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The pressure-induced rotationalabsorption spectrum of hydrogen: IIJ.P. Colpa a b & J.A.A. Ketelaar aa Laboratory for General and Inorganic Chemistry of theUniversity of Amsterdam, The Netherlandsb Koninklijke/Shell-Laboratorium, Amsterdam

Version of record first published: 23 Aug 2006.

To cite this article: J.P. Colpa & J.A.A. Ketelaar (1958): The pressure-induced rotationalabsorption spectrum of hydrogen: II, Molecular Physics: An International Journal at the InterfaceBetween Chemistry and Physics, 1:4, 343-357

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The pressure-induced rotational absorption spectrum of hydrogen : II

by J. P. COLPA t and J. A. A. KETELAAR

Laboratory for General and Inorganic Chemistry of the University of Amsterdam, The Netherlands

(Received 14 May 1958)

A fo rmula is der ived for the in tegra ted in tens i ty of p re s su re - induced ro ta t ional t rans i t ions in pu re gases and gas mix tures at modera t e pressures . T h i s fo rmula is appl ied to the pressure induced rota t ional spec t rum of hydrogen , descr ibed in a p rev ious article. A good agreement be tween the observed and the calculated in tens i t ies can be ob ta ined if i t is a ssumed tha t the induc t ion of the t rans i t ion m o m e n t s is main ly caused b y the quadrupo le field of the hydrogen molecule. In an analogous way, the in tens i ty of the i nduced s imul taneous v ibra t iona l - ro ta t iona l t rans i t ion in mix tu res of ca rbon monox ide and hydrogen is calculated.

1. A GENERAL FORMALISM FOR THE INTEGRATED INTENSITY OF PRESSURE-

INDUCED TRANSITIONS IN GASES OF MODERATE PRESSURES

For a general discussion of the theory of pressure-induced infra-red absorption spectra we refer to the publications of Van Kranendonk [1, 2]. In the present article we will give a simple derivation of the formula for the intensity of pressure-induced bands at moderate gas densities, by a generalization of the intensity formula for infra-red active transitions. At moderate densities from 1-100 Am only binary interactions are of importance, as was the case under the conditions of our experiments described ill our previous article [3].

The integrated specific absorption ~i (shortly called intensity) is defined by

(1)

in which v is the wave number and ~ is the absorption coefficient as defined by Lambert 's law. For infra-red active transitons, the intensity associated with a transition from a di-fold degenerate level i to a dz-fold degenerate level f is

8 a iz (Fi

(see Herzberg [4]). In this formula r and s number the degenerate sublevels of the lower state i and the upper state f. The summation is extended over all possible combinations of upper and lower sublevels; IVy*r/8] is the matrix element of the electric dipole moment; n is the total number of molecules per cm~; vii is the wave number at which the transition is observed; F i and F I give the fraction of the molecules present in state i respectively statef. The other symbols have their usual meaning. For normal infra-red work Fi/d i is used instead of {F,/d,-Ez/d~} in (2).

Presen t address : Kon ink l i jke /She l l -Labora to r ium, A m s t e r d a m .

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344 J . P . Colpa and J. A. A. Ketelaar

However, if the upper level has a non-negligible population the term Fdd C- Fg/df must be included in order to correct for the induced emission [5]. For pure rotational spectra this refinement is always necessary. Suppose the matrix elements in (2) are zero for the unperturbed molecules in a gas of low density, and magnetic dipole radiation and electric quadrupole radiation may be neglected~

At higher densities dipole moments may be induced by intermolecular forces. At moderate pressures we only need to consider binary interactions. The induced' moment depends on the distance R between the centres of two inter- acting molecules, on the vibrational normal coordinates r 1 and r 2 and on the orientations 81, r and 82, r relative to the intermolecular axis (figure 1)

/z = f (R , Y1~1r t"2~2r For the calculation of the matrix elements we assume that the total nuclear wave function of the pair of molecules before and after the absorption is the product of the unperturbed rotational and vibrational wave functions of the molecules 1 and 2. This approximation may be justified by the fact that the experimentally observed frequencies are nearly independent of pressure, and coincide almost exactly with the frequencies of the isolated molecules. We define a matrix element (&(R)) by the following equation:

(~i~]s(R)) : fiCv "~ (F1)iCv • (Y2)iCrot "x" (81r ~ (82r R, Yl, Y2, 81r 82r

X .fCV(rl)fCv(?/2)fCrot(81r162 dT 1 dT 2. (3 } The subscripts v and rot refer to vibrational and rotational wave functions. This matrix element (3) is still a function of R.

The contribution (dA) of the pairs with intermolecular distance R to the intensity may be calculated substituting [~](R)I as given by (3) for [[xi,'l~ I in (2), in which for n has to be taken the number of pairs per cm a with intermolecular distance between R and R + dR.

Statistical mechanics gives for this density of pairs [19]

�89 2 . 47rR 2 exp [ - E(R)/kT] dR. (4 a)

The density of pairs consisting of molecules of two different types is

nln24rrR 2 exp [ - E( R )/k T) ] dR, (4 b)

in which n 1 and n~ are the densities (number of molecules per cm 3) of the gases 1 and 2. Exp [ - E(R)/kT] is the approximation for the radial distribution func- tion for binary interactions, in which E(R) is defined so that for R-> ov E->0. Substituting (4a) for n in (2) and (3) for IN in (2) we find for the contribution of the pairs with intermolecular distance R to the absorption intensity

dA=8~3vu12(Fi Ff) [_E(R)/kT]4~R2dR" 3h~ 2 ~ ~ ~[ix(R)i~z,]2exp (5)

Integrating (5) we find for the intensity caused by self-induction

Ao- -8~3vnI~ : (~:' Ff~C4~R2exp[-E(R)/kT]~,I~(R)i#sl2dR. (6a)

If a transition in species 1 is also induced by a foreign gas 2 we find for the part of the intensity caused by foreign induction

A ~ - 8zr% ~,( f i F---f~[47rR2exp[-E(R)/kT]Y~l~(R)i,'/,i2dR. (6b) 3hc nln2 ~ d i d~ ] J ,.,

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Rotational absorption in hydrogen: H 345

means that for a calculation of the total intensity a summation must be made B over all transitions occurring in the absorption band. In (6 a) and (6 b) Fdd~ and FSd / for a pair are equal to the products of the analogous quantities o f the individual molecules in the pair ; they specify the probability that the two inter- acting molecules each have a certain energy level.

2. A FORMALISM FOR THE INTENSITY OF PRESSURE-INDUCED ROTATIONAL

SPECTRA OF MOLECULES WITH D~ol~ SYMMETRY. SELECTION RULES

2.1. A series expansion for the induced dipole moment

The coordinate system we use is given in figure 1. In our calculations we also use the coordinates }@2(x+iy); }~/2(x- iy) and z; they will be denoted by + , - a n d 0 .

4 - R - D

The components of the induced moment may be expanded as a function of the vibrational coordinates /-L~c( R, fl, r2, 31r 32r =/'60~:( R, rl = re, r2 = ro 31r 32r

+ (r 2 - re) (7) + ( r l - re)

( re=equi l ibr ium distance) K indicates the component + , - or 0. The first term in (7) has to be used for the description of the pure rotational spectrum, the second and the third term give rise to matrix elements for induced vibrational transitions [1] and need not be considered for the pure rotational spectra. We expand the angular dependence of the first term of (7) as a series of products of spherical harmonics

/~0~(R~lr162 = 27r E D~@l/~lA2/~2) YI@*/~I)Y2(A2/z2)" (8) Al~t hetz2

The coefficients D 'r are functions of the intermolecular distance R. Ys(A~./~3.) is a spherical harmonic, normalized to unity, j= 1, 2 ;

1 Y(A/x) = Oa.,.,(3 ) ~ exp (i/xr

in which 0~.l. I (3) is a normalized associated Legendre polynomial (for a list of these functions see Pauling and Wilson [6]).

2A:~

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346 J .P . Colpa and J. A. A. Ketelaar

2.2. A series expansion for the intensity of induced rotational transitions For rotational transitions, the matrix element (3) has the form

f~r (81r (8~r 3~r 8~r162162162162 dr~ dry. (9)

We assume that the functions (}rot are the unperturbed rotational wave functions, :so that for r may be substituted a spherical harmonic Y(J, m) in which J and m are the rotational and the magnetic quantum number.

We describe the initial state of the interacting pair by the quantum numbers J~mtJ~mz and the final state by d~'m~'dz'mz'. When ~ is given by (8) the matrix element (9) for a transition Jtm~d~rn~->J~'m~'J~'m~' (abbreviated by i~f) is

[S;-,:(R)I=2~ Z l D ~ ( a l t Z X A z > z ) [ l ( J * m * l A i t q J ~ ' m l ' ) l l ( a z m d a ~ t z ~ J = ' m ~ ' ) l �9 (10)

The symbol <JmlAtzg'm') is used for J Y~(Jm)Y(Atz)Y(J'm' ) dr.

For the total intensity of a transition J~Je-+J,'J~' (10) must be squared and then a summation must be carried out over the degenerate states mlmi'mam~'. In

=two types of terms occur.

1. Quadratic terms : 4=~ID'~(A,~a=~=)I ~

]/~%f(R)p

[(Jlmdh~l~Jl"rn~')l~ Z [(J:dA~J~'m~'>l ~. (11)

2. Cross terms:

4~r 2{D ~(A~/x~;~2/x~)}{D ~(A~'/xl'A 2'/x 2'))*

x ~, (JlmllAllXlJl'ml')(g~rn~lA~'tq'J~'m 1' )~ m l m l '

• ~ (J2m~l,~2tx2J2'm 2' )(g~m2],X2'tz~'J2'm2' )% (12) Ngs~ ~"

Matrix elements like (JmlAizJ'm' > have been extensively studied by Racah [7]. We shall use the following properties of these elements. A summation

Z (JmlA~J'm')(Jm[k'l ~'J'm' >~ (13) ram"

is zero, unless/~ = bd and A =)~'. If A = A' and/x =/z' the value of the summation :is independent of/x. We use the following abbreviation

1 [(jmlZl~J,rn, )12= ~ Sa(J,J'). (14) q??, ?n"

As a consequence of these properties the cross terms (12) vanish. Using the ,definition (14) we get

E I/*~-,f(R)l e= E lD~(h*b@2txe)l=Sa,(J1,J*')Sa=(J~,ae ') (15)

in which the coefficients D ~ are functions of R. Substituting (15) in (6 a) we find

A o - 8rravn, 2 y' f F(J1) F(J2) " F(J,') F(J2') \ 3he 2 7g~2J~+l 2J~+1 2 J , ' + l 2 J ~ ' + l J

• f 4~rR2 (~ ID"(,~tz,Pt21x2)l~'Sa,(J,,J~')Sa,(J2,J~')}exp[-E(R)/kT]dR. 12htz,

21~g.a K

(15a)

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Rotational absorption in hydrogen: H 347

For foreign induced absorption a completely analogous formula may be der ive4 starting f rom (6 b).

8~rSVnln2 ~; f F(J1) F(Jz) F(JI' ) F(J~') "~ A,~ - 3hc ~ L ( 2 J ~ ) (2J2 + 1 ) (2J~' + 1) (2J~ '+ 1) J

x F 4~R2~ 2 [D~('~lt~l~t21x2)[2Sa,(J1,Jl')Sa,(J2,J2')~exp( - E(R)/kT) dR. a k ~i~ J

~'~ (16 b) K

The application of formulae (16 a) and (16 b) is not restricted to infra-red spectra ; they apply also to rotational transitions that may be expected in the microwave region. Probably the microwave absorption in pure CO~, observed b y Birnbaum and Maryot t [8], is of this kind.

2.3. Selection rules for pressure-induced rotational transitions The coefficients S a (d, d')

For a discussion of the selection rules we consider the expression (15) for the square of the matrix element. We make first a remark about the possible values of A 1 and A2. For a pair of H 2 molecules (or any other pair of molecules both having D| symmetry) the dipole moment must be left unchanged when 3 is converted to z r - 3 and r to r This means that in the series expansion (8) the values for A 1 and ;~2 must be even or zero. For pairs with two similar molecules an angle-independent term D K (0000) in (8) must be zero. For pairs with two different molecules this te rm may occur, but even then one need not consider this term, for it does not occur in the description of rotational transitions. T h e first important coefficients are D K (2/z, 00) and D ~ (00, 2t@ Th e selection rules are determined by the values for A J = J ' - J for which Sa(d,J' ) # O. Applying recursion formulae and orthogonality relations for Legendre Polynomials we obtained the following results:

So(J,J')=O u n l e s s J = J ' , So(J,J)=�89 l ). S2(J,J')=O unless A J = + 2 or A J = 0 , J = J ' r ~ (17),

S2(J,J + 2)=27r ~ I(dml2t~J+2,m'~l~=3 (a + 1) (__J_+ 2) ] m~' " 4 2 J + 3 "

3 J ( J - 1) (18), s2(J,d-2)=2= Z ](Jml2~ d-z, m')12- 4 2 J - 1 "

1g(g + l)(2J + l) S2(J'J)=2rr~. ](Jml21~Jm')[z= 2 (Tf-~- 1)(2J + 3) "

In general s . (J ,J , ) r for a is even, if IJ ' -J l=0 , 2, 4.. . a. As in our calcula- tions A 1 and A 2 are 0 or 2, we shall not consider formulae for S4(J, J') etc. in detail'S.

The terms with D ~ (2~ 00) resp. D '~ (00 2/z) give the selection rules AJ 1= 2, AJ 2 = 0 and AJ 1 = 0, AJ 2 = 2 respectively. If indices 1 and 2 both refer to H2, these two rules give rise to equivalent transitions ; but if index 2 refers to N2, AJ 1 = 2, zXJ z = 0 describes induced H z transitions and A J 2 = 0, AJ 2 = 2 describes induced N z rotational transitions.

We also meet in our calculations coefficients D ~ (2/x12/xz). T h e y contr ibute to the intensity when A J 1 = 2 ; A J 2 = 0 and AJ~=0 , A J e = 2 , but also when AJ 1 = 2, zXJ 2 = 2. These transitions, in which both molecules of a pair make a

t cf. w 3.2, Table 1.

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348 J . P . Colpa and J. A. A. Keletaar

transition absorbing only one quantum, are the simultaneous transitions or double transitions. For a fur ther discussion of the selection rules for induced rotational transitions we refer to Galatry and Vodar [17], and for induced simultaneous vibrational transitions to Hooge and Ketelaar [18].

3, CALCULATION OF THE INTENSITY OF THE PRESSURE-REDUCED ROTATIONAL

ABSORPTION SPECTRUM OF HYDROGEN

3.1. The nature of the inducing forces

In order to evaluate the coefficients D~(A1/xlA2/~2) we will discuss the forces that contribute to the induction of the dipoles.

1. Dipole moments may be induced by the electric fields of each of the molecules of a pair.

2. Induct ion may be caused by overlap forces. For the calculation of the intensity of the induced vibrational spectrum of

hydrogen both kinds of forces appeared to be important as was shown by Van Kranendonk [1]. We shall show that for the induced rotational spectrum to a good approximation the intensity may be calculated on the basis of the assumption that the electric fields cause the induction and that overlap forces may be neglected.

(a) The electric field of a D oot~ molecule may be described as a quadrupole field. When such a molecule is close to another molecule, the field of the first one induces a dipole in the second one. Direction and magnitude of the induced moment depend on orientation and intermolecular distance. Classically we get the following picture:

A rotating H 2 molecule induces, e.g. in a helium or argon atom, a vibrating dipole field. The f requency of the induced moment is twice the rotational f requency of the H 2 molecule. The factor 2 comes in because of the D~h symmetry of the quadrupole field and is the classical analogue of the selection rule AJ = 2. Although the induced moment is located in the noble gas atom, it depends on the rotational coordinates of the H a molecule and gives rise to absorption at a hydrogen rotational frequency. In induced absorption, the two molecules behave as one complex in their interaction with the electromagnetic radiation field. One can easily see that the induced intensity caused by this effect depends on the quadrupole moment of H a and on the polarizability of the foreign gas molecule, or for self-induced absorption on the polarizability of a hydrogen molecule.

(b) In our example (a) the foreign molecule itself had no inducing field, but when we take, e.g. N 2 or HC1 as a foreign molecule, we have a second effect that contr ibutes to the induction. The quadrupole field of a rotating N 2 or the dipole field of a rotating HC1 molecule induces a vibrating dipole in a H a molecule. When we assume the polarizability of hydrogen to be isotropic, this induced moment can explain an induction of N 2 ~otational transitions or an enhancement of the intensity of the rotational spec t rum of HC1. When we consider the fields of the Nz or HC1 molecule to be vibrating, we can see that the vibrational spectrum of N z may be induced, or the intensity of a HC1 vibrational band may be enhanced.

However, when we take into account the anisotropy of the polarizability of Ha, we see that the moment induced by Nz or HC1 in Hz also depends on the

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Rotational absorption in hydrogen: H 349

rotational coordinates of the H a molecule and gives a contribution to the intensity of transitions characteristic for hydrogen. Pure rotational transitions of hydrogen may be induced and also simultaneous transitions in which take part a H 2 rotational transition and a N 2 or HClvibrational or rotational transition (see also w 4). In these cases the moment is induced by the foreign molecule in hydrogen, contrary to the effect under (a).

Usually the effect mentioned under (a) gives a much larger contribution to pure rotational absorption bands than the effect (b). We gave the description for the foreign-induced absorption. Of course both effects are present for the self-induced absorption. From the above-mentioned model it is clear that the induced moment for self-induced absorption is to a first approximation propor- tional to the polarizability of H 2 and for foreign-induced absorption to the polarizability of the foreign molecule. The intensity must be roughly propor- tional to the square of the polarizability. This is only true if overlap forces may be neglected. In the induced vibrational spectrum of H a, the Q-branch (An = 1, A J = 0) is mainly due to overlap forces, whereas the S-branch (An= 1, A J = 2 ) is caused by quadrupole fields (see Van Kranendonk [1]). Because of the very small polarizability of He we see that the S-branch in the vibrational spectrum induced by He has a very low intensity, whereas the Q-branch induced by He, and due to overlap forces, is as strong as the self-induced Q-branch (see Welsh [9]). In the case of the induced rotational spectrum we see that He has very little effect in inducing transitions [3, 10]. This is in accordance with the assumption that only electric field effects are of importance. In the formulae derived in the next sections, we neglected the contribution of overlap forces.

3.2. Formulae for the intensity of induced rotational transitions, neglecting the anisotropy of the polarizability

Molecule 1 (figure 1) has a quadrupole moment Q1. The components of the quadrupole field of molecule 1 at the position of molecule 2 are :

3Q1 F1, x = - R 4 cos 31 sin 51 cos r

3Q1 F1, .v- R4 cos 51sin31 sin r (19)

3Q1 Fl, z= -}- ~ ~(3 cos2~1 -- 1).

Instead of F~, F~j, Fz we use F =~ = 1 ~/2(Fx + iFv) and F~ F~. This leads to :

-3 Q1 FI+ = W ~/2 ~ i cos 51 sill 51 exp ( @ ir ) = -- ~r @305 R 4Q1 y1(2 ' + 1 ),

~/3o Q1 Y1(2, - 1), (20) FI-= 5

3Q1 FI~ + ~ ~(3 cos2 31- 1)= + V 2 . ~/90 Q1 y1(2 ' 0). 5 R 4 I

In (20) we expressed the angular dependence in spherical harmonics. When

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350 J . P . Colpa and J. A. A. Keteiaar

molecule 2 has a quadrupole moment Q2, the components of the field of this moleeule at the position of molecule 1 are

V30 Q~ Y~(2, 1), F~+=@(2,O 5 R~ I

@30 92 y2(2 ' _ 1), I F2-=@(Z~r) 5 R ' ~ (21) i

/90 Q2 Y2(2, 0) 1 F 0=_ 5 J

For the calculation of the induced moments, the polarizability tensor has been brought into a suitable form (see Appendix). An eIectric field F induces in a cylindrically symmetrical molecule a dipole ; the components of its moment ~ are :

, , , ~v/lO tz+=F+{~/(2~)@2 Y(O, O)o~-A@t,~Tr)-fff-Y(2,0)} ] +F-f A@(2~r)~f-~-55 y(2, +2)}+F~ +I)} ]

- = F+{ A@(27r) 2@15 Y(2, - 2) } + F-{@(2~)@2 Y(0, 0)~

-AV(27r) @--~ Y(2, 0 ) } + F ~ AV(2~r) -T5@30 Y ( 2 , - 1 ) } (22)

/,0 = F+{A@(2~r)-~-@30 Y ( 2 , - 1)} + F - { A @ ( 2 ~ r ) C V ( 2 , - 1 ) }

+F~ @(2~r)@2 Y(0, 0)~+A@(27r)2 l ~ g ( 2 , 0 ) }

c~ is the average polarizability, (% + 2%)/3 and A = % - ~ . is the anisotropy of the polarizability. The angular dependence is expressed in normalized spherical harmonics. The c~ in the diagonal elements of (22) formally has the factor

Y(0, 0)= 1 in order to express all elements in spherical harmonics. The total induced moment ~ is the vector sum of the moments bh and I*~

induced in the molecules 1 and 2.

b~ = bh + b% = ~1F2 + ~2 Ft , ( 2 3 )

in which ~ represents the polarizability tensor as given by (22) and F 1 and F 2 are the electric fields whose components are expressed by (20) and (21). After making the necessary substitutions and collecting the terms with the same product of spherical harmonics we obtain the coefficients D~(kdqk2/,2) as defined by (8). These coefficients are collected in Table 1.

The first six coefficients, viz. those proportional to % and % we call the isotropic coefficients; the other ones are called anisotropic coefficients.

As a first approximation we consider the polarizability to be isotropic. The correction for the apisotropy can be calculated separately by evaluating the contribution of the anisotropic terms af{er substituting these terms in (16a) or (16 b).

According to the discussion of the selection rules (w we expect in the approximation in which only the isotropic coefficients are used, the selection rules A J1=2 , k J 2 = 0 and A J1=0, A J==2. For self-induced absorption we have Q~ = Q2 and % = % so that O~(2 ~ 0 0) = D~(0 0 2 ~).

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Rotational absorption in hydrogen : H 35I

D• +100)= Q~R~ ~= 2x/15 - 5

D~ 0 0 0 ) - - Q1~2 6 V 5 R 4 5

D• T- 1 2 + 2) = QIAo. 2V2 - R 4 5

Q2el 2~/15 D • R~ 5

Q~ 6~/5 D~ 0 2 0 = R a 5

Q~A 12~/2 D• _+22-T-1)= R~ 5

D• + 1 2 0)= 2~/3 [QIA2_ 3Q2A1} - 15R ~

D• 0 2 + 1)= 2~/3 (3Q~A2- Q2A~}

D~ - 1 2 + l ) = D ~ +1 2 - 1 ) = 2 5R 4 {01A2 -- Q2AI}

DO(2 0 2 0) = 4

Table 1.

For the calculation of the intensity of an induced transition with AJ = 2, we- only calculate the intensity for A J I = 2, AJ 2 = 0 and multiply afterwards the intensity so obtained by a factor 2. Doing this we only need the three coefficients D=( 2 t~ 0 0) of Table 1 for substitution in (16 a). Further we substitute J l ' --Jx + 2 ; J (=J2 ; S~(J1, J2 +2) and So(J2, J2) as given by (18) and (17). The summation ~ becomes ~ ; and ~ F(J2)= 1. We finally obtain for the

B J~ J~ intensity of a transition AJ = 2 in this approximation:

Ao 48~% ~F(J) F(J+2) '~(J+l ) (J+2)r ) %~ 2 = hc n~22 ~2---ff~ ] 2d + 5 J 2J + 3 ~J~, la~

(~o exp [-E(R)/kT] RG dR. (24) •

J 0 For the intensity caused by foreign-induction we obtain an analogous result.

We denote the H 2 molecule by index 1 ; the foreign molecule by 2. If molecule 2 is for example a N 2 molecule, the transitions AJ~=0, Ad2=2 are rotational transitions corresponding to the N 2 rotational frequencies in the microwave region. We can omit therefore these transitions for the calculation of the intensity in the infra-red region, and as a consequence we also omit the coefficients D=(0 0 2 ~). In (16 b) we make the same substitutions as was done in (16a). This leads to the result :

Am - 487r% f,F(d) F(J+2) '~(J+I) (J+2) r) %;~

exp 1 - E(R)/kT] x ~ R6 dR. (25),

As in the derivation of (25) the anisotropy of the polarizability and the quadrupole field of molecule 2 have been neglected, it is immediately clear that (25) may also be applied to the foreign induction by noble gases.

The integral ~exp [ - E(R)/kT]R-6 dR has been evaluated numerically.. For E(R) we used a Lennard-Jones potential

=

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352 J . p . Colpa and J. A. A. Ketelaar

For the calculation of the integral in (25) we made the usual approximation q , z ~ / ( q e 2 ) and da, 2=�89 Numerical values for ~ and d have been taken from J-Iirschfelder et al. [11]. (It appeared that for pure hydrogen and hydrogen mixtures the integral, neglecting the relatively small temperature dependence, may be approximated roughly by �89 -5. For our calculations we used the exact values.) The values for c~ and A have been taken from the tables in Landolt-BSrnstein [12]. For the quadrupole moment of H a we used the theoretical value of James and Coolidge, Q = 0"6 x 10 -26 e.s.u. [13].

For a comparison of the calculated and the experimental values [3], we calculated the sum of the intensities of the transitions J = l + J = 3 ; J = 2 § = 4 ; J = 3-+J = 5. In order to obtain the integrated quadratic absorption coefficient used in our experimental work [3] we substituted for nH~ and n 2 the value for a density of 1 Amagat, viz. 2-69 x 1019. The values for F(J) are to be found in Table 2 of our first article [3]. In Table 2 the intensities obtained from (24) and {25) are collected and compared with the experimental values.

~t, heor, - 6 0 ~ I~exi).

l~theor./I~exp.

I~theor. + 25 ~ ]~exp.

Ptheor./Pexp.

]~theor. + 80~ Fex >

~'theor./l~cxp.

H2-H2

1"34 1-64 0'82

1-49 1"88 0"80

1"57 2"05 0"77

H2-He

0"17 ~0-2 ~0'85

H2-N2

3'22 4'0 0.80

3-5 4-5 0-78

3"6 4"7 0"77

H2-A

3"5 4.6 0"76

3"75 5"1 0-74

3"9 5"3 0.74

H~-CO

4-0 5"5 0-73

Table 2. Integrated quadratic absorption coefficients in 10 a cm-~ Am-2.

3.3. Correction on the intensity calculation for the anisotropy of the polarizability

The calculations made above may be refined for H2-H2, H2-N 2 and Ha-CO considering the anisotropy of the polarizability and the electrostatic fields of the N 2 and CO molecules. The correction can be made by substituting the coefficients D"(2/xt2/x2) of Table 1 in (16a) and (16b).

For the self-induced spectrum we have QI= 02; AI= A2; % = % . Calcu- lating the correction for transitions AJ 1 = 2, AJz = 0 and AJ 1 = 0, AJ z = 2 we find by the methods outlined above that for self-induced absorption the intensity as given by (24) must be multiplied by a correction factor

{16(AI '~2~F(j1) 1' Jl(JIJcl) } 1+ ~\c~--~/ j, 2 ( 2 J , + 3 ) ( 2 d , - 1 ) "

With the values F ( J ) for hydrogen [3] this factor becomes {1 +0.057(A/0c)}. In the spectral region in which we measffted the induced spectrum we must .expect the following simultaneous rotational transitions

J l = l , J 2 = l - + d a=3, Y 2=3 a t1173cm -I J l = l , d 2 = 0 + d l = 3 , J2 = 2 } J l = 0 , J 2 = l - ~ J ~ = 2 , J 2 = 3 at 940cm -1

J l = 0 , J 2 = 0 - + J l = 2 , J z = 2 at 708cm -1.

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For a good compar ison between the experimental intensi ty in the region o f 4 0 0 c m -1 to 1300cm -1 and the calculated intensity we mus t include in our calculations the contr ibut ion of these s imultaneous transitions. All e lements necessary for a calculation of the intensity of these transit ions are ment ioned in the previous sections, T h e results are summar ized in Tab l e 3. Compar ing

Table 3.

Frequency

- 60~ Intensity 25~

80~

708 cm -1

0.0005

940 cm -1 1173 cm -1

0"0041 [ 0"0068 0-0027 [ 0.0055 0'0022 0'0050

I _J Calculated intensities of some simultaneous rotational transitions in

10-a cm-a Am-2.

T a b l e 3 with Tab l e 2, we see that these simultaneous transitions are very weak ; consequent ly we could not observe t hem separately besides the stronger broad bands at 814 cm -1 and 1034 cm -1 [3]. We expect, however, that at lower t empera - tures at which the intensity of the bands at 814 cm -1 and 1034 cm -1 vanishes, the bands due to the simultaneous transit ions are observable.

T h e correction of the foreign induced intensity for anisotropy is somewhat more complicated, since A l # A 2 and Q I # Q2. T h e CO molecule besides a quadrupole m o m e n t has a small dipole moment . For mixtures of CO and H= we extended the intensity calculations, including in the series expansion (8) also te rms describing the influence of the dipole m o m e n t of CO. However , it appeared that the influence of the weak dipole field of CO is considerably less t han that of the relatively strong quadrupole field; in the vibrational g round state the CO molecule behaves as a quadrupole molecule. We need not discuss the calculations in detail ; the corrected values for the intensities are summar ized in Tab le 4. In the calculated intensities for H 2 - N 2 and H ~ - C O mixtures we included also the contr ibution of the simultaneous transitions A J = 2 falling in the region of our spectrum. For the quadrupole moment s see Hirschfelder ,et al. [11] and Feeny et al. [14].

~calc. ._60Oe Fexp '

~PcaIc./Pexp.

~calc. q-25~ Fexp.

I'c~le./Pex>

I~oMe. +80~ Fexp.

l~ealc./l~exp.

H~-H2

1"35 1"62

0-83

1-51

1 "88 0-81

1 "62 2"05 0"79

H2-N2

3"35 4'0 0'84

3.64 4.5 0.81

3"74 4"7 0"80

Ha-CO

4"16 5"5 0"76

'Table 4. Integrated specific absorption coefficients in 10 -a cm -2 Am -~.

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354 J. P, Colpa and J. A. A. Ketelaar

w 4. THE INTENSITY OF THE SIMULTANEOUS VIBRATIONAL--ROTATIONAL TRANSITION IN MIXTURES OF CO AND H 2

In our previous article [3] we described a band at 2730cm -I observed ilt mixtures of CO and H 2 and interpreted it as due to a simultaneous transition ilk which the CO vibrational transition and the J - 1 - ~ J = 3 rotational transition. take part. We will discuss here the intensity of this band.

In a pair consisting of a H 2 molecule (index 1) and a CO molecule (index 2} the induced moment may be caused by the following effects:

1. The dipole field of CO induces a dipole in H 2.

2. The quadrupole field of CO induces a dipole in H 2.

3.~ The quadrupole field of Hz induces a dipole in CO.

4. Overlap forces induce a dipole.

There is a large amount of similarity in the physical properties of the • electronic molecules CO and N e. One of the most important differences is that: only the CO molecule has a dipole field. In mixtures of N 2 and H 2 we could not observe the simultaneous vibrational-rotational transition analogous to tha t in CO-H~ mixtures. Therefore we assume that the dipole field of CO plays. an important role in the induction (effect 1). The dipole moment is small (N 0. 1 D), but as we have here a vibrational transition in the CO molecule, we are not interested in the moment itself but in the derivative with respect to the internuclear distance (8i~/3r) ~.=~, ; for CO this quantity is very large (3.14 x 10 -1~

e.s.u. [15]). We make the assumption that the simultaneous transition is mainly due to the vibrating dipole field of the CO molecule. This field induces in a H 2 molecule a dipole that depends on the rotational coordinates of the H e molecules because of the anisotropy of the polarizability of H e.

Using the definition of F • and F ~ as given in w 2.2 we find for the components, of the dipole field of molecule 2 (CO) at the position of molecule 1 (He)

,/6~er,(1, +_1) t Fe• = - V(27r)} - - R~ (26),

2 / ~ ~ 2 Fz ~ = + ~/(2~) ~ v o K5 Ye(1, 0). J

In (26) we have expressed again the angular dependence in spherical harmonics ; /23 is the dipole moment of CO; it is a function of the internuclear distance re: in CO.

The components of the induced moment /x are obtained by substituting (26) in the tensor expression (22). We can omit in (22) the angle-independent terms. c~, because they are of no use in the description of transitions in which the rotational quantum number of molecule 1 changes. After substitution we see that the relevant terms of/~• and/x ~ may be expressed in a series expansion of the following type :

2~ tz~(re) zxl ~ G"(2tz~I~2)Y~(2,~,)Ye(1,~). (27) bLl/~2

The coefficients G ~ are easily obt~iined from (22) and (26). In the calculatioa we only need the sum of their squares ; we found

16 2 IG(2~I~)I~= ~-. (28>

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'The square of the matrix element for a transition

J1, ml, J2, mn, nn = O--->Jl'ml'J2'm2 ', nn = 1 after summation over mlml'mnm n' is given by

Aln 3 _ [/x2]~ ~ [G~(21~lt~212Sn(Jl, J~')Sl(J~,Jn'). (29)

rn~m2'

{As mentioned in w the cross terms vanish on summation.) In (29) we have

I,n] = fCv(nn = O)/xnCv(nn = 1)dr

:this is the matrix element describing the vibrational transition in CO:

S~(J2Jn')= 2rr ~. I(Jnm2ll,2Jn' mn')l n.

Applying the recursion formula for Legendre polynomials one easily proves that Sl(Jn,Jn' ) r 0 only when AJ n= + 1. A calculation gives

S~(Jn, J2+ 1)= �89 1); SI(J2,J 2- 1)= �89 (30)

S~(J1JI' ) is discussed in w 2.2. Substituting (18), (28) and (30) in (29) and the result in (6b) we find the

:intensity for the transitions An n = 1, AJ n = + 1, AJ~ = + 2 ; Ff in (6 b) equals zero, .all the CO molecules are in the vibrational groundstate;

F~ F(JI) F(gn) d i (2J~ + 1)" (2J n + 1 )

For the calculation of the total intensity of the simultaneous transition we have to ~:arry out a summation of the above mentioned intensity for AJ n = 1 and AJ 2 = - 1 for all values of Jn. The result is

8rrav F ( J i ) �9 2' '22 (gl- t -1)(J~+2) rr 'exp E(R)/hT] A = - j - ~ - n H . n c o 2 j ~ a ~ I/*21 -5 -(2J-77-3 ~ 4 J [ - R6 dR. (31)

We can give this expression a simpler form. The intensity of the infra-red active vibrational band of CO is

8~rav~' n 2 n 2 = ~ n/~2 �9 ( 3 2 )

Substi tuting (32) in (31) we find for the simultaneous vibrational-rotational transition an intensity

A =An v_ n~A~ 2 F(Jx) 8~v ( J l + 1 ) ( J a + 2 ) ; exp [-E(R)/kT] 2J1+1 3 (2J +3) _ d R . (33)

We see that the intensity of the simultaneous transitions is proportional to the intensity of the infra-red active transition of CO. A n has been measured by Penner and Weber [15] and is found to be 258 cm-2Am -1. After the necessary .calculations we find for the integrated quadratic absorption coefficient

F.~ = 1-2 x 10 -5 cm-2 Am -n.

T h e experimental value was 1.5 x 10-acm-2Am -2 [3, 10], so that our approxi- mate theory gives a very reasonable account of the intensity of this induced band. An expression for the intensity of simultaneous vibrational transitions [10] will be published elsewhere [20].

The authors wish to express their thanks to Dr. J. van Kranendonk for helpful discussions.

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356 J .P . Colpa and J. A. A. Ketelaar

This work is part of the research programme of the Stichting voor Fundamenteel Onderzoek der Materie (Foundation for Fundamental Research on Matter, 'F.O.M.) and was made possible by financial support from the Neder- landse Organisatie voor Zuiver-Wetenschappelijk Onderzoek (Netherlands Organization for Pure Research, Z.W.O.).

APPENDIX

The polarizability tensor We indicate the polarizability of a cylindrically symmetrical molecule in the

direction of the molecular axis by % and in the direction perpendicular to the axis by o~• ; we define the anisotropy A by % - ~. The components of the dipole moment P induced by an electric field F may be calculated according to the method outlined, e.g. by Stuart [16]. Using the coordinate directions of figure 1, the result is Px = Fx{~. + A sin 23 cos ~ r + Fy{A sin ~ 6 cos r sin r + F~{A sin 3 cos 3 cos r Pu = Fx{A sin23 sin r cos r + Fy{~• + A sin S 3 sin 2 r + F~{A sin 3 cos 3 sin r P~ = Fx{A sin 6 cos 6 cos r + Fy{A sin 3 cos 6 sin r + F~{% + A cos 26}.

We preferably use a system with complex coordinates F•189 F~

= � 8 9 + i G ) po =

Using these coordinates we find p+ = F+{% + �89 sin ~ 6} + F-{�89 sin 26 exp ( + 2ir + F~ sin 3 cos 3

x exp ( + ir P - = F+(�89 sin 26 exp ( - 2ir + F-{~. + 1A sin 23} + F~189 sin 3 cos 3

x exp ( - i r p0 = F+{~r A sin 3 cos 3 exp ( - i6)} + F-{@(�89 sin S cos 3 exp (ir

+ F~ + A cos 26}. We now introduce the average polarizability c~=�89189 and

express the elements of the polarizability tensor in ~, A and normalized spherical harmonics Y(A,/x) (for a table of these functions we refer to Pauling and Wilson [6]). The final result is:

p+_-F + c~_A~/2rr ~r 0 ) } + F - { A ~ / 2 r ~ 2 ~ +2)} ~ Y(2, Y(2,

+ F~ Y(2,+ 1)},

P-=F+{A.v/2~r2~5y(2, + 2)} + F - { ~ - A.V'2~" ~/10 Y(2,0)} (34)

+FO {A-v/2~r-- ~ Y ( 2 , - l)'~,j J

o 2 2~/10 + F @ + A~/ ~r--]-g-- Y(2, 0) 5.

In our calculations we used ~/(2~r) ~/2 Y(0, 0)instead of ~, in order to get more symmetrical formulae. With the expression (34) the polarizability tensor may

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Rotational absorption in hydrogen: H 357

be s imply separa ted into the isotropic or spherical par t of the polarizabil i ty and the comple te ly anisotropic part.

On a deriv6 une formule dormant l'intensit6 integr6e des transitions de rotation induites par pression dans les gaz pu r se t les m61anges gazeux h des pressions mod6r6es. Cette formule a 6t6 appliqu6e au spectre de rotation induit par pression de l'hydrog~ne, d~crite dans une publicaidon ant6rieure. On obtient un bon accord entre les intensit6s mesur6es et calcul6es en supposant que l'induction des moments de transition est dfi principalement au champ quadrupolaire de la molecule d'hydrogbne.

L'intensit~ d'une transition de eombinaison entre la transition de vibration du monoxyde de earbone et une transition de rotation de l'hydroggne se calcule d'une mani6re analogue.

Far die integrale Intensitgt yon druckinduzierten Rotationsfiberg~ingen in reinen Gasen, sowie in Gasgemischen wird eine Formel abgeleitet. Sie wird auf das druckinduzierte Rotationsspektrum yon Wasserstoff angewandt, das in einer friiheren Arbeit beschrieben ist. Eine gute Uebereinstimmung zwischen den beobachteten und den berechneten Intensit~iten l~isst sich erreichen wenn man annimmt, dass die Induktion der Uebergangs- momente im wesentlichen durch das Quadrupolfeld der Wasserstoffmolekel verursacht wird.

In analoger Weise wird die Intensitgt des induzierten simultanen Rotations- Schwingungsfibergangs f~r Mischungen yon Kohlenmonoxyd und Wasserstoff berechnet.

REFERENCES

[1] VAN I~RANENDONK, J., 1952, Thesis, Amsterdam, VAN KRANENDONK, J., and BIRD, R. B., 1951, Physica, 17, 953.

[2] VAN KRANENDONK, J., 1957, Physica, 23, 825, 1958, Ibid., 24, 347. [3] COLPA, J. P., and KETELAAR, J. A. A., 1958, Mol. Phys., 1, 14. [4] HERZBERG, J., 1950, Spectra of Diatomic Molecules (New York: Van Nostrand). [5] WILSON, E. B., DECIUS, J. C., and CROSS, P. C., 1955, Molecular Vibrations (New York:

McGraw-Hill). [6] PAIdLING, L., and WILSON, E. B., 1935, Introduction to Quantum Mechanics (New York:

McGraw-Hill). [7] RACAH, G., 1942, Phys. Rev., 61, 186; 62, 438 and 1943, 63, 367. [8] BIRNBAUM~ G., MARYOTT, A. A., and WACHER, P. F., 1954, J. chem. Phys., 22, 1782. [9] CRAWFORD, M. F., WELSH, H. L., McDONALD, J. C. F., and LOCKE, J. L., 1950, Phys.

Rev., 80, 469. [10] COLPA, J. P., 1957, Thesis, Amsterdam. [11] HIRSCHFELDER, J. O., CURTISS, C. F., and BIND, R. B., 1954, Molecular Theory of Gases

and Liquids (New York : Wiley). [12] LANDOLT--B6RNSTEIN, 1951, Zahlenwerte und Funhtionen I l I (Berlin). [13] JAMES, H. M., and COOLIDaE, A. S., 1938, Astrophys. J., 87, 438. [14] FEENY, H., MADIGOSKY, W., and WINDERS, B., 1957, J . chem. Phys., 27, 898. [15] PENNER, S. S., and WEBER, D., 1951, y. chem. Phys., 19, 807. [16] STUART, H. A., 1952, Die Struhtur des freien Molehills (Berlin: Springer Verlag). [17] GALATRu L., and VODa_n, B., 1956, C.R. Acad. Sci., Paris, 242, 1871. [18] HoocE, F. N., and KETELAAR, J. A. A., 1957, Physica, 23, 423. [19] FOWLER, R. H., and GUGCENHEIM, E. A., 1939, Statistical Thermodynamics (Cambridge :

University Press). [20] Co5PA, J. P., and KETELAA~, J. A. A., Physica (to be published).

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