Positronium Chemistry in Aqueous KMnO4 SolutionsThomas L. Williams and Hans J. Ache Citation: The Journal of Chemical Physics 50, 4493 (1969); doi: 10.1063/1.1670921 View online: http://dx.doi.org/10.1063/1.1670921 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/50/10?ver=pdfcov Published by the AIP Publishing Articles you may be interested in A versatile electrochemical method to produce nanoparticles of manganese oxides by KMnO4electrolysis AIP Conf. Proc. 1586, 124 (2014); 10.1063/1.4866744 Positronium formation in aqueous micellar solutions of sodium dodecylsulfate J. Chem. Phys. 71, 2083 (1979); 10.1063/1.438578 Positron annihilation in aqueous solutions of KMnO4 and KI J. Chem. Phys. 61, 1257 (1974); 10.1063/1.1682010 Flash Heating and Kinetic Spectroscopy of KMnO4 J. Chem. Phys. 45, 2698 (1966); 10.1063/1.1727995 Radiation Chemistry of Aqueous Benzene Solutions J. Chem. Phys. 23, 604 (1955); 10.1063/1.1742061

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ELECTRON-IMPACT SPECTRUM OF MERCURY VAPOR 4493

6p' 1 P transition which, according to the data in Table II, shows a much stronger energy dependence than that m1p series. This type of behavior generally indi­cates a very rapid change of the generalized oscillator strength at small values of (..1P) 2, and, by definition /=/0 only for (..1P) 2 =0. For 6p'1P even the small value of (..1P)2 shown in Table III may be too far from zero for the assumptionJ""/o to hold. At present, we cannot estimate the magnitude of this error.

The two transitions to 6P'1P1 and 71S behave in unexpected ways and further study of these transitions as a function of scattering angle would be of obvious interest. This study has been postponed until the de­termination (by J. P. Bromberg) of absolute elastic collision cross sections for mercury have been com­pleted. Once these cross sections are available the

THE JOURNAL OF CHEMICAL PHYSICS

generalized oscillator strengths can be determined as a function of (tlP)2 and extrapolated to tlP=O thus providing an independent check on the value of Lurio.6

This study is of special interest since Kessler and collaborators22 have shown that substantial spin polari­zation occurs in the elastic scattering of electrons by mercury. Conceivably such polarization could occur in inelastic scattering as well but no account has been taken of this factor in assuming that lim/ =/0 where / is calculated from Eq, (3). It would be of consider­able interest to test this point by experiment, but considerable time will be required for these investi­gations.

II W. Eitel, K. Jost, and J. Kessler, Proc. Intern. Conf. Phys. Electron. At. Collisions 5th, Leningrad, U.S.S.R., 1967, 549 (1967) .

VOLUME SO, NUMBER 10 IS MAY 1969

Positronium Chemistry in Aqueous KMnO 4 Solutions*

THOMAS L. WILLIAMS AND HA.>;s J. ACHE

Department of Chemistry, Virginia Polytechnic Institute, Blacksburg, Virginia 24061

(Received 9 January 1969)

The reaction of positronium was studied in aqueous KMnO. solutions at various concentrations and pH. Lifetime measurements show a distinct decrease of 1'2 and increase of 12 with increasing temperatures, as expected for chemical reactions in which thermalized positronium is involved. No dependence on the pH of the solution could be observed. The activation energy calculated from the temperature dependence of 1'2

is rather small and about 0.1-{l.2 eV. The free positron annihilation lifetime was found to be approximately 0.55 nsec, which is in good agreement with the value obtained by using Dirac's formula, 0.5 nsec for water.

INTRODUCTION

Positrons emitted in the radioactive decay of certain nuclides can pair with an electron through an electro­static attraction forming a hydrogenlike system called positronium (Ps). Two states can be observed, singlet (spins antiparallel) and triplet (spins parallel). The interactions of the positronium have been studied in a great number of systems.1 A survey of the chemistry of positronium in aqueous solutions can be found in recent review articles by McGervey,2 Goldanskii,8 and Tao and Green.4

In aqueous solutions three types of interactions of the positronium with its chemical environment are of special interest: (1) conversion of one Ps state to

• This work was supported by the U.S. Atomic Energy Com­mission and by the Petroleum Research Fund.

I For recent reviews see (a) J. Green and J. W. Lee, Posi­tronium Chemistry (Academic Press Inc., New York, 1964); (b) Positronium Annihilation, A. T. Stewart and L. O. Roellig, Eds. (Academic Press Inc., New York, 1967).

2 J. D. McGervey, in general, Ref. 1(b), p. 143. 8 V. I. Goldanskii, in general, Ref. 1 (b), p. 183. • S. J. Tao and J. H. Green, J. Chern. Soc. 1968,408.

another, (2) oxidation by electron transfer; and (3) formation of a positron compound.

There are generally three methods which can be used to study these reactions: (1) Lifetime measurements of T2, the triplet Ps lifetime, including the measurement of I, the percentage of the positrons which decay via this lifetime. (2) Measurements of the three-quantum annihilation rate, which are approximately propor­tional to the product of T2 and the fraction of the positrons which form triplet Ps. (3) Measurements of the angular correlation of the annihilation quanta, where the narrow component is a measure of the singlet Ps present and of the average momentum of the singlet Ps at the moment of its annihilation. From the chemi­cal point of view, one of the most intriguing aspects which these methods offer is the possibility of studying the fundamental mechanisms of oxidation reactions by the observation of the positronium quenching.

Another source of useful chemical data can be the investigation of positron compounds.

Studies by Horstman6 on the temperature depen-

5 H. Horstman, J. Inorg. Nuc\. Chern. 27,1191 (1965).

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4494 T. L. WILLIAMS AND H. J. ACHE

TABLE 1. Mean lifes for positron annihilation in aqueous KMn04 solutions at various temperatures.

T(°C) 30 45 60 75

Cone '7"2 12 '7"2 12 '7"2 12 '7"2 12 pH (n) (nsec) (%) (nsec) (%) (nsee) (%) (nsec) (%)

2 0.1 0.41±0.02 0.4O±0.02 0.41±0.02 0.41±0.02

7 0.1 0.42±0.02 0.43±0.02 0.42±0.02 0.42±0.02

11 0.1 0.4O±0.02 0.42±0.02 0.41±0.02 0.46±0.02

2 0.01 1. 16±0.03 31.1 1.05±0.03 34.1 0.99±0.04 35.1 0.94±0.04 38.4

7 0.01 1. 16±0.03 31.9 1. 03±0. 03 33.2 0.92±0.03 40.0 0.88±0.04 42.5

11 0.01 1. 16±0.03 32.1 1.05±0.04 32.8 0.95±0.07 35.0 0.83±0.04 45.0

2 0.001 1. 73±0.03 26.6 1. 70±0.04 26.1 1.M±O.04 29.1 1. 52±0.04 31.4

7 0.001 1. 69±0.03 26.0 1. 63±0.04 27.9 1. 62±0.04 27.7 1. 58±0.04 28.6

11 0.001 1. 68±0.03 26.7 1. 61±0.04 26.7 1. 57±0.04 26.7 1.50±0.03 28.9

dence of 72 and 12 in oxidating solutions have shown that T2 decreases with increasing temperature, as ex­pected for chemical reactions involving thermalized positronium. The observed temperature dependence of 12 also provides evidence of compound formation.

In the following a systematic investigation was made to determine temperature variations of the oxidation probability of positronium in KMn04 solutions, by studying 72 and h An attempt was made to evaluate the activation energy for the oxidation reaction and the lifetime of free positrons in solution.

EXPERIMENTAL

The measurement of positronium lifetimes was ac­complished by the usual delayed coincidence method.6

The radioactive decay of 22Na, Fig. 1, to the ground state of 22Ne by the emission of a positron and subse­quent emission of a 1.3-MeV gamma serves not only as a source of positrons but also as a means of detecting the birth of the positron itself. A block diagram of the electronics is presented in Fig. 2. Two RCA 8575 photomultipliers, optically coupled to two 1 in.X 1 in. Naton 136 plastic scintillators, were each, respectively, biased to accept the 1.3- and O.511-MeV gammas. The relative time difference between the starting 1.3-MeV gamma and the stopping O.511-MeV gamma was measured by a time-pulse-height converter (TPHC) whose output amplitude is proportional to the time difference between the two gammas. The number of events vs pulse amplitude was collected by an ND 2200 multichannel analyzer using a 256-channel range. The TPHC and intermediate electronics were of ORTEC design.

The resolution of the system, measured by the prompt time distribution of a 6OCO source and without changing the 1.3- and O.511-MeV bias, was found to be

• A. Schwarzschild, Nuel. Instr. Methods 21, 1 (1963).

O.5-nsec, FWHM. The mean life (T) of the slope of the prompt distribution measured less than 10-10 sec. Consequently, the slope method was used to determine all lifetimes in this experiment.

All solutions measured were prepared in the following manner. Five microCuries of 22Na obtained from New England Nuclear Corporation were added in form of N aCI to each KMn04 solution contained in small cy­lindrical glass vials 30 mm long and to-mm i.d. The vials were then sealed under vacuum. No correction was attempted for the small amount of positron an­nihilation (2%) occurring in the walls of the vial.

Each measurement was made over a period of ap­proximately 11 h immediately followed by a shorter run of the 6OCO prompt spectrum. The centroid of the 6OCO prompt distribution determined the zero of the time distribution spectrum. Time calibration of the system was accomplished by inserting a known amount of delay cable and observing the resulting amplitude shift of the 6OCO spectrum.

Half-lives were determined directly from a plot of the time distribution data. These values for the half­lives later served as initial guesses for a least-squares fit to the data.

A description of the least-squares method and the

2.60y

3 ps ..=2;...+ __ J...;.;. 2:;;..;7~4:...;:6;...M"

E2 0+

FlO. 1. Decay scheme for UNa.

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P 0 SIT RON I U M C HEM 1ST R YIN A QUE 0 U S K M nO. SOL UTI 0 N S 4495

DYNODE 420 >----t-t TIMING t-------,

S.C.A.

ANODE 417 26!5 P.M. BASE FAST

DISC RCA 8575 PHOTO-MULT TUBE

437 Ell SOURCE .. NATON 136 (l"xl")

T.P.H.C. LINEAR

RCA 8575 PHOTO-MULT TUBE GATE

265 P. M. BASE 417 425 FAST DELAY

ANODE DISC.

MULTI-CHANNEL

420 ANALYZER TIMING NUCLEAR

DYNODE S.C.A. DATA 2200

FIG. 2. Fast-slow coincidence system for lifetime measurements.

method of calculating the intensity (12) of the long­lived component is offered in the Appendix.

RESULTS AND DISCUSSION

Previous investigations using both lifetime and narrow-component measurements have shown that a correlation exists between the chemical oxidation po­tential and the positronium oxidation rate.HI

Table I shows the results of lifetime measurements in KMnO. solutions of various concentrations and various pH. Although the chemical oxidation-reduction potential in aqueous KMnO. solutions differs greatly with the pH of the solution, practically no change could be observed for 1'2 and h In line with these re­sults are the findings of Goldanskii et al.8.12 who could not notice any variations of the three quantum annihi­lation under these conditions either. The oxidation of a positronium 'differs in more than one respect from ordinary chemical oxidation reactions. That is due mainly to the rapid annihilation of positrons which does not permit the chemical reaction to come to equilibrium. Consequently the oxidation of the posi­tronium proves to be irreversible and the chemical

7 J. E. Jackson and J. D. McGervey, 1. Chem. Phys. 38, 300 (1963).

8 J. D. McGervey, H. Horstman, and S. DeBenedetti, Phys. Rev. 124, 1113 (1961).

8 V.!. Goldanskii, T. A. Solonyenko, and V. P. Shantarovich, Dokl. Akad. Nauk SSSR 151, 608 (1963).

10 G. Trumpy, Phys. Rev. 118, 668 (1960). 11 S.]. Tao and]. M. Green, NRS 21 (1964). 11 V. 1. Goldanskii, O. A. Karpukhin, and G. G. Petrov, Zh.

Eksp. Teor. Fiz. 39, 1477 (1960).

oxidation-reduction potential which would show a dis­tinct dependence on the pH can be used only as a rough approximation for the power of an ion to oxidize Ps insofar as it reflects the electron affinity of a single io~ which could destroy Ps. In Figs. 3 and 4 the life­times 1'2 and the corresponding intensities 12 of O.OlM aqueous KMnO. solutions are plotted as a function of the temperature, with the pH of the solution as the third parameter. Again no pH dependence could be observed throughout the whole temperature range from 30°-75°C.

Both 1'2 as well as 12 show a distinct temperature dependence, 1'2 decreases sharply with an increase in temperature, whereas 12 shows the opposite trend.

The same kind of temperature dependence of 1'2 and 12 has been observed by Horstman5 in oxidizing solu­tions containing Hg2+, Sn*, Sb3+, and Pd2+ ions. These temperature studies allow the evaluation of the activa­tion energies involved in the positronium oxidation re­actions. If we consider only the two major processes, pick. off and oxidation, occurring in the solution, the total annihilation probability can be found as the sum of the probabilities for each decay mechanism.s

For sufficiently long times the observed decay con­stant X2 is

(1)

if Xo+Xl'<XF , where Xo is the oxidation probability, Xl' the pickoff annihilation probability, and XI' the free positron annihilation probability, all per unit time.s Xl' was measured in pure water and showed (see results in Table II) no temperature dependence. By subtract-

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4496 T. L. WILLIAMS AND H. J. ACHE

1.2

\ OOA KMn04 O.OIM

pH = 2 0

pH = 7 0

..... u CP til

1.1

.5. 1.0

0.9

pH = II A

o FIG. 3. Mean lives vs temperatures

for O.OlM KMnO. solutions at various pH. pH=2, 0; pH=7, 0; pH=l1, b,..

0.8~----~------~------~------~----~ 30 45 60

ing Ap =O.57 nsec-1 from the observed rate (A2) the oxidation rates (AO) at various temperatures and con­centrations could be derived. In Fig. 5 logAo for O.OlM KMnO, solutions (pH of 11) is plotted vs ]'-1 (T is the absolute temperature); from the resulting straight

75

line and Arrhenius rate law,

Ao=A exp( -E/kT) ,

the activation energies for the reactions can be deter­mined. The results are shown in Table III. The activa-

TABLE II. Mean lines for positron annihilation in water at various temperature.

30°C 45°C 60°C 75°C

1"2 I, 1", Is 1"2 12 1"1 Is (nsec) (%) (nsec) (%) (nsec) (%) (nsec) (%)

1. 7S±O.OS 22.4 1. 72±O.04 24.0 1. 7S±O.04 22.6 1. 76±O.OS 22.0

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P 0 SIT RON I U M C HEM 1ST R YIN A QUE 0 U S K M nO, SOL UTI 0 N S 4497

46

44 KMn04 O.OIM

pH = 2 0 pH = 7 0

42 pH = II ~

40

-FIG. 4. Intensity vs temperature for .... '" O.OlM KMn04 solutions at various pH. 38 pH=2, 0; pH=7, 0; pH=l1, f:::,..

36

0 ~

34 0

32 ~~

0

30 30 45 60 75

T(OC)

TABLE III. Activation energies, N'l'°, and free positron mean lives tion energies are of the order of 0.2 eV and close to those obtained by Horstman6 in other oxidizing solu­tions. The initial fraction of positrons which form triplet positronium (NTO) is not necessarily equal to h By using a few fundamental decay equations, in which NT is the number of triplet Ps atoms, N. the number of singlet Ps, and NF the number of free positrons, XF the annihilation rate for free positrons in solution, X, conversion rate for singlet Ps, and XT is the conver­sion rate for triplet Ps, an expression can be derived for NTo which includes the measured values for 12 and X2.6 The initial values of NTo, Nso, and NFo are assumed to be temperature independent and Xo is the only

in aqueous KMnO. solutions.

Cone E N'l'° -rl' pH (n) (eV) (%) (nsec)

2 0.01 0.12 24.4 0.55 7 0.01 0.14 24.1 0.54

11 0.01 0.16 24.6 0.50

2 0.001 0.44 26.3 0.93 7 0.001 0.15 24.4 0.94

11 0.001 0.11

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4498 T. L. WILLIAMS AND H. J. ACHE

.... ~ lit C -o ..c

0.6

0.5

0.4

0.3

, , , '.

FIG. 5. Log of oxidation rate vs reciprocal of the absolute temperature for O.OIM KMnO, solutions, pH = 11.

0.2~----------~------------~------L---~ 2.8 3.0 3.2 3.3

I/T(xI0-3 °lel}

temperature-dependent rate,

dNT/dt= - ('Ao+Ap) NT, (2)

dNs/dt= - (A.+Ao+Al')Ns, (3)

dNF/dt=-AFNF+Ao(NT+Ns). (4)

Solution of these equations while considering that the two-quantum rate can be written as

XpNT+X.N.+ApN.+XFNF (5) and

A2='}.,0+'}.,17 (1) yields5

12= {1 + [Ao/ (AF-A2) ]}NTO (6) or

1/ (12)-1= 1/ (NTO)-I_[AO/ (AF-AP)NTO]. (7)

In Fig. 6 (12)-1 is plotted as a function of Ao at various temperatures. The result is a straight line, with an intercept of (NTO)-1 and a slope [(AF-Ap)NT°J-l. The ratio of the latter two yields (AF-Ap). From the pick­off annihilation rate, taken from Table II, AF and NTo can be calculated. The results in Table III show NTo to be close to 24%. There is very little indication

for a variation of NTo with the concentration. The free positron annihilation lifetime obtained from Eq. (7) by using observed values of 12, Ao, and Xl' ranges from 0.5-0.9 nsec. This agrees quite well with lifetimes of 0.5 nsec for water calculated by using Dirac's formula, considering that only the outer-shell electrons con­tribute to the electron density. Previous investigations by Goldanskii et aI.a have shown a temperature de­pendence of positronium reactions with ions such as Cu2+ in various solvents. A linear correlation was found between the reciprocal rate constant of posi­tronium reactions and the viscosity of the solution. This direct relationship of the rate constants with the viscosity of the solvent was taken as evidence of the occurrence of these reactions in the diffusion region.

Similar observations have been made in the present investigation. In Fig. 7 where the reciprocal of Ao is plotted vs the viscosity of the solution (viscosity of the KMn04=viscosity of H20) a straight line is ob­tained which confirms the conclusion (vide supra), that the triplet positronium is completely thermalized before it undergoes reactions in oxidizing solutions. That is also in line with the observed values of the energy of activation, 0.1-0.2 eV, which is rather close to the

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POSITRONIUM CHEMISTRY IN AQUEOUS KMnO. SOLUTIONS 4499

FIG. 6. Reciprocal of the intensity vs oxidation rate for O.OIM KMnO. solutions, pH = 11.

'" .... ......

4

- 3

2

h

o

activation energy for viscosity which is in the case of water approximately 0.06 eV, again indicating that the oxidation of positronium is overwhelmingly a diffusion­controlled process.

4 ...

30 / 1 c

/ -~ ~2D

1.0 I . I I . aD 4.0 s.o 6.0 7.0 8.0 9.0

l'\, (Hlc) X , O' PO I SE

FIG. 7. Reciprocal of hO vs viscosity of solution for O.OIM KMnO. solution at pH 11.

0.2 0.4 0.6

APPENDIX

In the study of positronium annihilations, the data of the counting rate vs time was fitted by the method of least squares to determine the annihilation life­times.18

The multicomponent annihilation spectrum obeys the exponential decay law,

m

/'= L.x'rAj/i. (Al) ,..1

Where for each of the i data points, fi is the counting rate at time t. The contribution to the counting rate fi is given by the summation of the j-independent annihilation components.

The multicomponent decay curve program, coded in FORTRAN for CDC 6600 by Cumming,18 was slightly modified to be run on our IBM 360-40/50 computer. The data processing followed the outline previously described.18

Ill' B. Cumming, "CLSO, the Brookhaven Decay Curve Ana ysis Program," BNL 6470, was slightly modified for the use with an IBM 360/50. The authors wish to thank Dr. J. B. Cumming for valuable comments and assistance.

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4500 T. L. WILLIAMS AND H. J. ACHE

f/) ~ z ;:) o u

,\

l \x " \ ' \ , f

\ \ o

\ \

\ \ \

~

o

FIG. 8. Time distribution spectrum for positron annihilations in O.OlM KMnO. solutions, pH=l1. Dashed line is the lOCO prompt distribution.

\ -t o T2 = 1.16x 10 sec \ o~

\ o 2 3

TIME( nSe(\)

In addition to the data on half-lives, fit, etc., cards are also outputed which contain information in the appropriate format to be used in the intensity program.

The intensity of the long-lived component of the annihilation spectrum was determined by first estab­lishing the zero point on the time axis. This was taken to be the centroid of the 6OCO prompt spectrum (see Fig. 8).

The intensity, 12, is given by the area under the exponential component divided by the total area under the curve. Simpson's rule was used to determine the area from point B to point D,

Area 1 =l~X( YB +4YB+l+2YB+2+·· ·4Y0-1+ YD ).

The area from point D to 00 was determined by in-

tegrating the expression

Finally, the area under the entire exponential com­ponent was determined by integrating from P to 00,

Area 3 = A LCD exp( - A2t) dt.

The intensity (12) is then given by the expression

12 = (Area l+Area 2)/ Area 3.

A short program, coded in FORTRAN, was written to evaluate 12 by this method. From the cards obtained from the least-squares analysis, the program reads an

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P 0 SIT RON I U M C HEM 1ST R YIN A QUE 0 U S K M n 0 4 SOL UTI 0 N S 4501

identification card and a control card. The control card contains the value of the intercept (A) at point P, the decay constant >'2, the background counts, and the number X whose value is relative to the point P=O. The program now reads data points from point B to (X-1) and performs background subtraction.

THE JOURNAL OF CHEMICAL PHYSICS

The data points from the least-squares curve XD are now read in and compared point by point with curve PD until the point D is determined. The program now proceeds to find Area 1, Area 2, Area 3, and the in­tensity 12• The results are printed out and the program proceeds to the next data set.

VOLUME 50, NUMBER 10 15MAY1969

Absorptivity of Ice I in the Range 4000-30 cm-1* J. E. BERTIE

Chemistry Department, University of Alberta, Edmonton, Canada AND

H. J. LABBi AND E. WHALLEY

Division of Applied Chemistry, National Research Council, Ottawa, Canada (Received 13 December 1968)

The absorbance of several samples of ice Ih has been measured in the range 4000-30 em-I, and scaled to that of a particular film of unknown thickness. The thickness of the film has been calculated by two methods, first from the known absorptivity at 4940 em-I, and second by equating the appropriate Kramers-Kronig integral to the known infrared contribution to the microwave refractive index. The two thicknesses agreed well and allowed the absorptivity to be obtained in the range 4000-30 em-I. The complex refractive index and permittivity and the normal incidence reflectivity have been calculated from the absorptivity. About three-quarters of the infrared contribution to the microwave refractive index is caused by the transla­tional lattice vibrations and about 15% by the rotational vibrations; the o-H stretching bands which absorb very strongly contribute relatively little. The maximum of the density of states in the transverse acoustic branch is at 65 em-I rather than below 50 em-I as reported earlier. Below 50 cm-I the absorptivity is roughly proportional to the fourth power of the frequency. This arises because the vibrations here are short-wavelength sound waves with a density approximately proportional to the square of the frequency, and the integrated intensity of absorption by one vibration is proportional to the square of the frequency. A theory of the contribution of the translational lattice vibrations to the microwave permittivity is given based on the theory of the absorption by orientationally disordered crystals given in an earlier paper. From the theory and the experimental measurements reported in this paper the dipole-moment derivative for the relative displacement of two water molecules in ice along their line of centers (or equivalently the effective charge of a water molecule) is about 0.3 electronic charges.

I. INTRODUCTION

Few measurements of the infrared absorptivity of ice in the region of the fundamental vibrations have been made, and none has been made over the whole infrared region. partly because of the experimental difficulties. A recent detailed review1 of both the ab­sorption and reflection measurements well illustrates these difficulties. Near 3300 cm-I films about 1 }lm

thick are required if the sample is not to be black at the peak, whereas at 30 cm-I films several millimeters thick are required. It is not easy to measure accurately the thickness of all the films required for the range 4000-30 cm-I •

The optical constants of ice are of considerable practical interest in connection with the scattering of infrared light from ice particles in the atmosphere, as well as being crucial to an understanding of the elec­trical properties of both the intramolecular and inter­molecular motions. An outstanding problem, for exam­ple, is the origin of the large difference between the

• N.R.C. No. 10663. 1 W. M. Irvine and J. B. Pollack, Icarus 8,324 (1968).

microwave2 and visible3 refractive indexes of about 0.47. This corresponds to an infrared polarizability of 1.71 A3 molecule-t,4 and the origin is unknown al­though the absorption due to the rotational vibrations centered near 840 cm-1 has been suggested.5 A qualita­tive solution to this problem can in fact be obtained from very simple considerations and it is given in Sec. II.

Both the problems of the measurement of the ab­sorptivity and of the origin of the infrared polarization at low frequencies can be solved simultaneously as follows. According to the Kramers-Kronig relations8

the contribution An of an absorption band to the low-

I J. Lamb and A. Tumey, Proc. Phys. Soc. (London) B62,272 (1949) .

a Landolt-Bomstein, Zahlenwerte und Funktionen. (Springer Verlag, Berlin, 1962), 6th ed. Vol. 2, Pt. 8, p. 2-234.

4 E. Whalley, J. B. R. Heath, and D. W. Davidson, J. Chern. Phys. 48, 2362 (1968).

6 L. Onsager and M. Dupuis, Proc. Intern. Symp. Electrolyte Solutions, Trieste, 1959, 27 (1962).

• See, for example, F. Stem, Solid State Phys. 15,331 (1963); T. S. Moss, Optical Properties of Semiconductors (Butterworths Scientific Publications, Ltd., London, 1959), p. 27; S. Maeda and P. N. Schatz, J. Chem. Phys. 36, 571 (1962).

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