On the submillimetre wave refraction of liquids

Volume

2. number 7

ON

THE

CHEMICAL

SUBMILLIMETRE

PHYSICS

WAVE

LETTERS

November

REFRACTION

OF

1968

LIQUIDS

J. CHAMBERLAIN Division

of EZectricaL Science, Teddington, Received

National Physical Middlesex.

Laboratory,

UK

27 August 1968

It is shown that the refractive index associated with a heaviIy damped Lorentzian oscillator of low resonant frequency is radically different in spectral form from that found for the more familiar lightly damped oscillator. The complex refraction spectrum of a crude model consisting of three such overlapping oscillators of equal strengths and linearly spaced resonant frequencies shows a striking qualitative

resemblance to the obzzved

submillimetre spectra of liquids. The consistency of the agreement for both

refraction and absorpt
Hill [I] has recently calcuiated the profiles of the optical absorption coefficient Q!and the dielectric loss E” when these variables represent the synthetic case of three overlapping damped resonances centred at low frequencies. The peak value of (Y lies above the mean frequency of the system while that of E” lies below. A quantitative similarity between these profiles and those recently experimentally observed for both polar and nonpolar liquids [2-81 was indicated with the suggestion that the liquid state absorption might be composed of a continuously distributed number of overlapping daml. zd resonances. The nature of !he experimentally observed curves is such that it is difficult to be sure from inspection of the attenuation profiles alone whether the processes are relaxational or resonant. For example, the E w profiie for nonpolar liquids has some aspects that can be identified with a relaxational mode [83. The real part of the complex permittivity Zz(or complex refractive index tt) should provide some additional clue to the problem although it must be remembered that the real and imaginary parts of Z (or fi) are not independent. The alternative (real) representation may, however, be more readily recognisable as typical of a familiar resonant or relaxational process. The refractive index variations associated with a relaxational process and with a resonant process (exemplified by a molecular vibrational mode) are :;hown in fig. 1. In the former case the index fal1.s smoothly with negative slope while ln the latt jr case the variation is not monotonic and the ce is a turning point on either side of the rescnant frequency, The application of Fourier

the

spectroscopy to the measurement of R for liquids [9] has yielded curves that are similar in form for both polar and nonpolar systems [7] but the real refractive index II is not immediately recogr&sable 2s characteristic of either resonance or relaxation. The values at low frequencies decrease monotonically and fairly rapidly until a shallow minimum is reached prior to a slow monotonic increase at higher frequencies (see fig. 2). Recent experimental developments have provided results over a range sufficiently wide to overlap with conventional microwave data and ensure that no turning point or shoulder has been missed at the lower frequencies. We shall now investigate the refractive index variations associated with a heavily damped resonance. The complex permittivity c = e’ - ic” at wave number ais related to the complex refractive index A=n-i4$

by so that 2 or

and

?.z(Y E” =2G

Volume

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CHEMICAL

7

I

I

PHYSICS

November

LETTERS

lSG8

I

1.54 -

/ \ 1.53 -

1

1

I

I

I

0.1

1

10

100

I

akm-9

(8)

1.52

II ,

I

I

20

40

Fig. 2. Refractive feature illustrated

i

1.63 80

120

100

1443

Fig. 1. (a) Refractive index variation for a relaxational feature exemplified by the microwave dispersion of liquid water at room temperature. Experimental data fii the curve to about 10 cm-l beyond which frequency deviation from true Debye behaviour is evident. Data taken from J. Chamberlain, G. W. Chantry, H. A. Gebbie. N. W. B. Stone, T.B.Taylor and G. Wyllie, Nature 210 (1936) 790. (b) Refractive index variation for a resonant feature exemplified by the dispersion associated with the absorption at 112 cm-l in liquid tetrabromoethane [S]. The feature, which is aImost identical to a Lorentzian, has a damping factor rr = 2yi/o; = 0.07.

-I

I

100

1

120

%p i(l+

- 02,2 + (Zy,&

(2) l

where 2~~ is a measure of the damping and Ci is a constant related to the strength of the oscillator. If we convert to dimensionless frequency xi = ~/ffi and introduce a damping constant ri = = 2yi/oi then eq. (2) becomes EI - 1 t= %

a(crrr’)

W

=c

80

index variation for a submillimetre by spectrum of liquid chlorobenzene [71-

1.64

E”

,

60

1 - x; (I -&2z

f (Y&)2

(3) where

Si = Ci/$_

dependence on xi of ni and @i is simpfer than indicated by eq. (1) if the assumption ‘I >> E: can be made. When this is so The

and We assume electromagnetic radiation of frequency (wave number) cr to interact with a single classical Lorentzian oscillator of resonant frequency ci, then [lo] E! - 1 =ci 2

since

I+ - 02 q -

0.52@Yiq2 -I-

For a given oscillator

the quantities

-

and 465

Volume

are

2. number

CHEMICAL

7

a measure of the complex permittivity

PHYSICS

while

=E; and

= Ei”xi ;LTe a measure of the complex refractive index* provided that the oscillate; strength is sufficientIy small to ensure tFlat variation of ni with y can be ignored (the variation of ni - 1 is, of course, significant). We note that in this approximation the form of E’ is identical to the form of LZ(although numerically different). Consequently, E’ will not be considered separately in the following account. In fig. 3 is shown the dependence of Ni on xi for a singIe resonance with a series of values for the damping facior. The behaviour of q and E; for ~-i = 1 to 3 has been discussed.by Hill [l]; when ?-f = 0.1 or 0.5 the peak value of cyi and E; is at the resonant frequency but neither of the profiles is symmetrical. The quantity 3 _hasboth a maximum and a minimum turning point, such as are generally associated with a resonance, only when the damping is small. It is easily shown, by

0.5

1.0

i.5

20

25x

Fig. 3. Calculated values of Ni (proportional to nf - 1) for isolated Lorentzian resonances with damping f&ctors rf equal to 0.1. 0.5. 1. 2 and 3. * ET 3s equi;alent

466 _-

to X Ld

NT tc XX in ref. 111.

1968

differentiation, that i$ has a maximum or minimum value wheti Xi =+Jirri_ (6) For small damping these values occur at zi = = 1 r $i equidistant from xi = 1 but~when ri > 1 there is only one real root to eq. (6). This corresponds to a minimum of N;: and is given by xi = = + m. Thus when the damping is large the peak values of (Y and (particularly) E” are displaced from the resonant frequency and n varies in a manner intermediate between the relaxationaI and resonant forms. The spectral form of the radiation interaction with a range of damped resonant oscillators may be approximately represented by arbitrarily superimposing three moderately damped oscillators of equal strengths So = Sl = S2 and having frequencies uo, ul, 0~ reIated by cro = 0.5~1 and ~2 = = 1.5a. We then have 2 2 2 ui - u E’ - 1 =zF* ci (($ - 02)2 + (2% a)2 A

and

which

E’ -

become I

=

Sl 0.25 - X;

0

November

LETTERS

1 - X;

2.25 - X;

= (0.25 - $)2 c<

+ (1- x$~ +A$ •+(2.25 - 4c2 +4

x1 = (o-25-<)2+<

x1 +(1-<)2+<

x1 +(2.25-<)2+<

** The expression here for E R is slightly different from that used by Hill who replaced xi by x/O.5 and Xi/l.5 in eq. (3) for the lower and upper resonances. suznmed the three terms for each xf and divided the total by 3. The most important difference is that Hill’s damping parameter is the same for each of the three resonances whereas here the half-width is the same but the damping parameter decreases: {l/02). Since the asro : 3-l : 3-2 = (l/co): (l/cl): sumption of three discrete resonances is unrealistic and Hill’s attenuation curves are qualitatively similar to those here this difference is not of great significance. Strictly speaking, however, the three terms of Hill are not of ‘equal amplitude’ (or strength).

Volume

2, number

7

CHEMICAL

PHYSICS

November

1966

ism in alternative ways as either E” or (Y. No ciifferent basic information is carried by these two functions but in practice one may be more useful than the other when clues on the mechanism are being sought since certain aspects of this may be more dominant in one representation than in the other. The behaviour of N’ (and hence n) is also strikingly similar to that ob served in the submillimetre refraction spectra of liquids. There is no turning point which is a maximum but only a shallow minimum situated at about XI = 1.7. In spite of an apparent lack of similarity between the observed submillimetre refraction spectra of liquids and the spectrum that might have been expected for a resonant oscillator it is clear that if the damping of the oscillator is relatively high and a number of such oscillators are overlapping the observed spectra are compIetely compatible with a resonance phenomenon. It is worth noting that if we crudely relate the simple model to reality by putting UI = 40 cm-1 the halfwidth of each of the three component resonant features is 40 cm-l. The minimum in MI occurs at 70 cm-l, the maximum of Nz at 50 cm-1 and the total width of N: at half-maximum is 15 cm-l. The similarity bet6een these numbers * and the Rocard-Powles corrected data for chlorobenzene [7], for example, is striking.

r

t

ACKNOWLEDGEMENT

I

I

0

LETTERS

05

1.0

15

,

2.0

25

x

Fig. 4. Calculated values of (a) Ni (proportional to a- 1); (b) NT (proportional to CY): (c) ET (proportional to E “) for three overlapping Lorentzian resonances of equal strength and half-width situated at %I = 0.5, 1.0 and 1.5.

sumptions as formerly we can evaluate as functions of xl the expressions h$, i$ and E; defined analogously to eqs. (4) and (5). Taking x1 = 1 as the central frequency of the system we see, with Hill [l], that the maximum of E; (and hence E”) is displaced to a lower frequency xl = 0.25 while the maximum of $ (and hence o) is displaced to a higher frequency x - 1.3 and thus neither peak is representative of d- e mean resonant frequency of the system. The curves here are smoother than those of Hill and are, consequently, more like those observed for liquids. It must be remembered that we are representing the same mechan-

I am nicating tion and some of ‘This gramme

grateful to Dr. N. E. Hi1 both for commuthe contents of ref. [I] prior to publicafor explaining privately the details of the aspects contained therein. work forms part of the research proof the Division of Electrical Science of

NPL.

* It must be remembered that N; is proportional o&y to the contribution from the three resonances assumed empirically to crudely represent the submillimetre absorption in liquids. In practice. of course. the measured dispersion also includes contributions from the Debye reltintional process at lower frequencies and the molecular modes at higher frequencies. The sum of these contributions wili, however. be practically constant in the region of the submiffimetre feature since the contribution from lower frequencies falls as u increases while that frcm higher frequencies rises.

Volume

2.

number 7

CHEMICAL PHYSICS LETTERS

REFERENCES

[l] N.E.Hill. Chem.Phys. Letters 2 (1968) [2] G. W. Chantry and H. A. Gebbie, Nature 378. (31 Y. Leroy and E. Constant, Compt.Rend. 1391. [4] G. W. Chantry, H. A. Gebbie, B. Lsssier lie, Nature 214 (1967) 163.

468

5. 208 (1965) 262 (1966) and G. Wyl-

November

1968

[5] S. G.Kroon and 3. van der Eisken, Chem. Phys. Letters 1 (1967) 285. (61 J. Chamberlain, H. A. Gebbie. G. W. F. Pardoe and M. Davies, Chem. Phys. Letters 1 (1968) 523. [7j M. Davies, G. W. F.Pardoe, J. Chamberlain and H.A. Cebbie, Trans. Faraday Sot. 64 (1968) 847. [6J M. Davies, G. W. F. Pardoe, J. Chamberlain and H. A. Gebbie, Chem. Phys. Letters 1 (1968) 523. [9] J. Chamberlain, A.E. CostIey and H. A. Gebbie, Spectrochim. Acts 23 A (1967) 2255. [lo] See, for example, S. A. Korff and G. Breit, Rev. Mod.Phys. 4 (1932) 471.