Modelling of dielectric functions of glasses by convolution

$

] O O R N A L OF

ELSEVIER

Journal of Non-Crystalline Solids 195 (1996) 64-71

Modelling of dielectric functions of glasses by convolution H. Hobert *, H.H. Dunken Institute of Physical Chemistry, Friedrich-Schiller-University Jena, Lessingstrasse 10, 07743 Jena, Germany Received 15 March 1995; revised 28 June 1995

Abstract

Dielectric functions used to model infrared spectra of amorphous solids that treat the vibrational modes as the sum of damped harmonic oscillators can be improved by convolution with a broadening function accounting for a distribution of the vibrational frequencies. Such a broadening can be applied to each oscillator or to the entire dielectric function. The Fourier transformation is an effective algorithm to carry out the convolution. Examples show global and local folding on a sodium-lime glass and an application on glassy and crystallized Li2Si205.

1. I n t r o d u c t i o n

The complex dielectric function, ~ = e' + i E", is a key function in the evaluation and interpretation of reflection spectra of crystalline or glassy solids. This function can be obtained by a dispersion analysis which models the sample as a system of damped, harmonic oscillators: s ~2j ~(~) =~+

E j=l ~02j-

fit a glass spectrum by a dielectric function derived from such a model gives unsatisfactory results. More useful model functions can be obtained, however, by a sum of broadened oscillator functions, for instance, as

J

× f~_ ~ e - ~x- ~o,)2/2 ~?

~'2+iyi~'

where ~ is the wavenumber (in c m - ' ) , ~0 is the resonance wavenumber (in c m - 1), ~p is the intensity parameter (in c m - l ) , y is the damping constant (in cm -~) and e~ is the dielectric constant at high frequencies. This function is suited to describe the dielectric behavior of ordered solids. However, the attempt to

* Corresponding author. Tel: +49-3641 635 873. Telefax: + 49-3641 636 118. E-mail: [email protected].

1

~]~

X 2 __ ~2 ..~

iyj~, d x. (1)

x is now the variable oscillator wavenumber; o) express its distribution of the wavenumbers of the jth oscillator around its mean value, 90i- Such a function (using Sj instead of 9fj) was suggested first by Efimov and co-workers [ 1] and numerically treated by approximations. Similar proposals were made by other groups [2,3]. Brendel and Borman [4] used the same function and gave an analytical solution of Eq. (1) which includes the complex error function.

0022-3093/96/$15.00 © 1996 Elsevier Science B.V. All fights reserved SSDI 0 0 2 2 - 3 0 9 3 ( 9 5 ) 0 0 5 1 7 - X

H. Hobert, H.H. Dunken / Journal of Non-Crystalline Solids 195 (1996) 64-71

This equation represents a folding or convolution

65

with

process which can be expressed also by j ~2j

g = e - ~/2'~/

(2)

as the convolution function. To simplify the development of evaluation programs and to avoid the numer-

h. ioo

,.g °°

A

J,L 0400

~0

800

6O0

8(~

(WI0

800

,°

Y

°°

~

-io

.'30

S

10

f,

30

I

F

400

600

~ifF

I

800

400

H"

F"

1.0.., G X ,.40

H"

j, -30

". -10

10

X 4+

.I

-vl

X

~ F'I

F,

30

), ,,v,mld~[~

X 40

M

.,ill "'tp

Fig. 1. Convolution process: the complex function, h, forms by a Fourier transformation (3r ) the Fourier transform, H, which is multiplied (~tr) with the Fourier transform of the convolution function, G. The product function, F, is transformed by an inverse Fourier transformation ( ~ - - 1) into the convoluted function, f.

H. Hobert, H.H. Dunken / Journal of Non-Crystalline Solids 195 (1996) 64-71

66

ical difficulties disturbing the computation of the complex error function at larger arguments, we propose the application of the Fourier transform route to treat this problem.

2. Methodical part 2.1. Convolution by Fourier transformation

tion of a g function with an appropriate scaling factor equal to l / G 0 : F = G H / G o.

(G O means the value of G at x = 0.) If the folding function is a Gaussian function, the computations can be shortened immediately using G, which is also a Gaussian function, instead of g as input. It is possible to use the G in its simplest form, (3)

G = e -Ax2,

The discrete sample function - the spectrum, containing N = 2 n + 1 values - signified in the following by h and the equal-sized folding function by g is carried out by

a = ( ' r r o - ' / N ) 2.

N

f=h*g

or f , =

with A as a variation parameter. This G has the advantage that G O is always equal to 1. The relation between A and tr' is

Y'. g , - m h m . m=O

The same result can be obtained by a multiplication of the Fourier transformed function, H = .gr{h} and G = 9-{g}, which gives the Fourier transform, F, of the convoluted function, f,

Fig. 1 illustrates the convolution with a simple example, a one-oscillator function, h, generated with N = 401 points in the range [400,800] using ~0--600 c m - i, ~p = 500 c m - 1, Y = 10 c m - 1 and a folding function

F = GH.

~ / ~ - _71,e-X2/~ '2'

The asked function, f , is obtained by the inverse Fourier transformation, .9r - l, applied on F: f=sr-l{Sr{g}Sr{h}}

=...~- l { F}.

Practical realization requires the consideration of some rules. To avoid a phase shift between the real and the imaginary part of h, the folding function, g, should be a real, symmetric function. Because the discrete Fourier transform considers the sample function as a repetition unit of a cyclic function and gives the first point the index, i = 0, and the last point the index, i = - 1, the folding function which is also cyclic should have its symmetry center at go. A disadvantage of the cyclicity of h is a deformation of f at the connection between the first (i = 0) and the last (i = - 1) points if its values are different. It is possible to diminish this distortion by an elongation of the spectral data vector by a puffer region which connects the endpoints by a ramp function. The deformation is concentrated in this region which is removed at the end. To avoid shifts in the relative intensities of the oscillators which may occur if different trj are used in a model, the Fourier transform of the folding function, G, should be normalized to 1. This can be achieved either by the applica-

g=

Or

with the broadening parameter t r ' = 20 c m - J . The Fourier transform of g is G which could be computed, according to Eq. (3), also immediately by G

=

e -0'0245512X2.

The resulting function, f , especially f ' , shows the deformation at the boundaries, mentioned above. 2.2. Local versus global folding

The folding can be carried out either as a local process applied individually to each oscillator or as a global procedure applied only once to the oscillator sum. The local variant transforms each oscillator, p j, into a broadened oscillator, q j, qj = g) * pj, and adds these to the local folded dielectric function El = ~ + E

qj

J

(we omit in the following the ^ to characterize the as a complex function). Each oscillator can be characterized by an individual distribution parameter, trj.

H. Hobert, H.H. Dunken /Journal of Non-Crystalline Solids 195 (1996) 64-71

67

Table 1 Parameters of global and local folding of sodium-lime glass (all values with exception of ~ are in cm ~) Global folding

Local folding

No.

Vo

vp

3'

NO.

Vo

vp

1 2 3 4 5 6 7 8

457.9 532.9 642.3 772.4 971.1 1028.9 1100.5 1166.0

343.0 129.2 143.0 156.5 414.7 476.9 293.8 193.5

9.6 16.8 81.5 9.6 9.6 12.2 26.1 24.7

1 2 3 4 5 6 7 8

456.0 488.6 656.7 776.4 968.7 1026.9 1104.8 1 ! 72.9

330.4 193.7 137.7 147.5 407.9 492.2 283.7 174.6

O"

29.8 47.7 19.7 9.6 9.6 9.6 9.6 9.6

21.4 34.8 67.5 25.3 30.4 32.3 34.9 32.4

Global folding: ~ = 2.19, cr = 29.4. Local folding: ~ = 2.20.

20[

(a)

25

ibi

15 2O

10

15

5

0

-5 5_ -10 0 400

. 600

800

, 1000

1200

,

.

i

,

1400 v" / c m - '

,

,

,

1600

6

6 ~

i

i

4-00

600

800

1000

1200

1400 ; /cm-'

1600

,

400

400

l

t

,

,

600

'

'

i

600

i

,

i

,

800

,

,

,

,

800

I

,

,

1000

,

,

,

i

1000

I

,

,

,

I

b

I,

1600

1200 1400 v" / c m - ' '

'

'

,

1200

'

,

,

~

1400 ;/cm"

,

,

1600

Fig. 2. Comparison of the global ( ) and the local (. - - ) folded dielectric functions of a sodium-lime glass: (a) the imaginary part of the dielectric function before (E *") and after (¢") convolution and (b) the real part of the dielectric function before (~ "') and after (~') convolution.

H. Hobert, H.H. Dunken/ Journal of Non-Crystalline Solids 195 (1996) 64-71

68

0.30

The global folding computes at first the sum of the oscillator functions, p j, which is folded with a single g function Eg=~

0.25

+ ~'~pj* g.

0.20~ a~ u c

J

015

This prescription gives all oscillators the same distribution parameter, ~r. The advantage of this method is the lower number of parameters needed to model the dielectric function. The unfolded dielectric functions, ~ *, formed by the unfolded oscillators,

pj, ~*=~+EP;

,,& OAO

OO5

O.OC , 4OO

, 600

800

1000

1200

1400

1600

~"/cm"

J

are also interesting because they show the oscillators used in a model with higher resolution and allow a better comparison of different samples than the broadened functions, e. However, only these functions E are of physical relevance. 2.3. Spectra The IR reflection spectra were scanned with a FTIR spectrometer (Bruker IFS 66) in a Seagull cell (Harrick Scientific Instruments) at a definite angle of incidence, 0, and with TE-polarized IR radiation. A spectrum of an A1 mirror was used as reference. The parameterization was carried out with an 800-point part of the spectrum between 400 and 1940 cm-~, which was extended by a 160-point ramp function to interpolate between both ends of the spectrum. The dispersion analysis was programmed as an iterative random optimization process, controlled by diminishing the sum of the squared deviations between the experimental spectrum (or spectra set) and the modelled spectrum (or spectra set). The goodness-of-fit was expressed by X 2 Y'-~0_-01(Ri,exp -- Ri,mod)2/800. The spectra were computed from the dielectric functions by the Fresnel equations [5] assuming a simple two-media system (air/glass). The broadening parameters, o-, are used in the sense of Eq. (2). =

3. Results

As a first example, the modelling of a reflection spectrum of a sodium-lime glass (a slide for mi-

0.005

?

400

:

600

800

1000

1200

-- 14Q0 v /cm '

1600

Fig. 3. Reflection spectrum of sodium-lime glass ( - - ) and the modelled spectra obtained by local ( - . - ) and global ( - - - ) folding. The lower part gives the difference, A, of the experimental and the modelled spectra.

croscopy) ( 0 = 10°, TE polarization) was treated. The model with eight oscillators was developed in an attempt to find a good representation of the spectrum with a minimal set of oscillators and parameters. Table 1 gives the optimized parameters obtained with the local and the global approach. The imaginary parts of the unfolded and folded dielectric functions are shown in Fig. 2. Fig. 3 allows the comparison between the experimental spectrum and the computed spectra. An other example allows the comparison of two samples with the same composition (91.5 mass% Li2Si205) but with a different degree of order: a glass and a crystallized sample obtained from the glass by tempering. Their reflection spectra were taken with 0 = 20 ° and TE polarization. The glass was modelled with 12 oscillators (a larger number could not be drawn from the spectrum). The crystallized sample allowed the assumption of 17 oscillators. Figs. 4 and 5 show the imaginary parts of E * and e (global folding). The experimental spectra and

H. Hobert, H.H. Dunken /Journal of Non-Crystalline Solids 195 (1996) 64-71 .

=

'

,

,

,

'

i

,

,

•

,

.

.

.

,

,

,

.

,

•

,

,

20

16

,o

0

,

•

400

6

600

'

'

'

i

800

,

'

'

p

1000

'

'

i

~ '

1200

,

,

,

,

•

'

1400

1600

;"/cm" ,

'

'

i

'

,

'

4

69

with 33 parameters (four per oscillator and e=), the global folding should be favored in this and similar systems as a first approximation. It is obvious that the oscillators in this model are of different quality as concern the width of the spectral features. This difference is indicated in the different values of the 7 parameters. Some of these values are in the neighborhood of 9.6 cm- i (five times the point distance), a constraint built into the program to avoid too small 3' values which would disturb the further optimization of the discrete data set. The comparison of global convolution parameters of the glassy and the crystallized Li2Si205 shows, as the main effect, the expected difference of the distribution values: tr = 34.2 cm-1 in the glass but o~-11.7 cm-~ in the crystallized sample. In the glass, the achieved fit ( g 2 = 1.5 × 10 -6) was similar to the fit of the sodium-lime glass (2.5 × 10-6), but 20

2

15

1

0 4 0 0

,

,

I

600

,

,

,

,

800

,

,

1000

,

,

,

1200

"

1400

;/~-'

1 6 0 0

10

Fig. 4. Li2Si205, glass: imaginary part of the dielectric function before (E "") and after (E") the global folding. 5

their computed counterparts are given in Figs. 6 and 7.

0 400

600

800

1000

1200

1400

1600

;/¢m-' '

'

i

,

,

,

i

,

,

,

14

4. Discussion 12

The first example allows a comparison of the local and the global convolution. The unfolded E * functions of the soda lime glass, given in Fig. 2, show some differences reflecting the low degree of uniqueness of the solutions. However, it is apparent that both methods, nevertheless, give similar results concerning the broadened E and the approximated spectra. The goodness-of-fit values, X 2, were, respectively, 5.4 x 10 -6 and 2.5 x 10 -6. Because this result was achieved in the global folding with 26 parameters (three parameters per oscillator: 9pj, ~0j, "yj and additional E= and o-) and in the local folding

10

8

400

600

800

1000

1200

1400

1600

7, Icm-'

Fig. 5. Li2Si205, crystallized: imaginary part of the dielectric function before (~ *") and after (~") the global folding.

H. Hobert, H.H. Dunken / Journal of Non-Crystalline Solids 195 (1996) 64-71

70

040 I 0.50

tO.30 I

~

0 40 m

o= 0.30

3 0.20

010

l 0. t0

000 400

,

,

I

600

,

,

,

i

800

,

,

,

]

I ooo

0.00

,

1200

! 400

;"/cm"

) 600

&

600

400

I000

800

1200

1400

1600

;/cm-'

&

0.005

o.olo

0.000

o.ooo O.OLO

-0.005

:~ . . . . . . . . . . 600 80{3

400 400

600

8oo

lOOO

1200

_ 1400 v /era-'

,

,

1000

,

,

,

1200

Fig. 6. Li2Si20 s, glass: experimental ( ) and modelled (- - . ) spectrum. Lower part: difference spectrum.

the tendency to approximate the oscillator shape function by the Gaussian profile was lower. The local folding resulted in ~rj values between 28 and 46 cm - ] , indicating a considerable variation in the shape and width of the oscillator bands. This tendency was amplified in the crystallized sample: both approaches gave similar but larger X 2 values (1.5 × 1 0 - 5 / 2 . 0 × 10 -5 ) compared with the glass. The parameters 72 varied between 9.6 and 83 cm -~, the crj between 5.9 and 16.8 cm -]. This variation reflects some insufficiencies in the chosen model and indicates that such samples should be treated with a local folding. A comparison between the E" functions of the glassy and the crystallized sample is represented in Fig. 8, showing that the main silicate bands at 460, 780, 980 and 1050 cm -1 are contained in both spectra, although with different widths. However, many of the smaller bands which are seen in the polycrystalline sample have no counterpart in the glass. This place is not to discuss the structural

,

,

~

.

,

1400 ;'/¢~-'

16oo

1600

Fig. 7. L i 2 S i 2 0 s, crystallized: experimental ( ) and modelled ( . - - ) spectrum. Lower part: difference spectrum.

peculiarities of these samples, but the qualitative differences discussed by Lazarew et al. [6] are confirmed by these observations.

•

,

,

,

,

,

.

i

.

,

.

,

.

,

,

i

.

,

,

~

.

.

,

14

-~ 12 10

6

!

:

2

400

600

800

1000

1200

. V

1400 /crn"

1600

Fig. 8. Comparison of the imaginary parts of the folded dielectric functions of the glassy ( ) and the crystallized (- - . ) Li2Si205 sample..

H. Hobert, H.H. Dunken / Journal of Non-Crystalline Solids 195 (1996) 64-71

5. Conclusions The applied harmonic oscillator functions folded with a Gaussian function give models suited to represent the dielectric function of glasses. The application of the Fourier transformation is an effective way to implement the broadening of the oscillator functions in the computation and allows also the treatment of large data sets. Local and global folding give nearly the same results in the case of glasses. Therefore, the global folding should be preferred except when the number of oscillators is small. Partially ordered solids should be modelled by local or without any folding.

Acknowledgements This study was supported by Deutsche Forschungsgemeinschaft (Project SFB 196). The authors thank B. Durschang (Institute of Glass Chem-

71

istry, University Jena) for the lithium silicate samples and A.M. Efimov (St Petersburg) for valuable discussion.

References [1] A.M. Efimov and V.N. Khitrov, Fiz. Khim. Stekla 5 (1979) 583; A.M. Efimov and E.G. Makarova, in: Stekloobraznoe Sostoyanie (Vitreous State), Proc. 7th All-Union Conf., ed. E.A. Porai-Koshits (Nauka, Leningrad, 1983) p. 165; A.M. Efimov and E.G. Makarova, Fiz. Khim. Stekla 11 (1985) 385; A.M. Efimov and E.G. Makarova, Fiz. Khim. Stekla 15 (1989) 366. [2] M.L. Naiman, C.T. Kirk, R.P. Aucoin, F.L. Terry, R.W. Wyatt and S.D. Semturia, J. Electrochem. Soc. 131 (1984) 637. [3] A. Kucirkova and K. Navratil, Appl. Spectrosc. 48 (1994) 113. [4] R. Brendel and D. Bormann, J. Appl. Phys. 71 (1992) 1. [5] W.N. Hansen, J. Opt. Soc. Am. 58 (1968) 380. [6] A.N. Lazarew, A.P. Mirgorodskij and I.S. lgnatew, Kolebatelnye spektry sloshnych okislow (Vibrational Spectra of Complex Oxides)(Nauka, Leningrad, 1975).