Part 3

The theory of the phenomena just described has been dealt with by Airy,[27] C. Neumann,[28] Maxwell,[29] Fitzgerald,[30] Rowland,[31] H. A. Lorentz,[32] Voight,[33] Ketteler,[34] van Loghem,[35] Potier,[36] Basset,[37] Goldhammer,[38] Drude,[39] J. J. Thomson,[40] and Leatham;[41] for a critical discussion of many of these theories we refer the reader to Larmor's[42] British Association Report. Most of these theories have proceeded on the plan of adding to the expression for the electromotive force terms indicating a force similar in character to that discovered by Hall (see MAGNETISM) in metallic conductors carrying a current in a magnetic field, i.e. an electromotive force at right angles to the plane containing the magnetic force and the electric current, and proportional to the sine of the angle between these vectors. The introduction of a term of this kind gives rotation of the plane of polarization by transmission through all refracting substance, and by reflection from magnetized metals, and shows a fair agreement between the theoretical and experimental results. The simplest way of treating the questions seems, however, to be to go to the equations which represent the propagation of a wave travelling through a medium containing ions. A moving ion in a magnetic field will be acted upon by a mechanical force which is at right angles to its direction of motion, and also to the magnetic force, and is equal per unit charge to the product of these two vectors and the sine of the angle between them. For the sake of brevity we will take the special case of a wave travelling parallel to the magnetic force in the direction of the axis of z.

Then supposing that all the ions are of the same kind, and that there are _n_ of these each with mass _m_ and charge _e_ per unit volume, the equations representing the field are (see ELECTRIC WAVES):--

dX0 d[xi] d[beta] K0 --- + 4[pi]ne ----- = -------; dt dt dz

dX[0] d[beta] ----- = -------; dz dt

dY0 d[eta] d[alpha] K0 --- + 4[pi]ne ------ = - -------- dt dt dz

dY0 d[alpha] --- = - --------; dz dt

d²[xi] d[xi] / 4[pi] \ d[eta] m ------ + R1 ----- + a[xi] = ( X0 + ----- ne[xi] ) e + He ------ dt² dt \ 3 / dt

d²[eta] d[eta] / 4[pi] \ d[xi] m ------- + R1 ------ + a[eta] = ( Y0 + ----- ne[eta] ) e - He -----; dt² dt \ 3 / dt

where H is the external magnetic field, X0, Y0 the components of the part of the electric force in the wave not due to the charges on the atoms, [alpha] and [beta] the components of the magnetic force, [xi] and [eta] the co-ordinates of an ion, R1 the coefficient of resistance to the motion of the ions, and [alpha] the force at unit distance tending to bring the ion back to its position of equilibrium, K0 the specific inductive capacity of a vacuum. If the variables are proportional to [epsilon]^[l(pt - qz)] we find by substitution that q is given by the equation

4[pi]ne²p²P 4[pi]ne³Hp³ q² - K0p² - ----------- = ± -----------, P² - H²e²p² P² - H²e²p²

where

P = (a - (4/3)[pi]ne²) + R1[iota]p - mp²,

or, by neglecting R, P = m(s² - p²), where s is the period of the free ions. If, q1², q2² are the roots of this equation, then corresponding to q1 we have X0 = [iota]Y0 and to q2 X0 = -[iota]Y0. We thus get two oppositely circular-polarized rays travelling with the velocities p/q1 and p/q2 respectively. Hence if v1, v2 are these velocities, and v the velocity when there is no magnetic field, we obtain, if we neglect terms in H²,

1 1 4[pi]ne³Hp --- = -- + ------------, v1² v² m²(s² - p²)²

1 1 4[pi]ne³Hp --- = -- - ------------. v2² v² m²(s² - p²)²

The rotation r of the plane of polarization per unit length

/ 1 1 \ 2[pi]ne³Hp²v = ½p ( --- - --- ) = -------------. \ v1 v2 / m²(s² - p²)²

Since 1/v² = K0 + 4[pi]ne²/m(s² - p²), we have if µ is the refractive index for light of frequency p, and v0 the velocity of light in vacuo.

µ² - 1 = 4[pi]ne²v²0 / m(s² - p²) (1)

So that we may put

r = (µ² - 1)²p²H / s[pi]µne v0³ (2)

Becquerel (_Comptes rendus_, 125, p. 683) gives for r the expression

e H dµ ½ --- ---- ---------, m v0 d[lambda]

where [lambda] is the wave length. This is equivalent to (2) if µ is given by (1). He has shown that this expression is in good agreement with experiment. The sign of r depends on the sign of e, hence the rotation due to negative ions would be opposite to that for positive. For the great majority of substances the direction of rotation is that corresponding to the negation ion. We see from the equations that the rotation is very large for such a value of p as makes P = 0: this value corresponds to a free period of the ions, so that the rotation ought to be very large in the neighbourhood of an absorption band. This has been verified for sodium vapour by Macaluso and Corbino.[43]

If plane-polarized light falls normally on a plane face of the medium containing the ions, then if the electric force in the incident wave is parallel to x and is equal to the real part of A[epsilon]^[l(pt - qz)], if the reflected beam in which the electric force is parallel to x is represented by B[epsilon]^[l(pt + qz)] and the reflected beam in which the electric force is parallel to the axis of y by C[epsilon]^[l(pt + qz)], then the conditions that the magnetic force parallel to the surface is continuous, and that the electric forces parallel to the surface in the air are continuous with Y0, X0 in the medium, give

A B [iota]C ----------------- = ----------- = ---------- (q + q1) (q + q2) (q² - q1q2) q(q2 - q1)

or approximately, since q1 and q2 are nearly equal,

[iota]C q(q2 - q1) (µ² - 1)pH ------- = ---------- = ------------. B q² - q1² 4[pi]µne V0²

Thus in transparent bodies for which µ is real, C and B differ in phase by [pi]/2, and the reflected light is elliptically polarized, the major axis of the ellipse being in the plane of polarization of the incident light, so that in this case there is no rotation, but only elliptic polarization; when there is strong absorption so that µ contains an imaginary term, C/B will contain a real part so that the reflected light will be elliptically polarized, but the major axis is no longer in the plane of polarization of the incident light; we should thus have a rotation of the plane of polarization superposed on the elliptic polarization.

_Zeeman's Effect._--Faraday, after discovering the effect of a magnetic field on the plane of polarization of light, made numerous experiments to see if such a field influenced the nature of the light emitted by a luminous body, but without success. In 1885 Fievez,[44] a Belgian physicist, noticed that the spectrum of a sodium flame was changed slightly in appearance by a magnetic field; but his observation does not seem to have attracted much attention, and was probably ascribed to secondary effects. In 1896 Zeeman[45] saw a distinct broadening of the lines of lithium and sodium when the flames containing salts of these metals were between the poles of a powerful electromagnet; following up this observation, he obtained some exceedingly remarkable and interesting results, of which those observed with the blue-green cadmium line may be taken as typical. He found that in a strong magnetic field, when the lines of force are parallel to the direction of propagation of the light, the line is split up into a doublet, the constituents of which are on opposite sides of the undisturbed position of the line, and that the light in the constituents of this doublet is circularly polarized, the rotation in the two lines being in opposite directions. When the magnetic force is at right angles to the direction of propagation of the light, the line is resolved into a triplet, of which the middle line occupies the same position as the undisturbed line; all the constituents of this triplet are plane-polarized, the plane of polarization of the middle line being at right angles to the magnetic force, while the outside lines are polarized on a plane parallel to the lines of magnetic force. A great deal of light is thrown on this phenomenon by the following considerations due to H. A. Lorentz.[46]

Let us consider an ion attracted to a centre of force by a force proportional to the distance, and acted on by a magnetic force parallel to the axis of z: then if m is the mass of the particle and e its charge, the equations of motion are

d²x dy m --- + ax = -He --; dt² dt

d²y dx m --- + ay = He --; dt² dt

d²z m --- + ax = 0. dt²

The solution of these equations is

x = A cos (p1t + [beta]) + B cos (p2t + [beta]1)

y = A sin (p1t + [beta]) - B sin (p2t + [beta]1)

z = C cos (pt + [gamma])

where

a - mp1² = - He p1

a - mp2² = He p2

p² = [alpha]/m,

or approximately

He He p1 = p + ½ ---, p2 = p - ½ ---. m m

Thus the motion of the ion on the xy plane may be regarded as made up of two circular motions in opposite directions described with frequencies p1 and p2 respectively, while the motion along z has the period p, which is the frequency for all the vibrations when H = 0. Now suppose that the cadmium line is due to the motion of such an ion; then if the magnetic force is along the direction of propagation, the vibration in this direction has its period unaltered, but since the direction of vibration is perpendicular to the wave front, it does not give rise to light. Thus we are left with the two circular motions in the wave front with frequencies p1 and p2 giving the circularly polarized constituents of the doublet. Now suppose the magnetic force is at right angles to the direction of propagation of the light; then the vibration parallel to the magnetic force being in the wave front produces luminous effects and gives rise to a plane-polarized ray of undisturbed period (the middle line of the triplet), the plane of polarization being at right angles to the magnetic force. The components in the wave-front of the circular orbits at right angles to the magnetic force will be rectilinear motions of frequency p1 and p2 at right angles to the magnetic force--so that they will produce plane-polarized light, the plane of polarization being parallel to the magnetic force; these are the outer lines of the triplet.

If Zeeman's observations are interpreted from this point of view, the directions of rotation of the circularly-polarized light in the doublet observed along the lines of magnetic force show that the ions which produce the luminous vibrations are _negatively_ electrified, while the measurement of the charge of frequency due to the magnetic field shows that e/m is of the order 10^7. This result is of great interest, as this is the order of the value of e/m in the negatively electrified particles which constitute the Cathode Rays (see CONDUCTION, ELECTRIC III. _Through Gases_). Thus we infer that the "cathode particles" are found in bodies, even where not subject to the action of intense electrical fields, and are in fact an ordinary constituent of the molecule. Similar particles are found near an incandescent wire, and also near a metal plate illuminated by ultra-violet light. The value of e/m deduced from the Zeeman effect ranges from 10^7 to 3.4 × 10^7, the value of e/m for the particle in the cathode rays is 1.7 × 10^7. The majority of the determinations of e/m from the Zeeman effect give numbers larger than this, the maximum being about twice this value.

A more extended study of the behaviour of the spectroscopic lines has afforded examples in which the effects produced by a magnet are more complicated than those we have described, indeed the simple cases are much less numerous than the more complex. Thus Preston[47] and Cornu[48] have shown that under the action of a transverse magnetic field one of the D lines splits up into four, and the other into six lines; Preston has given many other examples of these quartets and sextets, and has shown that the change in the frequency, which, according to the simple theory indicated, should be the same for all lines, actually varies considerably from one line to another, many lines showing no appreciable displacement. The splitting up of a single line into a quartet or sextet indicates, from the point of view of the ion theory, that the line must have its origin in a system consisting of more than one ion. A single ion having only three degrees of freedom can only have three periods. When there is no magnetic force acting on the ion these periods are equal, but though under the action of a magnetic force they are separated, their number cannot be increased. When therefore we get four or more lines, the inference is that the system giving the lines must have at least four degrees of freedom, and therefore must consist of more than one ion. The theory of a system of ions mutually influencing each other shows, as we should expect, that the effects are more complex than in the case of a single ion, and that the change in the frequency is not necessarily the same for all systems (see J. J. Thomson, _Proc. Camb. Phil. Soc._ 13, p. 39). Preston[49] and Runge and Paschen have proved that, in some cases at any rate, the change in the frequency of the different lines is of such a character that they can be grouped into series such that each line in the series has the same change in frequency for the same magnetic force, and, moreover, that homologous lines in the spectra of different metals belonging to the same group have the same change in frequency.

A very remarkable case of the Zeeman effect has been discovered by H. Becquerel and Deslandres (_Comptes rendus_, 127, p. 18). They found lines in iron when the most deflected components are those polarized in the plane at right angles to the magnetic force. On the simple theory the light polarized in this way is not affected. Thus the behaviour of the spectrum in the magnetic field promises to throw great light on the nature of radiation, and perhaps on the constitution of the elements. The study of these effects has been greatly facilitated by the invention by Michelson[50] of the echelon spectroscope.

There are some interesting phenomena connected with the Zeeman effect which are more easily observed than the effect itself. Thus Cotton[51] found that if we have two Bunsen flames, A and B, coloured by the same salt, the absorption of the light of one by the other is diminished if either is placed between the poles of a magnet: this is at once explained by the Zeeman effect, for the times of vibration of the molecules of the flame in the magnetic field are not the same as those of the other flame, and thus the absorption is diminished. Similar considerations explain the phenomenon observed by Egoroff and Georgiewsky,[52] that the light emitted from a flame in a transverse field is partially polarized in a plane parallel to the magnetic force; and also Righi's[53] observation that if a sodium flame is placed in a longitudinal field between two crossed Nicols, and a ray of white light sent through one of the Nicols, then through the flame, and then through the second Nicol, the amount of light passing through the second Nicol is greater when the field is on than when it is off. Voight and Wiechert (_Wied. Ann._ 67, p. 345) detected the double refraction produced when light travels through a substance exposed to a magnetic field at right angles to the path of the light; this result had been predicted by Voight from theoretical considerations. Jean Becquerel has made some very interesting experiments on the effect of a magnetic field on the fine absorption bands produced by xenotime, a phosphate of yttrium and erbium, and tysonite, a fluoride of cerium, lanthanum and didymium, and has obtained effects which he ascribes to the presence of positive electrons. A very complete account of magneto- and electro-optics is contained in Voight's _Magneto- and Elektro-optik_.

FOOTNOTES:

[1] _Experimental Researches_, Series 19.

[2] _Comptes rendus_, 88, p. 709.

[3] _Wied. Ann._ 6, p. 332; 8, p. 278; 10, p. 257.

[4] _Wied. Ann._ 23, p. 228; 27, p. 191.

[5] _Wied. Ann._ 31, p. 941.

[6] _Phil. Trans._, A. 1885, Pt. 11, p. 343.

[7] _Wied. Ann._ 26, p. 456.

[8] _Phil. Trans._, A. 1895, Pt. 17, p. 621.

[9] _Wied. Ann._ 24, p. 161.

[10] _Wied. Ann._ 31, p. 970.

[11] _Comptes rendus_, 57, p. 670.

[12] _Comptes rendus_, 43, p. 529; 44, p. 1209.

[13] _Journ. Chem. Soc._ 1884, p. 421; 1886, p. 177; 1887, pp. 362 and 808; 1888, p. 561; 1889, pp. 680 and 750; 1891, p. 981; 1892, p. 800; 1893, pp. 75, 99 and 488.

[14] _Wied. Ann._ 44, p. 377.

[15] _Wied. Ann._ 43, p. 280.

[16] _Zeitschrift f. physikal. Chem._ 11, p. 753.

[17] _Phil. Mag._ [5] 3, p. 321.

[18] _Ann. de chim. et de phys._ [6] 4, p. 433; 9, p. 65; 10, p. 200.

[19] _Wied. Ann._ 23, p. 228; 27, p. 191.

[20] _Wied. Ann._ 39, p. 25.

[21] _Wied. Ann._ 42, p. 115.

[22] _Phil. Mag._ [5] 12, p. 171.

[23] _Journ. de Phys._ 1884, p. 360.

[24] _Beiblätter zu Wied. Ann._ 1885, p. 275.

[25] _Messungen über d. Kerr'sche Erscheinung._ Inaugural Dissert. Leiden, 1893.

[26] _Phil. Mag._ [5] 5, p. 161.

[27] _Phil. Mag._ [3] 28, p. 469.

[28] _Die Magn. Drehung d. Polarisationsebene des Lichts_, Halle, 1863.

[29] _Electricity and Magnetism_, chap. xxi.

[30] _Phil. Trans._ 1880 (2), p. 691.

[31] _Phil. Mag._ (5) 11, p. 254, 1881.

[32] _Arch. Néerl._ 19, p. 123.

[33] _Wied. Ann._ 23, p. 493; 67, p. 345.

[34] _Wied. Ann._ 24, p. 119.

[35] _Wied. Beiblätter_, 8, p. 869.

[36] _Comptes rendus_, 108, p. 510.

[37] _Phil. Trans._ 182, A. p. 371, 1892; _Physical Optics_, p. 393.

[38] _Wied. Ann._ 46, p. 71; 47, p. 345; 48, p. 740; 50, p. 722.

[39] _Wied. Ann._ 46, p. 353; 48, p. 122; 49, p. 690.

[40] _Recent Researches_, p. 489 et seq.

[41] _Phil. Trans._, A. 1897, p. 89.

[42] _Brit. Assoc. Report_, 1893.

[43] _Comptes rendus_, 127, p. 548.

[44] _Bull. de l'Acad. des Sciences Belg._ (3) 9, pp. 327, 381, 1885; 12 p. 30, 1886.

[45] _Communications from the Physical Laboratory_, Leiden, No. 33, 1896; Phil. Mag. 43, p. 226; 44, pp. 55 and 255; and 45, p. 197.

[46] _Arch. Néerl._ 25, p. 190.

[47] _Phil. Mag._ 45, p. 325; 47, p. 165.

[48] _Comptes rendus_, 126, p. 181.

[49] _Phil. Mag._ 46, p. 187.

[50] _Phil. Mag._ 45, p. 348.

[51] _Comptes rendus_, 125, p. 865.

[52] _Comptes rendus_, pp. 748 and 949, 1897.

[53] _Comptes rendus_, 127, p. 216; 128, p. 45.

(J. J. T.)

MAGNOLIA, the typical genus of the botanical order Magnoliaceae, named after Pierre Magnol (1638-1715), professor of medicine and botany at Montpellier. It contains about twenty species, distributed in Japan, China and the Himalayas, as well as in North America.

Magnolias are trees or shrubs with deciduous or rarely evergreen foliage. They bear conspicuous and often large, fragrant, white, rose or purple flowers. The sepals are three in number, the petals six to twelve, in two to four series of three in each, the stamens and carpels being numerous. The fruit consists of a number of follicles which are borne on a more or less conical receptacle, and dehisce along the outer edge to allow the scarlet or brown seeds to escape; the seeds however remain suspended by a long slender thread (the funicle). Of the old-world species, the earliest in cultivation appears to have been _M. Yulan_ (or _M. conspicua_) of China, of which the buds were preserved, as well as used medicinally and to season rice; together with the greenhouse species, _M. fuscata_, it was transported to Europe in 1789, and thence to North America, and is now cultivated in the Middle States. There are many fine forms of _M. conspicua_, the best being _Soulangeana_, white tinted with purple, _Lenné_ and _stricta_. Of the Japanese magnolias, _M. Kobus_ and the purple-flowered _M. obovata_ were met with by Kaempfer in 1690, and were introduced into England in 1709 and 1804 respectively. _M. pumila_, the dwarf magnolia, from the mountains of Amboyna, is nearly evergreen, and bears deliciously scented flowers; it was introduced in 1786. The Indian species are three in number, _M. globosa_, allied to _M. conspicua_ of Japan, _M. sphenocarpa_, and, the most magnificent of all magnolias, _M. Campbellii_, which forms a conspicuous feature in the scenery and vegetation of Darjeeling. It was discovered by Dr Griffith in Bhutan, and is a large forest tree, abounding on the outer ranges of Sikkim, 80 to 150 ft. high, and from 6 to 12 ft. in girth. The flowers are 6 to 10 in. across, appearing before the leaves, and vary from white to a deep rose colour.