II. MICROWAVE SPECTROSCOPY
Academic and Research Staff
Dr. T. E. McEnally, Jr.
Prof. M. W. P. Strandberg
Prof. R. L. Kyhl
J. G. Ingersoll
J. D. Kierstead
Graduate Students
A.
M. K. Maul
S. R. Reznek
B. N. Yung
J. U. Free
R. M. Langdon, Jr.
J. M. Ditz
R. L. Espino
L. R. Fox
VELOCITY SURFACES IN LITHIUM NIOBATE
The method for determining an analytical approximation for the velocity of sound sur1
faces for previously reported crystals has been applied to lithium niobate. This method
expresses an approximation to each of the three surfaces as a sum over functions of the
appropriate symmetries. The method determines the coefficients in the sum by a leastsquares fit to the exact surface. Lithium niobate has the symmetry of quartz or sapphire.
The basis functions are those listed in Quarterly Progress Report No. 84. Table II-1
). ( = 1, 2, 3 for each of the
CW(,
lists the coefficients in the expansion V(0, 4) =
three surfaces.)
Table II-1.
Coefficients C n .
=3
-=2
=1
n
1.4484
1
.7920
.8610
2
-.003786
-.02802
.07939
3
-.06458
-.01514
.04267
4
.04947
-.04637
-.001434
5
.04148
-.03404
-.004503
6
.02470
-.02417
.1117 X 10-
7
8
-. 1190 x lo6
.009004
6
-.008396
-.001100
-.006630
-. 6717 X 10- 8
-.001841
9
10
.006779
.70618 X 10-
11
.0003993
-.0007653
12
.0007973
-.0007792
13
-89
.1125 X 10- 8
-.8639 X 10
14
.1676 X 10- 10
-. 1638 X 10- 10
8
-.006694
.3242 X 108
-. 9156 X 10- 9
.0004249
-.0002178
9
-9
-.2138 X 10
.1692 X 10- 1Z
SThis work was supported by the Joint Services Electronics Programs (U.S. Army,
U.S. Navy, and U.S. Air Force) under Contract DA 28-043-AMC-02536(E).
QPR No. 85
5
LITHIUM NIOBATE
260
270
280
. TRUE
340
I~U
-
350
180
0
170
10
.60
160
20
084
150
30
1.08
140
40
1.32
130
50
1.56
120
60
I10
100
Fig. II-1.
QPR No. 85
1.80
90
70
80
True and approximate values for the velocity of sound
in Lithium Niobate.
260
LITHIUM NIOBATE
0= 240
270
280
290
250
* TRUE
240
300
x APPROXIMATE
\
230
310
220
320
/
330
210
340
200
.600
1
160
20
84
150
30
1.08
40
140
50
130
1.56
60
120
110
100
1.80
90
Fig. II-1.
QPR No.
85
70
80
Continued.
LITHIUM NIOBATE
260
270
280
. TRUE
340
350
0
10
0.60
20
8784
1.08
1.32
40
\50
1.56
60
70
1.80
100
Fig. II-1.
QPR No. 85
90
80
Continued.
30
LITHIUM NIOBATE
260
270
280
9 TRUE
350
100
Fig. II-1.
QPR No. 85
90
Continued.
LITHIUM NIOBATE
260
270
280
. TRUE
340
350
0
1O
00
Fig. II-1.
QPR No. 85
90
80
Continued.
LITHIUM NIOBATE
270
260
TRUE
280
240
300
x APPROXIMATE
230
310
220
320
210
330
200
340
190
180
350
-
-
-
170
10
O/60
160
20
0.84
150
1.0
/
30
1.08
\'
130
50
1.56
120
60
110
100
1.80
90
Fig. II-1.
QPR No. 85
70
80
Concluded.
(II. MICROWAVE SPECTROSCOPY)
Figure II-1 shows the surfaces computed exactly and those computed by using the
analytical approximations.
S. R. Reznek
References
1.
S. R. Reznek, "Expansion of Velocity Surfaces in Spherical Harmonics," Quarterly
Progress Report No. 84, Research Laboratory of Electronics, M.I.T., January 15,
1967, pp. 17-26.
QPR No. 85