Quantitative analysis in diffuse reflectance spectrometry: A modified Kubelka-Munk equation

ONAL SPECTROSCOPY ELSEVIER

Vibrational Spectroscopy9 (1995) 19-27

Quantitative analysis in diffuse reflectance spectrometry: A modified Kubelka-Munk equation Alfred A. Christy a,*, Olav M. Kvalheim a, Rance A. Velapoldi b a Department of Chemistry, University of Bergen, N-5007, Bergen, Norway b National Institute of Standards and Technology, Gaithersburg, MD 20899, USA

Received 7 July 1994

Abstract The behaviour of the Kubelka-Munk equation with particle size is analysed by using mono-disperse polystyrene particles. Based on the experimental results a model is proposed to explain the diffuse reflectance spectra of powdered samples in a quantitative manner. The new model explains quantitatively the behaviour of the diffuse reflectance spectra of mono-disperse polystyrene particles in KBr. Furthermore, the results show that the model can be used to explain the diffuse reflectance spectra of mixtures of mono-disperse polystyrene spheres in KBr and of a coal sample mixture containing a distribution of particle sizes and KBr. It appears that the proposed model can be used for dilute samples in a quantitative manner. Keywords: Diffuse reflectance spectrometry;Kubelka-Munkequation; Particle size; Polystyrene

1. Introduction Diffuse reflectance spectrometry has established its effectiveness in qualitative and semi-quantitative analysis. It provides an excellent alternative to transmission spectrometry in measuring the infrared spectrum of samples which are dark in colour and highly absorbing or samples which are in one way or another not suitable for the preparation of a transparent KBr disc for analysis. Quantitative analysis using the diffuse reflectance technique requires a certain amount of expertise on the part of the user. Diffuse reflectance spectrometry has several factors adherent to the technique that make quantitative analysis difficult. These should be taken care of in order to use the

* Corresponding author.

technique well for quantitative analysis. Griffiths and coworkers [1,2], and Christy and coworkers [3-5] have analysed these problems in several of their articles on diffuse reflectance spectrometry and shown some possible measures one could take in order to make the technique suitable for quantitative analysis. It is now realized that one of the important factors affecting the diffuse reflectance spectrum of a sample is the particle size. Fuller and Griffiths [1] have shown the effect of the particle size on diffuse reflectance spectra of powdered samples. Christy et al. [5] have investigated diffuse reflectance spectra of mono-disperse polystyrene spheres and qualitatively explained the effect of particle size on diffuse reflectance spectra. Furthermore, an attempt has been made to introduce particle size in the Kubelka-Munk equation to explain the effect of particle size in a qualitative manner.

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20

A.4. Christy et al. / Vibrational Spectroscopy 9 (1995) 19-27

The aim of this paper is to investigate the effects of particle size on diffuse reflectance spectra and make a modification in the Kubelka-Munk equation, to include particle size so that the equation could be used in a quantitative manner. We will also look at the Kubelka-Munk model and Simmon's modified Kubelka-Munk model from a theoretical point of view and examine whether they explain the behaviour of diffusely reflecting samples containing particles of definite sizes.

sent radiation in the downward and the upward direction, respectively. The downward flux through a layer of thickness d x is decreased by absorption and scattering processes and increased by scattering process of the light travelling upward towards the surface. Then one can write the following differential equations for both fluxes I and J: d//dx = -(k'+s)I+sJ

(1)

and (2)

d J / d x = ( k' + s ) J - sI

2. Kubelka-Munk equation and modifications Diffuse reflectance spectrometry concerns one of the two components of reflected radiation from an irradiated sample, namely specular reflected radiation and diffusely reflected radiation. The diffusely reflected radiation collected and detected by the detector is converted into a spectrum that is similar to absorbance spectrum in transmittance spectrometry. Several models have been proposed to describe the diffuse reflectance phenomena [6,7]. Most of these use basic optical properties of the sample. However, these are not widely used in the analyses of infrared spectra of diffusely reflecting samples. They lack linearity over a wide range of sample concentrations. The model put forward by Kubelka and Munk in 1931 [8] to describe diffusely reflected radiation of paint layers is widely used and accepted in diffuse reflectance infrared spectrometry. The Kubelka-Munk theory is based on a continuum model (see Fig. 1) where the reflectance properties are described by differential equations for infinitesimally small layers. The letters I and J repre-

J(o)/~ ~I(o) x=0 illurmnated surface ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::

i i

......................... !:!?ili(;::::,dx

where k is a sort of absorption coefficient (this is to differentiate the absorption coefficient of the material in transmission spectrometry) and s is the scattering coefficient. The above equations with simple substitution a = (s + k ) / s reduce to the form dI/sdx=

(3)

-aI + J

and

(4)

d J / s d x = aJ - I

Reflectance from the sample is analogous to the transmittance in the Beer-Lambert equation and is defined by (leaving intensity)/(entering intensity) (5)

R(x) =J(x)/I(x)

Eqs. 4 and 5 can be written as a single differential equation d R / ( R 2 - 2aR + 1) = s d x

(6)

The above equation can be integrated using the limits x = 0 to x = x. This yields sx= [{1/(p-q)}ln{(R-p)/(R-q)}]

o

(7)

where p and q are roots of the equation R 2 - 2aR + 1 [p = a + ~/(a2 - 1) and q = a - v/(a2 - 1)]. When the depth of the sample is infinite (x ~ oo), the reflectance at infinite depth Ro~ approaches to q = a - ~/(a2 - 1). This can be solved for a and the Kubelka-Munk function f(R~) becomes f(Roo) -- (1 -R®)2/2Roo = a - 1 = k / s

(8)

and Roo = [ k + s - ~ / ( k + 2ks)]

urnllurmnated surface Fig. 1. Model for the Kubelka-Munk analysis.

(9)

Since R= lies between 0 and 1, the negative root is chosen as the solution for R®.

A.A. Christy et al. / Vibrational Spectroscopy 9 (1995) 19-27

The Kubelka-Munk model was proposed to explain diffuse reflection of paint layers. However, the same model was extended to describe the diffuse reflection in infrared spectrometry where the samples contain particles of definite sizes. They do not fulfil the assumption of the Kubelka-Munk model. However, the theory has been found to be more useful than many other theories that involve the particular nature of the materials in their models. In diffuse reflectance infrared spectrometry the samples are generally prepared as a mixture in a non-absorbing and effectively scattering medium such as KBr. Therefore, the absolute reflectance at infinite depth in the Kubelka-Munk function is replaced by reflectance of the sample relative to the non-absorbing medium, and R= therefore represents the relative reflectance at infinite depth. Samples that have constant concentration of the absorbing material with different particle size distributions give different diffuse reflectance spectra in the Kubelka-Munk format (when the other factors remain constant). Christy and coworkers [5,9,10] have analyzed the effect of particle size on diffuse reflectance infrared spectrometry and shown that the Kubelka- Munk function has an inverse relationship with the particle size and proposed a qualitative model to explain the behaviour of the spectra with particle size. This equation can be simply written in the following form. (Note that the k'c in Ref. [5] is replaced here by k* in this article.) k* f(R~) =

(ss+sr)d

(10)

where k * is the particle-size-independent absorption coefficient (cm-1), d the particle diameter, s s the scattering of the sample (cm-1), sr the scattering of the non-absorbing medium (cm-1), and s = ss + s r. It appears that the absorption coefficient k in the Kubelka-Munk equation has to be redefined as the particle-size-dependent absorption coefficient. According to the above equation, if the particle size is given in cm, then f(R=) will have the unit cm -1. In order to preserve the units of the absorption and scattering coefficients, we have redefined d as the particle size factor, which is equivalent to the particle size in /xm without units. Furthermore, the relationship between the particle-size-dependent absorption coef-

21

ficient (k) and particle-size-independent absorption coefficient (k * ) can then be extracted from Eq. 10 as follows (11)

k = k*/d

At low concentrations the scattering of the absorbing medium can be neglected and the scattering of the non-absorbing medium containing a similar particle size distribution can be assumed constant. Then k * will be proportional to the concentration of the sample and the above equation can be written as (1 - R ~ ) 2 f(R~) =

2R~

= Idc/sd + F

(12)

where c is the sample concentration (wt.%) and k' the absorptivity of the sample (cm -1 wt.%-l). The linear correlation between f(R®) and inverse of particle size factor may not pass through the origin and therefore an additional term F has been introduced in Eq. 12. Simmons [11] derived a modified equation for f(R~). The assumption made in the Kubelka-Munk theory that sample layers can be considered as infinitesimally small layers is not valid for powdered samples where the samples have definite particle sizes. If we assume that the layers are of finite thickness d (average particle size), then the differential Eqs. 1 and 2 can be modified to include the average particle size d as follows: (/i+1 - Ii) / / d = - ( k ..at-s ) I i + sJi+ 1

(13)

(4+1 --4)/d

(14)

= ( k + s)J/+ 1

-

-

SIi

Then for an infinitely thick powdered sample R= can be written as R= = J , / I i = Ji+ l/Ii+ a

(15)

Eqs. 14 and 15 can be solved to give R 2 - 211 + k / s - kd - k 2 d / Z s ] R = + 1 = 0

(16)

and R=

= {(s + k -

kZd/2)

(17)

A.A. Christy et al. / Vibrational Spectroscopy 9 (1995) 19-27

22

A comparison of the Kubelka-Munk equation R 2 211 +f(R~)]R® + 1 = 0 with Eq. 17 gives f ( R=) = k / s - kd - k 2 d / 2 s

(18)

This equation explains partly the deviations of a plot of f(R=) vs. k obtained from transmission measurements at high k values [11]. Even the modified model of Simmon does not explain the behaviour of the Kubelka-Munk function with particle size variation. This is because the particle size dependency of the absorption coefficient has not been taken into consideration. However, Kortiim [12] and many other authors [1,13] have indicated the inverse relationship of the absorption coefficient with particle size. The absorption coefficients decrease as the particle diameter increases. By substituting k as shown previously, the above equation can be written as follows f ( R~) = k* / s d - k* - ( k* ) 2 / 2 s d

(19)

This again can be written as f ( R~) = e c / s d - k' c - ( k' c ) 2 / 2 s d

(20)

For mixtures diluted in KBr containing the same particle size distribution, s becomes a constant. For dilute samples, contribution from the last two terms may be small and the above equation can be written as

f(R®) = k'c/sd - O

(21)

For a sample containing particles of n definite particle sizes in a KBr mixture, one could assume additivity [14-16] and c / d for different particle sizes can be summed up, i.e. n

f ( R ~ ) = ( k ' / s ) ~_, ( c i / d i ) - E

(22)

1

and for a natural sample the summation has to be replaced by integration f ( R~ ) = ( k ' / s ) f ( c i / d i ) - F

- F

Particle size (~tm)

Mixture A

4.5 10 20 45

1.9

Mixture B 2.85 1.05

2.05

3. Experimental 3.1. Samples Dry polystyrene beads of sizes 4.5, 10, 20 and 45 /~m were purchased from Polysciences Inc., Warrington, USA. KBr used in these experiments was spectroscopic grade and purchased from Merck. It was dried and sieved to contain only particles below 45/xm. 1 and 4% ( w / w ) mixtures of the polystyrene beads in KBr were prepared carefully by weighing potassium bromide and required polystyrene beads directly in a Wig-L-Bug capsule. The mixtures were mixed thoroughly using a Wig-L-Bug vibrator without steel balls. Two more, approximately 4% ( w / w ) samples were prepared to contain two of the polystyrene beads in KBr. The particle sizes and the weights used in the mixtures are shown in Table 1. Furthermore, a matured coal sample was ground and particles in the range < 25/xm, 2 5 - 3 2 / z m and 3245/zm were sieved using sieves with pore sizes 25, 32 and 45/zm. 10% ( w / w ) sample mixtures in KBr were prepared by directly weighing KBr and sieved coal particles in a Wig-L-Bug steel capsule. The mixtures were then thoroughly mixed using the vibrator without steel balls in the capsule.

4. Spectra

( 23 )

When the concentrations c i are known, the above equation can be solved for f(R~). However, the simplest way to achieve an approximate solution is to take the average of the particle size distribution and the equation becomes f ( R~) = ( k ' / s ) c / d *

Table 1 Compositions (wt.%) of mixtures A and B made from two different particle sizes

(24)

where d* is the particle size factor equal to the average particle size of a natural sample.

Diffuse reflectance spectra of the mixtures were recorded using a Perkin Elmer 1720X FF-IR spectrometer equipped with a diffuse reflectance accessory (Spectra-Tech Inc., Stamford, CT) and a DTGS detector. The diffuse reflectance infrared spectra of the coal samples were recorded using a Nicolet 800 FT-IR instrument. The samples were filled in a 4-mm sample cup without using any pressure. Their surfaces were drawn using a sharp edged spatula. A to-

A.A. Christy et al. / Vibrational Spectroscopy 9 (1995) 19-27

23

0.37

0.;10

I'~I4.5 pm

/

0.30

O. 16

0.23 FI

0 • 12

~

0.08.

i

I li

0.16

~

0.09

10 pm

-

N o.o4

•

Wavenumber

7

.

.

¢m-t

Fig. 2. Diffuse reflectance spectrum of 4% (w/w) sample of 10/zm polystyrene particles in KBr.

tal of 20 scans was measured in the range from 4000 to 600 cm-1 at a resolution of 4 cm-1 using previously scanned KBr as the background. Diffuse reflectance spectra of the samples were measured for three different packings and their average was taken as the representative spectrum of a sample. The spectra in reflectance format were then transferred to a Microvax computer for Kubelka-Munk transformation and further processing.

.

W a v e . u m b e r cm-a

Fig. 3. Diffuse reflectance spectra of 4% (w/w) samples of polystyrene spheres in KBr in the region 3300-2700 cm -1.

intensities with particle size is clear. Figs. 5 and 6 show that the correlations are excellent (r is better than 0.98 in all cases) for all peaks. It is also obvious that all the linear correlations meet the y axis below zero when extrapolated. These linear correlations can be described by Eqs. 12 and 21. Correlation between particle size and f(Ro~) can also be studied for a wide spectral range with the help of chemometrics. For this purpose, we used partial

!'i/.5,-

5. Results and discussion The diffuse reflectance spectrum of 4% ( w / w ) polystyrene spheres (10 ~m) in KBr is shown in Fig. 2. We have avoided 2-/zm polystyrene particles because we found in our previous work [5] that the intensity maxima of the infrared absorption bands for the polystyrene particles mixture in KBr may lie inbetween a particle size of 2 and 4 ]zm. The particle sizes above 45 g m were also not used. This is to avoid analysing spectral bands that are severely distorted. The infrared absorption bands of the polystyrene particles/KBr spectra in the region 3300-2700 and 1700-1300 cm -1, in Kubelka-Munk format for 4% ( w / w ) samples are shown in Figs. 3 and 4, respectively. The tendency of an inverse relationship of the

0.24

o

45 p m ,,,t

/f ~'~ ,, .

!

•

!'~ll,

~' ~ lOFm

III

B ,,y

, I, ~

' ~._

,'l,",ll

~,'., ' \ ~ J u p m l

'~,i~' !i~

fq

,~ "~'.~/ ~.

~1

Wavenumber cm-~

Fig. 4. Diffuse reflectance spectra of 4% (w/w) samples of polystyrene spheres in KBr in the region 3300-2700 cm-1.

24

A.4. Christy et aL / Vibrational Spectroscopy 9 (1995) 19-27

0.10'

0.10 "

0,08

0,08 "

0,06

0,06 "

0.04

0,04

0,02

0,02

~.

,

0,00 0,0

•

,

.

0.10 •

c:)

0,08"

A 0,06 S

•

0,00

0,3

0,2

0,1

/

0,0

O,lO

0,08 :

,

.

0,2

0,3

0,I lld

0,2

013

d)

0,06 : 0,04:

0,02

0,02 -

.

"

0,04 /

0,00 0,0

,

0,1

0,00

,

0,2

0,I

0:3

0,0

lid

Fig. 5. Linearity plots showing the peak intensities in f(R~) format vs. the inverse particle size factor for peaks at (a) 3026, (b) 2923, (c) 1497 and (d) 1454 era-1, respectively. The sample is 1% (w/w) polystyrene particles in KBr.

least squares (PLS) calibration [17] of the spectral profiles of the polystyrene spectra inj Kubelka-Munk format against their respective particle sizes. Two PLS components described 100 and 98% of the variances in f(R~) and particle size, respectively. Parts

of the target projected component [17] that is correlated with particle size in the spectral regions 33002700 and 1700-1300 cm -1 are shown in Figs. 7 and 8, respectively. These two plots clearly show that the spectral profiles have a complete negative correlation

0,4 "

0,4"

0,3

0,3'

.e 0,2

0.2 •

0,1

0,1"

0,0

'

0.0

0,4

0,1

0,2

0,3

0,00.0

0,2

0,3

0,1

0,2

0,3

0.4"

-

0.3"

0,3-

0,2

0,2

0,1

0.1

0.0 0,0

0,1

0,!

0.2 I/d

0,3

0,0 0.0

I Id

Fig. 6. Linearity plots showing the peak intensities in f(R~.) vs. the inverse particle size factor for peaks at (a) 3026, (b) 2923, (c) 1497 and (d) 1454 cm -1, respectively. The sample is 4% (w/w) polystyrene particles in KBr.

A.A. Christy et al. / Vibrational Spectroscopy 9 (1995) 19-27

25

0.00

-0.03

•- 0 . 0 6

--0.09

at "-C.12

--0 ,.15

-0.J8

W a v e n u m b e r cm-(

Fig. 7. Part of the target projected component correlating with particle size in the region 3300-2700 cm -

with particle size. They also show that all absorption bands in these regions have negative correlations with particle size which supports the models shown by Eqs. 12 and 21.

1.

How can a mixture containing two particle sizes such as shown in Table 1 behave? Can the additivity suggested by Eq. 22 explain the behaviour of the spectrum obtained? In order to answer these ques-

0.00

-0.04

-0.08

at -0.12

-0.16

•- 0 . 2 0

W a v e n u m b e r cm.4

Fig. 8. Part of the target projected component correlating with particle size in the region 1700-1300 cm-1.

A.A. Christy et al. / Vibrational Spectroscopy 9 (1995) 19-27

26

Table 2 Calculated and measured intensities of absorption in KubelkaMunk units from linear regression models of 4% (w/w) samples

r--

it,-

Sample

Peak position (cm-~)

Intensity calculated

Intensity measured

A

3020 2923 1497 1453

0.161 0.166 0.176 0.170

0.172 0.180 0.183 0.177

B

3020 2923 1497 1453

0.129 0.133 0.120 0.118

0.135 0.130 0.162 0.161

tions, we proceeded from the linear correlations between spectral intensities and inverse of particle sizes. These correlations should explain the diffuse reflectance spectrum of a sample containing any particle size in-between 4.5 and 45/zm. The intensities of the infrared absorption bands in Kubelka-Munk format were calculated from their individual contributions. For example, intensities of the bands in Kubelka-Munk format for mixture 1 were calculated from the contribution of the 4.5- and 45-/~m polystyrene particles/KBr mixtures. Calculations were made using regression models from 1 and 4% of polystyrene particles in KBr. The calculated and measured intensities are given in Tables 2 and 3. It appears that the additivity holds and that Eq. 22 can be applied to samples containing mixtures of definite particle sizes. There are deviations in some of the calculated values. These imprecisions may be partly due to the deviations in the measured values. In the next stage we tested the behaviour of the coal sample mixtures containing specific particle size

~

~9

1~90

1~1

ligz

lt~,3 169~ ~AVENUHBER

16~5

§96

~7

1~9e

Fig, 9. Diffuse reflectance spectrum of 10% (w/w) coal sample (particle size < 25 ~m).

0,4"

0,3"

ee

0,2'

0,1" 0,0

-

,

-

0,02

0 , 6

"

0,4"

0,3"

e~

0,2"

0,1 0,0

,

.

,

•

0,04

,

•

,

•

0,06

,

-

,

0,08

// •

i

-

0,02

v

•

J

•

0,04

,

.

,

.

0,06

,

.

i

-

,

0,08

0,6"

Table 3 Calculated and measured intensities of absorption in KubelkaMunk units from linear regression models of 1% (w/w) samples

0,5"

0 , 4 "

Sample A

B

Peak position (cm- 1)

Intensity calculated

Intensity measured

3020 2923 1497 1453

0.190 0.210 0.215 0.200

0.172 0.180 0.183 0.177

3020 2923 1497 1453

0.138 0.149 0.153 0.150

0.135 0.130 0.162 0.161

gg

0,3" 0,2" 0,1" 0~0 0,02

-

,

-

,

-

,

0,04

-

, 0,06

-

,

-

,

Or00

l/d Fig. 10. Linearity plots showing the peak intensities in f(R®) format vs. the inverse of average particle size factor for peaks at (a) 1239, (b) 1023 and (e) 1009 c m - 1 respectively. The sample is 10% (w/w) coal sample particles in KBr.

A.A. Christy et al. / Vibrational Spectroscopy 9 (1995) 19-27

distributions. The diffuse reflectance spectrum of the 10% ( w / w ) coal sample containing particle size distribution < 25 ~m, in the region 1339-900 cm -1, is shown in Fig. 9. The coal sample is a matured sample and there is not much structure in the region 4000-2000 cm- 1. As expected, the intensities of the bands decreased with increasing particle size. Plots of the intensities of three of the absorption bands against the inverse of the particle size factor (average particle size 12.5, 28.5 and 38.5 ~m) are shown in Fig. 10a, b and c, respectively. A good linear correlation is evident and supports the model defined by Eq. 24. This equation may be valid for the spectrum of a sample containing a narrow particle size distribution. However, the validity of the equation for the spectrum of a sample containing a wider particle size distribution is questionable. The spectral distortion and inverse effect will induce a high relative error in the calculation.

6. Conclusion

In this article we have shown the relationship between particle size and f(R~) in a quantitative manner. Our chemometric approach clearly shows that the model can be used for absorption bands over a wide spectral region. Behaviour of the Kubelka-Munk function with particle size variations of mono-disperse polystyrene spheres and similar observations made by other authors [1,12,13] with systems other than mono-disperse polystyrene spheres show that our model can be used to explain the behaviour of the diffuse reflectance spectra in f(R~) format in a quantitative manner. We have also shown that in the case of mixtures containing mono-disperse particles, intensities of absorptions can be calculated from a relationship between intensities of absorption and particle size. Since the relationship is in the form shown in Eq. 12, one

27

needs at least two particle sizes and the absorption intensities in their diffuse reflectance spectra. The model proposed in this paper has been tested for a coal sample containing a definite particle size distribution. The model appears to hold and describe the variation in the diffuse reflectance spectra quantitatively. The model proposed in this paper is based on experimental data and observations. A theoretical basis to explain the particle size effect in diffuse reflectance spectrometry has yet to be formulated.

References [1] M.P. Fuller and P.R. Griffiths, Anal. Chem., 50 (1978) 1906. [2] S.A. Yeboah, S.H. Wang and P.R. Griffiths, Appl. Spectrosc., 38 (1984) 259. [3] A.A. Christy, J.E. Tvedt, T.V. Karstang and R.A. Velapoldi, Rev. Sci. Inst., 59 (1988) 423. [4] R.A. Velapoldi, J.E. Tvedt and A.A. Christy, Rev. Sci. Inst., 58 (1987) 1126. [5] A.A. Christy, Y.Z. Liang, C. Hui, O.M. Kvalheim and R.A. Velapoidi, Vib. Spectrosc., 5 (1993) 233. [6] N.T. Melamed, J. Appl. Phys., 34 (1963) 560. [7] P.D. Johnson, J. Appl. Phys., 35 (1964) 334. [8] P. Kubelka and Z. Munk, Z. Tech. Phys., 12 (1931) 593. [9] A.A. Christy and O.M. Kvalheim, Proceedings of the European Seminar on Vibrational Spectroscopy, Lyon, 1993. [10] A.A. Christy, Y.Z. Liang, C. Hui, O.M. Kvalheim and R.A. Velapoldi, Vib. Spectrosc., 6 (1994) 388 (erratum). [11] E.L. Simmons, Appl. Optics, 14 (1975) 1380. [12] G. Kortiim, Reflectance Spectroscopy, Springer Verlag, Berlin, 1969. [13] J.M. Chalmers and M.W. Mackenzie, in Advances in Applied Fourier Transform Infrared Spectroscopy, Wiley, New York, 1988. [14] W.J. Foote, Pap. Trade J., 122 (1946) 35. [15] M.E. Everhard, D.A. Dickcius and F.W. Goodhart, J. Pharm. Sci., 53 (1964) 173. [16] R.W. Frei, D.E. Ryan and V.T. Lieu, Can. J. Chem., 44 (1966) 1945. [17] A.A. Christy, O.M. Kvalheim, F.O. Libnau, G. Aksnes and J. Toft, Vib. Spectrosc., 6 (1993) 1.