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O*(1S) lifetime shortening in solid argon
Volume
118.
number
0*(‘S)
3
LRXTIME
Daniel
SHORTENING
MAILLARD
hboraroire
IN
SOLID
26 July 1985
LEl-rF,Rs
ARGON
’
de Spectrochimie
4 Place Irrrsreu, 75230 Pars
Claude
PHYsIcs
cHEhi1cAL
Molkularre. Cedex
1008. Uniwrsrtl
ERA
Pierre
et Marie
Cm-e,
OS. France
GIRARDET
Loboraroire de Physique Molkdaire. La Bouioie. i5030 E~an~n Ceder.
ERA 834. UniuersitP de Eesaqon, France
and Janine FOURNIER bzborafowe
de Specrrarcopie
de Tra&aIro&
UniuersitL Paris Sd
Bdrimenr
474
91404 Orsqy.
France
A model LO interprel the shortening of the 0(‘S) lifetime in matricrr; IS propc-sal. It r&es oxygen by olher precursormolecules(concenirarion-dependent effect) and (ii) via the dwotion
OP (i) the pa+rbaiion
of tie mati.
of the
The subsequent
inductron of initially forbidden uansition cbpole elements is shown LOconsIderablyaffect the desxciration processof the 0(‘S)
level.
1. In~oduetion
palm
in
both the gas phase and perfect matrix [4],
with a corresponding long lifetime, but becomes d@oZm It has been remarked the lifetune
in several
of the fluorescence
is dramatically
shortened
rare gas matrices.
instances
[I]
that
lines of simple systems
by concentration
effects
in
This is the case of the atom-like
O(1.S + lD) fluorescence in argon where oxygen is obtained by m situ vacuum W photolysis of CO, or N,O: the reduction covers five orders of magnitude (i.e. from 0560 s [2] to 2.4 ps 131) in the relatively narrow concentration range from l/1000 to l/100. A widely accepted general explanation for such a reduction is the migratron of the excitation among isolated atoms, until an atom with a nearby lying quencher is met, with the eventual emrssron of a photon. Another explanation could be the modification of the nature of the o(lS 4 ID) transition, which 1s qzma!m-
as the malooses its perfectness. Accordingly the transition elements increase by a factor ==l‘b7 [S] and thelifetime decreasesin the same ratio. Only the small-ness of the induced transition dipole reduces these seven orders of magnitude to fiie. The purpose of this Letter is to outline the model corresponding to the second explanation, for a quantitative treatment of the causes for such a shortening of the lifetime. New experimental results are summarized in section 2. The framework of the model is described in section 3, the calculations are developed in section L and preliminary quantitative results are given in section 5.
2. Experimental ’ Also at Laboratoirc de Paris 7. Pans. France.
348
Physique
Mol&ula~re.
Universit6
and results
Most of the expedmental 0 009-2614/85/S (North-Holland
apparatus has been pub-
03.30 @ EIsevier Science Publishers B.V. Physics Pu bkbing Division)
Volume 116, number 3
CHEMICAL PHYSICS LEITERS
Table 1
Lifetime of the 0(‘S) + O(lD) emission. mwmred at 562.5
nm, as a function of CO2 precursor concentiatlon co2/Ar (46)
r (ms)
0.1 0.2 0.3 0.4 0.45 0.5
560 t 10
1
<1o-3
100a5
59i3 28a2 lOi
1-i 0.5
lished previously [ 61, we will therefore only briefly describe the main points. The mati sample, deposited on a gold copper plate held at 45 Kin a 7 I ‘TAir fiquide” polycryostat, is irradiated by a pulsed krypton resonance lamp (10 ev). The light emitted by the sample is analyzed through a HRS gratmg monochromator (Jobin-Yvon) by a Hamamatsu (R585) photomultiplier. Lifeme analyses are performed, using a least-squares fitting procedure, by a Commodore 4032 microcomputer. When CO2 trapped in argon is excited by 10 eV photons, the following dissociative process occurs: co2+co+q1s) (thermodynamic
threshold
= 9.27 eV) _
The lifetime of the O(l S) emission corresponding to the atom-like O(lS)-O(lD) transitionis determined as a function of the CO2 concentration. As suspected earlier the lifetime of this transition is dramatically mdependent on the precursor (CO2 in the present case) concentiation (table 1). On the other hand, this lifetime does not depend, within expenmental accuracy, on the photolysLs duration *. These experimental evidences constitute the starting point of the present (and future) work.
3. Framework
of the made1
The model lies on two main points: *1thzbeEnverifedtbatthesp-
intensity, and thers
fore the 0 mncentration, does increase linearly wItl~ photolysis duration
26 July 1985
(i) the lifetime shortening is solely due to the interaction of the oxygen atom with another precursor molecule, and (ii) &ii interaction takes place through rnati distortion (indirect channel). The first point answers the questions what the solute oxygen atom considers as mati imperfectness. The matrix is actually imperfect for several reasons. It contains (even if pure, i.e. not doped) many mono- and multi-vacancies [7], but as the concentration of these is not expected to vary with precursor concentration, this is not the type of effect we are lookmg for. Another imperfection is that the in situ formation of an oxygen atom in the matrix requires the smnlltaneous forrnation of another photofragment (CO in the case of CO, photoiysis). It is often accepted that neither molecules nor
atoms can diffuse a considerable distance through rare gas solids. In this line there will be a CO partner ia the close vicuuty of the O* atom. For this O* atom to be sensitive (through its radiative lifetime) to other CO2 molecules, m concentration as low as 1 to 500, it is neceq that the CO photofragment be “transparent” to the 0’ molecule - i.e. that its presence be not resented - to a very high level. We believe that this is not realistic. First because such a nice “transparency” would be fortuitous and not persist when the photofrapent is changed (from CO to N2 and more seldom 02); all observations show that, in the very dilute expenmen& 0’ hfetime is essentially the same, whatever the precursor used and therefore the photofrag ment released. Second because we have verified that 0’ does resent the N2 molecule [ 81; where N, is added, in variable concentiati&, to C02/Ar samples, the 0’ lifetime is reduced when N2 concentration is above = 2 X 1 0B3, therefore 3 N2 photofragment could not maintain at distance maHer than the average O--N, distance above (==25 A). We believe therefore that it is much more satisfactory to accept longdistance diffusion ofthe two photofragments, as proven by the existence of thermoluminescence as bow as 6 K. in Ar matrix [9]. Anyhow, the photofiagment influence does not depend on the precursor concentration. This is in contradiction with experiment, and eliminates this hypothesis as a possible explanation. The only reasonable remaiaing possibtity is therefore the perturbation of the oxygen atom by precursor molecules themselves (we neglect the existence of the 349
Volume 118, number3
CHEMICAL PHYSICS LEITERS
oligomers of precursor molecules). The second point answers the question how the precursor perturbs the 0’ lifetime through matrix imperfectness. The problem has already been considered in the closely related case of the anisotropic mteraction energy between two far lying hydracid molecules [IO]. The energy perturbation to the pure, perfect rare gas crystal was separated into four components: the self, the direct, the indirect and the (weak) coupling terms. Transposition from energy to lifetime is not straightforward but the concepts can be used to determine the channel which dominates the others. It cannot be the selfterm that would account for the influence on the 0’ lifetime of the inclusion itself, a concenkation independent effect. The 0’ lifetime shortening could in principle be due to the direct influence of the CO2 precursor concentration increase via a med . idenncal in all aspect to that prevalent in the gas phase, ie. the induction of a non-zero transition dipole mo- _ ment by the mere existence of a perturbation. However the transition drpole moment falls off very rapidly with interspacresdistance # and rts magnitude (at the mean distances expected at the concentrations used) 1s far too small to account for the observed effect. If we disregard the carplirrg term, which is nonnegligible only when both the duect and indirect terms are large [lo], the only remaining possibility is the indirecf influence of the CO2 precursor on the O* atom In this channel, a CO2 precursor durtortsthe matnx m its vicinity. This distortion is propagated (and damped) by the matrix in the neighborhood of the oxygen atoms, whose environment is no longer symmetrical (Jahn-Teller effect), allowing the appear-h ante of an induced transihon dipole moment.
4. Calculation of the 0.
lifetime
The determination of the 0’ lifetime is performed in three steps. Firstly, we determine what the distortion induced by a foreign dopant (the precursor), located at an arbrtiary distance of the oxygen atom, is. This distance, on the average, is of course related with the precursor concentration. Secondly, the total induced transition dipole moment is obtained by local** The induced Mtion dipole is usually asswned to de crea6emore or lessexponentially withiuterspe-cies distance 1111. 350
26 July 1985 l
+
t
w
l
F& 1. Schematic two4h1ensional representationof the relative posttion of the dopant Co, (0) in substitutionalsite, of the sample 0 atom (0) ia interstitltlal potion and of one of its ne.Ighbo~sI (+I. ly surnmmg pair transition moments. The lifetime is
thirdly obtained using the Fermi golden rule. 4.1. Distortion The distortion vector of matrix atom I can, in the isotropic crystal approxtmation, be written as 1121:
f
(Rd
5=-RI,
(1)
RI
where RI is the distance between atom 1 and the precursor (see fig. 1). Determination of the function f is not straightforward in the general case. In this Letter, we describe the lattice as a continuum, and we write [9]r f(R) = (5/4&r) &R
-&AR-2,
(2)
where a0 is the lattrce parameter (531 A for argon), and l a constant characterizing the defect. What is relevantis the local structure around the oxygen atom, that is the relative distortion vector 5 01 with respect to the oxygen:
(31 where the subscript 0 refers to the oxygen. Considering that Rl.=R,+
ri,
and developing in successive powers of rl/Rs yields
hme
118, number 3
26 JuIy 19’35
CHEhfEGjL PHYSICS LETrERs
where the summations extend over the necessary number of neighboring atoms. ir describes the varration with distance q of the induced transition dipole moment in the pair O’-A.r and can be extracted from Dunning and Hay’s work [ 111, z is the direction where the
photon is emitted. Due to the fast dampmg of h, summation over the six nearest neighbors at a‘distance r from the 0’ site center is sufficient. In fact there are in the pair two different dipole active de-excitation channeJ.s, namely the 2 lZZ++ 1 W g. 2. Twodimc~Iort~~I rqxegentation of the 0 atom first ell distortion, vectono, b and c correspondto the three rmsof eq. (4) expansion
(parallel transition)
and the 2 1X+ + III (perpendicular transition). The latter is however much less intense (and therefore much less efficient) and is neglected in this preliminary study. After introduction of eq. (4) into eq. (6) and some algebraical m~p~a~ons, we get: IZA-A0
-A-A0 31 - -R,-3 Ri
+A3
RS here
Y,, = “r~(Rs-Z> A R;
S
G(q) Rs
Cl - 3 G(q)1 rl + higher orders,
(4)
+yAL2
(RFzx
RS 5 0,, the first term does not depend on atom 2 and us describes the translation as a whole of the undermed surrounding matrix minus that (with different qnitude) of the central oxygen. The second term ~ntainsthe gradient of ~~la~ment , and is such that e matrix compression (OSdilatation) around the Of om due to the other defect along Rs decreases as e distance increases. The third term accounts for e fact that compression (dilatation) is a 3-D phenom.on and thus depends on the relative positions of the
+Rsyz * +R$Z) _
(7)
At this point, we do not know the direction z where the photon is emitted and we leave this determination for a forthcoming publication [ 133. Here we assume that z is collinear with Rs, then
4.3. Lifetime of the IS level
:ygen atom, atom 1 and the other defect. Fig. 2 dn+ iys these comments.
The mean hfe of exerted 0 in its atom-like 1 S level is given by [S ] :
The induced transition dipole moment is written 31 as
where bI,
are the Einstein transition probabilities of spontaneous emkion for electronic transitions initiating in the atom-like IS Iwel. d coefficients yielding transitions to the atom-like 3P ground-state levels can be neglected for such captions
remain strongly spin351
Volume
CHEMICAL
118. number 3
PHYSICS
forbidden in the matrix. There remain two coefficients, one linked to the induced transition dipole moment given by: Ar = $ (AG/ttV)
4,
LETTERS
cursor molecule is now collinear to Z?S and defined as u&-&_o + Vk_&
Results and d&m&on
The fust problem to solve is to detennine coefflcients A and A0 appearing in eq. (4). In fact we shah use A and K ‘A/A0 (eq. (8)). A characterizes the force defect due to the precursor inclusion. Any matrix atom Z belonging to the fust shell around the oxygen atom thus feels a relative force collinear to RZ and proportional to A (or l) (eq. (2)) e is given as - V~-&Yff&+&
I
(10
where v is a pair potential and v’ and u” its first and second denvatives calculated at the distance P between nearest neighbor matrix atoms. For Lermard-Jones potentials, the force defect E may be approximated as the ratio (17~0~ - ~,&/a~ (cf. table 2) In a similar way, one can dete-mrineA0 (or eo) connected to the force defect due to the inclusion of 0 and C02. The relative force acting on 0’ atom and due to the preTable 2 values of the constans used
distancer tianntion dipole moment h 0)
where E, K and eO are calculated from the values of the potential parameters [14] (cf. table 2). The induced transition dipole moment function can
352
61 z
from
Dunning
and Hay’s
work
Ill].
In fact, only the value of the parallel dipole and that of its first derivative are used and these values are needed at the 0-A.r equilibrium distance in the matnx (cf. table 1). This distance has been determined in a previous publication [4] to be 0.23 Alarger than that in the perfect crystal. Finally the transition energy AE between the atom-like 1s and lD levels is measured to be 2.223 eV in the gas phase [15]. Its value in the dilute argon crystal and the natural quadrupohu Einstein coefficient A 2 are given in table 1. The calculations, performed following the scheme described above, yield results hsted in table 2, for four typical perturber distance Rs. These results are plotted
__----
0.176
_
present work present work
2.88 A
-0.422 2.200
W/d+
transition energy ti natural quadrupolar Emstein coef&lent
Ref_
_
0.145 -0.17
equilibrium0-m
)
0
Value
fr
- “b-A,
(12)
be determined
E = (v&-J,-~
- V&_&
and lO is expressed as
(10)
and the second 5Q related to the natural quadrupolar lrfetime. We do not know how Uus second coeffkient is affected by the perturbation. But as its influence on the lifetime is expected much smaller than that of the dipole, we shall retain the infiitely dflute matrix value in the discussion.
5.
26 July 1985
141 D
Cl11
DA-’ eV
1111 1231
1.79 s-1
121
Fig. 3. Comparison between the arcpm&me.ntal&sults (refs [2,3.13]; continuous curve) md the present calculation (o)_ The ordinate de is in log form whereas the abvissa usesthe or A’p representation. SW azB0table 2. POP(M/R)=*
Vohune118, number 3 Table 3 Calculated 0(‘S)
sa) 3 17
i; 63
26 July 1985
CHEMICAL PHYSICS LETI’ERS
lifetimes
Nsb) 14 168 490 298
al000
T (a
rJRs= 1.09S'n
6.99 x lo3 52.60
1.43 x 10-Q 1.84 x 10-Z
0.63 0.26
19.12 8.33
9.88 x 10” 5.23
o-21 0.18
2.70
2.23 X lo-’
0.14
4
W1)
A precursor located at &stanceR,y lies on the Sth shell S 18 sucfi ss RS = &a Sxn. b) Ns is the number of matrix atoms at d&tan= R Q Rs from the oxygen a)
in fig. 3. The comparison with the experimental data is excellent where the approximations developed in section 3 are expected to hold, i.e. at the higher dihrtions. In fact, at shorter distances, some of the assump tions become unvalid: (i) The second-order expannon of eq. (4) ISexpected to converge as (~/Rs)~. For the shorter distances, convergence is obviously not expected, as the values ofr/Rs listed in table 3 show. (ii) The cont~uum model involves a monotonic increase of the drsplacements as RS decreases. Some oscillations take place in the discrete model [ 121. Such oscillations may considerably affect the distortions [ 131 and therefore the lifetime, more particularly at the shortest drstances (lower dilutions). (iii) The matrix-dopant interaction is ~hema~ed by an isotropic Lennard-Jones potential. Clearly CO2 should be treated more like a prolate ellipsoid. This effect is expected greater at the shortest distances where the true shape of the perturber is not scrambled by the matrix_ Other approbation are expected not to be of so crucial importance: (iv) The self distortion of the matrix by the oxygen and the indirect one due to the dopant are assumed additive, which is true only in first-order approxrmation. (v) The assumption that the dire&on ofpolarization z is along R, is questionable. Though rt is not expected to act on the order of magnitude scale, consideration and determination of the actual z dkection could bring improvement to the model [l l]_ (vi) AIso, the solid medium is able to shorten the above lifetimes by a factor that depends on its index of refraction 1161. The factor is about l/2 to l/3,
depending on the rare gas, and will be taken into account in future work.
6. c.mlclusion
As it is, the model outlined in section 2 provides a satisfactory conceptual framework to understand the mechanisms by which natural lifetime is shortened in matrices. ~provements of rts predic~ons will be brought in forthcoming publications where the dis-
crete nature of the crystal and precise numerical determinations of the distortions 5 and of the polarization direction z will be performed. A satisfactory statistical treatment of trapping will also be considered. Lastly, since lifetime shorterrmg acts on five orders of magnitude we note that experiments involving various dopants (not only precursor molecules) could be performed. Such experunents would allow a classification of the dopants with respect to their abtity to distort and perturb a given matrix.
References [I] E.U. I-KM. J. MoL Spectry- 3 (1959) 425; ftP. Fresh and G. W. Robinson, J_ Chem. E%ys 41 (1964) 367; G-W. R&inson, J. MoL Sgectxy- 6 (1961) 58; J. Fomnier, HB Mohammed, J. DesDn. C. Vermeil and J- ab=ps, J. them. Phys. 73 (1980) 6039. [2] J. Fownk, H&I. Mohammed, J. Deson and D. MaiLk& Chem. Phys 70 (1962) 39. [ 31 RV. Taylor, W. Scott., P.R.. Findiey. Z. Wu. W.C w*-
and KIM. Monahan,J. Chem. Phys 74 (1981) 3718; W-C. Walker. XV. Taylor and KM. Monaban, (&em. Phys. Letterti 84 (1981) 288.
353
Vohune [4]
118. number 3
CHEMICAL.
PHYSICS
D.MaiUar& J. Foumier, H.H. Mohammed and C. Girardet, J. Chem Phys 78 (1983) 5480. [5] G. Henberg, Spezha of dntomicmolecules (van Nostrand, Princeton, 1950) [6] J. Foumier. J_ Deson and C. Vermeil, J. Phys E9 (1976) 879. [7] M.L. Klein and J.k Venables. eds.. Rare gas solids (Academic Press, New York, 1977). [8] J. Fournier and J. Deson, unpublished results. [9] J. Fournier. J. Deson, C. Vermea and G.C Pimentel. J. Chem. Phys 70 (19793 5726. [lo] C. Girardet and D_ b-d. J. Chem. Phys 77 (1982) 5923,594l.
354
[ll]
LErrERs
26 July 1985
T-H. Dunning and P-J. Hay, J. Chem Phys 66 (1977) 3767. [12] P-A Flinn and Ilk Mandud& Ann.Phyr 18 (1962) 81. [13] C. Girardet.D.M~dandJ.Fo~er.tobepublished 1141 J9. Hirschfelder, CF. Curtiss and RB. Bird_ Molecular theory of gases and liquids CWXey, New York, 1964). [15] ELA. Young, G. Black and T.L. Slanger, J. Chem. Phys. 50 (1969) 309; BA Ridley. R A-son and K-H. Welge. J. Chem. Phys. 58 (1973) 3878 1163 RI... Fulton, J. Chem. Phys 61(1974) 4141_
118.
number
0*(‘S)
3
LRXTIME
Daniel
SHORTENING
MAILLARD
hboraroire
IN
SOLID
26 July 1985
LEl-rF,Rs
ARGON
’
de Spectrochimie
4 Place Irrrsreu, 75230 Pars
Claude
PHYsIcs
cHEhi1cAL
Molkularre. Cedex
1008. Uniwrsrtl
ERA
Pierre
et Marie
Cm-e,
OS. France
GIRARDET
Loboraroire de Physique Molkdaire. La Bouioie. i5030 E~an~n Ceder.
ERA 834. UniuersitP de Eesaqon, France
and Janine FOURNIER bzborafowe
de Specrrarcopie
de Tra&aIro&
UniuersitL Paris Sd
Bdrimenr
474
91404 Orsqy.
France
A model LO interprel the shortening of the 0(‘S) lifetime in matricrr; IS propc-sal. It r&es oxygen by olher precursormolecules(concenirarion-dependent effect) and (ii) via the dwotion
OP (i) the pa+rbaiion
of tie mati.
of the
The subsequent
inductron of initially forbidden uansition cbpole elements is shown LOconsIderablyaffect the desxciration processof the 0(‘S)
level.
1. In~oduetion
palm
in
both the gas phase and perfect matrix [4],
with a corresponding long lifetime, but becomes d@oZm It has been remarked the lifetune
in several
of the fluorescence
is dramatically
shortened
rare gas matrices.
instances
[I]
that
lines of simple systems
by concentration
effects
in
This is the case of the atom-like
O(1.S + lD) fluorescence in argon where oxygen is obtained by m situ vacuum W photolysis of CO, or N,O: the reduction covers five orders of magnitude (i.e. from 0560 s [2] to 2.4 ps 131) in the relatively narrow concentration range from l/1000 to l/100. A widely accepted general explanation for such a reduction is the migratron of the excitation among isolated atoms, until an atom with a nearby lying quencher is met, with the eventual emrssron of a photon. Another explanation could be the modification of the nature of the o(lS 4 ID) transition, which 1s qzma!m-
as the malooses its perfectness. Accordingly the transition elements increase by a factor ==l‘b7 [S] and thelifetime decreasesin the same ratio. Only the small-ness of the induced transition dipole reduces these seven orders of magnitude to fiie. The purpose of this Letter is to outline the model corresponding to the second explanation, for a quantitative treatment of the causes for such a shortening of the lifetime. New experimental results are summarized in section 2. The framework of the model is described in section 3, the calculations are developed in section L and preliminary quantitative results are given in section 5.
2. Experimental ’ Also at Laboratoirc de Paris 7. Pans. France.
348
Physique
Mol&ula~re.
Universit6
and results
Most of the expedmental 0 009-2614/85/S (North-Holland
apparatus has been pub-
03.30 @ EIsevier Science Publishers B.V. Physics Pu bkbing Division)
Volume 116, number 3
CHEMICAL PHYSICS LEITERS
Table 1
Lifetime of the 0(‘S) + O(lD) emission. mwmred at 562.5
nm, as a function of CO2 precursor concentiatlon co2/Ar (46)
r (ms)
0.1 0.2 0.3 0.4 0.45 0.5
560 t 10
1
<1o-3
100a5
59i3 28a2 lOi
1-i 0.5
lished previously [ 61, we will therefore only briefly describe the main points. The mati sample, deposited on a gold copper plate held at 45 Kin a 7 I ‘TAir fiquide” polycryostat, is irradiated by a pulsed krypton resonance lamp (10 ev). The light emitted by the sample is analyzed through a HRS gratmg monochromator (Jobin-Yvon) by a Hamamatsu (R585) photomultiplier. Lifeme analyses are performed, using a least-squares fitting procedure, by a Commodore 4032 microcomputer. When CO2 trapped in argon is excited by 10 eV photons, the following dissociative process occurs: co2+co+q1s) (thermodynamic
threshold
= 9.27 eV) _
The lifetime of the O(l S) emission corresponding to the atom-like O(lS)-O(lD) transitionis determined as a function of the CO2 concentration. As suspected earlier the lifetime of this transition is dramatically mdependent on the precursor (CO2 in the present case) concentiation (table 1). On the other hand, this lifetime does not depend, within expenmental accuracy, on the photolysLs duration *. These experimental evidences constitute the starting point of the present (and future) work.
3. Framework
of the made1
The model lies on two main points: *1thzbeEnverifedtbatthesp-
intensity, and thers
fore the 0 mncentration, does increase linearly wItl~ photolysis duration
26 July 1985
(i) the lifetime shortening is solely due to the interaction of the oxygen atom with another precursor molecule, and (ii) &ii interaction takes place through rnati distortion (indirect channel). The first point answers the questions what the solute oxygen atom considers as mati imperfectness. The matrix is actually imperfect for several reasons. It contains (even if pure, i.e. not doped) many mono- and multi-vacancies [7], but as the concentration of these is not expected to vary with precursor concentration, this is not the type of effect we are lookmg for. Another imperfection is that the in situ formation of an oxygen atom in the matrix requires the smnlltaneous forrnation of another photofragment (CO in the case of CO, photoiysis). It is often accepted that neither molecules nor
atoms can diffuse a considerable distance through rare gas solids. In this line there will be a CO partner ia the close vicuuty of the O* atom. For this O* atom to be sensitive (through its radiative lifetime) to other CO2 molecules, m concentration as low as 1 to 500, it is neceq that the CO photofragment be “transparent” to the 0’ molecule - i.e. that its presence be not resented - to a very high level. We believe that this is not realistic. First because such a nice “transparency” would be fortuitous and not persist when the photofrapent is changed (from CO to N2 and more seldom 02); all observations show that, in the very dilute expenmen& 0’ hfetime is essentially the same, whatever the precursor used and therefore the photofrag ment released. Second because we have verified that 0’ does resent the N2 molecule [ 81; where N, is added, in variable concentiati&, to C02/Ar samples, the 0’ lifetime is reduced when N2 concentration is above = 2 X 1 0B3, therefore 3 N2 photofragment could not maintain at distance maHer than the average O--N, distance above (==25 A). We believe therefore that it is much more satisfactory to accept longdistance diffusion ofthe two photofragments, as proven by the existence of thermoluminescence as bow as 6 K. in Ar matrix [9]. Anyhow, the photofiagment influence does not depend on the precursor concentration. This is in contradiction with experiment, and eliminates this hypothesis as a possible explanation. The only reasonable remaiaing possibtity is therefore the perturbation of the oxygen atom by precursor molecules themselves (we neglect the existence of the 349
Volume 118, number3
CHEMICAL PHYSICS LEITERS
oligomers of precursor molecules). The second point answers the question how the precursor perturbs the 0’ lifetime through matrix imperfectness. The problem has already been considered in the closely related case of the anisotropic mteraction energy between two far lying hydracid molecules [IO]. The energy perturbation to the pure, perfect rare gas crystal was separated into four components: the self, the direct, the indirect and the (weak) coupling terms. Transposition from energy to lifetime is not straightforward but the concepts can be used to determine the channel which dominates the others. It cannot be the selfterm that would account for the influence on the 0’ lifetime of the inclusion itself, a concenkation independent effect. The 0’ lifetime shortening could in principle be due to the direct influence of the CO2 precursor concentration increase via a med . idenncal in all aspect to that prevalent in the gas phase, ie. the induction of a non-zero transition dipole mo- _ ment by the mere existence of a perturbation. However the transition drpole moment falls off very rapidly with interspacresdistance # and rts magnitude (at the mean distances expected at the concentrations used) 1s far too small to account for the observed effect. If we disregard the carplirrg term, which is nonnegligible only when both the duect and indirect terms are large [lo], the only remaining possibility is the indirecf influence of the CO2 precursor on the O* atom In this channel, a CO2 precursor durtortsthe matnx m its vicinity. This distortion is propagated (and damped) by the matrix in the neighborhood of the oxygen atoms, whose environment is no longer symmetrical (Jahn-Teller effect), allowing the appear-h ante of an induced transihon dipole moment.
4. Calculation of the 0.
lifetime
The determination of the 0’ lifetime is performed in three steps. Firstly, we determine what the distortion induced by a foreign dopant (the precursor), located at an arbrtiary distance of the oxygen atom, is. This distance, on the average, is of course related with the precursor concentration. Secondly, the total induced transition dipole moment is obtained by local** The induced Mtion dipole is usually asswned to de crea6emore or lessexponentially withiuterspe-cies distance 1111. 350
26 July 1985 l
+
t
w
l
F& 1. Schematic two4h1ensional representationof the relative posttion of the dopant Co, (0) in substitutionalsite, of the sample 0 atom (0) ia interstitltlal potion and of one of its ne.Ighbo~sI (+I. ly surnmmg pair transition moments. The lifetime is
thirdly obtained using the Fermi golden rule. 4.1. Distortion The distortion vector of matrix atom I can, in the isotropic crystal approxtmation, be written as 1121:
f
(Rd
5=-RI,
(1)
RI
where RI is the distance between atom 1 and the precursor (see fig. 1). Determination of the function f is not straightforward in the general case. In this Letter, we describe the lattice as a continuum, and we write [9]r f(R) = (5/4&r) &R
-&AR-2,
(2)
where a0 is the lattrce parameter (531 A for argon), and l a constant characterizing the defect. What is relevantis the local structure around the oxygen atom, that is the relative distortion vector 5 01 with respect to the oxygen:
(31 where the subscript 0 refers to the oxygen. Considering that Rl.=R,+
ri,
and developing in successive powers of rl/Rs yields
hme
118, number 3
26 JuIy 19’35
CHEhfEGjL PHYSICS LETrERs
where the summations extend over the necessary number of neighboring atoms. ir describes the varration with distance q of the induced transition dipole moment in the pair O’-A.r and can be extracted from Dunning and Hay’s work [ 111, z is the direction where the
photon is emitted. Due to the fast dampmg of h, summation over the six nearest neighbors at a‘distance r from the 0’ site center is sufficient. In fact there are in the pair two different dipole active de-excitation channeJ.s, namely the 2 lZZ++ 1 W g. 2. Twodimc~Iort~~I rqxegentation of the 0 atom first ell distortion, vectono, b and c correspondto the three rmsof eq. (4) expansion
(parallel transition)
and the 2 1X+ + III (perpendicular transition). The latter is however much less intense (and therefore much less efficient) and is neglected in this preliminary study. After introduction of eq. (4) into eq. (6) and some algebraical m~p~a~ons, we get: IZA-A0
-A-A0 31 - -R,-3 Ri
+A3
RS here
Y,, = “r~(Rs-Z> A R;
S
G(q) Rs
Cl - 3 G(q)1 rl + higher orders,
(4)
+yAL2
(RFzx
RS 5 0,, the first term does not depend on atom 2 and us describes the translation as a whole of the undermed surrounding matrix minus that (with different qnitude) of the central oxygen. The second term ~ntainsthe gradient of ~~la~ment , and is such that e matrix compression (OSdilatation) around the Of om due to the other defect along Rs decreases as e distance increases. The third term accounts for e fact that compression (dilatation) is a 3-D phenom.on and thus depends on the relative positions of the
+Rsyz * +R$Z) _
(7)
At this point, we do not know the direction z where the photon is emitted and we leave this determination for a forthcoming publication [ 133. Here we assume that z is collinear with Rs, then
4.3. Lifetime of the IS level
:ygen atom, atom 1 and the other defect. Fig. 2 dn+ iys these comments.
The mean hfe of exerted 0 in its atom-like 1 S level is given by [S ] :
The induced transition dipole moment is written 31 as
where bI,
are the Einstein transition probabilities of spontaneous emkion for electronic transitions initiating in the atom-like IS Iwel. d coefficients yielding transitions to the atom-like 3P ground-state levels can be neglected for such captions
remain strongly spin351
Volume
CHEMICAL
118. number 3
PHYSICS
forbidden in the matrix. There remain two coefficients, one linked to the induced transition dipole moment given by: Ar = $ (AG/ttV)
4,
LETTERS
cursor molecule is now collinear to Z?S and defined as u&-&_o + Vk_&
Results and d&m&on
The fust problem to solve is to detennine coefflcients A and A0 appearing in eq. (4). In fact we shah use A and K ‘A/A0 (eq. (8)). A characterizes the force defect due to the precursor inclusion. Any matrix atom Z belonging to the fust shell around the oxygen atom thus feels a relative force collinear to RZ and proportional to A (or l) (eq. (2)) e is given as - V~-&Yff&+&
I
(10
where v is a pair potential and v’ and u” its first and second denvatives calculated at the distance P between nearest neighbor matrix atoms. For Lermard-Jones potentials, the force defect E may be approximated as the ratio (17~0~ - ~,&/a~ (cf. table 2) In a similar way, one can dete-mrineA0 (or eo) connected to the force defect due to the inclusion of 0 and C02. The relative force acting on 0’ atom and due to the preTable 2 values of the constans used
distancer tianntion dipole moment h 0)
where E, K and eO are calculated from the values of the potential parameters [14] (cf. table 2). The induced transition dipole moment function can
352
61 z
from
Dunning
and Hay’s
work
Ill].
In fact, only the value of the parallel dipole and that of its first derivative are used and these values are needed at the 0-A.r equilibrium distance in the matnx (cf. table 1). This distance has been determined in a previous publication [4] to be 0.23 Alarger than that in the perfect crystal. Finally the transition energy AE between the atom-like 1s and lD levels is measured to be 2.223 eV in the gas phase [15]. Its value in the dilute argon crystal and the natural quadrupohu Einstein coefficient A 2 are given in table 1. The calculations, performed following the scheme described above, yield results hsted in table 2, for four typical perturber distance Rs. These results are plotted
__----
0.176
_
present work present work
2.88 A
-0.422 2.200
W/d+
transition energy ti natural quadrupolar Emstein coef&lent
Ref_
_
0.145 -0.17
equilibrium0-m
)
0
Value
fr
- “b-A,
(12)
be determined
E = (v&-J,-~
- V&_&
and lO is expressed as
(10)
and the second 5Q related to the natural quadrupolar lrfetime. We do not know how Uus second coeffkient is affected by the perturbation. But as its influence on the lifetime is expected much smaller than that of the dipole, we shall retain the infiitely dflute matrix value in the discussion.
5.
26 July 1985
141 D
Cl11
DA-’ eV
1111 1231
1.79 s-1
121
Fig. 3. Comparison between the arcpm&me.ntal&sults (refs [2,3.13]; continuous curve) md the present calculation (o)_ The ordinate de is in log form whereas the abvissa usesthe or A’p representation. SW azB0table 2. POP(M/R)=*
Vohune118, number 3 Table 3 Calculated 0(‘S)
sa) 3 17
i; 63
26 July 1985
CHEMICAL PHYSICS LETI’ERS
lifetimes
Nsb) 14 168 490 298
al000
T (a
rJRs= 1.09S'n
6.99 x lo3 52.60
1.43 x 10-Q 1.84 x 10-Z
0.63 0.26
19.12 8.33
9.88 x 10” 5.23
o-21 0.18
2.70
2.23 X lo-’
0.14
4
W1)
A precursor located at &stanceR,y lies on the Sth shell S 18 sucfi ss RS = &a Sxn. b) Ns is the number of matrix atoms at d&tan= R Q Rs from the oxygen a)
in fig. 3. The comparison with the experimental data is excellent where the approximations developed in section 3 are expected to hold, i.e. at the higher dihrtions. In fact, at shorter distances, some of the assump tions become unvalid: (i) The second-order expannon of eq. (4) ISexpected to converge as (~/Rs)~. For the shorter distances, convergence is obviously not expected, as the values ofr/Rs listed in table 3 show. (ii) The cont~uum model involves a monotonic increase of the drsplacements as RS decreases. Some oscillations take place in the discrete model [ 121. Such oscillations may considerably affect the distortions [ 131 and therefore the lifetime, more particularly at the shortest drstances (lower dilutions). (iii) The matrix-dopant interaction is ~hema~ed by an isotropic Lennard-Jones potential. Clearly CO2 should be treated more like a prolate ellipsoid. This effect is expected greater at the shortest distances where the true shape of the perturber is not scrambled by the matrix_ Other approbation are expected not to be of so crucial importance: (iv) The self distortion of the matrix by the oxygen and the indirect one due to the dopant are assumed additive, which is true only in first-order approxrmation. (v) The assumption that the dire&on ofpolarization z is along R, is questionable. Though rt is not expected to act on the order of magnitude scale, consideration and determination of the actual z dkection could bring improvement to the model [l l]_ (vi) AIso, the solid medium is able to shorten the above lifetimes by a factor that depends on its index of refraction 1161. The factor is about l/2 to l/3,
depending on the rare gas, and will be taken into account in future work.
6. c.mlclusion
As it is, the model outlined in section 2 provides a satisfactory conceptual framework to understand the mechanisms by which natural lifetime is shortened in matrices. ~provements of rts predic~ons will be brought in forthcoming publications where the dis-
crete nature of the crystal and precise numerical determinations of the distortions 5 and of the polarization direction z will be performed. A satisfactory statistical treatment of trapping will also be considered. Lastly, since lifetime shorterrmg acts on five orders of magnitude we note that experiments involving various dopants (not only precursor molecules) could be performed. Such experunents would allow a classification of the dopants with respect to their abtity to distort and perturb a given matrix.
References [I] E.U. I-KM. J. MoL Spectry- 3 (1959) 425; ftP. Fresh and G. W. Robinson, J_ Chem. E%ys 41 (1964) 367; G-W. R&inson, J. MoL Sgectxy- 6 (1961) 58; J. Fomnier, HB Mohammed, J. DesDn. C. Vermeil and J- ab=ps, J. them. Phys. 73 (1980) 6039. [2] J. Fownk, H&I. Mohammed, J. Deson and D. MaiLk& Chem. Phys 70 (1962) 39. [ 31 RV. Taylor, W. Scott., P.R.. Findiey. Z. Wu. W.C w*-
and KIM. Monahan,J. Chem. Phys 74 (1981) 3718; W-C. Walker. XV. Taylor and KM. Monaban, (&em. Phys. Letterti 84 (1981) 288.
353
Vohune [4]
118. number 3
CHEMICAL.
PHYSICS
D.MaiUar& J. Foumier, H.H. Mohammed and C. Girardet, J. Chem Phys 78 (1983) 5480. [5] G. Henberg, Spezha of dntomicmolecules (van Nostrand, Princeton, 1950) [6] J. Foumier. J_ Deson and C. Vermeil, J. Phys E9 (1976) 879. [7] M.L. Klein and J.k Venables. eds.. Rare gas solids (Academic Press, New York, 1977). [8] J. Fournier and J. Deson, unpublished results. [9] J. Fournier. J. Deson, C. Vermea and G.C Pimentel. J. Chem. Phys 70 (19793 5726. [lo] C. Girardet and D_ b-d. J. Chem. Phys 77 (1982) 5923,594l.
354
[ll]
LErrERs
26 July 1985
T-H. Dunning and P-J. Hay, J. Chem Phys 66 (1977) 3767. [12] P-A Flinn and Ilk Mandud& Ann.Phyr 18 (1962) 81. [13] C. Girardet.D.M~dandJ.Fo~er.tobepublished 1141 J9. Hirschfelder, CF. Curtiss and RB. Bird_ Molecular theory of gases and liquids CWXey, New York, 1964). [15] ELA. Young, G. Black and T.L. Slanger, J. Chem. Phys. 50 (1969) 309; BA Ridley. R A-son and K-H. Welge. J. Chem. Phys. 58 (1973) 3878 1163 RI... Fulton, J. Chem. Phys 61(1974) 4141_