[ SPECTROSCOPIC

BBE TECHNIQUES

Accurate OH Bond Angle Determination in Trigonal Crystals

A N T H O N Y R. M O O N a n d M A T Y H E W R. P H I L L I P S * Department of Applied Physics, University of Tech- nology, Sydney, P.O. Box 123, Sydney, Australia 2007

Index Headings: O H bonds; Bond angle; IR spectroscopy; Sapphire.

INTRODUCTION

Polarized optical absorption spectroscopy is a stan- dard technique used for determining the bond angle of an OH dipole aligned at an angle, 0, to the threefold c-axis of a trigonal crystal. Optical absorption spectra are measured with the electric vector, E, of the incident light both perpendicular and parallel to the trigonal sym- metry axis. The dipole bond angle is usually calculated with the use of the following expression:

a_Lc 1.5 sin20 1 . . . . . tan20 allc 3 cos20 2

where 0 is the OH dipole bond angle to the c-axis, and c~_Lc and allc are the absorption coefficients of the mea- sured IR bands with E_Lc and EJlc, respectively.

Application of this method can lead to large errors when, for example, the OH bond is approximately per- pendicular to the c-axis and the IR absorption with EII c is weak.

A more accurate technique involves measuring the OH absorption band intensity as the E-vector angle to the c-axis is varied from 90 ° (E_Lc) to 0 ° (Ellc) in 5 ° incre- ments.

T H E O R Y

An expression for the absorption band intensity as a function of polarizer angle can be developed with the use of simple rotation matrix and vector analysis techniques.

Consider three vectors, each representing an OH bond direction and each at an angle of 0 to the c-axis but

rotated at 120 ° to each other about the trigonal axis. In matrix notation, these vectors may be represented in an orthonormal crystal space x, y, z (with x [I a-axis and z H c- axis) as

ro ~- ; r~ ~- 0.866 sin ; \ C O S COS

/ / -0.500 sin 0 \ =- sin 0 ) r_l ~ -0 .866 ,

cos 0 /

provided ro is in the x-z plane. If ro is not in the x-z plane but is rotated by ~ about the z-axis, then each of the above vectors r is replaced by Sr where

Consider another space, the polarizer or laboratory space, x', y', z', in which the polarizer is fixed along the x' direction. If the crystal space is rotated (anticlockwise) through ¢ about the y-y' axis, then vector components in crystal space, X, may be calculated in polarizer space, X', through X' = PX, where

P = 1 0 . \ - c o s ¢ 0 sin

The electric field components along each of the three crystal vectors to, r~, and r_l may thus be calculated as

eo = (100)PSro el = (100)PSrl

e_~ = (100)PSr_x

where (100) is the unit vector along the polarizer direc- tion. The measured dipole absorption am is thus given by

a m = (eo 2 --F e l 2 + e_12 )ab (1)

where ab is the absorption coefficient when the field is aligned along one of the OH bonds. After matrix mul- tiplication, Eq. 1 can be written as

Received 20 Feb rua ry 1991. * Au tho r to whom correspondence should be sent . O/m = (3/2 sin2¢ sin20 + 3 cos2¢ cos20)ab (2)

Volume 45, Number 6, 1991 0003-7028/91/4506-105152.00/0 APPLIED SPECTROSCOPY 1051 © 1991 Society for Applied Spectroscopy

5.00 4.00 mlumnlull 3309 cm -~ o o o o o 3232 cm-' n n n n n 3309 cm- '

3187 am -t ~ oOOOO 3232 cm -~ 4.00 ~ 3187 c m - '

3.00

2.00

1.00 1.00

0.00 -r-,-, , , m , , , , , , , , , , , u , , n , , , , 0.00 , , , I , i ' , , -15 0 15 30 45 so 75 90 105 0.0 0.2 0.4 o.s 0.5

E Vector Angle to C-Axis cosZ(¢) (radians)

FIG. 1. OH band intensity as a function of E vector angle to the c-axis. The symbols represent the experimental data points and the solid lines the theoretical values.

1.0

FIG. 2. OH band intensity vs. the square of the cosine of the polarizer E vector angle to the c-axis. The symbols represent the data points, and the straight lines are lines of best fit determined by least-squares analysis.

where ¢ is the angle between the crystal c-axis and the polarizer axis, and 0 is the angle the OH bonds make with the c-axis. The resul t is independen t of the angle 8, defined above. In addit ion, for 4~ = lr/2, a _ L c = 1.5 sin205~, and for ¢ = 0, a[]c = 3 cos205b, so tha t

a .Lc 1.5 sin20 1 . . . . . tan20 all c 3 cos20 2

as expected. Re-arranging Eq. 2 gives

am = (%(1 - cos2~b)sin20 + 3 c0s24, cos20)ab = ~sin20ab + (3 cos20 -- %sin20)abcos24~. (3)

The graph of a m versus cos% will be of the form y = G x

+ L where the gradient G = (3 cos20 - % sin20)ab and the in te rcept I = %sin20ab. The values of G and I can be accurately determined with the use of linear least-squares analysis.

T h e OH bond angle can be calculated according to the following expression:

where

0 = t an - l (g )

" = (4)

As G and I are interrelated, the uncer ta in ty in 0 can be d0 dtt

de te rmined via A0 = - - - - AG, dg dG

a 0 = 4/2 \ 1 + g 2 ]

where AG is the error in the gradient and

~ y / ~ x i y = i and 2 = - ! - - -

N N

(5)

E X P E R I M E N T A L

Th e OH bonds at 3309, 3232, and 3187 cm -1, which appear in hydrogena ted na tura l and synthet ic sapphire [a-A12Os(R3c)] 1 containing T i impurit ies, were used to evaluate the technique presen ted in this paper. Crystal sections (1-3 mm) were accurately cut parallel to the c-axis of the a-A1203 crystal specimens. Th e section was m o u n ted on a sample jig which allowed rota t ion of the sample th rough 360 ° in 5 ° steps. T h e sample was orien- t a t ed in the spec t rometer so tha t the wave vector of the incident polarized light was perpendicular to the crystal c-axis plane. Room- tempera tu re polarized IR optical ab- sorpt ion spectra were measured on a Pe rk in -E lmer 1600 series F T - I R spec t rometer f i t ted with a Cambridge Phys- ical Sciences wire grid polarizer (Model IGP-227).

T h e polarizer was or iented to give max imum signal on the pyroelectr ic de tec tor and fixed in t h a t direction. T h e sample chamber was cont inuously purged with dry de- carbona ted air during the acquisi t ion of all background and sample spectra.

R E S U L T S A N D D I S C U S S I O N

The results of a rotation experiment are presented in

Fig. 1. The symbols represent experimental data, and the

solid lines represent the theoretical values of Eq. 2 from

1052 Vo lume 45, Number 6, 1991

TABLE I. Calculation of OH bond angles in sapphire.

Linear least-squares me thod Conventional method

r2~±cI tan- [ ~ ?

cm -1 0 ° A0O R 2 0 ° A0O

3309 76.6 ±0.5 0.999 75.4 ± 1.1 3232 78.3 ±0.5 0.999 79.0 ±3.6 3187 76.7 ±1.7 0.994 79.9 ±11.8

the data in Table I. The plots of am versus cos2¢ for each OH band are shown in Fig. 2. The symbols are the ex- perimental data, and the straight lines are the lines of best fit determined by linear least-squares analysis. The gradient and intercept of these lines of best fit have been used in Eqs. 4 and 5 to determine 0 and A0 for each OH bond. The results of this analysis are given in Table I together with the OH bond angles determined with the use of the conventional E_J_c and Ellc technique.

A comparison between techniques clearly shows that the analysis procedure presented in this paper yields the more accurate result, especially when the IR absorption intensity is "weak." However, the OH bond angles and associated errors determined with the conventional two- measurement technique are quite acceptable when the OH absorption intensity is "strong." This is a pleasing result considering that many polarized optical absorp- tion spectroscopy measurements are conducted at low temperature in cryostats which may only allow the ac- quisition of E_J_c and Ellc spectra.

CONCLUSION

The analysis technique described in this paper pro- vides a method for accurately determining the OH bond angle in trigonal crystals. The procedure could be par- ticularly useful for obtaining OH bond angles when the IR absorption intensity is "weak."

1. A.R. Moon and M. R. Phillips, "Defect Clustering in H, Ti: ot-Al203," accepted for publication in J. Phys. Chem. Solids.

Effect of Optical-Fiber Length on the Width of a Transmitted Laser Pulse: Comparison of Theoretical Calculations and Experimentally Measured Pulses

M A R Y K. C A R R O L L and GARY M. H I E F T J E * Indiana University, Department of Chemistry, Bloomington, Indiana 47405

Index Headings: Fiber optics; Time resolution; Spectroscopic tech- niques; Instrumentation, fiber optics.

INTRODUCTION

Optical fibers have several characteristics that make them attractive for sensing applications. They are com- pact and rugged, can carry light over great distances (enabling measurements external to the laboratory), and are not subject to electrical interferences. There have been numerous review articles detailing the use of optical fibers in the development of chemical sensors. (A rapid Chemical Abs trac ts on-line search turned up 125 review articles in the area of fiber-optic sensors, 80 of which have been published since 1984.) Several particularly good reviews that emphasize luminescence-based sensors are noted here. 1-3

One type of experiment that has been performed through optical fibers is time-resolved luminescence spectroscopy. 4-9 In such an experiment, the attainable time resolution is limited by the duration of the incident light pulse which exits the illuminating fiber. In prepa- ration for a series of time-resolved experiments in our laboratory, it was therefore deemed prudent to investi- gate the extent to which a laser pulse is broadened during transit through optical fibers of various diameters and lengths. Temporal broadening of laser pulses in sensor and other spectroscopic applications occurs mainly be- cause light travels through an optical fiber via an infinite number of optical paths, each with a discrete pathlength. The extent of temporal broadening depends on the re- sulting variation in optical pathlength, which is related to the length of optical fiber and the effective numerical aperture of the optical system. In other areas, especially in the communications industry, broadening from dis- persion (changes in refractive index with wavelength) can limit the attainable pulse width or modulation frequency.

Pulse widths calculated from theory are presented here and compared with experimental results. Several differ- ent lengths of fiber and various core-diameter fibers were used in this study. Within the limits of the measure- ments, the incident 310-ps pulses are not broadened as a result of passage through fibers of length 60 m or less.

Received 6 March 1991. * Author to whom correspondence should be sent.

Volume 45, Number 6, 1991 0003-7028/91/4506-I05352.00/0 APPLIED SPECTROSCOPY 1053 © 1991 Society for Applied Spectroscopy