1 For light-waves traveling in the direction making an angle & with their original direction, AB equals 1 wave-length, say λ; hence λ = d sin θ. For the second-order spectra, 2 λ = d sin θ. (28) (29) In an experiment, d and the angle o are measured; then the wave-length x can be computed from equation (28) or (29). The following table gives the results of the measurement of a few wave-lengths: Approximate limit of visible red, Fraunhofer line A Blue hydrogen line, Fraunhofer line F Approximate limit of visible violet, Fraunhofer line K 0.00007594 cm. 0.00006563 cm. 658. An alternative explanation of the grating. There is a relatively simple explanation of the diffraction grating based on Huygens' principle (sect. 570). Each of the slits of a diffraction grating may be regarded as a source of waves. When plane waves (in the last sections the discussion was limited to plane waves) fall on the grating, these sources will all be in phase. In Fig. 448 is shown a portion of a grating with wavelets emitted from each slit as a source. According to Huygens' principle the resultant wave is the surface which is tangent to the wavelets. In this figure the lines which are drawn tangent to the wavelets represent the waves which form the central image. But these are not the only tangent lines which can be drawn. Fig. 449 shows a line which is tangent to the first wave from slit 1, to the second from slit 2, to the third from slit 3, and so on. The position of such tangent lines can be traced out in Fig. 448, but for the sake of simplicity a group of them is shown in Fig. 449. These parallel lines represent the waves producing a first-order spectrum. The waves giving the second-order spectra are shown in Fig. 450. They are the lines formed by drawing a tangent to the second wavelet from slit 1, to the fourth from slit 2, to the sixth from slit 3, and so on. 659. X-ray spectra. Spectra of light-waves are produced by gratings in which the distance between the slits is of the order of magnitude of a wavelength of light. But X-rays are so much shorter than light-waves that no grating modeled after the one used with light-waves can be made. Furthermore, the absence of refraction of X-rays by prisms prevents the other common method of producing spectra. For a long time it seemed that no method would be devised for measuring their wave-lengths. But Laue and Bragg, by slightly different methods, showed that in the regular symmetrical arrangement of crystallographic units in crystals a natural diffraction grating was available (the crystallographic units are ordinarily atoms, but in organic crystals they are molecules). In Fig. 451 is shown the arrangement of atoms in a simple cubic structure such as that possessed by crystals of common salt (sodium chloride), potassium chloride, potassium bromide, and potassium iodide. In the case of common salt there are equal numbers of sodium and chlorine atoms. The black circles in the figure represent sodium atoms, and the light circles chlorine atoms. In any structure such as this it is possible to find a number of planes, taken in different FIG. 451 directions through the crystal, each of which is studded with atoms. The planes formed by the atoms are probably the most perfect that occur in nature. One of these will reflect X-rays, and the angle of incidence will always be equal to the angle of reflection. The reason for this equality can the direction indicated (this is another example of Huygens' principle). It can be shown that in other directions the waves from the atoms do not meet in phase, and hence tend to destroy each other by interference. The reflection, as may be seen from the figure, obeys the usual law of regular reflection, the angle of incidence being equal to the angle of reflection. If there were only one layer, or plane, containing atoms, as in Fig. 452, X-rays would be reflected from this plane at all angles of incidence; but there are vast numbers of parallel and equally spaced planes in any crystal of finite size. The reflection from these planes is somewhat similar to the reflection of light from the surfaces of thin films. The A 0 C FIG. 453 radiation reflected by one plane interferes with that from another except at certain angles of incidence. As shown in Fig. 453, when a beam of X-rays is reflected by two layers of atoms, the rays which are reflected from the second layer travel farther than those from the first, by a distance equal to BC + CD. If this distance is equal to 1, 2, or 3 wave-lengths, the two beams will reënforce each other. From the definition of the sine of an angle (sect. 590), or BC sin BAC = AC BC = AC sin BAC. The distance AC is the distance between the layers of atoms. We shall denote this distance by d. By construction the angle BAC is equal to the angle CAD and to 6, the angle of "grazing incidence." Then BC = d sin 0. But the path-difference of the two beams is BC + CD. Since BC = CD, the total path-difference is 2 BC. X-rays will be reflected, provided that Hence x = BC + CD = 2 BC. λ = 2 d sin 0. (30) X-rays will also be reflected when the path-difference is 20 or 30. The rays are reflected not only by two layers but by thousands of equally spaced ones. It can be seen that if the path-difference between the first and second layers is 1 wave-length, that between the first and third will be 2 wave-lengths, that between the first and fourth will be 3 wave-lengths, and so on; in other words, the rays reflected from all these layers will reënforce each other for the directions in which equation (30) is true. The method of measuring the wave-length of X-rays consists of two steps: (1) to find d, the distance between the layers of atoms; (2) to measure the angle o at which the rays are reflected. The determination of the distance d is not so difficult as appears at first sight. The masses of atoms are now quite accurately known. Since we know the density of a given crystal (that is, the mass of 1 cubic centimeter), the number of atoms in a cubic centimeter is found by dividing the density by the mass of 1 atom. If the arrangement of the atoms is known, the distances apart can be computed from the number in a cubic centimeter. Fortunately, the atomic arrangement of some crystals is simple and known. The value of d which has served as a standard for wave-length measurements has been either that of calcite or that of rock salt (between the planes known as "100"). For calcite, d is equal to 3.029 × 10-8 cm. For rock salt, d is equal to 2.814 × 10-8 cm. The wave-lengths of X-rays, as measured by this method, range from 0.1×10-8 to 12 × 10-8 centimeters. The wave-lengths depend on what substance is used to radiate the X-rays: the elements having more massive atoms radiate shorter waves. When the wave-length of the X-rays used is known, the distance between the planes in a crystal of unknown structure can be determined by the aid of equation (30). Hence, by reflecting X-rays from crystals, the most intricate systems of crystals are now being worked out. This method has furnished in recent years an entirely new and most fruitful method of working out the atomic arrangements in crystals. CHAPTER XLVII POLARIZATION OF LIGHT Introduction, 660. Polarization of light by reflection, 661. Polarization by tourmaline, 662. An explanation of polarization, 663. Polarization of light proves that light-waves are transverse, 664. Double refraction, 665. Explanation of double refraction, 666. The Nicol prism, 667. Rotation of the plane of vibration, 668. The direction of vibration of plane-polarized light, and the electromagnetic theory, 669. 660. Introduction. It will be shown in this chapter that, sometimes a beam of light possesses a property in one direction which it does not have in another. When this is true, the beam is said to be polarized. This term is used because the word polarization may be applied to anything which has a property in one direction that it does not have in another. A magnet tends to set itself in an approximately north-and-south direction. It has a property in one direction that it does not have in another; hence it is said to be polarized. When an electric current flows between two pieces of platinum which are immersed in acidulated water, a counter electromotive force is produced which acts in only one direction: it opposes the current flow. This is also a case of polarization. The subject of polarization of light is a large and important branch of the science of optics; however, only a few of the simpler facts will be presented in this book. 661. Polarization of light by reflection. Ordinary light can be reflected by a piece of plate glass in any desired direction. But reflected light may partly lose this ability: there may be directions in which it cannot be reflected by a second piece of glass. Let light incident at an angle of about 57 degrees be reflected from a mirror M of unsilvered glass (Fig. 454). A second mirror, M', of unsilvered glass is set so that the angle of incidence is the same as that of the first mirror. In the positions shown in the figure both mirrors reflect light; but when M' is rotated (using |