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Raman spectroscopy of powders: Effects of light absorption and scattering
Vol. 50A, No. 11, pp. 1833-1840, I994 Copyright@ 1994Elscvicr Science Ltd Printedin Great Britain.All rightsrcscrvcd
Specrrochimica Acta.
Pergamon 0584-8539(94)Eoo26-7
0584~8539/94 $7.00 + 0.00
Raman spectroscopy of powders: effects of light absorption and scattering DAVID N. WATERS Department of Chemistry, Brunel University, Uxbridge UB8 3PH, U.K. Abstract-New expressions are derived from Kubelka-Munk theory to describe the Raman intensities observed by back-scattering from powder samples. The equations relate the Raman intensity to the diffuse reflectance R, of the sample for two cases: (i) for a series of samples having constant values of the scattering coefficient, s, but which vary in their values of the absorption coefficient, k; and (ii) for a series of samples having constant values of k, but differing in their values of s. The predicted intensity dependences are compared with the results of experiment. Keywords: Raman, NIR, diffuse reflectance, powders, Kubelka-Munk.
RAMAN SPECTROSCOPYis increasingly being used for the investigation of powder samples, and for the quantitative analysis of mixtures in the form of powders and compacted powders. Such materials are in general characterized by both absorption and scattering, and it is well known that both of these properties influence the observed Raman intensities. It may be generalized that increasing absorption, for a set of samples having constant scattering power, leads to decreasing observed intensities; and increasing scattering, for a set of samples having constant absorption coefficient, leads also to a lowering of intensities. An account of the Raman intensities of powder samples, exhibiting both absorption and scattering, has been given by SCHRADER and BERGMANN [l] as an extension of the theory due to KUBELKA and MUNK (K-M theory), originally developed [2-4] to describe the transmittance and diffuse reflectance of powdered solids. The treatment [l] considers the four differential equations which describe absorption and scattering of the primary and the Raman-shifted radiation within the solid, and solves these to give intensity expressions both for transmission through the powder layer (0” scattering) and back-scattering from the front surface (180” scattering). The expressions, of course, involve d, the thickness of the layer, and a primary result of the treatment is to show the dependence of the Raman intensities upon this thickness [l, 51. There is a possible alternative approach to the application of K-M theory to the description of Raman intensities from powdered solids. This is based on work by POLLAK [6,7] on the calculation of fluorescence intensities from diffusing media. This work is directed particularly to applications in thin-layer chromatography, for which fluorimetry offers an attractive method of quantitation. Results of this work [7] have been frequently cited in writings on chromatography, e.g. Refs [&lo]. There is a formal analogy between Raman scattering and fluorescence as far as this intensity problem is concerned. The two treatments [l, 71, however, differ considerably. It has not been shown that they lead to the same results, and indeed, as published, they do not. There is a need to resolve this discrepancy, and this provides part of the rationale for this work. Additionally, it can be shown that the results of either treatment (once they have been brought into convergence) can be recast into forms which are particularly amenable to testing by experiment and to comparisons with published data, and this too is part of our purpose.
EXPERIMENTAL
Diffuse reflectance spectra were measured using a Perkin-Elmer model Lambda 9 spectrophotometer equipped with an integrating sphere accessory. Kodak “white reflectance standard” was used as the reference material. Reported values of R, are for the wavelength region 1.0-1.2 pm, 1833
D. N. WATERS
1834
x=0
x
x=d
Fig. 1. Geometry of the Kubelka and Munk model. which is the region containing the measured Raman spectrum. Raman spectra were excited at 1.064 pm and were observed in the back-scattering geometry with a Perkin-Elmer model 1760X
IT-Raman spectrometer. Care was taken to ensure constant laser power and reproducible sample presentation.
DISCUSSION
In our view, the approach taken in Ref. [7] offers a good physical insight into the intensity problem. Unfortunately this treatment [7] contains an error which leads to wrong expressions for the fluorescence (and Raman) intensities. For these two reasons we shall outline a correct presentation of the intensity analysis according to this approach, leading ultimately to the new relations. We shall then indicate how a result from Ref. [l] can be re-expressed in this form. Some results of the original K-M theory, i.e. as applied to diffuse reflectance, are required first. We use the nomenclature of FREI
WI.
A plane parallel layer of thickness d is irradiated in the positive x-direction with a beam of intensity I,,. The radiation flow inside the layer in the positive x-direction is represented by Z, and the radiation flow in the negative x-direction (caused by scattering) is represented by J,. The subscripts indicate that the radiation flows are functions of x, which is the distance from the front face of the layer (Fig. 1). The absorption coefficient is represented by k and the scattering coefficient by s. Both of these quantities have dimensions of reciprocal length. It is found [12] that: Z,=A(l-B)e”+B(l+/I)e-“,
(I)
J,=A(1+/3)e”+B(l-fi)e-“,
(2)
where: (T=V%(G%),
(3)
B =~kl(k-t%),
(4)
(1-p) e-Od A=-(I+8)2eod_(I_B)2e-odzo’ (1 +j9) cad ‘=(I+~)2e~d_(I_~)2e-~dzo’ The transmittance
of the layer is: &=d 48 Td=~=(I+~)2e~d_(I_~)2e-~d
and the diffuse reflectance:
Jx=o
Rd=----= IO
(1 -@‘)(e”-
e-“)
(1+/3)2e”-(1-/3)2e-“’
O-9
Raman spectroscopy of powders
183.5
For s = 0 and k > 0, /I = 1, Td= emrd(which is Lambert’s law), and Rd= 0. For infinite Td=T,=O, and &=R,= layer thickness, i.e. d+ a, we obtain A =O, B=Z,l(l+/l), (l-/3)/(1 +/l). The well-known Kubelka-Munk function F(R) is then obtained:
The quantity R, is a particularly convenient parameter for the characterization of powdered solids, being a quantity directly accessible to experiment through diffise reflectance spectrometry [ 11,131. It is our aim to relate relative Raman intensities to this parameter. Therefore we now consider Raman generation. Both Z, and .Z, contribute to the Raman light produced at a given point in the medium. Let p be the coefficient of Raman generation. That is, within an infinitesimal lamina of thickness dx at distance x from the front face of the sample, we consider the Raman flux generated to be: di, = djX= p(ZX+.Z,) dx.
(10)
Here, di, denotes the forward-generated light and djXthe backward-generated light, both equal on account of the symmetry of the radiation pattern of a dipole. (Of course, Raman light is in fact generated in all directions; the abstraction that only two directions need be considered is analogous to the corresponding procedure for the primary light in K-M theory, and is permissible if the angular distribution of the Raman light within the sample can be assumed to be the same as that of the primary light.) It should be noted that Eqn (10) contains the implicit assumption that p is small, so that Z, and .Z, can be allotted the values given by Eqns (1) and (2), i.e. that the Raman conversion process does not diminish these values. This will always be true for Raman conversion; extension of the treatment to fluorescence similarly requires that the fluorescence quantum yield should not be too large, and in many cases this too will be a satisfactory approximation. The observed Raman light originating within the lamina dx is that which reaches the front face of the sample. Both djX and di, contribute to this. In calculating the contributions, we note that di,, propagating in the forward direction, is partly backscattered in the part of the medium between x and d and enters the front part of the medium where it travels together with the backward component dj,, towards the observer. Similarly, part of this light is scattered forwards in the front part and adds to the component di,. Therefore, crossing the dividing plane at x from the two sides, respectively, are the Raman fluxes: dqX = di, + Rx dly, ,
(11)
dll?, = djX+ Rd_ dqX.
02)
Here, dqX and dqX are the total Raman fluxes in the forward and backward directions, at x, originating from the lamina dx. R, is the diffuse reflectance of a layer of thickness x (the front part) and Rd_ is the reflectance of a layer of thickness d-x (the back part). Solving for d&, djX+ Rd_ di, dly, =
1 - R,_,R,
*
(13)
In its passage to the front face, this has to be multiplied by TXto obtain the contribution to the observed intensity. An analogous contribution is made by laminae at other distances. The total observed intensity yd from the thickness d is therefore:
Although this expression is not limited to a particular thickness of sample, the restriction is now imposed that d--r 03, i.e. that the sample can be considered “infinitely thick”. An “infinitely thick” sample in the context of K-M theory means a layer of sufficient thickness that the diffuse reflectance is not measurably changed if the thickness is further increased, and for fine powders this is achieved in practice for layers of thickness
D. N. WATERS
1836
cu l-2 mm. Under these conditions the measured value of the reflectance is R,. It is usual in Raman spectroscopy to use samples of at least such a thickness, and therefore this is the case having greatest practical importance for Raman work. With this restriction the integral (14) is simplified considerably. Writing di, and djX in terms of quantities defined earlier, it becomes:
I
- (Z*+JJ. 0
Y*=p
ll_+RR; T*d.x m
x
=
T,dX.
(1% (16)
In Eqn (15), R, replaces Rd+ which is in order for the limiting case d+ ~0. Equations corresponding to Eqns (14) and (15) are given in Ref. [7], but an error in this paper* leads to incorrect evaluation of the integral. Substituting for TXand R, from Eqns (7) and (8), with d replaced by X, we obtain after simplification: Y,,=pZ,,*(l+R,)’
me-2”‘&. I 0
(17)
The result is:
(18)
‘Y,=$(l+R,)’ pZo R,(l +Rco) PZ, =-. =s-G(R), s (l-R,)
(19)
or: pZo (1-R:) m=-. yk 2
PZO =k - H(R).
Equations (19) and (20) are alternative expressions of the result. They are obtained from Eqn (18) with the aid of Eqns (3) and (9). The functions G(R) and H(R), defined by Eqns (19) and (20), respectively, are especially useful for the discussion of Raman intensities. We note that H(R)IG(R) = F(R). The relations given in Eqns (19) and (20) may be extracted from the results of SCHRADER and BERGMANN [l]. These authors derive approximate formulae for the observed intensities of primary and Raman light in a number of limiting cases, e.g. for special or limiting values of k, s, 0 or d (our nomenclature). Thus, for the case (ad> 3), the following result is given for the Raman intensity observed from the front face of the sample: pal,
k+s+a-2sade-od (k+s+a)*
il+k.
*
(21)
As d+ 00, this becomes: Y,=
Paz0 k(k+s+a)’
(22)
Use of Eqns (3) and (9) permits the right-hand side of Eqn (22) to be expressed in terms of k and R, only. The result is Eqn (20); Eqn (19) then follows. The functions G(R) and H(R), plotted in Fig. 2, are now discussed. The function H(R). Equation (20), which contains the absorption coefficient k as a parameter, is appropriate to the discussion of the intensities arising from a series of * Equation (9) of Ref. [7], defining the diffuse reflectance, should be multiplied on the right-hand side by - 1.
Raman spectroscopy of powders
1837
Fig. 2. The functions X(R) = F(R) x 10, G(R), and H(R) X 100, plotted against R,.
samples having constant values of k, but differing in their values of R, (as a result of varying values of s). That is, for a series of samples of constant k, the relative Raman intensities (of a given component of the sample) should be proportional to the quantity H(R) for the sample. The equation can in principle be tested if such a set of samples can be found. It is known that for a crystal powder the scattering coefficient is approximately inversely proportional to the diameter of the particles [5,13]. A few literature reports describe the dependence of Raman intensity on particle size; e.g. it has been shown [5, 141 for a number of materials that the Raman intensity falls as particle size decreases, i.e. as s (and therefore R,) increases. This is in qualitative accord with Eqn (20). The predicted dependence of Raman intensity upon R, is not strong. The function G(R). Equation (19), containing s as a parameter, predicts the relative Raman intensities from a series of samples having constant values of the scattering coefficient, but differing in their R, values (as a result of varying values of k). There are many qualitative examples available of this dependence. In considering a test of Eqn (19), it should be remembered that there are a number of implicit assumptions contained in its derivation. Among these is the assumption that the diffuse reflectance value is the same at both the exciting line and the Raman line wavelengths, i.e. that the sample can be considered as “grey” throughout this region. Secondly, K-M theory postulates diffuse incident radiation at the sample surface, whereas in Raman spectroscopy collimated (laser) illumination is usually used. It has been shown, however, that for ordinary diffuse reflectance spectroscopy, R, values are reliably obtained when collimated incident light is used [ 15,161, and it may reasonably be inferred that the use of laser excitation does not seriously impair the validity of the theory in its extended form. Lastly it is possible that there may be significant “edge” effects at the illuminated sample area (which is typically small for laser irradiation) where the “radiation balance” conditions of K-M theory are not well approximated. On the assumption that the limitations just mentioned are not too serious, we now describe an experiment which was devised as an attempt to examine the dependence of Raman intensity upon diffuse reflectance, for a component in a series of samples of approximately constant s. Very small amounts of graphite powder were intimately mixed with titanium dioxide (anatase) by prolonged light grinding in an agate mortar. A portion of anatase was similarly ground in order to ensure as far as possible that the particle sixes of all samples were similar. The mixtures were a light grey in colour, and gave R, values in the range cu 0.4-0.95. The pure anatase sample gave an R, value of 0.975. Figure 3 shows as plotted points (triangles) the relative observed intensities of the 635 cm-’ band of TiO*
D. N.
1838
WATERS
2lo-....,....,....,..~~...,...~,....,..,.,....,.... 0.1 0.2 0.3 0
0.4
0.5
0.6
0.7
0.8
0.9
R,
Fig. 3. Plotted points: relative observed Raman intensities of 635cm-’ line of anatase from anatase-graphite mixtures (triangles) and from vanadialtitania catalysts (squares), recorded under constant conditions. Continuous curve: the function G(R) scaled to pass through the data point at R, =0.90.
for these samples. The continuous curve in the figure is the function G(R), scaled to pass through the experimental point at R, = 0.90. The agreement between theory and experiment is overall very satisfactory. Two other points are plotted in this figure. These are for vanadialtitania catalysts (“EUROCAT” samples [17] ELlOVl and ELlOV8), containing 1 and 8% by weight V,05, respectively, on anatase supports. Figure 4 shows the Raman spectra of these
ai
01
1200
lob0
860
a60
‘lb0
Fig. 4. FT-Raman spectra of (a) anatase, (b) EUROCAT ELlOVl, (c) EUROCAT ELlOVS, recorded under constant conditions.
Raman spectroscopy of powders
1839
coloured materials, and the considerable fall in the TiOz band intensities compared with the intensities of pure Ti02 (noted also by others [l&19]) is apparent. The plotted points (squares in Fig. 3) represent the 635 cm-’ anatase band intensities, divided by 0.99 and 0.92, respectively, to correct for the weight contribution of the minor component (V,O,) to the composition of the materials. These additional points also lie acceptably near the curve. In conclusion we note some consequences of the above discussion for analytical Raman spectrometry. We comment briefly on (a) the possible advantages (or otherwise) of diluting an absorbing powdered solid with a white dispersant in order hopefully to improve the signal intensity, and (b) the application of Raman (or fluorescence) spectrometry to the quantitative analysis of mixtures. (a) Dilution of absorbing samples. The question is sometimes asked, “can the dispersal of a highly coloured substance in a nonabsorbing (white) matrix be advantageous for the detection of the Raman spectrum. 3” Since the dependence of R, upon c (the concentration) on the one hand, and of Y, upon R, on the other, are both highly nonlinear, the answer may not be obvious. We require the quantity (dYYm/W,)I(-dcIc): if for some range of R, this is positive, then there will be for such values of R, an intensity advantage. Since Y, 0~pG(R), for constant Z,,, and p = c a F(R), the required quantity becomes [d(F(R)G(R))l(F(R)G(R))]/[-dF(R)/F(R)]. This reduces to: --.
1
G(R)
d(F(R)G(R))
dF(R)
1
dH(R)
=-G(R)‘dF(R) 2R2, =-(1+R,)2.
This is negative for all values of R, >O, and therefore there is never an intensity advantage to be gained from sample dilution. (b) Quantitative analysis of mixtures. In quantitative analysis by Raman spectroscopy [20] the assumption is usually made that the measured intensity given by a component of a mixture is proportional to its concentration. This is unexceptionable if the diffuse reflectance values, R,, of the mixtures are independent of concentration and are thus constant for all samples and standards in the batch. This independence should hold at both exciting line and Raman line wavelengths. Equation (19) shows that Y, ap for constant R,. If R, varies with concentration, as a result of absorption either by a component being measured or by some other component of the mixture, then the proportionality between Y, and p (and therefore c) fails. For materials of high or fairly high values of R, , even small changes in reflectance can be a significant source of error. However, if the analytical method employs an internal standard, whereby a Raman band of a second component of the mixture, present in constant amount, is used to provide an intensity reference, then the effects of varying diffuse reflectance are annulled. For fluorescence measurements, the use of an internal standard is normally not practicable. Acknowledgements-The catalyst samples were provided Engineering Research Council is thanked for its support.
by Professor
G. C. Bond. The Science and
REFERENCES [l] (21 [3] [4] [5] [6] [7] [8]
B. Schrader and G. Bergmann, Z. anal. Chem. 225,230 (1967). P. Kubelka and F. Munk, Z. fech. Phys. 12, 593 (1931). P. Kubelka, J. Opt. Sot. Am. 38, 448 (1948). F. A. Steele, Puper Trude J. 100, 37 (1935). B. Schrader, A. Hoffmann and S. Keller, Spectrochim. Actu 47A, 1135 (1991). V. Pollak and A. A. Boulton, J. Chromatog. 72,231 (1972). V. Pollak, Opticu Actu 21, 51 (1974). J. C. Touchstone and J. Sherma (Eds), Densitometry in Thin Layer Chromutography. Pructice and Applications. Wiley, New York (1979).
VI(A) 50:11-C
1840
D. N.
WATERS
[9] J. N. Miller, Pure Appl. Chem. 57, 515 (1985). [lo] J. Sherma and B. Fried (Eds), Handbook of Thin-Layer Chromatography. Dekker, New York (1990). [ll] R. W. Frei and J. D. MacNeil, Diffuse RejYectanceSpecfroscopy in Environmental Problem-Solving. CRC Press, Cleveland, OH (1973). [12] Ref. [ll], Appendix 1. [13] G. Korttim, ReflectanceSpectroscopy. Springer, Berlin (1%9). [14] P. Hendra, C. Jones and G. Warnes, Fourier Transform Raman Spectroscopy, p. 152. Ellis Horwood, New York (1991). [15] G. Kortiim and G. Schreyer, Z. Naturforsch. He, 1018 (1956). [16] K. IUier, CataL Rev. 1,207 (1968). [17] G. Busca and A. Zecchina, Catalysb Today 20,61 (1994). [18] F. Roozeboom, M. C. Mittelmiejer-Haxeleger, J. A. Moulijn, J. Medema, V. H. J. deBeer and P. J. Gellings, 1. Phys. Chem. 84,2783 (1980). [19] G. C. Bond, J. Perez Zurita, S. Flamerz, P. J. Gellings, H. Bosch, J. G. Van Ommen and B. J. Kip, Appl. Catal. 22,361 (1986). [20] P. Hendra, C. Jones and G. Warnes, Fourier Transform Raman Specfroscopy, Chapter 6. Ellis Honvood, New York (1991).
Specrrochimica Acta.
Pergamon 0584-8539(94)Eoo26-7
0584~8539/94 $7.00 + 0.00
Raman spectroscopy of powders: effects of light absorption and scattering DAVID N. WATERS Department of Chemistry, Brunel University, Uxbridge UB8 3PH, U.K. Abstract-New expressions are derived from Kubelka-Munk theory to describe the Raman intensities observed by back-scattering from powder samples. The equations relate the Raman intensity to the diffuse reflectance R, of the sample for two cases: (i) for a series of samples having constant values of the scattering coefficient, s, but which vary in their values of the absorption coefficient, k; and (ii) for a series of samples having constant values of k, but differing in their values of s. The predicted intensity dependences are compared with the results of experiment. Keywords: Raman, NIR, diffuse reflectance, powders, Kubelka-Munk.
RAMAN SPECTROSCOPYis increasingly being used for the investigation of powder samples, and for the quantitative analysis of mixtures in the form of powders and compacted powders. Such materials are in general characterized by both absorption and scattering, and it is well known that both of these properties influence the observed Raman intensities. It may be generalized that increasing absorption, for a set of samples having constant scattering power, leads to decreasing observed intensities; and increasing scattering, for a set of samples having constant absorption coefficient, leads also to a lowering of intensities. An account of the Raman intensities of powder samples, exhibiting both absorption and scattering, has been given by SCHRADER and BERGMANN [l] as an extension of the theory due to KUBELKA and MUNK (K-M theory), originally developed [2-4] to describe the transmittance and diffuse reflectance of powdered solids. The treatment [l] considers the four differential equations which describe absorption and scattering of the primary and the Raman-shifted radiation within the solid, and solves these to give intensity expressions both for transmission through the powder layer (0” scattering) and back-scattering from the front surface (180” scattering). The expressions, of course, involve d, the thickness of the layer, and a primary result of the treatment is to show the dependence of the Raman intensities upon this thickness [l, 51. There is a possible alternative approach to the application of K-M theory to the description of Raman intensities from powdered solids. This is based on work by POLLAK [6,7] on the calculation of fluorescence intensities from diffusing media. This work is directed particularly to applications in thin-layer chromatography, for which fluorimetry offers an attractive method of quantitation. Results of this work [7] have been frequently cited in writings on chromatography, e.g. Refs [&lo]. There is a formal analogy between Raman scattering and fluorescence as far as this intensity problem is concerned. The two treatments [l, 71, however, differ considerably. It has not been shown that they lead to the same results, and indeed, as published, they do not. There is a need to resolve this discrepancy, and this provides part of the rationale for this work. Additionally, it can be shown that the results of either treatment (once they have been brought into convergence) can be recast into forms which are particularly amenable to testing by experiment and to comparisons with published data, and this too is part of our purpose.
EXPERIMENTAL
Diffuse reflectance spectra were measured using a Perkin-Elmer model Lambda 9 spectrophotometer equipped with an integrating sphere accessory. Kodak “white reflectance standard” was used as the reference material. Reported values of R, are for the wavelength region 1.0-1.2 pm, 1833
D. N. WATERS
1834
x=0
x
x=d
Fig. 1. Geometry of the Kubelka and Munk model. which is the region containing the measured Raman spectrum. Raman spectra were excited at 1.064 pm and were observed in the back-scattering geometry with a Perkin-Elmer model 1760X
IT-Raman spectrometer. Care was taken to ensure constant laser power and reproducible sample presentation.
DISCUSSION
In our view, the approach taken in Ref. [7] offers a good physical insight into the intensity problem. Unfortunately this treatment [7] contains an error which leads to wrong expressions for the fluorescence (and Raman) intensities. For these two reasons we shall outline a correct presentation of the intensity analysis according to this approach, leading ultimately to the new relations. We shall then indicate how a result from Ref. [l] can be re-expressed in this form. Some results of the original K-M theory, i.e. as applied to diffuse reflectance, are required first. We use the nomenclature of FREI
WI.
A plane parallel layer of thickness d is irradiated in the positive x-direction with a beam of intensity I,,. The radiation flow inside the layer in the positive x-direction is represented by Z, and the radiation flow in the negative x-direction (caused by scattering) is represented by J,. The subscripts indicate that the radiation flows are functions of x, which is the distance from the front face of the layer (Fig. 1). The absorption coefficient is represented by k and the scattering coefficient by s. Both of these quantities have dimensions of reciprocal length. It is found [12] that: Z,=A(l-B)e”+B(l+/I)e-“,
(I)
J,=A(1+/3)e”+B(l-fi)e-“,
(2)
where: (T=V%(G%),
(3)
B =~kl(k-t%),
(4)
(1-p) e-Od A=-(I+8)2eod_(I_B)2e-odzo’ (1 +j9) cad ‘=(I+~)2e~d_(I_~)2e-~dzo’ The transmittance
of the layer is: &=d 48 Td=~=(I+~)2e~d_(I_~)2e-~d
and the diffuse reflectance:
Jx=o
Rd=----= IO
(1 -@‘)(e”-
e-“)
(1+/3)2e”-(1-/3)2e-“’
O-9
Raman spectroscopy of powders
183.5
For s = 0 and k > 0, /I = 1, Td= emrd(which is Lambert’s law), and Rd= 0. For infinite Td=T,=O, and &=R,= layer thickness, i.e. d+ a, we obtain A =O, B=Z,l(l+/l), (l-/3)/(1 +/l). The well-known Kubelka-Munk function F(R) is then obtained:
The quantity R, is a particularly convenient parameter for the characterization of powdered solids, being a quantity directly accessible to experiment through diffise reflectance spectrometry [ 11,131. It is our aim to relate relative Raman intensities to this parameter. Therefore we now consider Raman generation. Both Z, and .Z, contribute to the Raman light produced at a given point in the medium. Let p be the coefficient of Raman generation. That is, within an infinitesimal lamina of thickness dx at distance x from the front face of the sample, we consider the Raman flux generated to be: di, = djX= p(ZX+.Z,) dx.
(10)
Here, di, denotes the forward-generated light and djXthe backward-generated light, both equal on account of the symmetry of the radiation pattern of a dipole. (Of course, Raman light is in fact generated in all directions; the abstraction that only two directions need be considered is analogous to the corresponding procedure for the primary light in K-M theory, and is permissible if the angular distribution of the Raman light within the sample can be assumed to be the same as that of the primary light.) It should be noted that Eqn (10) contains the implicit assumption that p is small, so that Z, and .Z, can be allotted the values given by Eqns (1) and (2), i.e. that the Raman conversion process does not diminish these values. This will always be true for Raman conversion; extension of the treatment to fluorescence similarly requires that the fluorescence quantum yield should not be too large, and in many cases this too will be a satisfactory approximation. The observed Raman light originating within the lamina dx is that which reaches the front face of the sample. Both djX and di, contribute to this. In calculating the contributions, we note that di,, propagating in the forward direction, is partly backscattered in the part of the medium between x and d and enters the front part of the medium where it travels together with the backward component dj,, towards the observer. Similarly, part of this light is scattered forwards in the front part and adds to the component di,. Therefore, crossing the dividing plane at x from the two sides, respectively, are the Raman fluxes: dqX = di, + Rx dly, ,
(11)
dll?, = djX+ Rd_ dqX.
02)
Here, dqX and dqX are the total Raman fluxes in the forward and backward directions, at x, originating from the lamina dx. R, is the diffuse reflectance of a layer of thickness x (the front part) and Rd_ is the reflectance of a layer of thickness d-x (the back part). Solving for d&, djX+ Rd_ di, dly, =
1 - R,_,R,
*
(13)
In its passage to the front face, this has to be multiplied by TXto obtain the contribution to the observed intensity. An analogous contribution is made by laminae at other distances. The total observed intensity yd from the thickness d is therefore:
Although this expression is not limited to a particular thickness of sample, the restriction is now imposed that d--r 03, i.e. that the sample can be considered “infinitely thick”. An “infinitely thick” sample in the context of K-M theory means a layer of sufficient thickness that the diffuse reflectance is not measurably changed if the thickness is further increased, and for fine powders this is achieved in practice for layers of thickness
D. N. WATERS
1836
cu l-2 mm. Under these conditions the measured value of the reflectance is R,. It is usual in Raman spectroscopy to use samples of at least such a thickness, and therefore this is the case having greatest practical importance for Raman work. With this restriction the integral (14) is simplified considerably. Writing di, and djX in terms of quantities defined earlier, it becomes:
I
- (Z*+JJ. 0
Y*=p
ll_+RR; T*d.x m
x
=
T,dX.
(1% (16)
In Eqn (15), R, replaces Rd+ which is in order for the limiting case d+ ~0. Equations corresponding to Eqns (14) and (15) are given in Ref. [7], but an error in this paper* leads to incorrect evaluation of the integral. Substituting for TXand R, from Eqns (7) and (8), with d replaced by X, we obtain after simplification: Y,,=pZ,,*(l+R,)’
me-2”‘&. I 0
(17)
The result is:
(18)
‘Y,=$(l+R,)’ pZo R,(l +Rco) PZ, =-. =s-G(R), s (l-R,)
(19)
or: pZo (1-R:) m=-. yk 2
PZO =k - H(R).
Equations (19) and (20) are alternative expressions of the result. They are obtained from Eqn (18) with the aid of Eqns (3) and (9). The functions G(R) and H(R), defined by Eqns (19) and (20), respectively, are especially useful for the discussion of Raman intensities. We note that H(R)IG(R) = F(R). The relations given in Eqns (19) and (20) may be extracted from the results of SCHRADER and BERGMANN [l]. These authors derive approximate formulae for the observed intensities of primary and Raman light in a number of limiting cases, e.g. for special or limiting values of k, s, 0 or d (our nomenclature). Thus, for the case (ad> 3), the following result is given for the Raman intensity observed from the front face of the sample: pal,
k+s+a-2sade-od (k+s+a)*
il+k.
*
(21)
As d+ 00, this becomes: Y,=
Paz0 k(k+s+a)’
(22)
Use of Eqns (3) and (9) permits the right-hand side of Eqn (22) to be expressed in terms of k and R, only. The result is Eqn (20); Eqn (19) then follows. The functions G(R) and H(R), plotted in Fig. 2, are now discussed. The function H(R). Equation (20), which contains the absorption coefficient k as a parameter, is appropriate to the discussion of the intensities arising from a series of * Equation (9) of Ref. [7], defining the diffuse reflectance, should be multiplied on the right-hand side by - 1.
Raman spectroscopy of powders
1837
Fig. 2. The functions X(R) = F(R) x 10, G(R), and H(R) X 100, plotted against R,.
samples having constant values of k, but differing in their values of R, (as a result of varying values of s). That is, for a series of samples of constant k, the relative Raman intensities (of a given component of the sample) should be proportional to the quantity H(R) for the sample. The equation can in principle be tested if such a set of samples can be found. It is known that for a crystal powder the scattering coefficient is approximately inversely proportional to the diameter of the particles [5,13]. A few literature reports describe the dependence of Raman intensity on particle size; e.g. it has been shown [5, 141 for a number of materials that the Raman intensity falls as particle size decreases, i.e. as s (and therefore R,) increases. This is in qualitative accord with Eqn (20). The predicted dependence of Raman intensity upon R, is not strong. The function G(R). Equation (19), containing s as a parameter, predicts the relative Raman intensities from a series of samples having constant values of the scattering coefficient, but differing in their R, values (as a result of varying values of k). There are many qualitative examples available of this dependence. In considering a test of Eqn (19), it should be remembered that there are a number of implicit assumptions contained in its derivation. Among these is the assumption that the diffuse reflectance value is the same at both the exciting line and the Raman line wavelengths, i.e. that the sample can be considered as “grey” throughout this region. Secondly, K-M theory postulates diffuse incident radiation at the sample surface, whereas in Raman spectroscopy collimated (laser) illumination is usually used. It has been shown, however, that for ordinary diffuse reflectance spectroscopy, R, values are reliably obtained when collimated incident light is used [ 15,161, and it may reasonably be inferred that the use of laser excitation does not seriously impair the validity of the theory in its extended form. Lastly it is possible that there may be significant “edge” effects at the illuminated sample area (which is typically small for laser irradiation) where the “radiation balance” conditions of K-M theory are not well approximated. On the assumption that the limitations just mentioned are not too serious, we now describe an experiment which was devised as an attempt to examine the dependence of Raman intensity upon diffuse reflectance, for a component in a series of samples of approximately constant s. Very small amounts of graphite powder were intimately mixed with titanium dioxide (anatase) by prolonged light grinding in an agate mortar. A portion of anatase was similarly ground in order to ensure as far as possible that the particle sixes of all samples were similar. The mixtures were a light grey in colour, and gave R, values in the range cu 0.4-0.95. The pure anatase sample gave an R, value of 0.975. Figure 3 shows as plotted points (triangles) the relative observed intensities of the 635 cm-’ band of TiO*
D. N.
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WATERS
2lo-....,....,....,..~~...,...~,....,..,.,....,.... 0.1 0.2 0.3 0
0.4
0.5
0.6
0.7
0.8
0.9
R,
Fig. 3. Plotted points: relative observed Raman intensities of 635cm-’ line of anatase from anatase-graphite mixtures (triangles) and from vanadialtitania catalysts (squares), recorded under constant conditions. Continuous curve: the function G(R) scaled to pass through the data point at R, =0.90.
for these samples. The continuous curve in the figure is the function G(R), scaled to pass through the experimental point at R, = 0.90. The agreement between theory and experiment is overall very satisfactory. Two other points are plotted in this figure. These are for vanadialtitania catalysts (“EUROCAT” samples [17] ELlOVl and ELlOV8), containing 1 and 8% by weight V,05, respectively, on anatase supports. Figure 4 shows the Raman spectra of these
ai
01
1200
lob0
860
a60
‘lb0
Fig. 4. FT-Raman spectra of (a) anatase, (b) EUROCAT ELlOVl, (c) EUROCAT ELlOVS, recorded under constant conditions.
Raman spectroscopy of powders
1839
coloured materials, and the considerable fall in the TiOz band intensities compared with the intensities of pure Ti02 (noted also by others [l&19]) is apparent. The plotted points (squares in Fig. 3) represent the 635 cm-’ anatase band intensities, divided by 0.99 and 0.92, respectively, to correct for the weight contribution of the minor component (V,O,) to the composition of the materials. These additional points also lie acceptably near the curve. In conclusion we note some consequences of the above discussion for analytical Raman spectrometry. We comment briefly on (a) the possible advantages (or otherwise) of diluting an absorbing powdered solid with a white dispersant in order hopefully to improve the signal intensity, and (b) the application of Raman (or fluorescence) spectrometry to the quantitative analysis of mixtures. (a) Dilution of absorbing samples. The question is sometimes asked, “can the dispersal of a highly coloured substance in a nonabsorbing (white) matrix be advantageous for the detection of the Raman spectrum. 3” Since the dependence of R, upon c (the concentration) on the one hand, and of Y, upon R, on the other, are both highly nonlinear, the answer may not be obvious. We require the quantity (dYYm/W,)I(-dcIc): if for some range of R, this is positive, then there will be for such values of R, an intensity advantage. Since Y, 0~pG(R), for constant Z,,, and p = c a F(R), the required quantity becomes [d(F(R)G(R))l(F(R)G(R))]/[-dF(R)/F(R)]. This reduces to: --.
1
G(R)
d(F(R)G(R))
dF(R)
1
dH(R)
=-G(R)‘dF(R) 2R2, =-(1+R,)2.
This is negative for all values of R, >O, and therefore there is never an intensity advantage to be gained from sample dilution. (b) Quantitative analysis of mixtures. In quantitative analysis by Raman spectroscopy [20] the assumption is usually made that the measured intensity given by a component of a mixture is proportional to its concentration. This is unexceptionable if the diffuse reflectance values, R,, of the mixtures are independent of concentration and are thus constant for all samples and standards in the batch. This independence should hold at both exciting line and Raman line wavelengths. Equation (19) shows that Y, ap for constant R,. If R, varies with concentration, as a result of absorption either by a component being measured or by some other component of the mixture, then the proportionality between Y, and p (and therefore c) fails. For materials of high or fairly high values of R, , even small changes in reflectance can be a significant source of error. However, if the analytical method employs an internal standard, whereby a Raman band of a second component of the mixture, present in constant amount, is used to provide an intensity reference, then the effects of varying diffuse reflectance are annulled. For fluorescence measurements, the use of an internal standard is normally not practicable. Acknowledgements-The catalyst samples were provided Engineering Research Council is thanked for its support.
by Professor
G. C. Bond. The Science and
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B. Schrader and G. Bergmann, Z. anal. Chem. 225,230 (1967). P. Kubelka and F. Munk, Z. fech. Phys. 12, 593 (1931). P. Kubelka, J. Opt. Sot. Am. 38, 448 (1948). F. A. Steele, Puper Trude J. 100, 37 (1935). B. Schrader, A. Hoffmann and S. Keller, Spectrochim. Actu 47A, 1135 (1991). V. Pollak and A. A. Boulton, J. Chromatog. 72,231 (1972). V. Pollak, Opticu Actu 21, 51 (1974). J. C. Touchstone and J. Sherma (Eds), Densitometry in Thin Layer Chromutography. Pructice and Applications. Wiley, New York (1979).
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[9] J. N. Miller, Pure Appl. Chem. 57, 515 (1985). [lo] J. Sherma and B. Fried (Eds), Handbook of Thin-Layer Chromatography. Dekker, New York (1990). [ll] R. W. Frei and J. D. MacNeil, Diffuse RejYectanceSpecfroscopy in Environmental Problem-Solving. CRC Press, Cleveland, OH (1973). [12] Ref. [ll], Appendix 1. [13] G. Korttim, ReflectanceSpectroscopy. Springer, Berlin (1%9). [14] P. Hendra, C. Jones and G. Warnes, Fourier Transform Raman Spectroscopy, p. 152. Ellis Horwood, New York (1991). [15] G. Kortiim and G. Schreyer, Z. Naturforsch. He, 1018 (1956). [16] K. IUier, CataL Rev. 1,207 (1968). [17] G. Busca and A. Zecchina, Catalysb Today 20,61 (1994). [18] F. Roozeboom, M. C. Mittelmiejer-Haxeleger, J. A. Moulijn, J. Medema, V. H. J. deBeer and P. J. Gellings, 1. Phys. Chem. 84,2783 (1980). [19] G. C. Bond, J. Perez Zurita, S. Flamerz, P. J. Gellings, H. Bosch, J. G. Van Ommen and B. J. Kip, Appl. Catal. 22,361 (1986). [20] P. Hendra, C. Jones and G. Warnes, Fourier Transform Raman Specfroscopy, Chapter 6. Ellis Honvood, New York (1991).