Refractive Index Measurements in Fused NaNO 3 and KNO3

by a Modified Thermooptic Technique

Silas E. Gustafsson and Ernest Karawacki

A thermooptic technique for studying the temperature dependence of refractive index of liquids at room

temperature has been modified and applied to the study of molten NaNOa and KNO3 within a tempera-

ture range of some 80 K above the melting point for five different wavelengths within the visible spectral

range. The experiments show a temperature dependence of the polarizability for the two salts as calculat-

ed from the Lorentz-Lorenz formula, which cannot be explained by experimental inaccuracy.

1. IntroductionThe intention of this work is to present a ther-

mooptic method for determination of the refractiveindex of liquids and its temperature and frequencydependence. The measurements can be performedboth at high and low temperatures and over a largetemperature interval. To demonstrate the useful-ness of the method, the measurements were appliedto the molten NaNO3 and KNO3 . The obtainedvalues of the refractive index were then used to cal-culate the molar refractivity of NaNO3 and KNO3 ac-cording to the Lorentz-Lorenz equation",2

R 4 47rNa = n 2 - MR 3 7Nal+ 2 p (1)

where n is the refractive index, p is the density, M isthe molar weight, a is the polarizability, NA is Avoga-dro's number, and R is the molar refractivity. Themolar refractivity values obtained in such a way showthe dependence on temperature which was indicatedin some publications both for molten salts3 4 and or-ganic liquids.5 6 Besides the description of the ex-perimental arrangement and procedure with refer-ence to NaNO3 and KNO3, the paper includes the re-sults from the measurements of the refractive indexof air and fused quartz at temperatures up to 500°C.

II. Experimental ProcedureThe order of interference in the pattern influenced

by the optical path difference between two rays-onepassing through the plane-parallel test plate placed

The authors are with the Physics Department, Chalmers Uni-

versity of Technology, S-402 20 Gothenburg 5, Sweden.

Received 17 January 1975.

perpendicular to the optic axis and the other throughthe surrounding medium-is given by

m(to) = [n(to) - n(to)]d(to)/X, (2)

where np(to) and n/(to) is the refractive index of thetest plate and the surrounding medium, respectively,d(to) is the thickness of the test plate, and X is thewavelength of the incident light. The change of thetemperature from to to t causes a change of the inter-ference order from m(to) to m(t) given by

m(t) = [n (t) - n(t)] d(to) + (t - to)]P ~~~~~X (3)

where a is the thermal expansion of the test plate.Combining Eqs. (2) and (3) one gets

n,(t = n,(t - nb(t 0 ) - n(to). _ zmX

1 = aL (t - to) d(to)[1 + a(t- t)]'(

where Am = [m(t) - m(to)] and

A, An,, + [n,(t + at)- n(t + At)1 aA t At 1 + (t- to)

AK A5'- d(t)[L1 + a(t - to)] '

where Anp = np(t + At) - np(t), Afns = ns(t + At) -

n,(t), and AK = m(t + At) - m(t). The expressionnp(t + At) - n(t + At) is to be calculated from Eq.(4).

Equations (4) and (5) can be used in two ways: (1)To calibrate the test plate, i.e., to determine np(t)and Anp/At if the refractive index of the surroundingmedium nS(to), n,(t), and An,/At are known. This isthe case when the surrounding medium is vacuum orair. (2) To determine the refractive index of the sur-rounding medium ns(t) and Ans/At when the testplate is calibrated.

May 1975 / Vol. 14, No. 5 / APPLIED OPTICS

7 I -

-N Fig. 1. (a) High temperature thermostat: A, the in-vestigated liquid; B, the platinum vessel; C, the fur-nace; D, the inner quartz window; E, the light port; F,

-R the pneumatic pressure mechanism; G, the support-ing bench; H, the glass container for salt drying andmelting; I, the feeding channel of the cell; J, the milli-pore filter; K, the connection to the vacuum pump; L,the cooling flanges of the light port; M, the test plate;N, the plate holder [shown in detail in Fig. 1(b)]; 0,the protecting quartz tube; and P, the riffle in theplate holder for the thermoelement. (b) The plateholder: N, connection to the goniometer; M, the testplate; R, pivots for the adjustment of the test plate;

KA and S, protecting platinum collar.

(b)

The experimental procedure thus can be describedby the following steps: step one: calibration of thetest plate, which includes (a) determination of the re-fractive index of the test plate np(to) for one fixedtemperature to; (b) determination of the thickness ofthe test plate for this temperature to; (c) determina-tion of the thermal expansion a of the test plate if itis unknown; (d) thermooptical measurements of therefractive index of the test plate, i.e., determinationof the np(t) and np/At for the temperature intervalof interest.

Step two: Determination of the refractive index ofthe surrounding medium n(to) for one fixed temper-ature to (not necessarily the same as in point a) andthermooptical measurements of the refractive indexof the surrounding medium. [All constants in Eqs.(4) and (5) are at this stage known from the previousmeasurements.] A detailed description of the suc-cessive steps follows after the discussion of the exper-imental arrangement.

Ill. Experimental ArrangementTwo gas lasers were used as light sources for the

experiments: a He-Ne laser (6328-A) Spectra-Phys-ics model 132 and an argon-ion laser (4765-A, 4880-A, 4965-A, and 5145-A) Spectra-Physics model 141.The interferometer consisted of a lens system and adouble Savart polariscope with the necessary polar-oids. The intensity was measured by a photodetec-tor connected through an amplifier to a Hewlett-Packard double channel recorder model 7100 B.The second channel of this recorder was coupledthrough a differential voltmeter to a thermocouple sothat the temperature t and intensity (i.e., the changeof the interference order Am) could be determinedsimultaneously.

The test sample of the liquid (A) was kept in aplatinum vessel (B) placed in the middle of the ther-mostat (C) as shown in Fig. 1(a). In order to com-plete the cell, two optically flat quartz windows (D)were placed perpendicular to the optic axis touching

the two circular openings of the platinum cell. Thewindows were kept in position by the pressure trans-mitted to the two light ports (E), one on each side ofthe thermostat, from a pneumatic pressure mecha-nism (F) firmly fixed to the optical bench (G). Themechanism acted on the cooling flanges (L) on thelight ports. The air from the light ports was evacu-ated by a vacuum pump to avoid undesirable opticalpath differences between rays passing through thewalls of the thermostat.7

Prior to the actual experiment, the salt was placedin a special glass container (H), the opening of whichwas fitted to the feeding channel of the cell (I). Inorder to remove the moisture from the salt the tem-perature of the thermostat with the solid salt waskept at about 1000C during a couple of days beforethe experiment. Then the temperature was raisedabove the melting point, and the melt was forcedthrough a millipore filter (J) at the bottom of thecontainer directly into the test cell.

The test plate (M) was mounted in the holder (N)as shown in Fig. 1(b). The directing pivots (R) of theplate holder made it possible to adjust the test plateperpendicular to the light beam. The plate holderwas coupled to a precision goniometer (Wild T2).The asbestos rings (T) served to stop the air convec-tion along the tube (0) separating the cell and theplate holder from the thermostat [Fig. 1(a)].

The temperature was measured with a Platinel IIthermocouple protected by a quartz tube and low-ered directly into the cell along a riffle (P) in theplate holder. During the thermooptical measure-ments, the temperature was changed very slowly tominimize the difference in temperature between thesurrounding medium, the test plate at the recordingpoint, and the thermocouple. Furthermore the tem-perature was both raised and lowered, and the posi-tion of the fringes vs the temperature was controlled.The shift of the fringe position showed that a minorcorrection of the temperature was needed. The opti-cal system was chosen so that the shearing of the

1106 APPLIED OPTICS / Vol. 14, No. 5 / May 1975

(a)

"I

wavefront was as small as possible to make sure thatthe interfering beams are passing practically in thesame temperature region. It should be emphasizedhere that the wavefront-shearing interferometergives direct information about temperature gradientsinside the cell, which in no instance could be ob-served.

A. Determination of the Refractive Index of the TestPlate for a Fixed Temperature to

To determine the refractive index np(to) of the testplate for a fixed temperature, the rotatable platemethod8 was used. The rotation of the test plate, agiven angle ai from a position perpendicular to theincident beam, gives the change in the order of inter-ference accordingly:

Amrn= ns(to)d(to). {[n 2(to) - sin2ail 2- nr(to)

- cosa + 1,

where

nr(tO) = n(to)/n,(to).

If we denote

Yi[ai,Amri;nr(to)] =

[nr2(t,) - sin2 auP /2 - (tp) - cosa + 1Ami

then

(6)

where Y[ai, Ammi; nr(to)] are the values calculatedfrom Eq. (8) with the experimental pairs (, Ami)and already obtained value nr(to). Calculations ofthe plate thickness, according to Eq. (10), are per-formed at the same temperature for all wavelengthsthat have been used, which serves as an excellentcontrol of the internal consistency of the measure-ments. The rotatable plate method gives an accura-cy of the plate thickness of the order of :l1 X 10-5cm.

C. Thermal Expansion of the Test Plate

The measurements described in Secs. III.A andIII.B can be repeated for several different tempera-tures, and then the linear expansion coefficient a canbe calculated from the measured values of the platethickness for these temperatures. The error of thethermal expansion of the test plate measured in thisway is of the order of 1 X 10-7 deg-1.

D. Thermooptical Measurements of the RefractiveIndex of the Test Plate

The test plate is placed perpendicular to the inci-dent beam, and the temperature is slowly changedstarting from to t cover the temperature interval ofinterest. The change of the temperature and of

(8) fringe order at a particular point in the fringe patternis recorded continuously, and the refractive index ofthe test plate is calculated from Eqs. (4) and (5).

Yi[ a, nzi;n(to) = A (9)

is constant independent on the variables (as, Ami) aslong as a particular experiment is considered at con-stant temperature and n,/to) is the ratio of the truevalues of the refractive index of the test plate to therefractive index of the surrounding medium.

At a typical determination of nr(to) about fifteenindependent readings were taken at angles rangingfrom 35° to 88°. For every angle cei (and correspond-ing Ami) and an arbitrarily chosen nr(to), Yi can becalculated from Eq. (8). A linear function is numeri-cally adjusted to Y, vs as by a least squares method.The parameter nr(to) is then varied so that the slopeof the linear function becomes zero. This particularparameter giving a zero slope is the desired nr(to).In reality the measured values ao or/and Ami are af-fected by experimental errors that give some statisti-cal random fluctuation of Y[aj, Ami; nr(to)] aroundthe value X [n(to) * d(to)J->. The magnitude of thisfluctuation can be used as a measure of the experi-mental accuracy. When nr(to) is determined the re-fractive index of the test plate np(to) can be obtainedfrom Eq. (7) as the refractive index of surroundingn/(to) (vacuum or air) is known.

B. Plate Thickness d(to)

From Eq. (9) one has

E. Measurements of the Refractive Index of the Liquid

The measurements described in Secs. III.A andIII.D are repeated with the liquid as surrounding me-dium. The refractive index is then calculated fromEqs. (7), (4), and (5) using values obtained from thepreviously performed calibrations of the test plate.

IV. Results and Discussion

A. Refractive Index of Air

The calibration of the test plate was performedwith the air as the surrounding medium. It meansthat the accuracy of the known values of the refrac-tive index of air affects the accuracy of the refractiveindex of both the test plate and the liquid.

The refractive index of air has been measured witha very high precision at room temperature over alarge frequency range.9 This makes it possible tocalculate the refractive index of air for all other tem-peratures. According to Eq. (1), one obtains forgases1

R n. ns(T) - 1 Red- T (13 P

where Rg is the gaseous constant, p is the pressure, Tis the temperature in Kelvin, and n,(T) is the refrac-tive index of the gas at temperature T. Assumingthat the molar refractivity for gases is temperatureindependent one gets for an isobaric process

d(t0 ) = it (t)YiaiAnzi;nr(to)] (1 0) ns2

(T) = 1 + TO[n2(T) - 1], (12)

May 1975 / Vol. 14, No. 5 / APPLIED OPTICS 1107

Table I. Refractive Index of Air for the 6328-A Wavelengtha

Temp. (nc, - 1) (n - 1) In,., - next!oC X 106 X 106 X 106

19.3 269 272 327.2 261 265 435.4 254 258 446.1 246 249 356.1 238 242 467.2 230 234 478.4 222 226 492.6 214 217 3

105.8 206 210 4122.2 198 201 3139.0 190 193 3158.1 182 184 2177.7 174 176 2201.3 166 168 2226.4 1.58 159 1258.2 150 150 0289.8 142 141 1340.6 134 130 4383.2 126 121 5435.8 118 112 6

a A comparison between refractive indices extrapolated (n,,t)with the Lorentz-Lorenz formula from earlier room tempera-ture values and the experimental results (nap.) of this investiga-tion.

where n(To) is the refractive index of air at roomtemperature. From this equation the refractiveindex for different temperatures T can be obtained.

To test this equation a separate experiment wasperformed at a wavelength of 6328 A. We used thesame optics and recording system as for the ther-mooptic measurements, but instead of the platinumvessel and the test plate a two-channel cell wasplaced in the middle of the thermostat. The cell wasabout 4 cm long. One of the channels of the cell wasconnected to the vacuum pump. To obtain the re-fractive index of the air at room temperature, thechange of the interference order was recorded vs thechange of pressure in accordance with

, _ [n (T)- 1]d(T) (13)

which allows the calculation of n(T). Then the airwas evacuated once more, and the temperature wasslowly increased while the change of the interferenceorder was recorded. It is evident that Eq. (4) can beused even in this case if one replaces np(t) and np(to)with unity. The results of the measurements for the6328-A wavelength are shown in Table I. The differ-ence between the experimental values and those cal-culated from Eq. (12) should not be greater than +4X 10-6, and this can thus be taken as the error of ourreference refractive index.

B. Refractive Index of Quartz

The test plate made of quartz Herasil I was cali-brated in the temperature interval from 200'C to

5000C. The thermal expansion of the test plateturned out to be 5 X 10-7 deg-1. The experimentalerrors of the refractive index of quartz calculatedfrom Eq. (4) was of the order of +0.5 X 10-6. Thecalculations of the refractive index from Eq. (4) weremade for a change of the interference order; accord-ingly, Am = 1, 2, 3 ... giving about fifty values of therefractive index. To this set of points a quadraticfunction was numerically adjusted. The coefficientsof the quadratic function in the temperature regionfrom 200'C to 500'C and for five measured wave-lengths are presented in Table II. It is seen that therefractive index of the quartz increases both withtemperature and frequency. The derivation of thequadratic equation gives dnp/dt, and it is interestingto compare these values of dnp/dt for quartz withthose obtained directly from Eq. (5) as shown in Fig.2. The points in the diagram were obtained by usingEq. (5), and the straight lines were received from thederivation of the quadratic equation with the coeffi-cient a and a2 from Table II. The diagram showsthat a nonlinear or nonquadratic variation of dnp/dtvalues for fused quartz does exist which confirms ear-lier observations.10

C. Refractive Index of the Molten NaNO 3 and KNO3The refractive index for molten NaNO3 was mea-

sured in the temperature interval between 310'C and390'C and for molten KNO3 between 340'C and4200C. Both NaNO3 and KNO3 were of analyticalgrade quality being commercially available. The ex-perimental errors of the absolute values of the refrac-tive index of fused NaNO3 and KNO3 can be estimat-ed to be of the order of +1 X 10-5. About 120 valuesfor the refractive index of NaNO3 and about 160values for KNO3 (with a thickness of the test plate ofabout 0.5 cm) were obtained. The coefficients of thequadratic and linear equations adopted to the refrac-tive index data are presented in Tables III and IV,respectively. The refractive index of molten NaNO3and KNO3 increases with frequency but decreaseswith temperature. The dns/dt values calculatedfrom Eq. (5) give a random fluctuation around thelines obtained from the derivation of the quadraticequations with coefficients from Tables III and IV.

The molar refractivity R was calculated from Eq.

Table II. Refractive Index of Fused Quartz Herasil I Expressedby the Equation n (t) = ao + at + a2t2, where t is in OC.a

X(A) ANr ao al X 105 a2 X 109 Or X 106

4765 4.5 1.463989 1.0411 4.673 2.84880 44 1.463263 1.0456 4.539 3.2496. 47 1.462700 1.0440 4.473 2.75145 46 1.461694 1.0461 4.393 2.86328 33 1.457300 0.9966 4.337 3.4

a A temperature range from 200'C to 500'C has been coveredwith the experiments. N is the number of experimental points,and is the standard error.

1108 APPLIED OPTICS / Vol. 14, No. 5 / May 1975

(1) for different frequencies but for the same temper-atures with the density values given by Janz.1" Theextrapolation to infinite wavelength was made on thebasis of the dispersion formula4

n2

- 1 c+ 2 - 2 no+ 2 == -(3)

(2)The calculated values of the molar refractivity can bepresented with an accuracy of +1 X 10-3 cm 3 whenthe experimental error of the density is of the orderof +1 X 10-4 g/cm3 and that of the refractive index of:1 X 10-5.

The results are presented in Tables V and VI.The observed variation of the molar refractivity as afunction of temperature is so large that it is far out-side the experimental inaccuracy. This effect will bethe subject of further investigations.

The absorption lines for the NO3- anion can be ob-tained from the equation4

R = N e2

( Pi2

- P+ 237r/i 'v - ,' V2 -

250 300 350 400 450 500 T [C]

Fig. 2. Variation of dnp/dt with temperature. The temperature

axis is the same for all wavelengths. The vertical axes are shifted

a certain distance to better elucidate the individual plots with the

beginning and the end clearly marked and numbered. The points

of the diagrams are obtained directly from the experiment accord-

ing to Eq. (5). The straight lines are calculated from the al and a2

values from Table II. The plot given by circles is from the mea-

surements at a 6328-A wavelength, the lower squares are from 5146

A, upper squares from 4965 A, filled triangles are from 4880 A, and

open triangles are from 4765 A.

where e and A are the charge and mass of the elec-tron, NA is Avogadro's number, R is the molar refrac-tivity, vi and pi are the vibration frequency and thenumber of the electrons with pir type of bond, V2 andP2 are the vibration frequency and the number ofother electrons, and VL is the frequency of the ap-plied field. For the NO3- anion Pi = 3 and P2 = 29.The calculations gave 128 mg and 21 mA for the mea-surements in molten NaNO3 and 130 mAt and 27 muin molten KNO3 , which is in good agreement withearlier observation.4

Table IlIl. Refractive Index of Molten NaNO 3 Expressed by the Two Equations n,(t) = a + bt and n,(t) = ao + alt + ast2 ,

where t is in OCa

X(A) N a -b X 104 a X 10 5 ao -a, X 10

4 a2 X 108 a X 105

4765 124 1.481895 1.4445 1.4 1.485114 1.6328 2.741 1.0

4880 121 1.480431 1.4376 1.8 1.486018 1.7644 4.759 1.1

4965 121 1.479377 1.4396 1.2 1.479797 1.4045 0.368 1.2

5145 114 1.477293 1.4383 1.8 1.480003 1.5970 2.316 1.3

6328 101 1.468581 1.4232 1.6 1.471192 1.5745 2.182 1.2

a Temperature interval 310°C to 390°C. N is the number of experimental points, and of is the standard error.

Table IV. Refractive Index of Molten KNO3 Expressed by Two Equations n.(t) = a + bt and n8 (t) = ao + ait + a2t2, where t is in 'Ca

x(A) N a -b X 104

a X 105 ao -a, X 10

4a2 X 10 a X 1

4765 177 1.475786 1.5744 1.5 1.478201 1.7016 1.665 0.9

4880 172 1.474426 1.5740 1.5 1.476863 1.7021 1.675 0.7

4965 170 1.473472 1.5698 1.5 1.474948 1.6473 1.013 1.1

5145 164 1.471528 1.5656 1.0 1.472543 1.6191 0.699 0.8

6328 73 1.464045 1.5635 2.3 1.476987 2.2824 9.964 2.0

a Temperature interval 340'C to 420'C. NY is the number of experimental points, and a is the standard error.

May 1975 / Vol. 14, No. 5 / APPLIED OPTICS 1109

(5)

dnl.1 0

dtlc

15.0

,15A

15.0

15.0

14.5

12.0 h5)

12.0

12.0

12.0

11.5 21

, (5)

i(4)

(3)

,(2)

{ (1)

(1)

)oo

-(4)ta

(14)

(2)oo °0

(1) I l I I I I

(15)

l 5 *t | l | a -

Ds

I -

)o

.

Table V. Temperature and Frequency Dependence of theMolar Refractivity of Molten NaNO 3 Calculated from the

Lorentz-Lorenz Equation

T('C) 4765 A 4880 A 496) A 5145 A 6328 A -

310 11.702 11.668 11.646 11.597 11.405 11.037330 11.722 11.688 11.666 11.618 11.424 11.053350 11.742 11.709 11.687 11.638 11.443 11.070370 11.764 11.730 11.707 11.659 11.462 11.087390 11.786 11.753 11.727 11.680 11.482 11.104

Table V. Temperature and Frequency Dependence of theMolar Refractivity of Molten KNO3 Calculated from the

Lorentz-Lorenz Equation

T('C) 4765 A 4880 A 4965 A 514.5 A 6328 Ai X

340 13.761 13.722 13.700 13.648 13.436 13.032360 13.778 13.739 13.717 13.665 13.450 13.042380 13.796 13.757 13.734 13.681 13.465 13.057400 13.814 13.774 13.751 13.699 13.483 13.076420 13.832 13.792 13.769 13.716 13.503 13.100

This work was financially supported by the Swed-ish Board for Technical Development and Carl-Ber-tel Nathhorsts vetenskapliga stiftelse.

References1. M. Born and E. Wolf, Principles of Optics (Pergamon, Oxford,

1970), p. 87.2. C. J. F. Bottchen, Theory of Electric Polarization (Elsevier,

Amsterdam, 1952), p. 238.3. L. Wendelv, S. Gustafsson, N. 0. Halling, and R. Kjellander,

Z. Naturforsch. 22a, 1363 (1967).A. H. R. Jindal, Dissertation (Temple University, Philadelphia,

1966).5. E. Reisler, H. Eisenberg, and A. Minton, J. Chem. Soc. Fara-

day Trans. II 68, 1001 (1972).6. R. H. Stokes, J. Chem. Thermodynam. 5, 374 (1973).7. S. Gustafsson, in Abstracts of Gothenburg Dissertations in

Science (1969), Vol. 12.8. S. Andreasson, S. Gustafsson, and N. 0. Halling, J. Opt. Soc.

Am. 61, 595 (1971).9. R. C. Weast, Ed. Handbook of Chemistry and Physics (Chemi-

cal Rubber Co., Cleveland, 1964).10. P. S. Narayanon, J. Indian Inst. Sci. A 35, 9 (1953).11. G. J. Janz, Molten Salts Handbook (Academic, New York,

1967), p. 42.

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1110 APPLIED OPTICS / Vol. 14, No. 5 / May 1975