alignment

^{on two} consecutive transitions of a cascade

N1 --> N2 --> N3 [20] :

In the case of ^« b »

coupling,

^the

degree

^of

alignment

^{for a} transition between two

triplet

^levels

(N’ --> N)

can be

expressed using

ts

given by ( 15) :

-

parahydrogen :

-

orthohydrogen :

Formulae

(17)

^and

(18)

^{allow one to deduce the}

cascade effects for

triplet

levels from that of

singlet

levels. The ratios

tT/ts

^{can be} obtained for the different lines from the values calculated for the direct excita- tion of _p

lllu (using

^formula

(15))

^{and of p}

3llu

(table 1, § 2.4).

The calculated relative values of the

degree

^of

alignment

^for

Q

^{and R branches} of the Fulcher

bands, taking

^{into account} the direct excitation of

3p 3llu

and the cascades from

d3IllJ,

^{are shown}

in

figures

^{3a and 3b. The} influence of each level

dl, dll,

^{dJ has been} considered

separately, assuming

that it

gives

^{one third of the cascade.}

The influence of cascades as estimated above is smaller than the

experimental

error,

except

^{for the}

Q(4)

^line which is the least intense. We recall that the

importance

of the cascade is

probably exaggerated.

4. Conclusion. ^- The agreement between the

experiment

^{and the}

theory

^is

remarkably good

ⁱⁿ

spite

^of the many

approximations

^{in the}

theory

^and

the _presence of a

light

cascade in the

experiment.

The relative values of the

polarization

^show very different features between the four

Q

lines and the four R lines with variations as great ^{as seven. The}

discrepancy

is however within the

experimental

^error

estimated to be about 15

%.

^{More refined measure-}

ments with a resolution in the electron _energy of a

few

10 - 2

V at

energies

^{less than} 1 eV above threshold would avoid the effect of

cascades,

^{but the resonances}

would

probably

^{make the}

experimental

results less

significant

^with respect ^{to the}

simple theory.

Acknowledgments.

^{- We are indebted to J. C. Jolivet}

who

helped

^{us to}

improve

^the

experimental

^set up and made the first measurements and to J. M. Caste-

jon

who made the

glass

^cells.

References [1] NEDELEC, O., BALTAYAN, P., ^J. Phys. B : ^{At. Mol.} Phys.

3 (1970) 1646.

[2] BALTAYAN, P., NEDELEC, O., ^J. Phys. B : ^{At. Mol.} Phys.

4 (1971) ^1332.

[3] LAMB, W. E., Phys. ^{Rev. 105} (1957) ^559.

[4] PERCIVAL, ^I. C., SEATON, ^M. J., ^{Phil. Trans.} Roy. ^{Soc. London}

A 251 (1958) ^113.

[5] ^MC FARLAND, R. H., SOLTYSIK, ^E. A., Phys. ^{Rev. 127} (1962) 290.

[6] JETTE, A. N., CAHILL, P., SCHWARTZ, R., ^{J. Chem.} Phys.

19 (1967) ^283.

[7] BALTAYAN, P., Thèse, Grenoble, ^1973.

[8] MARECHAL, ^M. A., Thèse, Grenoble, ^1973.

MARECHAL, ^M. A., JOST, R., LOMBARDI, M., Phys. ^Rev.

A 732 (1972) ^740.

[9] SCHULZ, G. J., ^Rev. Mod. Phys. ⁴⁵ (1973) ^378.

[10] HERZBERG, G., Spectra of diatomic molecules (Van ^Nostrand Co, New York) ^1950.

[11] VILLAREJO, D., STOCKBAUER, R., INGHRAM, ^M. G., ^{J. Chem.}

Phys. ⁵⁰ (1969) ^1754.

[12] HALMAN, M., LAULICHT, I., ^J. Q. ^{R. S. T. 8} (1968) ^935.

[13] MULLIKEN, ^R. S., CHRISTY, A., Phys. ^{Rev. 38} (1931) ^87.

[14] SLATER, J. C., Phys. ^{Rev. 36} (1930) ^57.

[15] MESSIAH, A., Mécanique Quantique (Dunod, Paris) ^1964.

[16] TEMKIN, A., VASAVADA, ^K. V., CHANG, ^E. S., SILVER, A.

Phys. ^{Rev. 186} (1969) ^57.

[17] SMITH, ^F. T., Phys. ^{Rev. 118} (1960) ^349.

[18] RUBIN, ^P. J., Opt. Spectr. ²⁰ (1966) ^325.

[19] OMONT, A., ^J. Physique ²⁶ (1965) ^26.

OMONT, A., Thèse, Paris, 1967.

OMONT, A., MEUNIER, J., Phys. ^{Rev. 169} (1968) ^92.

[20] NEDELEC, O., ^J. Physique ²⁷ (1967) ^660.

NEDELEC, O., Thèse, Grenoble, ^1966.

[21] D’YAKONOV, M. I., Sov. Phys. J. E. T. P. 20 (1965) ^1484.

D’YAKONOV, ^M. I., Opt. Spectr. ¹⁹ (1965) ^372.

[22] MILLER, ^T. A., FREUND, ^R. S., ^{J. Chem.} Phys. ⁵⁶ (1972) ^3165.

[23] FONTANA, ^R. P., Phys. ^{Rev. 125} (1962) ^220.

[24] CHIU, L. Y. C., ^J. Chem. Phys. ⁴⁰ (1964) ^2276.

[25] JETTE, ^A. N., CAHILL, P., Phys. ^{Rev. 160} (1967) ^35.

[26] LOMBARDI, M., J. Chem. Phys. ⁵⁸ (1973) ^797.

[27] EDEL, P., ^{Thèse de 3e} cycle, ^{Grenoble, 1972.}

[28] DIEKE, G. H., ^The hydrogen ^molecule wavelength ^tables (Wiley- Interscience, New York) ^1972.

[29] REICH, ^H. J., SCHMORANZER, H., ^Z. Phys. ¹⁸⁶ (1965) ^305.

[30] CORRIGAN, ^{S. J.} B., J. Chem. Phys. ⁴³ (1965) ^4381.

CARTWRIGHT, ^D. C., KUPPERMANN, A., Phys. ^{Rev. 163} (1967)

86.

KHARE, ^S. P., Phys. ^{Rev. 157} (1967) ^107.

PROK, G. M., MONNIN, C. F., HETTEL, ^H. J., ^J. Q. R. S. T.

9 (1969) ^361.