THE DOPPLER–FIZEAU PHENOMENON AND MOLECULAR MOTION

G. S. Landsberg

It is well known that the Doppler–Fizeau phenomenon, caused by the molecular motion of gas particles, is one of the causes of the broadening of spectral lines. At low pressures of a luminous gas (fractions of a millimeter), the Doppler–Fizeau effect is usually the principal cause of such broadening. It is known that when working with interference instruments of high resolving power, sources are used in which the luminous gas has as low a temperature as possible, or the glow of particles flying in one direction (a molecular beam) perpendicular to the direction of observation is used, in order to reduce the Doppler–Fizeau effect to a minimum.

In the scientific literature the question has repeatedly been raised of the influence of the molecular motion of the particles constituting a mirror on the width of the lines of light undergoing reflection from this mirror. The phenomenon of reflection (regular or diffuse) is based on the process of scattering of light waves by the atoms constituting the mirror. It has therefore been suggested that reflection of monochromatic light should be accompanied by broadening of the lines as a consequence of the Doppler effect in scattering by the randomly moving atoms of the mirror.

Special experiments have repeatedly been carried out to study this phenomenon, using optical methods characterized by very high resolving power. Experimental investigations of this kind were performed in laboratories enjoying well-deserved renown. Thus W. Rump¹, in J. Franck’s laboratory, carried out an extensive study of line broadening in the scattering of light by atoms of mercury vapor at low and high pressure (specular reflection), and also in reflection from an aluminum mirror. To analyze the monochromaticity of the light, whose source was a resonance mercury lamp \((\lambda = 2536.7\ \text{Å})\), Rump used the absorption of light in an absorption vessel filled with mercury vapor, whose temperature coincided with the temperature of the resonance lamp. When the light of the primary resonance lamp was scattered by the hotter mercury vapor of a secondary lamp, this scattered light was poorly absorbed in the absorption vessel. In this way a broadening of the lines of the scattered light was established, corresponding to the eff—

effect on mercury atoms, which possess more rapid thermal motion. However, when, by heating the secondary lamp, the vapor density in it was brought to a value corresponding to mirror reflection (Wood), then the light reflected from such dense and hot vapor was well absorbed in the absorption vessel, i.e. was characterized by a line width corresponding to the primary lamp. Likewise, no broadening of the lines was observed when light was reflected from an aluminum mirror, although the thermal velocities of the light aluminum atoms are considerably greater than those of mercury atoms, and the expected broadening should have been almost three times greater than the broadening in scattering by mercury vapor.

The same question was studied in Jean Cabannes’s first-class optical laboratory, both by Cabannes himself² and by some of his students (Rocard, Rothschild³). With the aid of interference instruments of high resolving power they studied the effect of reflection from polished materials (glass, silver, gold) and from matte surfaces (porcelain, wood, paper). Recently, Ryu and Gober⁴ attempted to detect the influence of the Doppler effect in twenty-fivefold reflection from a mirror, using a careful photometric analysis of the interference pattern obtained with a Michelson echelon. The result of these experiments, as of all the preceding ones, was invariably negative. It seems quite surprising to set up special experiments on the influence of the Doppler effect in reflection. If it existed, it should have manifested itself in the operation of interference instruments. In work with a Fabry–Perot etalon or a Lummer–Gehrcke plate, interference is observed of rays multiply reflected from silver or glass, and, of course, if the Doppler effect occurred in this case, these instruments could not fulfill their purpose.

Thus, the absence of any influence of reflection on the wavelength of the reflected light raises no doubt, and the question can only concern the theoretical foundations of this result.

In a recent paper by Wolffers⁵ an attempt was made at such an analysis, leading the author to strange and unexpected conclusions. Wolffers considers the influence exerted by the Doppler effect in the scattering of light by a single resonator, and obtains general formulas for this case. Then, proceeding from the well-founded experimental absence of the effect, he seeks the condition under which his formulas lead to such a result, and comes to the conclusion that the thermal velocities at the mirror surface must have only components parallel to this surface. The author attempts to make his conclusion more acceptable by supposing that thermal waves in the body form a node at its surface.

Wolffers’s reasoning is based on a misunderstanding connected with the circumstance that, in solving the problem of reflection of light from the surface of a mirror, he carries out the argument for a single resonator only.

Meanwhile, L. I. Mandelstam\(^6\) showed as early as 1907 that, when the action of an aggregate of resonators whose separations are small compared with the wavelength is taken into account, reasoning based on separate resonators leads to erroneous conclusions. Thus Rayleigh’s argument concerning the disturbance of the coherence of waves scattered by molecules in thermal motion is valid for isolated molecules and loses its force in scattering by an optically homogeneous medium, i.e., a medium in which one can isolate fixed volume elements containing molecules whose number is proportional to the volume element. In such cases the reasoning must be carried out for the aggregate of resonators, i.e., of the medium, and not for isolated molecules.

The simplest way to carry out such reasoning is by considering the thermal motion of molecules in the form of thermal waves, as Debye first did in his theory of specific heats. Along this path we also obtain an answer concerning the influence of thermal motion on the change in wavelength (the Doppler effect). However, in the case of condensed systems the determining role is played not by the velocities of molecular motion, but by something else.

In order to make this method of reasoning especially clear as applied to our problem of reflection at a surface, let us briefly recall the phenomena observed in scattering in the volume.

Scattering by fluctuation inhomogeneities in a solid body (and in a liquid) may be considered\(^{7,8}\) as reflection, in accordance with Bragg’s condition, from spatial gratings formed by Debye thermal waves. In this case, as is known, we also obtain the effect of a change in wavelength upon scattering, representing the influence of the Doppler effect on moving temperature inhomogeneities; moreover, the velocity determining the Doppler effect is the velocity of propagation of elastic waves. The necessary relation can be obtained by considering the phenomenon in terms of standing waves (the modulation effect) or in terms of equivalent traveling waves (the Doppler effect). Figs. 1 and 2 explain both methods of reasoning.

Standing waves. In the scattering of light of wavelength \(\lambda\), in the direction \(\theta\), there participates an ultrasonic wave \(\Lambda\), whose wavelength satisfies Bragg’s condition (see Fig. 1)

\[ 2\Lambda \sin \frac{\theta}{2}=\lambda . \]

The change in frequency \(\nu\left(=\frac{c}{\lambda}\right)\) occurs as a consequence of modulation with the ultrasonic frequency

\[ N\left(=\frac{v}{\Lambda}\right). \]

Thus, a change in frequency is observed

\[ \pm \frac{\Delta \nu}{\nu} = \pm \frac{N}{\nu} = \pm \frac{v}{\Lambda}\cdot\frac{\lambda}{c} = \pm 2\,\frac{v}{c}\sin\frac{\theta}{2} \]

(the Mandelstam–Brillouin doublet).

Traveling waves. Reflection from a mirror moving with velocity $\pm v$ corresponds to the motion of a source with velocity $\pm 2v$ in the direction $\sigma$, i.e., at an angle $\alpha=\dfrac{\pi}{2}-\dfrac{\theta}{2}$ to the direction of observation (Fig. 2). The Doppler effect produces a change in frequency equal to

\[ \frac{\Delta\nu}{\nu}=\pm 2\,\frac{v}{c}\cos\alpha = \pm 2\,\frac{v}{c}\sin\frac{\theta}{2}. \]

Both the one and the other mode of reasoning lead to the effect of a change in frequency expressed by the formula

\[ \frac{\Delta\nu}{\nu}=\pm 2\,\frac{v}{c}\sin\frac{\theta}{2}. \tag{1} \]

Fig. 1.

As is known, a similar Doppler effect, determined by the velocity of elastic waves, has repeatedly been observed both in solid and in liquid bodies[^9].

It follows from formula (1), in particular, that in the direction of the primary ray $(\theta=0)$ no change in wavelength will be observed:

\[ \frac{\Delta\nu}{\nu}=0. \]

Fig. 2.

The physical reason for this lies in the circumstance that “scattering” in the original direction occurs on elastic waves,

moving perpendicular*) to the direction of scattering.

The influence of thermal motion on reflection from a mirror can be considered in an analogous way. Under the action of thermal motion, an ideal mirror is transformed into a slightly rough surface, whose irregularities are small in comparison with the wavelength of light. The complete picture of reflection from such a surface can be obtained by Rayleigh’s method, which gives simultaneously both regular (specular) reflection and surface scattering.

According to Rayleigh,^10 the surface of the mirror may be described by the equation**)

\[ z=\sum_n\left(a_n\cos n\frac{2\pi}{L}x+b_n\sin n\frac{2\pi}{L}x\right), \]

i.e. represented in the form of a Fourier series; \(a_n\) and \(b_n\) are expansion coefficients, small in comparison with \(\lambda\); \(L\) is a parameter determining the linear

[In the figure: “Incident light”; “Scattered (reflected) light”.]

Fig. 3

dimensions of the reflecting surface, very large in comparison with \(\lambda\). Thus the surface of the mirror is represented as an aggregate of sinusoidal gratings of different wavelengths \(\Lambda_n=\dfrac{L}{n}\). A plane light wave \((\lambda)\), incident on this surface at an angle \(\theta\) in the \(zx\) plane, is scattered in all directions in the plane of incidence; the intensity of the light going in the direction \(\theta_n\) is determined by the values of the corresponding coefficients \(a_n\) and \(b_n\).

To each of the Rayleigh gratings of number \(n\) there correspond diffracted rays lying on both sides of the specularly reflected ray (spectra of order \(\pm 1\)), whose direction \(\theta_n\) is deter-

*) In describing the Doppler effect we used the ordinary formulas, and not the formulas of the theory of relativity. Therefore the transverse Doppler effect does not figure in our results. Its magnitude is, as is known, of the order

\[ \left(\frac{v}{c}\right)^2, \]

and practically it may be disregarded.

**) We restrict ourselves to the problem of scattering in the plane of incidence \(zx\). The more general case, scattering in all azimuths, was considered by L. I. Mandelstam [Ann. d. Phys. 41, 609 (1913)], where \(z=f(xy)\) is expanded into two-dimensional harmonic gratings.

follows from the diffraction condition (see Fig. 3): the path difference \(CD-AB=\pm\lambda\), or \(\Lambda_n\sin\theta_n-\Lambda_n\sin\theta=\pm\lambda\), where \(\Lambda_n=CB\) is the period of grating number \(n\). Thus, the direction of the diffracted rays \((\theta)\) is determined by the condition

\[ \sin\theta_n-\sin\theta=\pm\frac{\lambda}{\Lambda_n}=\pm\frac{\lambda}{L}\cdot n. \tag{II} \]

Regular (specular) reflection is a special case of expression (II) and corresponds to \(n=0\). The “grating” of number zero is \(z=a_0\), i.e. a plane surface producing specular reflection in the direction \(\theta_0\), which, according to (II), corresponds to the law of reflection \((\theta_0=\theta)\).

However, light reflected by our gratings, which arise as a result of the thermal motion of the reflecting surface, undergoes a change in wavelength which, similarly to the case of volume scattering considered above, can be obtained either by the standing-wave method (modulation) or by the traveling-wave method (Doppler effect).

Fig. 4

Standing waves. In the direction \(\theta_n\), light is sent by a grating \(\Lambda_n\) having frequency \(N_n=\dfrac{v_n}{\Lambda_n}\), where \(v_n\) is the velocity of the surface wave, generally speaking dependent on frequency. Thus the modulation of light takes place with frequency

\[ \frac{\Delta\nu}{\nu} = \pm\frac{N_n}{\nu} = \pm\frac{v_n}{\Lambda_n}\cdot\frac{\lambda}{c} = \pm\frac{v_n}{c}\cdot\frac{\lambda}{L}\cdot n = \pm\frac{v_n}{c}\,(\sin\theta_n-\sin\theta). \tag{III} \]

Traveling waves. Scattering in the direction \(\theta_n\) is reflection from the mirror \(MN\), whose normal \(OQ\) is the bisector

angle \(SOP\) (see Fig. 4). The mirror moves with velocity \(\pm v_n\) along the reflecting surface. Reflection from a moving mirror occurs as though the light source were moving with velocity

\[ 2v_n\sin\alpha = 2v_n\sin\frac{\theta_n-\theta}{2} \]

in the direction of the normal to the mirror, i.e., at an angle

\[ \frac{\theta_n+\theta}{2} \]

to the direction of observation.

The change in the frequency of light occurs as a consequence of the Doppler effect according to the formulas

\[ \frac{\Delta\nu}{\nu} = \pm \frac{2v_n}{c}\cdot \sin\frac{\theta_n-\theta}{2} \cos\frac{\theta_n+\theta}{2} = \frac{v_n}{c}\left(\sin\theta_n-\sin\theta\right), \]

which, of course, coincides with (III).

Thus, the change in wavelength upon reflection from a mirror surface undergoing thermal motion takes place according to the expression:

\[ \frac{\Delta\nu}{\nu} = \pm \frac{v_n}{c}\left(\sin\theta_n-\sin\theta\right). \]

Regular (specular) reflection corresponds to the value \(n=0\), i.e., it occurs in the direction determined by the condition \(\sin\theta_n=\sin\theta\). It corresponds to \(\frac{\Delta\nu}{\nu}=0\), i.e., for regular reflection the Doppler effect is absent, as is confirmed by all observations. In scattered light, in directions different from the direction of regular reflection, a change in wavelength due to molecular motion (the Doppler effect) should be observed, determined, however, not by the velocity of motion of individual molecules composing the mirror, but by the velocity of surface waves. These latter may have different origins: elastic transverse waves, capillary waves, gravitational waves. For the case of small amplitudes and waves of small wavelength that is of interest to us, the latter may always be neglected. For the surface of a liquid, waves of the first type are absent and only capillary waves remain. For a solid mirror, elastic waves may play the principal role. However, observation of molecular scattering at the surface of a solid mirror is hardly possible, since inevitable polishing defects produce much stronger “parasitic” scattering. Molecular scattering at the surface of a liquid, however, is quite accessible to observation \(^{11}\). In this case the velocity of capillary waves is expressed as:

\[ v_n= \left(\frac{2\pi K}{\rho+\rho'}\right)^{1/2} (\Lambda_n)^{-1/2}, \]

where \(K\) is the capillary constant, \(\rho\) is the density of the liquid, and \(\rho'\) is the density of the gas above it, i.e., a quantity that may be neglected in comparison with \(\rho\).

From relation (11) we have \(\Lambda_n = \dfrac{\lambda}{\sin \theta_n - \sin \theta}\), so that

\[ \frac{\Delta \nu}{\nu} = \pm \frac{1}{c} \sqrt{ \frac{2\pi K}{\rho} \cdot \frac{(\sin \theta_n - \sin \theta)^3}{\lambda} }. \]

For example, when observing light scattered by the surface of mercury at an angle \(\theta_n = 45^\circ\), with normal incidence of the light \((\theta = 0)\), taking \(\lambda = 5000\,\text{\AA}\), \(K = 500\,\text{erg}/\text{cm}\), \(\rho = 13.6\,\text{g}/\text{cm}^3\), \(c = 3 \cdot 10^{10}\,\text{cm/sec}\), we obtain: \(\dfrac{\Delta \nu}{\nu} = \pm 4 \cdot 10^{-8}\), i.e. a quantity inaccessible to observation, especially if one takes into account the weakness of the scattered light.

As is known,\(^{12}\) the intensity of surface-scattered light can be considerably enhanced if observation is carried out at the interface of two liquids near the temperature at which they mix. In this case, however, \(K\) tends to zero, and consequently the velocity of capillary waves and the Doppler shift also tend to zero.

NOTE ADDED IN PROOF

This note was written when two publications devoted to the same question appeared: R. Lenouier, C. R., 226, 708, 1948, and J. Cabannes, C. R., 226, 710, 1948. Both authors carry out a theoretical analysis of the phenomenon essentially similar to that set forth in my note, and arrive at the conclusions I obtained. It may be hoped that this will put an end to useless experiments and debates, which have been going on for quite a long time.

REFERENCES

1. W. Rump, Zeits. f. Phys. XXIX, 196 (1924).
2. J. Cabannes, La diffusion moléculaire, p. 84 (1929).
3. I. Rocard et Rotschild, C. R. 186, 313 (1928).
4. J. Roig et J. Gobert, C. R. 221, 620 (1945).
5. Wolfers, Journ. de Physique, VIII, 14 (1947).
6. L. Mandelstam, Ann. d. Phys. IV, 23, 626 (1907).
7. L. I. Mandelstam, Zh. R. F. Kh. O., phys. section 58, 381 ([1926]).
8. L. Brillouin, Ann. de Physique 9, XVII, 88 (1922).
9. E. Gross, Zeits. f. Phys. 63, 685 (1930); Nature 126, 201, 400, 603 (1930).
10. Rayleigh, Sci. Papers 5, 398.
11. V. Raman a. L. Ramdas, Proc. Roy. Soc. (A) 108, 561; 109, 150; 272 (1925).
12. L. Mandelstam, Ann. d. Phys. 41, 609 (1913); F. S. Barshchanskaya, Zh. E. T. F. VII, 51 (1937).