NBSIR 74-544

Kerr Coefficients of Nitrobenzene

and Water

Robert E. Hebner, Jr., Richard J. Sojka and Esther C. Cassidy

Electricity Division

Institute for Basic Standards

National Bureau of Standards

Washington, D. C. 20234

August 7, 1974

Final Report on

Project No. AFWL 74-067

USAF Technical Contact: Capt. Au-gust S. Sanchez

USAF Administrative Contact: Lt. Col. Anthony

Prepared for U. S. Air Force

Kirtland Air Force Base

Albuquerque, New Mexico 87117

NBSIR 74-544

KERR COEFFICIENTS OF NITROBENZENE

AND WATER

Robert E. Hebner, Jr., Richard J. Sojka and Esther C. Cassidy

Electricity Division

Institute for Basic Standards

National Bureau of Standards

Washington, D. C. 20234

August 7, 1974

Final Report on

Project Order No. AFWL 74-067

USAF Technical Contact: Capt. August S. Sanchez

USAF Administrative Contact: Lt. Col. Anthony

Prepared for U. S. Air Force

Kirtland Air Force Base

Albuquerque, New/ Mexico 87117

V. ^

U. S. DEPARTMENT OF COMMERCE, Frederick B. Dent, Secretary

NATIONAL BUREAU OF STANDARDS, Richard W. Roberts. Director
TABLE OF CONTENTS

I. INTRODUCTION 1

II. APPARATUS 2

III. EXPERIMENTAL APPROACH 8

A. Identification of Sources of Error 8 B. • "End-field free" Cell Technique 11

C. Two Cell Technique 12

D. Single Cell Technique 13

IV. RESULTS 14

A. Magnitude of the Kerr Coefficient of

Water and Nitrobenzene, 14

B. Temperature Dependence of the

Kerr Coefficients 20

C. Wavelength Dependence of the Kerr Coefficients 24

• V. SUMMARY AND CONCLUSIONS 28

REFERENCES...... 31
I. Introduction

The electro-optic Kerr effect is used routinely in a niomber of

laboratories throughout the world for the measurement of high voltage

pulses. Almost all work to date has used nitrobenzene as the

birefringent liquid. The primary reasons for the dominance of nitrobenzene are that this fluid exhibits an approximately flat frequency

response from very low frequencies to about 10^ Hz, and it has a larger

Kerr coefficient than any other liquid possessing a suitable frequency

response. There are a niomber of properties of nitrobenzene that

preclude its use as an insulant in conventional high voltage apparatus.

Primary among these is the fact that it is a good solvent. In addition

the resistivity of nitrobenzene is reduced by orders of magnitude by

impurities. For these reasons when it is used for voltage measurement

it is usually sealed in a cell constructed of glass, or other material

impervious to nitrobenzene, and the cell is connected to the high

voltage circuitry. Measurement of the phase shift between two

orthogonal components of the linearly polarized light beam incident

upon the calibrated cell provides the information necessary to measure

the pulse shape and/or the peak value of the applied pulse. This

optical coupling between the high voltage and detection circuits provides

both high noise immunity and can prevent the transfer of any high voltage

or high current surges between the pulse generator and sensitive

detection circuitry.

This report is directed toward an improvement of pulsed-voltage

measurement accuracy through a measurement of the temperature and

wavelength dependence of the Kerr coefficients of nitrobenzene and

water. Accurate values for magnitudes of these coefficients are significant because knowledge of the cell geometry and the Kerr

coefficient of the birefringent fluid permits measurement of voltage

with an uncertainty of a few percent. This is useful in cases in which

it is impractical to calibrate the Kerr system against another measuring

device. The main disadvantage of this approach ha^ been the large

uncertainty in the value of the Kerr coefficient. For example, a large

number of measurements of the Kerr coefficient of nitrobenzene have been

made. At a given temperature and wavelength, these values differ from

the average by ±40%. For water only one careful determination of the

Kerr coefficient was available in the literature. In presenting that

work, however, the authors made no estimate of the measurement error.

In this report the Kerr coefficient of nitrobenzene was measured with

an estimated uncertainty of ±6% and the measurements in water had an uncertainty of ±8%.

The measurement of the temperature dependence of the Kerr coefficient

is even more significant for accurate voltage measurement because the

cell calibration is temperature dependent. Calibration of a Kerr cell

for high voltage measurement consists of applying a known voltage,

measuring the phase shift between two components of the polarized light

beam, and calculating the voltage necessary to cause a phase shift of

1
IT radians. This voltage is called the cell constant and depends only

on the cell geometry and the Kerr coefficient of the birefringent

fluid. Accurate measurements of the temperature dependence of the Kerr

coefficient permit correction of the cell calibration. The temperature

dependence of the Kerr coefficients of both water and nitrobenzene are presented in this report.

The experimental approach was to repeat and refine the measurements

previously made of the Kerr coefficient of nitrobenzene. Measurements

in nitrobenzene would serve to calibrate the apparatus used for the

measurement in water. Furthermore this redetermination of the Kerr

coefficient of nitrobenzene would identify any systematic errors which

may have arisen. Using the same cells used in the nitrobenzene study,

the ratio of the Kerr coefficient of water to the Kerr coefficient of

nitrobenzene was determined. To verify the accuracy of these measurements, the Kerr coefficient of water was determined independently from

measurements of phase shift, cell geometry and voltage.

A description of the apparatus used in the course of these measurements

is presented in Section II. The various techniques used in voltage

measurement and the necessary equations for data reduction and error analysis are presented in Section III. Section IV presents the results

of this investigation and compares them to those obtained by other workers.

The report is summarized in Section V.

II. Apparatus

The purpose of this section is to provide a detailed report of the

apparatus and measurement techniques used. The apparatus is shown

schematically in Fig. 1. The high voltage pulse generator was a Marxtype generator, i.e., capacitors charged in parallel, discharged in

series, with appropriate wave shaping inductors to yield an approximately

rectangular pulse with rise and fall times of approximately 1 ys and a duration of about 10 us". The peak output voltage was approximately

65 kV. The Kerr cells used had spacing as small as 0,25 cm. This

resulted in the ability to generate field strengths of order 250 kV/cm

which was sufficient to cause electrical breakdown in the cells used.

The high voltage divider was a resistive divider, total resistance

nominally 500 fi. In addition to serving as a method of measuring the

maximum pulse voltage, the divider is used to define the burden for the

pulse generator. It should be noted, however, that with the largest

capacitance Xiirater cell the RC time constant of the burden approached the

designed risetime of the pulse generator. This was demonstrated

experimentally by a lengthening of the pulse rise and fall times to about

2 jis. Conventional practice is to insure accurate pulse voltage measurement by using both a calibrated Kerr cell and the pulse divider to

measure the voltage pulse. For most of the data presented in this

report this practice was followed. In cases when measurements using

both devices were obviously superfluous, e.g., when a large number of

2
c

(U

•1-1 o

•l-l ^-^ iw

0)oo

i-i u

«4-l oCo

•H

4-1 toc

•H

>-l (U

4-1 (U

(U

4-J

(U

U)3

CO3u

tou

CO

ex. ato • 0)

0) c

si (U U NC

<4-l <U O XIo S

to 4J U -H bO C

COo

•H !-i 4J (U

COS to x:

C/D o

bO

•l-l 3
observations were made using exactly the same set-up, only the divider

measurement was recorded. In these cases the divider performance was verified before each series of data runs.

The light sources used for this study were a HeNe laser C632.8 nm) and an argon laser, A wavelength selector at the output of the argon laser

permitted measurements at 514.5, 496,5, 488.0 and 476,5 nm.

Four different Kerr cells were used during the course of this study.

These are shown in Fig, 2-4. Two of the cells, which have been discussed

previously, were constructed to be, except for the length of the plates,

geometrically identical. The plates in each cell were made of stainless

steel, were electro-polished after cell construction, and were 4 cm wide

and spaced 0.-5 cm apart. The length of the plates in one cell was 12 cm,

while the other had plates that were 24 cm long.

The third cell, also previously discussed, was designed so that the

birefringent liquid was confined to a central volume well removed from

electrode edges. In this cell, the liquid volume had dimensions:

width 1 cm, thickness 1 cm, length 4 cm. The electrodes were stainless

steel with dimensions: width 5 cm, length 14 cm and thickness 0.75 cm.

The fourth cell, which was used only for measurements in water, was specially constructed for this investigation. While the other cells

were constructed of glass, this was made from polymethylmethacrylate. In

order to provide an acceptable light path circular (diameter - 5.5 cm)

holes were cut in the end faces of the plastic box. Strain-free glass

was then cemented over these holes. The spacing between the aluminum

electrodes (4 cm wide, 30 cm long) was adjustable between 0.25 and 1 cm. The top plate of the cell was removable. When the cell was in use, it was filled so that the birefringent liquid covered the entire upper

electrode. The upper electrode then served as the grounded electrode,

and the high voltage pulse was applied to the lower electrode. The air

space between the top of the liquid and the cover of the box served as

a compressable volume so that the shock wave caused by electrical

breakdown would not rupture the glass end plates. During the course of

these investigations, a large number of electrical breakdowns occurred.

In no case was the cell damaged.

During most of this study, a photomultiplier tube (1P21) was used to

detect the modulated light beam. The output from the photomultiplier

was displayed on a conventional oscilloscope and the data was recorded

by photographing the oscilloscope trace whose sweep was synchronized

with the high voltage pulse.

The temperature of the cell was controlled through a control of the

temperature of the oil bath in which the cell was immersed. The heatingcooling unit was a commercially-available thermostated unit. The

temperature range covered was from 280 K to 322 K, Upon changing the

selected temperature, the system was given about three hours to stabilize.

For the temperature increments used, no change in the cell temperature as

4
to

<u

CL

CD QJ N x:

a_)

4-1 o CO _C

4-J • 1-4

(i;

p ,

_c QJ -u O O X

<u o

CJ CO I-I

4-1 4_) c

cu cu • r-l "Xl •H •

S_i

•4-1 1—

J

i-H -U o qj o

'H 0) M Si

i-i 4-* <u6 C O -I-I <u <u .c 00 <u 4J

(U (U 0 O co x:

'H

4-i CUC

m c t>o to •H

CO 00

i-i CU

C

QJU 0

4-J (—1 0) 0) 13 }-l CO

0) 03

(U -C (U

J-J

m o

1-J 'I-I

J-< o QJ 4-1 o

QJ -o (U )-i x: Q) H 3

CO

)-i QJ

CU o

<D 4-1 O CO QJ

)-i 5§

ID

CD c c

ca -H 5
Fig. 3 Kerr cell used in the determination of the Kerr constant of nitrobenzene. The birefringent liquid is confined to a volume

well removed from the electrode edges.

6
4-1 CD

1—1CL

01 x;H

0) in

3

-u O

e c

QJ

013

nj 4-1 01 01 e ^

S-i 01 O .-I

nj T3 4J

0) w

tn 3 D •'-1 T3

I—i n)

.—I0) w

O -H O Ml

-H C

4-1 -H

cn O

CO CO

i-H CX

P-l CO
large as 0,1 K was detected after a stabilization time of about 1.5 hrs.

A very limited amount of work was performed using the plastic cell at

various temperatures. In this case the temperature of the liquid was controlled by passing heated or cooled transformer oil through a loop

of copper tubing on the back side of the grounded electrode.

III. Experimental Approach

A) Identification of Sources of Error

The Kerr coefficient of a substance, B, can be defined by the

relationship

^11 ~ ^ "

'^^^^

where index of refraction for light polarized parallel

(perpendicular) to the direction of the applied field E and X is the

wavelength of the monochromatic light traversing the Kerr cell. Eq. (1) can be solved for the phase shift, cp, between the components of the light

beam polarized parallel to and perpendicular to the direction of the

applied field:

L

9 = 2n r BE^ dJi (2) ^0

In Eq, (2), L is the length of the optical path through the applied

field. Making the usual assumptions , Eq. 2 can be rewritten

cp = 2n BV^^e'd"^ , (3)

where J,' is an effective length which includes the effects of fringing

fields at the edges of the plates, V is the applied voltage and d is the

plate spacing. From (3),

B = cpd^/2n . (4)

Eq. 4 can be used to provide an estimate of the measurement uncertainty.

Assuming all uncertainties are small,

M^ + ^+ AT + oAd+oAV B"cp-X'-^d~ V'-

The measurement uncertainty of the pulsed high voltage is ±1% of the

peak value, so

2 ^= 0.02

8
The plate spacing can be controlled and measured only to within 0.01 cm,

so

2 ^ - 0.03

a With the cells used, the total contribution of the uncertainty of length

measurements is less than 1% even if the effects of fringing fields are included '

, so

The uncertainty is somewhat more difficult to express because the

phase shift is not directly measured. The experiment is arranged in

such a way that light polarized at an angle of 45° to the direction of

the applied field passes through the Kerr cell and then impinges upon

an analyzer oriented perpendicular to the direction of the polarization.

Under these conditions, the transmittance through the optical system is

described by the relationship *

I/I = sin^ [cp/2] (6)

where I is the instantaneous value of the transmitted light and Im

denotes the maximum amount of transmitted light. Eq. 6 can be solved to

yield

9

cos'-^IC-l)'' (l-2[l/l ± A I/I ])] - cos"^[(-l)''(l-2 I/I )] m m m

nrr + cos"-^ [(-1)'' (1-2 I/I )] m

(7)

In Eq. 7, n is an integer defined by the relationship:

^cp = n + a , n where a is the appropriate positive number between 0 and 1. The method

of determining the value^o^ n from the Kerr system response was discussed

in previous publications '

. In order to clarify the relationship

between the uncertainty in the relative transmittance and the uncertainty

in the phase shift, approximate values calculated from Eq. 7 are presented

in Table 1, In the preparation of this table it was assumed that

A(I/I ) = 0.1. The values for Acf/cp are not symmetrical about for a S3mmie?rical choice of ), i.e., ±0.1. For this reason the values

listed in Table 1 are approximate and only intended to indicate the trend

of measurement uncertainty with increasing phase shift.

9
Table 1. The approximate measurement uncertainty,"^, is tabulated as a function of the magnitude of the total phase shift. The phase shift is

presented in terms of n + (l/l^) because this is the quantity obtained

from the measurement.

Phase Shift

Ln + (I/I )] Acp/cp (%)

0.3 18

0.6 11

0.9 19

1.2 4.8

1.5 4.3

1.8 6.3

2.1 6.7

2.4 2.7

2.7 2.7

3.0 13

3.3 1.9

3.6 1.9

3.9 4.6

4.2 1.9

4.5 1.4

4.8 1.7

5.1 2.5

5.4 1.2

5.7 1.3

6.0 12

6.3 1.1

6.6 1.0

6.9 2.1

For measurements in nitrobenzene n >> 6, so the error due to A9

was negligible compared to other errors in the system. Most of the water

data were taken with [n + (l/l )] = 2.2 - 2.8. The estimated uncertainty

m

is then ±2% as 0.1 would be an upper limit to the uncertainty in I/I

in these experiments. This analysis leads to an uncertainty of ±6% for

nitrobenzene and ±8% for water.

In Eq. 5, the sources of error were identified and their relative

magnitudes have been estimated. To minimize the error a number of cells

were specially constructed and a variety of measurement techniques were

developed. Consistency between these approaches lends credibility to

10
the results. The three measurement techniques used can conveniently be

labeled "End-Field Free" Cell Technique, Two Cell Technique and Single

Cell Technique.

B) "End-field free" Cell Technique.

The "end-field free" cell. Fig. 3, used to make Kerr coefficient

measurements is designed specifically so that the applied electric field

is uniform over the liquid-filled segment of the light path, i.e., so that the edge field distortions are exterior to the portion of the light

path through the liquid. The cell was constructed with two parallel

plate electrodes overlapping a four-sided, rectangular, glass chamber

used to confine the nitrobenzene. The surfaces of the glass chamber

and electrodes were ground optically flat so as to allow glass-to-metal

sealing and the glass chamber was designed to provide a liquid volume

of 1 cm X 1 cm X 4 cm.

For a light path centrally located in the glass chamber and parallel

to the length of the electrodes the total phase retardation 9 is given

by

cp = a^ + cp ^ (7)

J 2 mech ^ d

where I is the interior length of the glass chamber (the length of the

nitrobenzene portion of the light path) and

^j^^^^j^

is the phase retardation

resulting from mechanical strain present in the glass windows at the^ends

of the chamber. During construction of more conventional Kerr cells

the cell is annealed to reduce to a negligible quantity. In this

case, however, the glass is subjected to mechanical stress so cp ^0. * ' ^ ^mech

The approach taken was then to assume that '^^^^j^ was independent of the

applied voltage. The strain-optic coefficients of the glass that was used

are not well enough known to predict the magnitude of the voltage

dependent phase shift. An order of magnitude estimate can be obtained,

however, using published typical values^. With the geometry of the cell

used, it would require a force in excess of 100 N to cause a phase shift

of 10~ TT, With the field strength used during this investigation, the

net attractive force should not exceed 15 N.

So under the assumption that '^^^^ is a constant, the electro-optic

coefficient can be determined from two separate experiments which

measure the phase retardations 9^ and at two different applied

voltages and V^, respectively. Using this information, both B and

9 1 can be calculated from Eq. 7: mech ^

(cp.-cp )d^

B = 2 — (8)

InJl (V^ -V^ )

11
and

Tr 2 , 2

^mech

V

^ V

^

1 "2

The total uncertainty in B depends on the uncertainties in measuring the

phase retardation, applied voltages and the geometry parameter d /£ .

C) Two Cell Technique

This technique utilizes the two Kerr cells of conventional design

shown in Fig. 2, For a centrally located light path between the

electrodes in each cell the applied electric field will vary along the

length of the light path due to the fringing fields at the ends of the

electrodes. Each cell is constructed to be identical with the other

except for the length of the electrodes where the respective lengths for

cell A and cell B are 24 cm and 12 cm.

From Eq, 3, the phase retardation for each cell can be written

V

2

cp^ = 2nB (—) V ^ , ^ (10)

-• , , A

V

2 ^B

= 2nB (-) . (11)

B

The cells are constructed so that d. = d„ and it is assumed that

A B

= + Ai (12)

= ji^ + Aji (13)

where AS. is the correction that must be added to the geometric length

of the plates to obtain the effective length. The basis for the

assumption that^Ail is independent of the plate length is the theoretical

work of Lemoine , Chaumont , and most recently by Thacher . In Thacher's

12
work he derived expressions for

2 ^2

< m > = J

Y. di . (14)

0

This is related to our £' through the equation

V = < > d^/v^ . (15)

For Rogowski profile electrodes and a central light path, he derives

£' = £ + d/TT , (16)

which is of the form of Eqs, 12 and 13. For electrodes which do not

conform to Rogowski profile and for non-central paths the correction

term is more complex, but is still independent of path length.

Using Eqs. 12 and 13 and assuming d = d = d, Eqs. 10 and 11 can be

solved to yield

2nV i^-^j

D) Single Cell Technique

To clarify the motivation for this approach, Eq, 3 can be rewritten

2 2 = ttV /V m where

(18)

A physical significance of V is that it is the lowest voltage

necessary to produce a transmittance maximum through the system. If

two different liquids are used in the same cell, the following

13
relationship is valid

V , ^ B-

/ mi s 2 (7-) =i- , (19)

m2 1

where the subscripts 1 and 2 refer to the two liquids.

This approach is convenient in that years of development have

demonstrated that V for a given cell can be determined with an m 9 uncertainty no larger than ±1%. If the Kerr constant of one substance

is known, then the measurement of the cell constants of the same cell

containing first the known liquid then the unknown liquid provides a measure of the Kerr constant without the associated problems of

geometrical measurement. This approach was used to determine the ratio

of the Kerr coefficient of water to the Kerr coefficient of nitrobenzene,

The final, and historically most coianion, method to determine B using a single cell is to measure the geometry of the cell, the applied voltage

and the phase shift and use Eq. 3,

, ; cp = 2nBi'V^/d^ ,

to calculate B. This approach was used but in a slightly modified form

to conform with established practice in the authors' laboratory. Using

Eqs. 3, 6, and 18, one obtains

I/I = sin^ [(n/2) (V/V )^] . m m

Then l/l and V are measured and V is calculated. Rewriting Eq. 18, m m

1 d

2 B = (—) , (20) m

is obtained. So from V , which contains the measurement of cp and V,

m

and the cell geometry, B is calculated.

IV Results

A) Magnitude of the Kerr Coefficient of Water and Nitrobenzene

A flow chart showing the measurement strategy and the interrelationships between the various experimental techniques is shown in

14
Fig. 5. This section describes the results obtained in each of the

diagramed measurements.

Using the "end-field-free" cell, the Kerr electro-optic coefficient, B, was measured at a number of different wavelengths. Values B of nitrobenzene were determined from a series of at least ten experiments at

each specified wavelength. Within each set of measurements no determination of B differed from the average by more than 3%. The

measurement technique consisted of recording the phase shift and the

applied voltage at two different voltage levels. Using these values in

combination with the cell geometry allows Eqs. 8 and 9 to be solved for

the Kerr coefficient B and for 9 , . The results are summarized in

Table 2, The electro-optic Kerr coefficient of nitrobenzene as measured

using the "End-Field Free" Technique at 296 K.

Wavelength (nm) B(m/V )

cp mech

632.8

-12

3.17 X 10 -0.46

514.5 4.22 X lO"-'-^ -0.56

496.5 4.43 X 10"^2 -0.68

488.0 4.59 X lO"-*-^ -0.58

476.5 4.70 X

10"-^^ -0.27

The results in Table 2 need two points of additional clarification.

First, data were taken at nominal voltage levels of 60, 80, and 100 kV.

Three sets of voltage pairs were then used at each wavelength to

compute B according to Eq. 8. No determination differed from the average

by more than 1.5% and no evidence of a voltage dependence of B could be

identified. This observation is significant, not because the Kerr

coefficient of nitrobenzene is expected to have a measurable voltage

coefficient at these field strengths, but because Eq. 8 was derived

assuming , was voltage independent to within the precision of this

" mech , . , , , experiment. It therefore seems reasonable that ^^^^-^ """^ projnarily due to

residual strain left in the glass during fabrication of the cell. The

variation in ^ would then be due small variations in the optical path

through the An additional source of evidence supporting this

interpretation is that expanded beam observations did show existence of

strain birefringence and that the magnitude of this birefringence is

indeed a function of position.

15
PHASE SHIFT + GEOMETRY + VOLTAGE

(

i »

SINGLE-CELL TWO-CELL END-FIELD FREE

TECHNIOUE TECHNIQUE TECHNIQUE \/

i

SINGLE-CELL TECHNIQUE

.

' »

B' B(H20)

B(C6H5N02) B(C6H5N02)

i

B(H20)

B'

I

BIHjO)

BICeHsNOj)

i

BIHzO) B(H20) B(H20)

Fig. 5. Flow chart for the various measurements made of the Kerr

coefficients of water and nitrobenzene. B' denotes the Kerr

coefficient of Halowax Oil. This fluid was used to transfer

measurements between two different cells.

1 6
Because the errors associated with cell geometry could lead to a •systematic error in the results, it was decided to repeat the measurements

using the two-cell technique. Using this technique, applying the same voltage pulse to both cells, and calculating B according to Eq. 17 led

to a value B = 3.24 x 10 mV at 296 K and a wavelength of 632.8 nm. The two methods therefore agree to about 2% which is well within the

predicted measurement uncertainty.

The cells that were used in the two cell technique were then used to

determine the ratio of the Kerr constant of water to the Kerr constant of

nitrobenzene. The measurement involved determining the cell constant

of the cell with longer electrodes when it was filled with nitrobenzene

and when it was filled with water. Using Eq. 19, it was determined that

B(H20)/B(C^H^NO2) = 1.058 x lO"^, at 295.6 K.

As a further check, it was decided to perform measurements of the Kerr

coefficient of water using a plastic cell with electrodes having

approximately a Rogowski profile. This would yield two different

determinations of the Kerr constant. One would be simply from measurements of the phase shift and geometry and the other would utilize the

single cell technique. Unfortunately nitrobenzene could not be used

in the plastic cell because nitrobenzene, a good solvent, would

destroy the cell's structural integrity. A less direct procedure was then used. First, a glass cell was filled with nitrobenzene and its cell

constant was measured. The cell was then filled with Halowax Oil,

a commercial blend of chlorinated naphalenes , and its cell constant

determined. From these observations, the following ratio was obtained

at 295.6 K,

B'/B(CgH^N02) = 4.209 x lO"^,

where B' represents the Kerr coefficient of the Halowax Oil. Now

Halowax Oil and water could be used in the plastic cell to determine

the ratio of their cell constants. The procedure used in this case was

first to fill the cell with water (water used was the house distilled

water), seal the cell, and determine its cell constant. Through a small

hole in the top of the cell, the water was then pumped out> the cell was dried and filled with Halowax Oil, This procedure was adapted to insure

that the plate spacing remained unchanged during the refilling process.

Using this technique, the following ratio was obtained at 295.6 K,

BCH20)/B' = 0.2499.

So,

17
B(H„0)/BCC,H,NO_) = 1.052 x lO'

i. o _) Z

at 295.6 K.

It should be noted that the two single cell techniques yield results

that differ by less than 0,6%. From the temperature data presented

in Table 3 . the Kerr constant of nitrobenzene at 295.6 K is

3.26 X 10" m/V . The two corresponding values of B(H20) are

BCH^O) = 3.45 X lO"-'"^ m/V^

and ,

"

B(H20) = 3.43 X lO"-^^ m/V^ .

From the cell constant and the geometry of the plastic cell an additional value of the Kerr coefficient of water can be derived.

Recall from Eq, 16 that

]l' = 2. + - .

IT Inserting this into Eq. 20, yields

2 ~ 2a + d/7T) W ' m

-3

In this case % = 0.2998 m, d = 6.32 x 10 m and V = 44,010 V. So,

ra

B = 3.42 x lO"-"-^ m/V^

The close agreement among these three values is thought to be

serendipity, because, as was discussed in connection with Eq, 5, the

estimated uncertainty in these measurements is ±8%,

An additional source of error was identified during the course of these

investigations. Under microsecond duration pulses the water was occasionally in sufficiently turbulent motion as to modulate I^, the .

18
maximum transmitted irradiance. This is in marked contrast to

experience by the authors with nitrobenzene and a-dichloronaphalene in

which no turbulence is detected under high voltage pulses of this

duration.

In order to put this anomalous behavior into perspective, Schlieren

studies have determined that if nitrobenzene is subjected to a step

voltage pulse, it takes a few milliseconds f oj^turbulence to become

established. In addition theoretical studies of unipolar injection

predict an approximately equal time interval for the initial injected

charge to traverse the gap from the injecting electrode to the collectin

electrode. The agreement between these two studies is to be expected if

the motion of charge through the fluid is the cause of turbulence.

One can assume in water, as in other polar fluids, that the primary

source of carriers at these field strengths is unipolar injection.

This would, however, require that the mobility of the carriers be three

orders of magnitude larger than the mobility in other polar fluids. Thi

possibility cannot be ruled out without more detailed knowledge of the

nature of the charge carriers in the two cases.

Another possibility is that the source of carrie^^ is not injection but

volume dissociation in the water. Previous work has indicated that

about one in 10 of the water molecules will be dissociated into ionic

carriers. It is therefore possible that the turbulence, rather than

propagating from one electrode to the other, is formed at approximately

the same time throughout the liquid.

A more detailed investigation of the conduction mechanisms in water is

beyond the scope of this work, but is an area for further investigation.

For completeness, it should be noted that the water used in this study

had a typical resistivity of 10 Q 'cm.

Previous results^ have indicated that accurate Kerr measurements can

be made even in the presence of turbulent motion. This was true because

in that case the ratio of the amount of scattered light to the amount of

transmitted light was undetectably small with the apparatus used. A

second effect occurs, however, when the scale of turbulent motion is of

order the size of the light beam. In this situation the fluid motion

causes density gradients in the fluid. The density gradients lead to a change in the index of refraction which serve to deflect the light beam.

As a consequence, the light beam "wanders" across the photo-sensitive

surface of the detector. Because of spatial variations in quantum

efficiency of the detectors used and/or because of the fact that a portion of the light beam may miss the photosensitive surface altogether

the photocurrent is amplitude modulated. This leads to an envelope on

the Kerr response and was first detected in this study as an apparent

voltage dependence of the cell constant. In this study the effects of

fluid motion were minimized in three ways. First the distance between

19
the photodetector and the Kerr cell was kept as short as possible. The

other two approaches exploited the fact that this larger scale turbulence was apparently modified by the geometry of the cell. In the glass cell

there was no detectable amplitude modulation during the first 6 ys of the

applied voltage pulse, Kerr constant measurements were then taken only

during this relatively quiescent period. In the plastic cell the time

interval between the rise of the voltage pulse and the onset of amplitude

modulation could be varied by varying the water level in the cell. By

a judicious choice of water level no amplitude modulation occurred during

the entire 10 ys voltage pulse,

A final word of caution is in order concerning the electric field

dependence of the Kerr coefficient of water. There is some evidence

that the dielectric constant of water is 0,7% lower when measured at a field strength of 250 kV/cm than when measured in weak fields. The

uncertainty in the present work precluded the observation of any

corresponding saturation effect, but the possibility of saturation

cannot be ignored if the present work serves as the basis of measurement

of voltage at significantly higher field strengths.

B) Temperature Dependence of the Kerr Coefficients

- 14 15 Experimental evidence and theoretical investigations , which

assume that the structure of the liquid molecules and their interactions

are independent of temperature and applied fields, predict the following

temperature dependence for the electro-optic Kerr coefficient

where the a's are constants.

This expression has been sufficient to describe the behavior of the

Kerr coefficient of nitrobenzene. Recent observations of the Kerr effect

in water"'" »"'"^, however, indicate an inflection point for the temperature

dependence of B at 308 K, Those authors suggest that in water

may be a more accurate functional form for the Kerr coefficient. The

values of the electro-optic coefficient measured at selected temperatures

are fitted by the method of least squares using Eq, 21 for nitrobenzene

and Eq. 22 for water. The a's and 6 *s are determined by this fitting

procedure. The functional form of the temperature dependence for both

fits are normalized by taking the ratio of B(T) to the value predicted

B(T) = Q;^ + (a^/T) + (a^/T^) , (21)

,3 (22)

by the fit at 298 K.

20
For the first observation of the temperature dependence in nitrobenzene,

the two cell technique was used over a temperature range from 296 K to

305 K. The measurement procedure was to heat the cells to approximately

310 K and then allow them to cool to room temperature. Measurements

of the electro-optic coefficient were made after every 1 K drop in

temperature as the cooling rate was approximately linear with a slope

of about 2 K/h. The fit to these measured values of B, when extrapolated

beyond the temperature range of the measurements, indicated a minimum in

the Kerr elec tro- optic coefficient at 350 K.

To verify that the minimum was only a result of the extrapolation and

as an additional check on the original data, the experiment was repeated

several months later using the single cell technique. In this case measurements were made at selected temperatures between 280 K and 315 K. Over a common . tempera ture range the measurements agreed to within ±17o. In addition, over the common temperature range between these measurements

and those of Gabler and Sokab''"'^, there was agreement of order ±27o. These

results are plotted in Fig. 6. It should also be noted that with this

extension of the measurements to a higher temperature, the minimum value

of Eq. 21 is shifted to a higher temperature. This implies that Eq . 21

is an accurate description of the temperature dependence of nitrobenzene

over the range the model is intended to be applied.

If the normalized temperature dependence is fitted to an equation of the

form

B/B(298 K) = Aq + (A^/T) + (A^/T^),

the following are the values for the constants

Aq = 8.085

A^ = -5.450 X 10"^ K

A^ = 9.954 X 10^

These constants are intended to serve only as guide for the

interpolation between the data points given. Experimental values for

B at selected temperatures, and at a wavelength of 632.8 nm are presented in Table 3. At least 5 determinations of B were made at each

temperature; each value of B differed by no more than 0.6% from its

average value.

21
1.5

1.0

oo

CD

CD

0.5

GABLER & SOKOB A = 546.1 nm

• PRESENT RESULTS A = 632.8 nm

J I I I I 1-

280 300 320 340 360 380

TEMPERATURE (K)

Fig. 6 Comparison of present and previously published data for the

temperature dependence of the Kerr coefficient of nitrobenzene.

Solid line represents least-squares fit to present data.

22
Table 3, The Kerr coefficient of nitrobenzene at selected temperatures.

B(m/V^) T(K)

3.89 X

-12

10 286

3.47 X 10-^2 294

3.39 X

10-12 295

3.22 X 10-12 299

3.04 X 10-12 304

2.86 X 10-12 308

2.61 X

10-12 314

Using the single cell technique (glass cell) the Kerr electro-optic

coefficient of water was also determined for selected temperatures

between 280 K and 315 K at a wavelength of 632.8 nm. The mean values

of B determined from a series of at least 5 experiments are compared

in Table 4 at each specified temperature. The values of B determined

in each series of experiments did not vary by more than 1.3% of the

mean value.

Table 4, The Kerr coefficient of water at selected temperatures.

B(m/v2) T(K)

3.56 X 10 280

3.70 X

10-1^ 286

3.64 X lO"!^ 288

3.45 X 10-1^ 296

3.21 X lo"-^^ 305

3.05 X lO--*-"^ 313

3.03 X 10-1^ 314

2.95 X 10-1^ 322

23
Again if the normalized temperature dependence is fitted to an equation

of the form

B/B(298 K) = Dq + (D^/T) + (D^/T^) + (D^/T^),

the values for the coefficients are

Dq = 255.46

D = -2.322 X 10^ K

= 7.038 X

10'^

= -7.087 X 10^

as was the case for the fit to the nitrobenzene data, these coefficients

are intended only to aid interpolating between experimental values.

In Fig,^^, these data are compared with the recent experimental data

of Chen . An analysis of the difference between these results is

presented following the discussion of the wavelength dependence of the

Kerr coefficient of water.

The temperature dependence data were taken using the glass cell because

for glass and metal the coefficient of linear expansion is of order

10 /K. The change in cell dimensions with temperature should, therefore, be negligible. To verify the accuracy of the values found for

the temperature dependence, however, measurements were made in the

plastic cell at 288, 295.5 and 304 K. These observations were within

±5% of the values obtained in the glass cell.

C) Wavelength Dependence of the Kerr Coefficients

The Kerr coefficient as a function of wavelength at a constant

temperature is given in Table 2. From the definition of the Kerr

coefficient, Eq. 1, it can be seen that in the absence of dispersion,

i.e., if the index of refraction is independent of wavelength, the

product XB should be a constant. From the data in Table 1 the average

value of BX is 2.171 x 10 (m/V) . No individual value differs by

more than 8% from this average value. For purposes of comparison,

the measurement of the wavelength dependence of the Kerr coefficient

of nitrobenzene by a number of workers is presented in Fig. 8. Note

that there is satisfactory agreement among all authors oyer the common

range of measurement.

The present determinations of the Kerr coefficient of water ^s^g

function of wavelength and previous work by Orttung and Meyers are

24


compared in Table 5. In the present work no measurement differed from

the average by more than 1,2%. In the work of Orttung and Meyers the

standard deviation of all measurements at each given wavelength was

less than ±4% of the average value.

Table 5. Wavelength Dependence of the Kerr Coefficient of Water

B (m/V^) Wavelength (nm) Temperature (K)

Present

-14

3.45 X 10 632.8 295.6

Results 4.17 X lO"-'-^ 514.5

Orttung

-14

4.14 X 10 436

and

-14

3.22 X 10 546 298.2

Meyers 3.03 X lO"-*-^ 578

For the present work the average value of BX is 2,16 x 10 (m/V)

and the two determinations were within ±1% of that. From the work of

18 —20 2

Orttung and Meyers a value of BA equal to 1,77 x 10 (ja/V) is

obtained. Their three readings agree to within ±2%, The difference

between the two determinations of BA is indicative of the fact that the

present data yield a value for B which is about 20% larger than that

obtained by Orttung and Meyers.

A disagreement of that magnitude is, however, consistent with the

cumulative uncertainty of the two experiments. The estimated uncertainty

of the present results is approximately ±8% as was discussed previously.

Orttung and Meyers make no estimate of the uncertainty of their measurements, but they do state that the precision of their measurements is of

order 3-5%. It is assumed that this is due to A^/^ in Eq. 5, because

this would be typical of the uncertainty in when measurements are made

below the first transmittance maximum. It is difficult to estimate their

uncertainty with only their publication as guidance but their dimensional

measurement uncertainty was somewhat less than that in the previous

work, so in their case

2 ^ - 0.01-0.02

d

^ = 0.01

27
From their description of their voltage measurement system, the measurement uncertainty is probably of order 0.03-0.05. If these estimates of

their uncertainty are valid, the two experiments agree in the sense that

their error bars "overlap".

A seemingly more serious problem is the substantial disagreement between

the l^gmperature dependence data presented here and that determined by

Chen . As an analogy to this situation, the wavelength dependence of

the Kerr coefficient of nitrobenzene was plotted in Fig. 8. The results

of six different investigators are presented and they all agree to

within a few percent. The unnormalized values of the Kerr coefficient

however differ by as much as 40%.

In this case neither the magnitude nor the shape of the temperature

dependence agree, A significant source of error in the measurement

of the temperature variation of the Kerr coefficient is a thermal change

in cell characteristics . This is the reason the results presented here

were verified using a completely different cell and a completely

different method of temperature control.

A final point to be considered is the role of turbulence as a source of error. When turbulence was present on a sufficiently large scale to

deflect the beam, there was significant error; typically a reduction

of the apparent value for B by 10-30%. Neither of the previous works

mentions the existence of turbulent motion in their fluid. In order to

use the Kerr effect in water for accurate voltage measurement, the

prediction and control of turbulent behavior in the liquid must be

accomplished

.

V. Summary and Conclusions

The Kerr coefficients of both water and nitrobenzene were measured

and the variation of these quantities with temperature and wavelength

was investigated. At a temperatureof 296 K and at a wavelength of

632.8 nm the following values for the Kerr coefficients were obtained

BCC^H^NO^) = 3.24 x lO"-*-^ m/V^, and

B(H20) = 3.43 X lO"-*"^ m/V^ .

The estimated uncertainty in the measurement in nitrobenzene is ±6%

while in water the estimated uncertainty is ±8%.

The agreement between this determination of the Kerr coefficient of

nitrobenzene and previous measurements is judged to be satisfactory.

The determination of the temperature and wavelength variations agreed.

28
within experimental uncertainty with all other measurements. The

magnitude of the Kerr coefficient as measured for this report lies

within the range of previously published data. The spread of published

values, approximately ±30% of an average value, discourages more

precise comparison.

The smaller Kerr constant of water causes a smaller phase shift per

unit field strength. The percentage uncertainty is then larger for

water because the percentage uncertainty in the phase shift is larger.

Not only were the results with water less precise than those with

nitrobenzene, but there was also an additional source of experimental

difficulty. In water there was substantial fluid motion during 10 ps

duration pulses. In contrast, nitrobenzene does not demonstrate any

fluid motion until the voltage has been applied for a few milliseconds.

Two qualitative explanations for this behavior were postulated. First,

if the production of charge carriers in water is due to unipolar

injection, as is probably the case in nitrobenzene, one can infer that

the effective mobility of the carriers in water is about three orders

of magnitude larger than it is in nitrobenzene. As no detailed knowledge

of the nature of the carriers exists in either case, this must be

accepted as a possible explanation. A second explanation might be that

bulk dissociation rather than unipolar injection is the primary source

of carriers. If one accepts the model that turbulence is caused by the

motion of charges through the fluid, it then follows that turbulence will

form more rapidly in a fluid if the carriers are formed by bulk

dissociation. This is because experimental studies have shown that

turbulence propagates from one electrode to the other in the case of

unipolar injection. Theoretical work has shown that the propagation rate

is approximately equal to the transit time of the injected charge

carriers. If the carriers are, on the other hand, uniformly distributed

through the interelectrode region it seems likely that the turbulence

would be formed at approximately the same time throughout the gap. To

the authors' knowledge, there have been no experimental investigations

of this phenomenon, and a more detailed study was beyond the scope of

this work. Turbulence in water does remain an area that merits further

study, because in Kerr effect measurements it is a source of error of

unknown magnitude at best and at worst makes Kerr results uninterpretable.

A final observation is that although the results in nitrobenzene

demonstrate the expected agreement with previously published data, this

is not true in water. The difference in magnitude of the Kerr coefficient

between this and previously published work can probably be attributed to

the cumulative error in the two experiments. The difference in the

temperature dependence is more striking and no convincing explanation

for the difference is apparent.

The primary conclusion is that although nitrobenzene can be used for

accurate measurement of a wide, and fairly well defined, variety of high

voltage pulses, the situation in water is more poorly understood. This

report indicates water can be used for the measurement of pulsed high

29
voltage, but it also identifies two unresolved areas of difficulty. The

first is the existence of turbulence under pulses of duration of a few

microseconds. The performance of accurate measurements in water

requires a more precise definition of the dependence of turbulence on both electric field strength and time. The second is the possibility

of a voltage dependence of the Kerr coefficient of water due to

saturation of the alignment of the molecules with the applied field. The

field strength at which this effect becomes significant must be

determined so that the range of applicability of the observations

presented in this report can be established.

Acknowledgments

The authors would like to thank Mr. William Bagley for his assistance

in the construction and operation of the experimental apparatus and

Miss June Jones for her help in the accumulation and analysis of the

data. The assistance of Mrs. Mary Cutkosky in the preparation of the

diagrams and the editorial and typographical help supplied by Miss J.

Cunningham are also gratefully acknowledged.

30
References

1) R. E. Hebner and E. C. Cassidy, "Measurement of 60 Hz Voltages Using

the Kerr Effect" Rev. Sci. Instru., vol. 43, pp. 1839-41, 1972.

2) E. C. Cassidy, W. E. Anderson, S. R. Booker, "Recent Refinements and

Developments in Kerr System Electrical Measurement Techniques"

IEEE Trans, on Instr. and Meas., vol. IM-21, pp. 504-510, 1972.

3) P. D. Thacher, private communication.

4) D. C. Wunsch and A. Erteza, "Kerr Cell Measuring System for High

Voltage Pulses" Rev. Sci. Instru., vol. 35, pp. 816-820, 1964.

5) E. C. Cassidy, R. E. Hebner, R. J. Sojka and M. Zahn, "Kerr-Effect

Studies of an Insulating Liquid Under Varied High-Voltage Conditions"

IEEE Trans. Elec. Insul . , vol. EI-9, pp. 43-56, 1974.

6) N. F. Borrelli and R. A. Miller, "Determination of the Individual

Strain-Optic Coefficients of Glass by an Ultrasonic Technique"

Appl. Optics, vol. 7, pp. 745-750, 1968.

7) J. Lemoine, "Verification de la Loi de Kerr-Measures Absolutes"

Compte Rend., vol. 122, pp. 835-837, 1896.

8) L. Chaumont, "Recherches Experimen tales sur le Phenomene Electrooptique de Kerr et sur les Methods Servant a 1 'Etude de la Luminere

Polarisee Elliptiquement" Ann. de Phys . , vol. 5, pp. 17-78, 1916.

9) E. C. Cassidy, "Development and Analysis of Techniques for

Calibration of Kerr Cell Pulse-Voltage Measuring Systems V" NBS

Report 10 493, Oct. 1971 and the references contained therein.

10) E. J. Hopfinger and J. P. Gosse, "Charge Transport by Self -Generated

Turbulence in Insulating Liquids Submitted to Unipolar Injection"

Phys. Fluids, vol. 14, pp. 1671-1682, 1971.

11) M. Zahn, C. F. Tsang, and S. Pao, "Transient Electric Field and

Space Charge Behavior for Unipolar Ion Conduction" J. Appl. Phys.,

vol. 45, pp. 2432-2440, 1974.

12) D. Eisenberg and W. Kauzmann, The Structure and Properties of Water.

Oxford Press, London, 1969.

13) J. B. Hasted, "Liquid Water: Dielectric Properties" Water A

Comprehensive Treatise , F. Franks, ed. , Plenum Press, New York,

vol. 1, p. 262, 1972.

14) F. Gabler and P. Sokob, "Die Temperaturabhangigkeit des

Kerreffektes am Nitrobenzol" Zeitschr. F. Techn. Physik, vol. 6,

pp. 197-201, 1936.

31
A. D. Buckingham and R. E. Raab , "A Molecular Theory of the

Electro-optic Kerr Effect in Liquids" J. Chem. Soc, pp. 2341-51,

1957.

Y. Chen, "Temperature Dependence of the Kerr Constants of Water

and Heavy Water" University Microfilms, Inc., Ann Arbor,

Michigan, #71-3874, 1970.

Y. Chen and W. H. Orttung, "Observation of a Minimum in the Kerr

Constants of Light and Heavy Water Near 30°" J. Phys. Chem.,

vol. 72, pp. 3069-70, 1968.

W. H. Orttung and J. A. Meyers, "The Kerr Constant of Water"

J. Phys. Chem., vol. 67, pp. 1905-1910, 1963.

M. A. Jeppeson, "Measurements of the Kerr Electro-optical Effect"

Am. J. Phys., vol. 24, pp. 623-25, 1956.

L. R. Ingersoll and W. R. Winch, "Dispersion of the Kerr Electrooptic Effect in the Short Infrared Spectrum" Phys. Rev., vol. 44,

pp. 399-405, 1933.

W. Ilberg, "Eine Methods zur Bestimmung der Kerr-Kons tante Schlecht

Isolierender Staffe mit Hilfe Elektrischer Wechself elder " Physik.

Zeitschr., vol. 29, pp. 670-78, 1928.

G. Szivessy, "Uber die Temperatura bhangigkeit der Dispersion des

Elektrooptischen Kerref f ektes " Z. Physik, vol. 2, pp. 30-49, 1920.

H. E. McComb, "Dispersion of Electric Double Refraction and

Ordinary Dispersion in Liquids" Phys. Rev., vol. 29, pp. 525-40,

1909.

32
NbS-II4A 1RS.V. 7-73)

U.S. CEPT. OF COMM.

RIRI 1 nr. P A PH 1 f" DATA

SHEET

1. PUBLICATION OR KtPOKT NO. NBSIR 74-5A4

2. Gov't Accession

No.

3. Recipient's .^ccci>iion No.

4. TITLE AND SUBTITLE

Kerr Coefficients of Nitrobenzene and Water

5. Publication Date August 7, 1974

6. Performing Organization Code

7. AUTHOR(S)

Robert E. Hebner. Jr.. Richard J. Sojka, and Esther C. Cas.qidy •

8. Performing OrgQ.0. Report No. NBSIR 74-544

9, PERFOR.MING ORGANIZATION NAME AND ADDRESS

NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

WASHINGTON. D.C. 20234

10. Project/Task/U^ork Unit No.

11. Contract/Grant No.

AFWL 74-067

12. Sponsoring Organization Name and Complece Address (Street, City, State, ZIP)

Air Force Weapons Laboratory

Kirtland Air Force Base, N.M. 87117

13. Type of Report & Period Covered Final

Sept. 17, '73- June 30, '74

14. Sponsoring Agency Code

IS. SUPPLEMENTARY NOTES

1'6. ABSTRACT (A 200-word or leaa (actual summary ol moat sigpilicant information. II document includes a signilicant

bibliography or literature aurrey, mention it here.) - The Kerr coefficients of both water and nitrobenzene were measured and the variation

of these quantities with temperature and wavelength was investigated. At a temperature

of 296 K and at a wavelength of 632.8 nm, the following values for the Kerr coefficients were obtained B(C^HcNO.) = 3.24 x lO"*^ m/V^, and B(H^O) = 3.43 x lo'^^ m/

The estimated uncertainty in the measurement in nitrobenzene is ±6% while in water

the estimated uncertainty is ±8%.

The primary conclusion drawn is that although nitrobenzene can be used for accurate

measurement of a wide, and fairly well defined, variety of high voltage pulses, the

situation in water is more poorly understood. The, primary areas of difficulty using

Water are the existence of turbulence under pulseS' of duration of a few microseconds

and the possibility of a voltage dependence of the Kerr coefficient due to a saturation

6f the alignment of the molecules with the applied field.

17. KEY WORDS (six to twelve entries; alphabetical order; capitalize only the first letter of the first key word unless a proper name; separated by aemicolona)

Dielectric fluids; electrical properties of fluids; high voltage measurements; Kerr

coefficient; Kerr effect; nitrobenzene; pulse measurements; water

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