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Theoretical approach of the infrared profile of molecular adsorbates on clean surfaces: inhomogeneous broadening
Chemical Physics 230 Ž1998. 67–81
Theoretical approach of the infrared profile of molecular adsorbates on clean surfaces: inhomogeneous broadening C. Girardet 1, J. Humbert, P.N.M. Hoang Laboratoire de Physique Moleculaire – UMR CNRS 6624, Faculte´ des Sciences, La Bouloie, UniÕersite´ de Franche-Comte, ´ ´ 25030 Besanc¸on Cedex, France Received 18 September 1997
Abstract A simple analytical approach is developed to interpret the infrared profile of molecules adsorbed on surfaces without defects. The homogeneous profile is convoluted by a potential distribution function which describes the radial and orientational arrangement of mutually interacting admolecules on the surface. The resulting profile shape is thus asymmetric and its behavior is discussed in terms of a reduced set of interaction potential coefficients and physical parameters Žtemperature and coverage.. Although the model is simple since it is based on a simplified expression of the binding and lateral interactions for the adsorbate–substrate system, it gives the main trends of the profile shape behavior for CO and NH 3 adsorbed on ionic substrates. q 1998 Elsevier Science B.V.
1. Introduction Among the non-destructive tools which can probe molecular adsorbates on a substrate, traditional and polarization infrared spectroscopies w1x, and more recently sum frequency generation experiments w2x, provide accurate information on the configuration and dynamics of the admolecules. The line profile in the infrared spectrum of the adspecies results from the conjunction of several phenomena which remain in general very difficult to analyze in a separate way by using only experimental arguments. If we consider that the experimental resolution is sufficiently small to have no influence on the profile, the mechanisms which govern the line broadening of adsorbates can be separated into two processes. The homogeneous processes are the consequence of pure dephasing mechanisms due to fluctuations of thermal motions of a system undergoing a vibrational transition, and of energy transfers Žvibrationrorientationrtranslation. from the admolecule to the substrate which plays the role of a phonon bath. They can be studied by coupling the molecular motions to the substrate dynamics through the adsorbatersubstrate interaction potential. The inhomogeneous broadenings are mainly due to the heterogeneous distribution of molecules on the surface and to the presence of surface defects. They can be analyzed in terms of the lateral
1
E-mail: [email protected].
0301-0104r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved. PII S 0 3 0 1 - 0 1 0 4 Ž 9 7 . 0 0 3 7 4 - 1
68
C. Girardet et al.r Chemical Physics 230 (1998) 67–81
interaction between the molecules and of the interaction between the admolecules and the defects more or less randomly distributed on a substrate. Estimation of the widths associated with phase and energy relaxation mechanisms can be obtained from infrared spectroscopy and pump-probe time resolved spectroscopy w3,4x. Many studies devoted to the infrared profile analysis of adspecies have been interpreted within the homogeneous scheme, on metals w5x, semiconductors w6x as well as on dielectrics w7x. On metals, the very fast vibrational energy relaxation leading to widths of few wavenumbers, as commonly measured for CO on CuŽ110., PtŽ111. and NiŽ111. w8–10x, are interpreted in terms of electron-hole pair excitations w11x. In contrast much longer relaxation times are observed for the Si-H stretch in the SiŽ111. Ž1 = 1.-H system w12x and for CO adsorbed on NaCl w13x, yielding negligible widths due to the vibrational energy decay via a multiphonon excitation process. On dielectric substrates ŽNaCl and MgO. the homogeneous widths associated with CO, CO 2 and NH 3 were shown to be due to the dephasing mechanism of the internal vibrations, although lifetime broadening could be also present when a low frequency molecular vibration is resonant with the optical phonon band w7x. The influence of inhomogeneous processes has also been studied for the Si-H stretch vibration of the SiŽ111. Ž1 = 1.-H system w6,14x, but most of the studies have been applied to the broadening of the infrared lines of CO chemisorbed on metals w3–6,15,16x. Elegant formulations implying the influence of lateral interaction in the interpretation of the CO infrared lineshape were developed w5x and the influence of phenomenological parameters responsible for the broadening was extensively discussed. However, at metallic surfaces, it appears that the inhomogeneous broadening is not generally the dominant contribution. In contrast for molecules physisorbed on insulators it seems that the lineshape depends crucially on the surface state and on the quality of the adsorbate preparation. Indeed, the inhomogeneous processes in this case play always a significant role in the analysis of the infrared profile and they can be the main cause of broadening. The groups of Ewing w17–23x and Heidberg w24–30x have independently analyzed the role of surface heterogeneities on the infrared profile of CO and CO 2 molecules adsorbed on single crystal NaClŽ100. surface as a function of temperature, adsorbate coverage, and cleanliness of the surface. Both groups arrive at the same conclusions regarding i. the very small width associated with the pure vibrational band ŽQ peak. of these species Žless than or about one wavenumber., and ii. the probable inhomogeneous nature of this width. Quantitative estimates of the homogeneous vs inhomogeneous widths have been recently given w23x by combining data obtained from 12 COr 13 CO isotopic dilution and from CO vibrational overtones, which corroborate the strong influence of inhomogeneities in the broadening mechanism of the admonolayer infrared spectrum. In this paper, we develop a theoretical approach to interpret the influence of the inhomogeneous broadening on the homogeneous line profile of admolecules. The molecules are adsorbed on identical sites and they are coupled to the substrate only, leading to homogeneous relaxation processes. The influence of the surface defects is not considered. These molecules are furthermore distributed at random above the surface and the lateral interactions are responsible for the inhomogeneous broadening. We assume that the coverage is sufficiently small in order to preclude the formation of islands or of a monolayer. Long range dipole–dipole interactions are thus the dominant contributions to the lateral potential. The emphasis is brought to the individual response of the admolecule. As a consequence the mutual polarization of the admolecules and the collective dynamics of the adsorbate which should be necessary included in the theoretical description of the monolayer spectrum w5x is disregarded here. Though the present approach is too simple to give a quantitative calculation of the infrared width for a given molecule–substrate species it contains some of the ingredients which are required to interpret the profile in terms of inhomogeneous versus homogeneous broadening and it could be extended to more general expressions of the potential with, however, a loss in the simplicity of the model. In the present model we discuss the broadening mechanisms in term of four quantities: the strenght of the binding and lateral interactions experienced by the admolecules, the temperature and the molecule coverage. We present in Section 2 the model used to calculate the inhomogeneous width resulting from a stochastic potential distribution, experienced by the admolecules. The infrared adsorption coefficient is then defined as an inhomogeneous distribution of lines, each having a homogeneous lorentzian shape determined in a previous
C. Girardet et al.r Chemical Physics 230 (1998) 67–81
69
paper w7x. In Section 3 we give an application to dipolar molecules adsorbed on dielectrics and discuss the total profile by comparing the theoretical results to some typical experimental data.
2. The model 2.1. Interaction potentials Let us consider a set of N molecules adsorbed on identical sites of a single crystal substrate. The interaction of a molecule with the substrate is written as VS s A S FS Ž V . ,
Ž 1.
where A S describes the magnitude of the interaction potential when the molecular center of mass is located at the equilibrium distance from the surface in the stable site. FS Ž V ., characterizes the angle dependent part of the interaction, with the Euler angles V defining the orientation of the molecular axis with respect to the absolute frame tied to the substrate ŽFig. 1.. VS schematizes in a general way the dispersion-repulsion plus electrostatic contributions to the moleculersubstrate interactions calculated for the translational equilibrium sites of the molecule. The interaction VM between two admolecules located at a mutual distance r and with respective orientations V and V X is expressed as VM s
A˜M rn
FM Ž V , V X ,W . ,
Ž 2.
A˜M is the amplitude of the lateral potential which is assumed to vary as the nth inverse power of the mutual distance r between the two molecules. FM characterizes the angle dependent part of the lateral interactions which also depends on the orientation W of the vector r parallel to the surface of the substrate. VM schematizes for instance the interaction between electrostatic multipolar moments Ž n s 3 for the dipole–dipole contribution, and so on .... 2.2. Inhomogeneous stochastic potential We assume that the number of admolecules on the surface is sufficiently small to consider the lateral interactions as a small perturbation of the total potential V s VS q VM . Each molecule experiences the interaction VS with the substrate which determines the adsorption site and defines the dynamical state of the
Fig. 1. Geometry of the system formed by N molecules adsorbed on a square substrate with rocksalt structure.
C. Girardet et al.r Chemical Physics 230 (1998) 67–81
70
admolecule, and an additional potential Õ, which behaves as a stochastic contribution. The latter potential accounts for the interaction of the molecule with its molecular neighborhood. This small stochastic contribution does not change the adsorption site of the molecule and provides only corrections to the total potential. It depends on the molecule distribution above the surface through the distribution function that is expressed as Ny1
JŽ Õ. s
Ł Hd R d RX d V d V X g Ž R y RX , V , V X . d Ž Õ y Ý VM . .
Ž 3.
i
The integrals are taken over the absolute positions RX in a plane parallel to the surface and orientations V X of the N y 1 molecules interacting with the molecule Ž R, V . and over the positions and orientations of this latter molecule. g characterizes the pair distribution function and the Dirac function defines the condition for the lateral potential summed over all the molecular pairs implying one molecule and every other N y 1 molecules to be equal to Õ. The pair distribution function g is written as g s exp yb A S FS Ž V . q A S FS Ž V X . q
ž
=
½H
X
X
A˜M rn
FM Ž V , V X ,W .
/
X
ž
d R d R d V d V exp yb A S FS Ž V . q A S FS Ž V . q
A˜M rn
y1 X
FM Ž V , V ,W .
/5
,
Ž 4.
where b s Ž kT .y1 and T is the temperature of the system. Then we transform d R d RX into d S rd r dW since the molecule–surface distance is constant, and replace the Dirac function by its exponential form for N y 1 , N. This leads to JŽ Õ. s
1
N
q`
H dr e 2p y`
ir Õ
ž Hrd r dW d V d V
X
fr , b Ž V , V X ,r ,W .
ž
yŽ b q i r .
/
.
Ž 5.
The function fr , b is defined as fr , b s exp yb A S FS Ž V . q FS Ž V X .
=
Hrd r dW d V d V
X
4 exp
A˜M rn X
exp yb A S FS Ž V . q FS Ž V .
FM Ž V , V X ,W .
4 exp
ž
yb
A˜M rn
/ y1 X
FM Ž V , V ,W .
/
.
Ž 6.
By using the cumulant expansion of the exponentials and replacing w31x the complex exponential expwyŽ b q i r .... by 1 y w1 y expwyŽ b q i r ....x one obtains the expression for J Ž Õ . as
° Hrd r dWexp ž yb Ž A˜ rr . ² F :/ 1 y exp ž yi r Ž A˜ rr . ² F :/ ¶ ~1 y •. ¢ ß Hr d r dW exp ž yb Ž A˜ rr . ² F : / N
JŽ Õ. s
1
n
q`
H dr e 2p y`
n
M
M
M
M
ir Õ
n
M
M
Ž 7. In Eq. Ž7., the bracket ² FM : stands for ² FM : s d V d V X exp yb A S Ž FS Ž V . q FS Ž V X . . FM Ž V , V X ,W .
H
y1
=
d V d V X exp yb A S Ž FS Ž V . q FS Ž V X . .
H
,
Ž 8.
C. Girardet et al.r Chemical Physics 230 (1998) 67–81
71
which characterizes the thermal average of the angle dependent potential function FM Ž V , V X ,W . over the molecule orientations. We then define the adsorbate coverage on the surface as l s NrŽ Sra2 . where S is the substrate area, a the lattice parameter of the substrate and Sra2 defines the number of available adsorption sites on the surface. Using the asymptotic behavior of the binomial distribution for large N values w31x finally leads to 1 q` JŽ Õ. s d r e i r Õ ey l B nŽ b , r . , Ž 9. 2p y`
H
where the complex function Bn is defined as
ž
Bn Ž b , r . s r d r dW 1 y exp yi r
H
=
Hr d r dW exp
ž
yb
A˜M
A˜M rn
r
n
² FM :
/ ž
exp yb
A˜M rn
² FM :
/
y1
² FM :
/
.
Ž 10 .
Its imaginary part gives the value of the resonance of the stochastic potential, i.e. the position of the maximum of J Ž Õ ., while the real part characterizes the width of the distribution. Note that Bn is a function of the temperature, of the magnitude of the lateral potential and of the nth inverse power of the distance between two admolecules which characterizes the radial dependence of the lateral interaction. The angle average leads to an additional temperature dependence through the molecule–surface potential ŽEq. Ž8... The twofold integrals occurring in the numerator and the denominator of Eq. Ž10. can be easily calculated if we assume that ² FM : does not depend on W ŽSection 3.. In this case, one gets Bn s BnR q iBnI , where the real part is defined as BnR
ž
Ž x , y . s pG 1 y
2
/ž
n
2rn
A˜M AS a
n
/
Ž ² FM : .
2rn
Ž x2 qy2 .
1rn
cos
ž
2 n
arctan
y
ž // x
y x 2r n ,
Ž 11 .
and the imaginary part, as BnI
ž
Ž x , y . s pG 1 y
2 n
/ž
2rn
A˜M AS a
n
/
Ž ² FM : .
2rn
Ž x2 qy2 .
1rn
sin
ž
2 n
arctan
y
ž // x
,
Ž 12 .
x s b A S and y s r A S are reduced dimensionless variables, G defines the gamma function and a is the lattice parameter of the substrate Žnearest neighbor distance between adsorption sites.. The potential distribution function J Ž Õ . is finally written as JŽ Õ. s
1
p AS
`
H0 d y cos
Õ
ž / AS
R
y y l BnI Ž x , y . ey l B n Ž x , y. .
Ž 13 .
It depends only on the temperature through x and the thermal average ² FM :, on the magnitude of the binding and lateral interactions through A S and A˜M , and on the adsorbate coverage l. A numerical integration of Eq. Ž13. is required in the general case. Note that a linear dependence of Bn with respect to y would lead to a Lorentzian profile for the potential distribution. When the potential distribution J Ž Õ . is introduced in the infrared profile, it accounts for the inhomogeneous frequency distribution, as explained in the next section. 2.3. The infrared profile Let us call I Ž v ,Õ . the absorption coefficient of a molecule which interacts with the substrate through the potential VS and experiences the stochastic potential Õ due to its neighbors. The resulting absorption coefficient
C. Girardet et al.r Chemical Physics 230 (1998) 67–81
72
for a set of such molecules identically adsorbed on the substrate and experiencing a potential Õ distributed according to J Ž Õ . ŽEq. Ž7.. is given as q`
IŽ v . s
Hy` J Ž Õ . I Ž v ,Õ . dÕ.
Ž 14 .
Within the impact approximation to describe the collisional process between the molecules and the substrate and the assumption of vanishingly small initial statistical coupling between the molecule and substrate motions, I Ž v ,Õ . has a lorentzian shape which characterizes the homogeneous broadening and shift of the infrared lines of each admolecule w7x. Moreover the fact that each molecule is perturbed by a stochastic potential Õ leads to additional broadenings which describe the inhomogeneous influence of the molecule distribution on the surface. Any vibrational transition from the ground state
8p 2 3"c
2
meg
v Geg
Ž v y veg q "y1 aeg Õ .
2
, q Geg2
Ž 15 .
where c is the light velocity and meg defines the vibrational transition dipole moment. The vibrational frequency veg takes into account the very small homogeneous shift while the corresponding homogeneous broadening is described by Geg . The stochastic dependence Õ of the adsorption coefficient which occurs in the Lorentzian function is multiplied by a dimensionless factor a eg which characterizes the vibrational dependence of Õ, as
a eg s
1 EÕ Õ Eh
heg ,
Ž 16 .
heg s ²e
3. Application to dipolar molecules adsorbed on dielectrics 3.1. Potential distribution To proceed, it is necessary to formalize the angle dependence of the binding Ž FS . and lateral Ž FM . interactions. We thus consider dipolar molecules which experience a potential due to the substrate written as a sum of an isotropic term not considered here and of an angle dependent term A S ) 0, as VS s yA S cos u .
Ž 17 .
The equilibrium orientation for each molecule is obtained when the molecular axis is perpendicular to the
2
Eq. Ž15. is valid within the assumption of non-polarized infrared probe. Extension to polarization infrared spectroscopy would be straightforward, as shown in Ref. w32x.
C. Girardet et al.r Chemical Physics 230 (1998) 67–81
73
surface Ž u s 0.. The lateral interaction between the dipolar molecules is the electrostatic contribution expressed in the absolute frame tied to the surface, as VM s
AM r
cos u cos u X q
3
AM r3
sin u sin u X cos Ž w y w X . y 3cos 2 W cos Ž w q w X .
y3sin w sin w X y 32 sin2W sin Ž w q w X . ,
Ž 18 .
X
where w and w define the azimuthal Euler angles for the first and second molecules. The calculation of ² FM : ŽEq. Ž8.. is straightforward and yields at the first order of the cumulant expansion an expression which does not depend on the angle W. We have then ² FM : s L 2 Ž b A S . ,
Ž 19 .
where L Ž x . s coth x y 1rx is the Langevin function. Bn is then determined for n s 3, as
ž
B3 Ž b , r . s rd r 1 y exp yi r
H
=
Hrd r exp
ž
yb
A˜M
A˜M r3
r
3
L 2 Ž b AS .
/ ž
exp yb
A˜M r3
L 2 Ž b AS .
/
y1 2
L Ž b AS .
/
.
Ž 20 .
The numerator and the denominator of Eq. Ž20. can be easily calculated and we obtain B3 as 4r3 B3 Ž b , r . s pG Ž 1r3 . A2r3 Ž b AS . M L
Ž b 2qr2 .
1r3
exp
ž
2i 3
r arctan
b
/
y b 2r3 ,
Ž 21 .
where A M s A˜M ra3. B3 is a complex function through the exponential term. Moreover it varies like the 2r3th power of the lateral interaction A M , while the magnitude A S of the angle dependent binding interaction VS is contained in the Langevin function. As T increases Ž b ™ 0., L Ž b A S . , b A Sr3 and B3 is expressed as ip r3 B3 Ž b ™ 0, r . s pG Ž 1r3 . r 2r3 A2r3 Ž b A Sr3 . M e
4r3
.
Ž 22 .
From Eq. Ž22. we see that B3 increases monotonically like r 2r3 and vanishes, as T rises, like Ty4r3. This latter feature is consistent with the vanishing angle dependent molecule–surface interaction by averaging over the orientation of the molecules. When T decreases Ž b increases., L Ž b A S . , 1, the molecular axes are thus frozen at their equilibrium orientation given by the substrate. The expression of B3 then becomes 2r3 B3 Ž b ™ `, r . s pG Ž 1r3 . A2r3 Ž 1 q Ž rrb . M b
2 1r3
.
exp
ž
2i 3
r arctan
b
/
y1 .
Ž 23 .
B3 increases like r 2 and it vanishes like T 4r3 when T is lowered. This feature can be interpreted as a loss of inhomogeneity due to the freezing of the whole adsorbed system which prevents the possibility for the admolecules to be, statistically, close neighbors for reasonably small values of the coverage l. As a result B3 has zero values for T s 0 and `. It thus reaches a maximum with respect to T and then monotonically decreases when T increases. The position and magnitude of the maximum depends on r since it is shifted towards lower T values and is enhanced when r increases. The maps of the real and imaginary parts of B3 Ž x, y ., which are responsible for the inhomogeneous shift and broadening of the infrared lines, display the trend discussed before, i.e. a behavior of B3R and B3I like y 2r3 ' Ž r A S . 2r3 for small x ' b A S values and rather like y 2 for large x values.
74
C. Girardet et al.r Chemical Physics 230 (1998) 67–81
3.2. BehaÕior of J(Õ) Õersus physical parameters The width of the infrared profile ŽEq. Ž14.. depends in a crucial way on the distribution function J Ž Õ .. The substitution of Eq. Ž21. into Eq. Ž13. for n s 3 allows us to discuss the shape of J Ž Õ . as a function, on one hand, of the interaction parameters A S and A M , and on the other hand, of the physical parameters T and l, ŽFigs. 2–4.. The selected values of the four quantities A S , A M , T and l correspond to realistic data for CO and NH 3 adsorbed on NaCl and MgO: A S ranges between 20 and 200 meV whereas A M is one order of magnitude smaller. Moreover, experiments are done at low temperature ŽT F 200 K. and coverage values ranging between 0.1 and 0.3 correspond to physical situations which are expected generally far from the formation of a layer. Fig. 2 displays the influence of the lateral interactions A M and of temperature T when the coverage is taken equal to 0.1 and the angle dependent binding potential A S is rather large Ž A S s 50 meV.. The potential distribution is clearly asymmetric in all situations. For a small lateral interaction Žabout 2.5 meV, curves a., the full width at half height ŽFWHH. is around 0.8 meV, but a significant temperature increase, from 10 K up to 100 K, gives rise to a shift of the distribution maximum toward smaller Õ values and to a concomitant broadening and higher asymmetry of the distribution. An increase of the lateral energy by a factor 2 Žcurves b. and 4 Žcurves c. leads to a more significant broadening ŽFWHH , 1.1 meV and 1.6 meV. with an enhanced influence on the maximum shift and on the distribution asymmetry when T rises. Within this range of values for A S and l, the shape of J Ž Õ . is that inferred from empirical arguments: the potential distribution broadens when the orientational order decreases Žas T increases. and when the lateral interactions increase. In Fig. 3, we discuss the influence of the magnitude A S of the binding potential on the distribution shape. The decrease of A S from 50 meV Žcurves a. to 10 meV Žcurves b. leads to an increase of the maximum shift and to a concomitant drastic increase of the broadening from 0.3 to 1.3 meV at T s 10 K. When T rises up to 100 K, the distribution broadening is still much more pronounced since it reaches a factor of about 25. Note that the narrowing of J Ž Õ . when temperature raises, is consistent with the increasing disorder created on the admolecule orientation. The effect is trivially enhanced when the thermal energy tends to be similar to the magnitude of A S which characterizes the orientational hindering of the molecule due to the substrate.
Fig. 2. Potential distribution function J Ž Õ . versus the strenght of the lateral interaction Õ ŽmeV. for two temperatures T s10 K Žfull curves. and T s100 K Žbroken curves.. Curves a, b and c correspond to the values of the ratio A M r A S equal to 0.05, 0.1 and 0.2, respectively. The values of the binding potential A S and of the coverage are equal to 50 meV and 0.1.
C. Girardet et al.r Chemical Physics 230 (1998) 67–81
75
Fig. 3. Potential distribution function J Ž Õ . versus the strenght of the binding interaction A S for two temperatures T s10 K Žfull curves. and T s100 K Žbroken curves.. Curves a and b correspond to values of A S equal to 50 meV and 10 meV, respectively. The ratio A M r A S is fixed and equal to 0.1, and l s 0.1.
In Fig. 4, we show the influence of coverage. An increase of the coverage from 0.1 Žcurve a. to 0.2 Žcurve e. broadens and shifts the distribution by a factor 2 when T s 10 K, and enhances significantly the broadening asymmetry at higher T values Žcurve b.. When the strenght of the lateral interaction increases by a factor 2, a similar behavior of the distribution shape is obtained when l increases from 0.1 Žcurve c. to 0.2 Žcurve f..
Fig. 4. Potential distribution function J Ž Õ . versus the coverage l for two temperatures T s10 K Žfull curves. and T s100 K Žbroken curves.. Curves a, b and e are drawn for A S s 50 meV and A M r A S s 0.1; the first two curves correspond to a coverage l s 0.1 while for the latter curve l s 0.2. Curves c, d and f are drawn for A S s 50 meV and AMr A S s 0.2; the first two curves correspond to l s 0.1 and the third to l s 0.2.
C. Girardet et al.r Chemical Physics 230 (1998) 67–81
76
3.3. Numerical results on the line profile Eqs. Ž14. and Ž15. are used to calculate the infrared profile of the dipolar molecules CO and NH 3 adsorbed on MgO and NaClŽ100. surfaces for which linewidth measurements are available. In addition to the four quantities discussed in Section 3.2 which occur in the determination of the potential distribution function J Ž Õ ., three other parameters are considered to define I Ž v ,Õ .. While veg can be taken as the pure infrared frequency since it was shown w7x that the homogeneous shift remains very small Žabout 1 cmy1 ., the estimate of Geg and a eg is somewhat more delicate. The homogeneous broadenings Geg were calculated in another paper w7x for the three molecular species CO, CO 2 and NH 3 and they may be satisfactorily compared to the experimental profiles, mainly for CO and CO 2 . Here, we use these calculated values although an uncertainty of at least 20–30% is possible on Geg . The parameter a eg ŽEq. Ž16.. depends on the vibrational transition dipole moment of the molecules after averaging over the first excited and ground vibrational states, since the lateral interaction is limited to the dipole–dipole electrostatic contribution between admolecules. Its accuracy Žsee footnote of Table 1. is thus highly connected to that of the vibrational dependence of the electric moment in the admolecule. Indeed, considering that n s 3 and A˜M s m2 in the lateral potential given by Eq. Ž2., the relative vibrational dependence of VM is given by 1
E VM
VM
Eh
hs
2 Em
m Eh
h,
Ž 24 .
where EmrEh characterizes the vibrational dependence of the dipole moment m taken at the internal equilibrium of the molecule. 3.3.1. CO adsorbed on MgO and NaCl(100) surfaces The stable adsorption site for the isolated CO molecule is the cation site ŽMg 2q or Naq .. The molecular axis is perpendicular to the MgO surface w34x, and upright or slightly tilted by less than 208 with respect to the normal above the NaCl surface w35x, depending on the potentials used to calculate the equilibrium configuration. At small but finite coverage, the lateral interactions remain weak, even when the molecules are adsorbed on adjacent cation sites and they do not influence the orientational equilibrium of the molecular axes. This indicates that the molecular orientations are mainly imposed by the molecule–substrate potential. When coverage increases up to the monolayer completion, the projections of two adjacent molecular axes onto the surface plane tend to be mutually antiparallel in order to reduce the influence of repulsive lateral interactions w34–37x. In all situations Žsmall or larger coverages., the molecular centers of mass are strongly trapped in the adsorption wells, performing only small amplitude oscillations around equilibrium and the molecular axes undergo librations. The resulting infrared spectrum exhibits w34,35x a pure vibrational peak with small additional Table 1 Parameters used in the calculation of I Ž v . ŽEq. Ž14.. Admolecule
veg a Žcmy1 .
a eg b
AS ŽmeV.
AM ŽmeV.
T ŽK.
Geg a Žcmy1 .
l
CO
2155 2155 1100 1100
3.7 3.7 y0.16 y0.16
20 20 200 200
1 1 20 20
10 55 50 200
5.= 10y3 0.07 0.1 2.0
0.05–0.2 0.05–0.2 0.12–0.3 0.12–0.3
NH 3 a
b a eg s Ž2rm .Ž EmrEh .heg . The difference between the length of the normal coordinate h in the excited and ground vibrational Ref. w7x. ˚ for CO and 3.4 = 10y3 A˚ for NH 3 . The value for CO is obtained from the data in the literature w38x while for states is heg s 8.2 = 10y3 A ˚ y1 for CO Žsee Ref. w19x. the bending mode of NH 3 , it has been evaluated from a valence force model. Moreover Ž1rm .Ž EmrEh . s 28.3 A ˚ y1 for NH 3 Žsee Eq. ŽA4. in Ref. w40x.. Note that the a eg value corresponds to Õ expressed in meV ŽEq. Ž15.. while v is in and y2.99 A cmy1 .
C. Girardet et al.r Chemical Physics 230 (1998) 67–81
77
Fig. 5. Infrared profile of CO adsorbed on NaCl or MgOŽ100. surface. The total profile including homogeneous and inhomogeneous broadenings is drawn for T s10 K and 55 K Žfull curves.. For comparison, the homogeneous lorentzian profile is shown at the same temperatures T s10 K Ždotted curve. and 55 K Žbroken curve.. The other parameters are given in the third and sixth rows of Table 2.
librational residues at about 140–160 cmy1 from the main peak frequency. The values of the potential coefficient A S are thus 15 and 20 meV on NaCl and MgO, respectively. The lateral dipolar interactions between CO molecules lead to values for A M around 0.5 and 1 meV. Recent calculations w7x have shown that the vibrational phase relaxation responsible for the homogeneous broadening of the pure vibrational peak leads in general to a very small FWHH for CO, less than 10y2 cmy1 at 10 K and only 0.14 cmy1 at 55 K. The quantity a eg is quite large for CO due to the small dipole versus large transition dipole values and we therefore choose several values of this parameter in the total width calculation. Finally, the coverage appears to be a quantity which remains difficult to measure and it will be studied from 0.05 to 0.35. The results are presented in Fig. 5 and Table 2. At T s 10 K, the calculated width varies from 0.06 to 1.1 cmy1 depending on the values of a eg w38x and l; the other parameters have been kept constant, since they are expected to be more accurately known. The agreement with the measured values seems to be better when l and Table 2 Infrared profiles for CO and NH 3 adsorbed on ionic substrates comparison between calculation and experiments Admolecule
veg Žcmy1 .
T ŽK.
G Žcmy1 .
AS ŽmeV.
AM ŽmeV.
a eg
l
D calc Žcmy1 .
CO
2155 2155 2155 2155 2155 2155 1100 1100 1100
10 10 10 55 55 55 50 200 200
0.01 0.01 0.01 0.14 0.14 0.14 0.5 2.0 2.0
20 20 20 20 20 20 200 200 200
1 1 1 1 1 1 20 20 20
3.7 1.0 1.0 3.7 3.7 1.0 y0.16 y0.16 y0.64
0.1 0.1 0.05 0.1 0.2 0.1 0.3 0.3 0.3
1.1 0.3 0.14 1.1 2.6 0.4 2.1 5.2 16.0
NH 3
a b
Ref. w23x; the measured values depend on the surface preparation, on the coverage. Ref. w41x; estimated from the data of the infrared profile.
D exp Žcmy1 .
0.12 a
0.4–1.0 a
12–20 b
78
C. Girardet et al.r Chemical Physics 230 (1998) 67–81
a eg are relatively small. Note that the experimental data given in Table 2 are only indicative of the peak width since the broadening strongly depends on the surface preparation and on the CO coverage. This feature will be discussed in Section 4. However an upper limit of about 0.07 cmy1 was set in Ref. w23x for the inhomogeneous linewidth of COrNaCl at 10 K. When the temperature raises up to 55 K, the calculated width increases from 0.4 to 2.6 cmy1 , still depending on the values of a eg and l. This set of results is in good agreement with the range of experimental data Ž0.4 to 1 cmy1 .. Two typical profile shapes issued from the values of Table 2 are shown in Fig. 5 and compared to the homogeneous Lorentzian profile. The influence of inhomogeneity is clearly seen, occurring as a nearly symmetric broadening, excepted at the linefoot, for the low temperature spectrum and as an asymmetric broadening at higher temperature. The corresponding frequency shift as T raises is about 1 cmy1 . 3.3.2. NH3 adsorbed on MgO(100) The ammonia molecule is adsorbed on the cation site, with its dipole moment along the C 3 symmetry axis standing upright and pointing opposite to the surface w39x. The orientational motion with respect to the equilibrium configuration is strongly hindered while the spinning motion around the equilibrium axis is nearly free. When coverage increases, the lateral interactions strongly increase since the value of the NH 3 dipole moment is large. They are repulsive and prevent adsorption of NH 3 molecules on adjacent Mg sites. As a result, ˚ The calculated profile the molecules are rather mutually located w39x at larger distances about 4–4.5 A. associated with the symmetric bending mode n 2 of ammonia exhibits w40x a pure vibrational peak around 1100 cmy1 Žinstead of 950 cmy1 in the gas phase.. The potential coefficients are much larger than for CO since A S and A M reach 200 and 20 meV, respectively. But it can be noted that the ratio A M rA S is roughly the same for the two molecular species. The homogeneous widths calculated w7x for 50 and 200 K are significantly larger than for CO. Finally, the parameter a eg has the opposite sign, consistent with a blue shift for the bending mode, and a significantly small absolute value when compared to the CO stretching mode. Its value is varied to study the influence of a eg on the total linewidth of the spectrum. The results are displayed in Fig. 6 and Table 2. At 50 K, the profile shape remains nearly symmetric with a
Fig. 6. Infrared profile of NH 3 adsorbed on MgOŽ100.. The total profile including homogeneous and inhomogeneous broadenings is drawn for T s 50 K and 200 K Žfull curves.. For comparison, the homogeneous lorentzian profile is shown at T s 50 K Ždotted curve. and 200 K Žbroken curve.. The parameters are given in the 7th and 9th rows of Table 2.
C. Girardet et al.r Chemical Physics 230 (1998) 67–81
79
width equal to 2 cmy1 , whereas the homogeneous lorentzian profile is clearly narrower. When T increases substantially, the profile significantly shifts toward the low frequencies and broadens. Moreover it becomes very asymmetric. The total width can reach 16 cmy1 while the homogeneous profile displayed by the lorentzian peaks remains narrow. This latter value is in fair agreement with an estimate of the width deduced from examination of the infrared spectrum of NH 3 adsorbed on MgO Ž100. given in Ref. w41x. Nevertheless an accurate estimate of the experimental width is very difficult due to the presence of defects connected to the preparation of the surface. Indeed MgO crystallites generate large heterogeneities and defects which affect the ammonia molecules.
4. Discussion The literature reporting on the study of the lineshape as a function of temperature and coverage has been mainly devoted to chemisorbed H atoms and CO molecules on single metal surfaces or semiconductors. The goal of the present paper is to show that only a few number of known quantities is sufficient to understand some features of the broadening and of the asymmetry of the infrared lines associated with molecules physisorbed on dielectrics. Thus a simple model restricted to dipolar molecules adsorbed on dielectric substrates has been developed in the low coverage limit which allows us to interpret the main features of the available spectra. Two main assumptions which are consistent with what we know about these adspecies have been done. First, the electric field of the substrate dictates the orientation of the admolecules leading to repulsive lateral interactions described by the dipole–dipole electrostatic contribution. Consequently two potential coefficients are required to schematize the influence of the substrate on one hand, and the influence of the potential due to the molecular surrounding, on the other hand. Second, the infrared profile appears as a simple convolution product of the absorption coefficient of each admolecule by the inhomogeneous potential distribution, regarded as a stochastic term. For simple potentials, the expressions connected to the shift and broadening of the infrared signals are analytical and they depend on the potential parameters and on the temperature in a nice way, while the coverage appears as a linear multiplicative factor. The leading parameters are those given in the homogeneous profile which can be independently determined. This quasi-analytic approach is used to discuss the profile shape behavior in simple terms since the set of implied parameters generally appears through power or exponential functions. For instance, it can be noted that the third inverse power dependence of the terms occurring in the shift and broadening function B3 is a characteristic of the dipole–dipole lateral interaction. The influence of other types of interactions such as the dispersion terms Ž n s 6. or higher multipolar electrostatic contributions Ž n s 4,5.... could be analyzed in a straightforward way within this scheme. Similarly, the temperature dependence through the Langevin function characterizes the orderingrdisordering process connected to the molecule motions. An advantage of the model is its flexibility since it could be extended to other situations such as a distribution of defects on the surface or a distribution of vacancies or impurities in a monolayer adsorbed on ionic substrates. In the first situation the broadening of the infrared signals connected to single admolecule perturbed by the presence of these defects could be calculated using the same approach. In the second situation, the signals of the monolayer Žfor instance the two CO molecules belonging to the unit cell of the low temperature Ž2 = 1. phase. would be broadened by the presence of defects in the layer. These studies could complement extensive calculations based on an excitonic approach and devoted to the study of chemisorbed species ŽH and CO. on metals and semiconductors w5,6x. Nevertheless, as it stands, this approach appears too simple for a direct quantitative comparison with available experimental data. Indeed, the best measurements of the vibrational peak width for CO adsorbed on NaCl have been obtained w19–23x at coverages close to the monolayer completion : the narrowest line ever obtained had a width of 0.07 cmy1 at 5 K. For the monolayer, static and dynamic couplings between adjacent molecules become important and are responsible for the occurrence of satellite peaks, frequency shifts and
C. Girardet et al.r Chemical Physics 230 (1998) 67–81
80
splittings in the infrared spectrum. An accurate description of the monolayer dynamics using an exitonic approach is required w5x which accounts for the line narrowing due to strong lateral interactions. Such a situation cannot be interpreted by the present calculations which tend to privilege the individual character of the admolecules. But, if the monolayer was considered as infinite and ideal Žwithout defects., the width of the peaks would be mainly due to the homogeneous broadening i.e. to the coupling between the adsorbate and the substrate. The line broadening due to these lateral couplings would be efficient only when domains and defects would limit the perfectness of the layer w19x. Such a situation is probably close to the experimental conditions since the measured broadening is about 0.07 cmy1 at 5 K but it can reach 0.4 cmy1 for the monolayer peaks, depending on the surface flatness. When coverage decreases down to 0.3, the bands were found severely broadened, being around 2 cmy1 at 55 K and it is very difficult to observe whether the corresponding profiles are asymmetric. This line asymmetry is directly related to the asymmetry of the distribution J Ž Õ . which is due itself to the form of lateral interactions. This tends to screen the high frequencies. Experiments at still lower coverages are difficult because the surface can be contaminated by unwanted species and because the signal to noise ratio considerably diminishes. Nevertheless it seems reasonable to assign homogeneous versus inhomogeneous width values for COrNaCl ranging between 0.05 and 2 cmy1 . We can thus conclude that the agreement between the calculated profiles and the experimental widths is satisfactory for the two adspecies CO and NH 3 , since temperature and coverage variations lead to significant changes on the broadening, as observed in the experimental spectra. Note finally that the present model should be slightly modified when the lateral interactions between molecules are attractive. This is the case of CO 2 admolecules which stand flat above the surface and are mutually perpendicular to optimize the attractive lateral interaction. Such a situation leads to other analytical developments which will be performed in a further work.
References w1x w2x w3x w4x w5x w6x w7x w8x w9x w10x w11x w12x w13x w14x w15x w16x w17x w18x w19x w20x w21x w22x w23x w24x w25x
Y.J. Chabal, Surf. Sci. Repts 8 Ž1988. 211. Y.R. Shen, Surf. Sci. 299r300 Ž1994. 551. R.G. Tobin, Surf. Sci. 183 Ž1987. 226. J.W. Gadzuk, A.C. Luntz, Surf. Sci. 144 Ž1984. 429. B.N.J. Persson, F.M. Hoffmann, R. Ryberg, Phys. Rev. B 34 Ž1986. 2266; Phys. Rev. B 34 Ž1986. 4354; J. Electron Spectr. Relat. Phenom. 45 Ž1987. 215. B.N.J. Persson, Phys. Rev. B 46 Ž1992. 12701. A. Lakhlifi, C. Girardet, J. Chem. Phys. 105 Ž1996. 2471. E. Schweizer, B.N.J. Persson, M. Tushaus, D. Hoge, A.M. Bradshaw, Surf. Sci. 213 Ž1989. 49. ¨ J.D. Beckerle, M.P. Casassa, R.R. Cavanagh, E.J. Heilweil, J.C. Stephenson, Phys. Rev. Lett. 64 Ž1990. 2090; A.L. Harris, L. Rothberg, L.H. Dubois, L. Dhar, Phys. Rev. Lett. 64 Ž1990. 2086. L.F. Sutcu, H.W. White, J.L. Wragg, Surf. Sci. 249 Ž1991. L343. B.N.J. Persson, R. Ryberg, Phys. Rev. B 40 Ž1989. 10273. P. Guyot-Sionnest, P. Dumas, Y.J. Chabal, G.S. Higashi, Phys. Rev. Lett. 64 Ž1990. 2156. H.C. Chang, G.E. Ewing, J. Electron Spectrosc. Relat. Phenom. 54r55 Ž1990. 39. P. Dumas, Y.J. Chabal, G.S. Higashi, Phys. Rev. Lett. 65 Ž1990. 1124. B.N.J. Persson, F.M. Hoffmann J. Chem. Phys. 88 Ž1988. 3349. P. Hollins, Surf. Sci. Repts. 16 Ž1992. 51. O. Berg, G.E. Ewing, Surf. Sci. 220 Ž1989. 207; R. Disselkamp, G.E. Ewing, J. Chem. Soc. Faraday Trans. 86 Ž1990. 2369. C. Noda, G.E. Ewing, Surf. Sci. 240 Ž1990. 181. R. Disselkamp, H.C. Chang, G.E. Ewing, Surf. Sci. 240 Ž1990. 193. C. Noda, H.H. Richardson, G.E. Ewing, J. Chem. Phys. 92 Ž1990. 2099. O. Berg, R. Disselkamp, G.E. Ewing, Surf. Sci. 277 Ž1992. 8. D.J. Dai, G.E. Ewing, Surf. Sci. 312 Ž1994. 239. H.C. Chang, D.J. Dai, G.E. Ewing, J. Chinese. Chem. Soc. 42 Ž1995. 317. J. Heidberg, E. Kampshoff, R. Kuhnemuth, O. Schonekas, ¨ ¨ ¨ M. Suhren, J. Electron Spectrosc. Relat. Phenom. 45 Ž1987. 87. J. Heidberg, E. Kampshoff, O. Schonekas, ¨ ¨ H. Stein, H. Weiss, Ber. Bunsenges. Phys. Chem. 94 Ž1990. 127.
C. Girardet et al.r Chemical Physics 230 (1998) 67–81 w26x w27x w28x w29x w30x w31x w32x w33x w34x w35x w36x w37x w38x w39x w40x w41x
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J. Heidberg, E. Kampshoff, R. Kuhnemuth, O. Schonekas, ¨ ¨ ¨ Surf. Sci. 251r252 Ž1991. 314; 226 Ž1990. L43. J. Heidberg, D. Meine, Surf. Sci. Lett. 279 Ž1992. L175. J. Heidberg, D. Meine, Ber. Bunsenges. Phys. Chem. 97 Ž1993. 211. J. Heidberg, D. Meine, B. Redlich, J. Electron. Spectroc. Relat. Phenom. 64r65 Ž1993. 599. J. Heidberg, E. Kampshoff, R. Kuhnemuth, O. Schonekas, ¨ ¨ ¨ Surf. Sci. 272 Ž1992. 306; J. Electron. Spectrosc. Relat. Phenom. 64r65, 803 Ž1993.; Can. J. Chem. 72 Ž1994. 795. H. Margenau, Phys. Rev. 82 Ž1951. 156. S. Picaud, S. Briquez, A. Lakhlifi, C. Girardet, J. Chem. Phys. 102 Ž1995. 7229. J. Humbert, C. Girardet, P.N.M. Hoang, to be published. P.N.M. Hoang, S. Picaud, C. Girardet, Surf. Sci. 360 Ž1996. 261. S. Picaud, P.N.M. Hoang, C. Girardet, A.W. Meredith, A.J. Stone, Surf. Sci. 294 Ž1992. 149. C. Girardet, P.N.M. Hoang, S. Picaud, Phys. Rev. B 53 Ž1996. 16615. P.N.M. Hoang, S. Picaud, C. Girardet, A.W. Meredith, J. Chem. Phys. 105 Ž1996. 8453. G. Zumofen, J. Chem. Phys. 68 Ž1978. 3747. S. Picaud, A. Lakhlifi, C. Girardet, J. Chem. Phys. 98 Ž1993. 3488; Y. Ferro, A. Allouche, F. Cora, C. Pisani, C. Girardet, Surf. Sci. 325 Ž1995. 139. A. Lakhlifi, S. Picaud, C. Girardet, A. Allouche, Chem. Phys. 201 Ž1995. 73. R. Echterhoff, E. Knozinger, Surf. Sci. 230 Ž1990. 237. ¨
Theoretical approach of the infrared profile of molecular adsorbates on clean surfaces: inhomogeneous broadening C. Girardet 1, J. Humbert, P.N.M. Hoang Laboratoire de Physique Moleculaire – UMR CNRS 6624, Faculte´ des Sciences, La Bouloie, UniÕersite´ de Franche-Comte, ´ ´ 25030 Besanc¸on Cedex, France Received 18 September 1997
Abstract A simple analytical approach is developed to interpret the infrared profile of molecules adsorbed on surfaces without defects. The homogeneous profile is convoluted by a potential distribution function which describes the radial and orientational arrangement of mutually interacting admolecules on the surface. The resulting profile shape is thus asymmetric and its behavior is discussed in terms of a reduced set of interaction potential coefficients and physical parameters Žtemperature and coverage.. Although the model is simple since it is based on a simplified expression of the binding and lateral interactions for the adsorbate–substrate system, it gives the main trends of the profile shape behavior for CO and NH 3 adsorbed on ionic substrates. q 1998 Elsevier Science B.V.
1. Introduction Among the non-destructive tools which can probe molecular adsorbates on a substrate, traditional and polarization infrared spectroscopies w1x, and more recently sum frequency generation experiments w2x, provide accurate information on the configuration and dynamics of the admolecules. The line profile in the infrared spectrum of the adspecies results from the conjunction of several phenomena which remain in general very difficult to analyze in a separate way by using only experimental arguments. If we consider that the experimental resolution is sufficiently small to have no influence on the profile, the mechanisms which govern the line broadening of adsorbates can be separated into two processes. The homogeneous processes are the consequence of pure dephasing mechanisms due to fluctuations of thermal motions of a system undergoing a vibrational transition, and of energy transfers Žvibrationrorientationrtranslation. from the admolecule to the substrate which plays the role of a phonon bath. They can be studied by coupling the molecular motions to the substrate dynamics through the adsorbatersubstrate interaction potential. The inhomogeneous broadenings are mainly due to the heterogeneous distribution of molecules on the surface and to the presence of surface defects. They can be analyzed in terms of the lateral
1
E-mail: [email protected].
0301-0104r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved. PII S 0 3 0 1 - 0 1 0 4 Ž 9 7 . 0 0 3 7 4 - 1
68
C. Girardet et al.r Chemical Physics 230 (1998) 67–81
interaction between the molecules and of the interaction between the admolecules and the defects more or less randomly distributed on a substrate. Estimation of the widths associated with phase and energy relaxation mechanisms can be obtained from infrared spectroscopy and pump-probe time resolved spectroscopy w3,4x. Many studies devoted to the infrared profile analysis of adspecies have been interpreted within the homogeneous scheme, on metals w5x, semiconductors w6x as well as on dielectrics w7x. On metals, the very fast vibrational energy relaxation leading to widths of few wavenumbers, as commonly measured for CO on CuŽ110., PtŽ111. and NiŽ111. w8–10x, are interpreted in terms of electron-hole pair excitations w11x. In contrast much longer relaxation times are observed for the Si-H stretch in the SiŽ111. Ž1 = 1.-H system w12x and for CO adsorbed on NaCl w13x, yielding negligible widths due to the vibrational energy decay via a multiphonon excitation process. On dielectric substrates ŽNaCl and MgO. the homogeneous widths associated with CO, CO 2 and NH 3 were shown to be due to the dephasing mechanism of the internal vibrations, although lifetime broadening could be also present when a low frequency molecular vibration is resonant with the optical phonon band w7x. The influence of inhomogeneous processes has also been studied for the Si-H stretch vibration of the SiŽ111. Ž1 = 1.-H system w6,14x, but most of the studies have been applied to the broadening of the infrared lines of CO chemisorbed on metals w3–6,15,16x. Elegant formulations implying the influence of lateral interaction in the interpretation of the CO infrared lineshape were developed w5x and the influence of phenomenological parameters responsible for the broadening was extensively discussed. However, at metallic surfaces, it appears that the inhomogeneous broadening is not generally the dominant contribution. In contrast for molecules physisorbed on insulators it seems that the lineshape depends crucially on the surface state and on the quality of the adsorbate preparation. Indeed, the inhomogeneous processes in this case play always a significant role in the analysis of the infrared profile and they can be the main cause of broadening. The groups of Ewing w17–23x and Heidberg w24–30x have independently analyzed the role of surface heterogeneities on the infrared profile of CO and CO 2 molecules adsorbed on single crystal NaClŽ100. surface as a function of temperature, adsorbate coverage, and cleanliness of the surface. Both groups arrive at the same conclusions regarding i. the very small width associated with the pure vibrational band ŽQ peak. of these species Žless than or about one wavenumber., and ii. the probable inhomogeneous nature of this width. Quantitative estimates of the homogeneous vs inhomogeneous widths have been recently given w23x by combining data obtained from 12 COr 13 CO isotopic dilution and from CO vibrational overtones, which corroborate the strong influence of inhomogeneities in the broadening mechanism of the admonolayer infrared spectrum. In this paper, we develop a theoretical approach to interpret the influence of the inhomogeneous broadening on the homogeneous line profile of admolecules. The molecules are adsorbed on identical sites and they are coupled to the substrate only, leading to homogeneous relaxation processes. The influence of the surface defects is not considered. These molecules are furthermore distributed at random above the surface and the lateral interactions are responsible for the inhomogeneous broadening. We assume that the coverage is sufficiently small in order to preclude the formation of islands or of a monolayer. Long range dipole–dipole interactions are thus the dominant contributions to the lateral potential. The emphasis is brought to the individual response of the admolecule. As a consequence the mutual polarization of the admolecules and the collective dynamics of the adsorbate which should be necessary included in the theoretical description of the monolayer spectrum w5x is disregarded here. Though the present approach is too simple to give a quantitative calculation of the infrared width for a given molecule–substrate species it contains some of the ingredients which are required to interpret the profile in terms of inhomogeneous versus homogeneous broadening and it could be extended to more general expressions of the potential with, however, a loss in the simplicity of the model. In the present model we discuss the broadening mechanisms in term of four quantities: the strenght of the binding and lateral interactions experienced by the admolecules, the temperature and the molecule coverage. We present in Section 2 the model used to calculate the inhomogeneous width resulting from a stochastic potential distribution, experienced by the admolecules. The infrared adsorption coefficient is then defined as an inhomogeneous distribution of lines, each having a homogeneous lorentzian shape determined in a previous
C. Girardet et al.r Chemical Physics 230 (1998) 67–81
69
paper w7x. In Section 3 we give an application to dipolar molecules adsorbed on dielectrics and discuss the total profile by comparing the theoretical results to some typical experimental data.
2. The model 2.1. Interaction potentials Let us consider a set of N molecules adsorbed on identical sites of a single crystal substrate. The interaction of a molecule with the substrate is written as VS s A S FS Ž V . ,
Ž 1.
where A S describes the magnitude of the interaction potential when the molecular center of mass is located at the equilibrium distance from the surface in the stable site. FS Ž V ., characterizes the angle dependent part of the interaction, with the Euler angles V defining the orientation of the molecular axis with respect to the absolute frame tied to the substrate ŽFig. 1.. VS schematizes in a general way the dispersion-repulsion plus electrostatic contributions to the moleculersubstrate interactions calculated for the translational equilibrium sites of the molecule. The interaction VM between two admolecules located at a mutual distance r and with respective orientations V and V X is expressed as VM s
A˜M rn
FM Ž V , V X ,W . ,
Ž 2.
A˜M is the amplitude of the lateral potential which is assumed to vary as the nth inverse power of the mutual distance r between the two molecules. FM characterizes the angle dependent part of the lateral interactions which also depends on the orientation W of the vector r parallel to the surface of the substrate. VM schematizes for instance the interaction between electrostatic multipolar moments Ž n s 3 for the dipole–dipole contribution, and so on .... 2.2. Inhomogeneous stochastic potential We assume that the number of admolecules on the surface is sufficiently small to consider the lateral interactions as a small perturbation of the total potential V s VS q VM . Each molecule experiences the interaction VS with the substrate which determines the adsorption site and defines the dynamical state of the
Fig. 1. Geometry of the system formed by N molecules adsorbed on a square substrate with rocksalt structure.
C. Girardet et al.r Chemical Physics 230 (1998) 67–81
70
admolecule, and an additional potential Õ, which behaves as a stochastic contribution. The latter potential accounts for the interaction of the molecule with its molecular neighborhood. This small stochastic contribution does not change the adsorption site of the molecule and provides only corrections to the total potential. It depends on the molecule distribution above the surface through the distribution function that is expressed as Ny1
JŽ Õ. s
Ł Hd R d RX d V d V X g Ž R y RX , V , V X . d Ž Õ y Ý VM . .
Ž 3.
i
The integrals are taken over the absolute positions RX in a plane parallel to the surface and orientations V X of the N y 1 molecules interacting with the molecule Ž R, V . and over the positions and orientations of this latter molecule. g characterizes the pair distribution function and the Dirac function defines the condition for the lateral potential summed over all the molecular pairs implying one molecule and every other N y 1 molecules to be equal to Õ. The pair distribution function g is written as g s exp yb A S FS Ž V . q A S FS Ž V X . q
ž
=
½H
X
X
A˜M rn
FM Ž V , V X ,W .
/
X
ž
d R d R d V d V exp yb A S FS Ž V . q A S FS Ž V . q
A˜M rn
y1 X
FM Ž V , V ,W .
/5
,
Ž 4.
where b s Ž kT .y1 and T is the temperature of the system. Then we transform d R d RX into d S rd r dW since the molecule–surface distance is constant, and replace the Dirac function by its exponential form for N y 1 , N. This leads to JŽ Õ. s
1
N
q`
H dr e 2p y`
ir Õ
ž Hrd r dW d V d V
X
fr , b Ž V , V X ,r ,W .
ž
yŽ b q i r .
/
.
Ž 5.
The function fr , b is defined as fr , b s exp yb A S FS Ž V . q FS Ž V X .
=
Hrd r dW d V d V
X
4 exp
A˜M rn X
exp yb A S FS Ž V . q FS Ž V .
FM Ž V , V X ,W .
4 exp
ž
yb
A˜M rn
/ y1 X
FM Ž V , V ,W .
/
.
Ž 6.
By using the cumulant expansion of the exponentials and replacing w31x the complex exponential expwyŽ b q i r .... by 1 y w1 y expwyŽ b q i r ....x one obtains the expression for J Ž Õ . as
° Hrd r dWexp ž yb Ž A˜ rr . ² F :/ 1 y exp ž yi r Ž A˜ rr . ² F :/ ¶ ~1 y •. ¢ ß Hr d r dW exp ž yb Ž A˜ rr . ² F : / N
JŽ Õ. s
1
n
q`
H dr e 2p y`
n
M
M
M
M
ir Õ
n
M
M
Ž 7. In Eq. Ž7., the bracket ² FM : stands for ² FM : s d V d V X exp yb A S Ž FS Ž V . q FS Ž V X . . FM Ž V , V X ,W .
H
y1
=
d V d V X exp yb A S Ž FS Ž V . q FS Ž V X . .
H
,
Ž 8.
C. Girardet et al.r Chemical Physics 230 (1998) 67–81
71
which characterizes the thermal average of the angle dependent potential function FM Ž V , V X ,W . over the molecule orientations. We then define the adsorbate coverage on the surface as l s NrŽ Sra2 . where S is the substrate area, a the lattice parameter of the substrate and Sra2 defines the number of available adsorption sites on the surface. Using the asymptotic behavior of the binomial distribution for large N values w31x finally leads to 1 q` JŽ Õ. s d r e i r Õ ey l B nŽ b , r . , Ž 9. 2p y`
H
where the complex function Bn is defined as
ž
Bn Ž b , r . s r d r dW 1 y exp yi r
H
=
Hr d r dW exp
ž
yb
A˜M
A˜M rn
r
n
² FM :
/ ž
exp yb
A˜M rn
² FM :
/
y1
² FM :
/
.
Ž 10 .
Its imaginary part gives the value of the resonance of the stochastic potential, i.e. the position of the maximum of J Ž Õ ., while the real part characterizes the width of the distribution. Note that Bn is a function of the temperature, of the magnitude of the lateral potential and of the nth inverse power of the distance between two admolecules which characterizes the radial dependence of the lateral interaction. The angle average leads to an additional temperature dependence through the molecule–surface potential ŽEq. Ž8... The twofold integrals occurring in the numerator and the denominator of Eq. Ž10. can be easily calculated if we assume that ² FM : does not depend on W ŽSection 3.. In this case, one gets Bn s BnR q iBnI , where the real part is defined as BnR
ž
Ž x , y . s pG 1 y
2
/ž
n
2rn
A˜M AS a
n
/
Ž ² FM : .
2rn
Ž x2 qy2 .
1rn
cos
ž
2 n
arctan
y
ž // x
y x 2r n ,
Ž 11 .
and the imaginary part, as BnI
ž
Ž x , y . s pG 1 y
2 n
/ž
2rn
A˜M AS a
n
/
Ž ² FM : .
2rn
Ž x2 qy2 .
1rn
sin
ž
2 n
arctan
y
ž // x
,
Ž 12 .
x s b A S and y s r A S are reduced dimensionless variables, G defines the gamma function and a is the lattice parameter of the substrate Žnearest neighbor distance between adsorption sites.. The potential distribution function J Ž Õ . is finally written as JŽ Õ. s
1
p AS
`
H0 d y cos
Õ
ž / AS
R
y y l BnI Ž x , y . ey l B n Ž x , y. .
Ž 13 .
It depends only on the temperature through x and the thermal average ² FM :, on the magnitude of the binding and lateral interactions through A S and A˜M , and on the adsorbate coverage l. A numerical integration of Eq. Ž13. is required in the general case. Note that a linear dependence of Bn with respect to y would lead to a Lorentzian profile for the potential distribution. When the potential distribution J Ž Õ . is introduced in the infrared profile, it accounts for the inhomogeneous frequency distribution, as explained in the next section. 2.3. The infrared profile Let us call I Ž v ,Õ . the absorption coefficient of a molecule which interacts with the substrate through the potential VS and experiences the stochastic potential Õ due to its neighbors. The resulting absorption coefficient
C. Girardet et al.r Chemical Physics 230 (1998) 67–81
72
for a set of such molecules identically adsorbed on the substrate and experiencing a potential Õ distributed according to J Ž Õ . ŽEq. Ž7.. is given as q`
IŽ v . s
Hy` J Ž Õ . I Ž v ,Õ . dÕ.
Ž 14 .
Within the impact approximation to describe the collisional process between the molecules and the substrate and the assumption of vanishingly small initial statistical coupling between the molecule and substrate motions, I Ž v ,Õ . has a lorentzian shape which characterizes the homogeneous broadening and shift of the infrared lines of each admolecule w7x. Moreover the fact that each molecule is perturbed by a stochastic potential Õ leads to additional broadenings which describe the inhomogeneous influence of the molecule distribution on the surface. Any vibrational transition from the ground state
8p 2 3"c
2
meg
v Geg
Ž v y veg q "y1 aeg Õ .
2
, q Geg2
Ž 15 .
where c is the light velocity and meg defines the vibrational transition dipole moment. The vibrational frequency veg takes into account the very small homogeneous shift while the corresponding homogeneous broadening is described by Geg . The stochastic dependence Õ of the adsorption coefficient which occurs in the Lorentzian function is multiplied by a dimensionless factor a eg which characterizes the vibrational dependence of Õ, as
a eg s
1 EÕ Õ Eh
heg ,
Ž 16 .
heg s ²e
3. Application to dipolar molecules adsorbed on dielectrics 3.1. Potential distribution To proceed, it is necessary to formalize the angle dependence of the binding Ž FS . and lateral Ž FM . interactions. We thus consider dipolar molecules which experience a potential due to the substrate written as a sum of an isotropic term not considered here and of an angle dependent term A S ) 0, as VS s yA S cos u .
Ž 17 .
The equilibrium orientation for each molecule is obtained when the molecular axis is perpendicular to the
2
Eq. Ž15. is valid within the assumption of non-polarized infrared probe. Extension to polarization infrared spectroscopy would be straightforward, as shown in Ref. w32x.
C. Girardet et al.r Chemical Physics 230 (1998) 67–81
73
surface Ž u s 0.. The lateral interaction between the dipolar molecules is the electrostatic contribution expressed in the absolute frame tied to the surface, as VM s
AM r
cos u cos u X q
3
AM r3
sin u sin u X cos Ž w y w X . y 3cos 2 W cos Ž w q w X .
y3sin w sin w X y 32 sin2W sin Ž w q w X . ,
Ž 18 .
X
where w and w define the azimuthal Euler angles for the first and second molecules. The calculation of ² FM : ŽEq. Ž8.. is straightforward and yields at the first order of the cumulant expansion an expression which does not depend on the angle W. We have then ² FM : s L 2 Ž b A S . ,
Ž 19 .
where L Ž x . s coth x y 1rx is the Langevin function. Bn is then determined for n s 3, as
ž
B3 Ž b , r . s rd r 1 y exp yi r
H
=
Hrd r exp
ž
yb
A˜M
A˜M r3
r
3
L 2 Ž b AS .
/ ž
exp yb
A˜M r3
L 2 Ž b AS .
/
y1 2
L Ž b AS .
/
.
Ž 20 .
The numerator and the denominator of Eq. Ž20. can be easily calculated and we obtain B3 as 4r3 B3 Ž b , r . s pG Ž 1r3 . A2r3 Ž b AS . M L
Ž b 2qr2 .
1r3
exp
ž
2i 3
r arctan
b
/
y b 2r3 ,
Ž 21 .
where A M s A˜M ra3. B3 is a complex function through the exponential term. Moreover it varies like the 2r3th power of the lateral interaction A M , while the magnitude A S of the angle dependent binding interaction VS is contained in the Langevin function. As T increases Ž b ™ 0., L Ž b A S . , b A Sr3 and B3 is expressed as ip r3 B3 Ž b ™ 0, r . s pG Ž 1r3 . r 2r3 A2r3 Ž b A Sr3 . M e
4r3
.
Ž 22 .
From Eq. Ž22. we see that B3 increases monotonically like r 2r3 and vanishes, as T rises, like Ty4r3. This latter feature is consistent with the vanishing angle dependent molecule–surface interaction by averaging over the orientation of the molecules. When T decreases Ž b increases., L Ž b A S . , 1, the molecular axes are thus frozen at their equilibrium orientation given by the substrate. The expression of B3 then becomes 2r3 B3 Ž b ™ `, r . s pG Ž 1r3 . A2r3 Ž 1 q Ž rrb . M b
2 1r3
.
exp
ž
2i 3
r arctan
b
/
y1 .
Ž 23 .
B3 increases like r 2 and it vanishes like T 4r3 when T is lowered. This feature can be interpreted as a loss of inhomogeneity due to the freezing of the whole adsorbed system which prevents the possibility for the admolecules to be, statistically, close neighbors for reasonably small values of the coverage l. As a result B3 has zero values for T s 0 and `. It thus reaches a maximum with respect to T and then monotonically decreases when T increases. The position and magnitude of the maximum depends on r since it is shifted towards lower T values and is enhanced when r increases. The maps of the real and imaginary parts of B3 Ž x, y ., which are responsible for the inhomogeneous shift and broadening of the infrared lines, display the trend discussed before, i.e. a behavior of B3R and B3I like y 2r3 ' Ž r A S . 2r3 for small x ' b A S values and rather like y 2 for large x values.
74
C. Girardet et al.r Chemical Physics 230 (1998) 67–81
3.2. BehaÕior of J(Õ) Õersus physical parameters The width of the infrared profile ŽEq. Ž14.. depends in a crucial way on the distribution function J Ž Õ .. The substitution of Eq. Ž21. into Eq. Ž13. for n s 3 allows us to discuss the shape of J Ž Õ . as a function, on one hand, of the interaction parameters A S and A M , and on the other hand, of the physical parameters T and l, ŽFigs. 2–4.. The selected values of the four quantities A S , A M , T and l correspond to realistic data for CO and NH 3 adsorbed on NaCl and MgO: A S ranges between 20 and 200 meV whereas A M is one order of magnitude smaller. Moreover, experiments are done at low temperature ŽT F 200 K. and coverage values ranging between 0.1 and 0.3 correspond to physical situations which are expected generally far from the formation of a layer. Fig. 2 displays the influence of the lateral interactions A M and of temperature T when the coverage is taken equal to 0.1 and the angle dependent binding potential A S is rather large Ž A S s 50 meV.. The potential distribution is clearly asymmetric in all situations. For a small lateral interaction Žabout 2.5 meV, curves a., the full width at half height ŽFWHH. is around 0.8 meV, but a significant temperature increase, from 10 K up to 100 K, gives rise to a shift of the distribution maximum toward smaller Õ values and to a concomitant broadening and higher asymmetry of the distribution. An increase of the lateral energy by a factor 2 Žcurves b. and 4 Žcurves c. leads to a more significant broadening ŽFWHH , 1.1 meV and 1.6 meV. with an enhanced influence on the maximum shift and on the distribution asymmetry when T rises. Within this range of values for A S and l, the shape of J Ž Õ . is that inferred from empirical arguments: the potential distribution broadens when the orientational order decreases Žas T increases. and when the lateral interactions increase. In Fig. 3, we discuss the influence of the magnitude A S of the binding potential on the distribution shape. The decrease of A S from 50 meV Žcurves a. to 10 meV Žcurves b. leads to an increase of the maximum shift and to a concomitant drastic increase of the broadening from 0.3 to 1.3 meV at T s 10 K. When T rises up to 100 K, the distribution broadening is still much more pronounced since it reaches a factor of about 25. Note that the narrowing of J Ž Õ . when temperature raises, is consistent with the increasing disorder created on the admolecule orientation. The effect is trivially enhanced when the thermal energy tends to be similar to the magnitude of A S which characterizes the orientational hindering of the molecule due to the substrate.
Fig. 2. Potential distribution function J Ž Õ . versus the strenght of the lateral interaction Õ ŽmeV. for two temperatures T s10 K Žfull curves. and T s100 K Žbroken curves.. Curves a, b and c correspond to the values of the ratio A M r A S equal to 0.05, 0.1 and 0.2, respectively. The values of the binding potential A S and of the coverage are equal to 50 meV and 0.1.
C. Girardet et al.r Chemical Physics 230 (1998) 67–81
75
Fig. 3. Potential distribution function J Ž Õ . versus the strenght of the binding interaction A S for two temperatures T s10 K Žfull curves. and T s100 K Žbroken curves.. Curves a and b correspond to values of A S equal to 50 meV and 10 meV, respectively. The ratio A M r A S is fixed and equal to 0.1, and l s 0.1.
In Fig. 4, we show the influence of coverage. An increase of the coverage from 0.1 Žcurve a. to 0.2 Žcurve e. broadens and shifts the distribution by a factor 2 when T s 10 K, and enhances significantly the broadening asymmetry at higher T values Žcurve b.. When the strenght of the lateral interaction increases by a factor 2, a similar behavior of the distribution shape is obtained when l increases from 0.1 Žcurve c. to 0.2 Žcurve f..
Fig. 4. Potential distribution function J Ž Õ . versus the coverage l for two temperatures T s10 K Žfull curves. and T s100 K Žbroken curves.. Curves a, b and e are drawn for A S s 50 meV and A M r A S s 0.1; the first two curves correspond to a coverage l s 0.1 while for the latter curve l s 0.2. Curves c, d and f are drawn for A S s 50 meV and AMr A S s 0.2; the first two curves correspond to l s 0.1 and the third to l s 0.2.
C. Girardet et al.r Chemical Physics 230 (1998) 67–81
76
3.3. Numerical results on the line profile Eqs. Ž14. and Ž15. are used to calculate the infrared profile of the dipolar molecules CO and NH 3 adsorbed on MgO and NaClŽ100. surfaces for which linewidth measurements are available. In addition to the four quantities discussed in Section 3.2 which occur in the determination of the potential distribution function J Ž Õ ., three other parameters are considered to define I Ž v ,Õ .. While veg can be taken as the pure infrared frequency since it was shown w7x that the homogeneous shift remains very small Žabout 1 cmy1 ., the estimate of Geg and a eg is somewhat more delicate. The homogeneous broadenings Geg were calculated in another paper w7x for the three molecular species CO, CO 2 and NH 3 and they may be satisfactorily compared to the experimental profiles, mainly for CO and CO 2 . Here, we use these calculated values although an uncertainty of at least 20–30% is possible on Geg . The parameter a eg ŽEq. Ž16.. depends on the vibrational transition dipole moment of the molecules after averaging over the first excited and ground vibrational states, since the lateral interaction is limited to the dipole–dipole electrostatic contribution between admolecules. Its accuracy Žsee footnote of Table 1. is thus highly connected to that of the vibrational dependence of the electric moment in the admolecule. Indeed, considering that n s 3 and A˜M s m2 in the lateral potential given by Eq. Ž2., the relative vibrational dependence of VM is given by 1
E VM
VM
Eh
hs
2 Em
m Eh
h,
Ž 24 .
where EmrEh characterizes the vibrational dependence of the dipole moment m taken at the internal equilibrium of the molecule. 3.3.1. CO adsorbed on MgO and NaCl(100) surfaces The stable adsorption site for the isolated CO molecule is the cation site ŽMg 2q or Naq .. The molecular axis is perpendicular to the MgO surface w34x, and upright or slightly tilted by less than 208 with respect to the normal above the NaCl surface w35x, depending on the potentials used to calculate the equilibrium configuration. At small but finite coverage, the lateral interactions remain weak, even when the molecules are adsorbed on adjacent cation sites and they do not influence the orientational equilibrium of the molecular axes. This indicates that the molecular orientations are mainly imposed by the molecule–substrate potential. When coverage increases up to the monolayer completion, the projections of two adjacent molecular axes onto the surface plane tend to be mutually antiparallel in order to reduce the influence of repulsive lateral interactions w34–37x. In all situations Žsmall or larger coverages., the molecular centers of mass are strongly trapped in the adsorption wells, performing only small amplitude oscillations around equilibrium and the molecular axes undergo librations. The resulting infrared spectrum exhibits w34,35x a pure vibrational peak with small additional Table 1 Parameters used in the calculation of I Ž v . ŽEq. Ž14.. Admolecule
veg a Žcmy1 .
a eg b
AS ŽmeV.
AM ŽmeV.
T ŽK.
Geg a Žcmy1 .
l
CO
2155 2155 1100 1100
3.7 3.7 y0.16 y0.16
20 20 200 200
1 1 20 20
10 55 50 200
5.= 10y3 0.07 0.1 2.0
0.05–0.2 0.05–0.2 0.12–0.3 0.12–0.3
NH 3 a
b a eg s Ž2rm .Ž EmrEh .heg . The difference between the length of the normal coordinate h in the excited and ground vibrational Ref. w7x. ˚ for CO and 3.4 = 10y3 A˚ for NH 3 . The value for CO is obtained from the data in the literature w38x while for states is heg s 8.2 = 10y3 A ˚ y1 for CO Žsee Ref. w19x. the bending mode of NH 3 , it has been evaluated from a valence force model. Moreover Ž1rm .Ž EmrEh . s 28.3 A ˚ y1 for NH 3 Žsee Eq. ŽA4. in Ref. w40x.. Note that the a eg value corresponds to Õ expressed in meV ŽEq. Ž15.. while v is in and y2.99 A cmy1 .
C. Girardet et al.r Chemical Physics 230 (1998) 67–81
77
Fig. 5. Infrared profile of CO adsorbed on NaCl or MgOŽ100. surface. The total profile including homogeneous and inhomogeneous broadenings is drawn for T s10 K and 55 K Žfull curves.. For comparison, the homogeneous lorentzian profile is shown at the same temperatures T s10 K Ždotted curve. and 55 K Žbroken curve.. The other parameters are given in the third and sixth rows of Table 2.
librational residues at about 140–160 cmy1 from the main peak frequency. The values of the potential coefficient A S are thus 15 and 20 meV on NaCl and MgO, respectively. The lateral dipolar interactions between CO molecules lead to values for A M around 0.5 and 1 meV. Recent calculations w7x have shown that the vibrational phase relaxation responsible for the homogeneous broadening of the pure vibrational peak leads in general to a very small FWHH for CO, less than 10y2 cmy1 at 10 K and only 0.14 cmy1 at 55 K. The quantity a eg is quite large for CO due to the small dipole versus large transition dipole values and we therefore choose several values of this parameter in the total width calculation. Finally, the coverage appears to be a quantity which remains difficult to measure and it will be studied from 0.05 to 0.35. The results are presented in Fig. 5 and Table 2. At T s 10 K, the calculated width varies from 0.06 to 1.1 cmy1 depending on the values of a eg w38x and l; the other parameters have been kept constant, since they are expected to be more accurately known. The agreement with the measured values seems to be better when l and Table 2 Infrared profiles for CO and NH 3 adsorbed on ionic substrates comparison between calculation and experiments Admolecule
veg Žcmy1 .
T ŽK.
G Žcmy1 .
AS ŽmeV.
AM ŽmeV.
a eg
l
D calc Žcmy1 .
CO
2155 2155 2155 2155 2155 2155 1100 1100 1100
10 10 10 55 55 55 50 200 200
0.01 0.01 0.01 0.14 0.14 0.14 0.5 2.0 2.0
20 20 20 20 20 20 200 200 200
1 1 1 1 1 1 20 20 20
3.7 1.0 1.0 3.7 3.7 1.0 y0.16 y0.16 y0.64
0.1 0.1 0.05 0.1 0.2 0.1 0.3 0.3 0.3
1.1 0.3 0.14 1.1 2.6 0.4 2.1 5.2 16.0
NH 3
a b
Ref. w23x; the measured values depend on the surface preparation, on the coverage. Ref. w41x; estimated from the data of the infrared profile.
D exp Žcmy1 .
0.12 a
0.4–1.0 a
12–20 b
78
C. Girardet et al.r Chemical Physics 230 (1998) 67–81
a eg are relatively small. Note that the experimental data given in Table 2 are only indicative of the peak width since the broadening strongly depends on the surface preparation and on the CO coverage. This feature will be discussed in Section 4. However an upper limit of about 0.07 cmy1 was set in Ref. w23x for the inhomogeneous linewidth of COrNaCl at 10 K. When the temperature raises up to 55 K, the calculated width increases from 0.4 to 2.6 cmy1 , still depending on the values of a eg and l. This set of results is in good agreement with the range of experimental data Ž0.4 to 1 cmy1 .. Two typical profile shapes issued from the values of Table 2 are shown in Fig. 5 and compared to the homogeneous Lorentzian profile. The influence of inhomogeneity is clearly seen, occurring as a nearly symmetric broadening, excepted at the linefoot, for the low temperature spectrum and as an asymmetric broadening at higher temperature. The corresponding frequency shift as T raises is about 1 cmy1 . 3.3.2. NH3 adsorbed on MgO(100) The ammonia molecule is adsorbed on the cation site, with its dipole moment along the C 3 symmetry axis standing upright and pointing opposite to the surface w39x. The orientational motion with respect to the equilibrium configuration is strongly hindered while the spinning motion around the equilibrium axis is nearly free. When coverage increases, the lateral interactions strongly increase since the value of the NH 3 dipole moment is large. They are repulsive and prevent adsorption of NH 3 molecules on adjacent Mg sites. As a result, ˚ The calculated profile the molecules are rather mutually located w39x at larger distances about 4–4.5 A. associated with the symmetric bending mode n 2 of ammonia exhibits w40x a pure vibrational peak around 1100 cmy1 Žinstead of 950 cmy1 in the gas phase.. The potential coefficients are much larger than for CO since A S and A M reach 200 and 20 meV, respectively. But it can be noted that the ratio A M rA S is roughly the same for the two molecular species. The homogeneous widths calculated w7x for 50 and 200 K are significantly larger than for CO. Finally, the parameter a eg has the opposite sign, consistent with a blue shift for the bending mode, and a significantly small absolute value when compared to the CO stretching mode. Its value is varied to study the influence of a eg on the total linewidth of the spectrum. The results are displayed in Fig. 6 and Table 2. At 50 K, the profile shape remains nearly symmetric with a
Fig. 6. Infrared profile of NH 3 adsorbed on MgOŽ100.. The total profile including homogeneous and inhomogeneous broadenings is drawn for T s 50 K and 200 K Žfull curves.. For comparison, the homogeneous lorentzian profile is shown at T s 50 K Ždotted curve. and 200 K Žbroken curve.. The parameters are given in the 7th and 9th rows of Table 2.
C. Girardet et al.r Chemical Physics 230 (1998) 67–81
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width equal to 2 cmy1 , whereas the homogeneous lorentzian profile is clearly narrower. When T increases substantially, the profile significantly shifts toward the low frequencies and broadens. Moreover it becomes very asymmetric. The total width can reach 16 cmy1 while the homogeneous profile displayed by the lorentzian peaks remains narrow. This latter value is in fair agreement with an estimate of the width deduced from examination of the infrared spectrum of NH 3 adsorbed on MgO Ž100. given in Ref. w41x. Nevertheless an accurate estimate of the experimental width is very difficult due to the presence of defects connected to the preparation of the surface. Indeed MgO crystallites generate large heterogeneities and defects which affect the ammonia molecules.
4. Discussion The literature reporting on the study of the lineshape as a function of temperature and coverage has been mainly devoted to chemisorbed H atoms and CO molecules on single metal surfaces or semiconductors. The goal of the present paper is to show that only a few number of known quantities is sufficient to understand some features of the broadening and of the asymmetry of the infrared lines associated with molecules physisorbed on dielectrics. Thus a simple model restricted to dipolar molecules adsorbed on dielectric substrates has been developed in the low coverage limit which allows us to interpret the main features of the available spectra. Two main assumptions which are consistent with what we know about these adspecies have been done. First, the electric field of the substrate dictates the orientation of the admolecules leading to repulsive lateral interactions described by the dipole–dipole electrostatic contribution. Consequently two potential coefficients are required to schematize the influence of the substrate on one hand, and the influence of the potential due to the molecular surrounding, on the other hand. Second, the infrared profile appears as a simple convolution product of the absorption coefficient of each admolecule by the inhomogeneous potential distribution, regarded as a stochastic term. For simple potentials, the expressions connected to the shift and broadening of the infrared signals are analytical and they depend on the potential parameters and on the temperature in a nice way, while the coverage appears as a linear multiplicative factor. The leading parameters are those given in the homogeneous profile which can be independently determined. This quasi-analytic approach is used to discuss the profile shape behavior in simple terms since the set of implied parameters generally appears through power or exponential functions. For instance, it can be noted that the third inverse power dependence of the terms occurring in the shift and broadening function B3 is a characteristic of the dipole–dipole lateral interaction. The influence of other types of interactions such as the dispersion terms Ž n s 6. or higher multipolar electrostatic contributions Ž n s 4,5.... could be analyzed in a straightforward way within this scheme. Similarly, the temperature dependence through the Langevin function characterizes the orderingrdisordering process connected to the molecule motions. An advantage of the model is its flexibility since it could be extended to other situations such as a distribution of defects on the surface or a distribution of vacancies or impurities in a monolayer adsorbed on ionic substrates. In the first situation the broadening of the infrared signals connected to single admolecule perturbed by the presence of these defects could be calculated using the same approach. In the second situation, the signals of the monolayer Žfor instance the two CO molecules belonging to the unit cell of the low temperature Ž2 = 1. phase. would be broadened by the presence of defects in the layer. These studies could complement extensive calculations based on an excitonic approach and devoted to the study of chemisorbed species ŽH and CO. on metals and semiconductors w5,6x. Nevertheless, as it stands, this approach appears too simple for a direct quantitative comparison with available experimental data. Indeed, the best measurements of the vibrational peak width for CO adsorbed on NaCl have been obtained w19–23x at coverages close to the monolayer completion : the narrowest line ever obtained had a width of 0.07 cmy1 at 5 K. For the monolayer, static and dynamic couplings between adjacent molecules become important and are responsible for the occurrence of satellite peaks, frequency shifts and
C. Girardet et al.r Chemical Physics 230 (1998) 67–81
80
splittings in the infrared spectrum. An accurate description of the monolayer dynamics using an exitonic approach is required w5x which accounts for the line narrowing due to strong lateral interactions. Such a situation cannot be interpreted by the present calculations which tend to privilege the individual character of the admolecules. But, if the monolayer was considered as infinite and ideal Žwithout defects., the width of the peaks would be mainly due to the homogeneous broadening i.e. to the coupling between the adsorbate and the substrate. The line broadening due to these lateral couplings would be efficient only when domains and defects would limit the perfectness of the layer w19x. Such a situation is probably close to the experimental conditions since the measured broadening is about 0.07 cmy1 at 5 K but it can reach 0.4 cmy1 for the monolayer peaks, depending on the surface flatness. When coverage decreases down to 0.3, the bands were found severely broadened, being around 2 cmy1 at 55 K and it is very difficult to observe whether the corresponding profiles are asymmetric. This line asymmetry is directly related to the asymmetry of the distribution J Ž Õ . which is due itself to the form of lateral interactions. This tends to screen the high frequencies. Experiments at still lower coverages are difficult because the surface can be contaminated by unwanted species and because the signal to noise ratio considerably diminishes. Nevertheless it seems reasonable to assign homogeneous versus inhomogeneous width values for COrNaCl ranging between 0.05 and 2 cmy1 . We can thus conclude that the agreement between the calculated profiles and the experimental widths is satisfactory for the two adspecies CO and NH 3 , since temperature and coverage variations lead to significant changes on the broadening, as observed in the experimental spectra. Note finally that the present model should be slightly modified when the lateral interactions between molecules are attractive. This is the case of CO 2 admolecules which stand flat above the surface and are mutually perpendicular to optimize the attractive lateral interaction. Such a situation leads to other analytical developments which will be performed in a further work.
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