December 30, 19 15] 



NATURE 



497 



particle is drifting, and depends also upon the force 

 which a molecule exerts on a particle when it comes 

 near it. When the particle is under the action of an 

 electric force X, the momentum communicated to it 

 by the force per second is Xe ; when it settles down to 

 a steady state, the gain in momentum from the force 

 must equal the loss from the impacts, so that ^u = Xe 

 or u = XeJI3, and k must be equal to e//3. Let us try 

 to estimate the value of k for metals, from the measure- 

 ments which have been made of the drift of electrons 

 in circumstances where it can be measured. In 

 conduction through flames the negative electricity is 

 carried by electrons, and at a temperature of 2000° C. 

 the speed of the electrons under a force of a volt per 

 centimetre is estimated at 10,000 centimetres per 

 second. This would make the speed of an electron 

 through air at normal pressure at 0° C. under the 

 same electric force about 3300 centimetres per second. 

 Now, in the air there are 275 xio^' molecules per 

 cubic centimetre, while in a cubic centimetre of silver 

 there are about 6x10'' atoms of silver, about 2200 

 times the number of air molecules, hence we should 

 expect the velocity of an electron in silver under a 

 volt per centimetre would be about 3300/2200 or 1-5 

 centimetre per second, k is tjie velocity for unit force 

 which is i/io* of a volt per centimetre, so that k 

 should be about 1-5x10-*, and since nfe = 3-9x10", 

 n would be 2-6 x 10^". This is about forty times the 

 number of silver atoms, so that the silver atoms would 

 have on the average a charge of forty units. This 

 number of electrons would exert a pressure of 100,000 

 atmospheres, and their specific heat in unit mass of 

 silver would be many times the actual specific heat of 

 silver. 



We can also estimate the number of electrons re- 

 quired on this theory by considering the resistance of 

 the metal under periodic instead of steady electric 

 forces. If the electric force is represented by X cos pt, 

 then we can show that if u is the velocity of drift, m 

 the mass of an electron, 



m~+0u = Xecospf, 



the solution of which is 



Thus the effective resistance under alternatmg cur- 

 rents is greater than that under steady currents in the 

 proportion of {i + (ni^p^l^^^)]i to i. Now one very 

 interesting case of alternating electric forces is that of 

 a wave of light. Rubens has investigated the resist- 

 ance of metals under the electric forces which occur 

 in light of various wave-lengths. He finds that when 

 the wave-length is greater than 2X10-* centimetres 

 the resistance is indistinguishable from that for steady 

 currents, and when the wave-length is 4X10-* centi- 

 metres the resistance is only 20 per cent, greater. It 

 is the second of these results which we shall use to 

 calculate )3. p for a wave-length of 4x10-* is 

 i-5xn-io**, and if the resistance is increased by 20 per 

 cent. m'p'/f3' = 2ls, or m/)/i3 = 2/n-, approximately; 

 thus jS = 7-5X io**xm; and since k = e//3, and 

 e/m= 1-7x10% fe = 2-2xio-*. As nfe = 3-9xio", 

 n = i-8xio'*. This is not greatly different from the 

 value we got by the preceding method, and leads 

 again to the conclusion that to explain an electrical 

 conductivity as large as that of silver requires a num- 

 ber of electrons so great that each atom of silver 

 would have on an average to lose 20 or 30 electrons, 

 and the specific heat of these would be far greater 

 than the actual specific heat of silver. 



A great deal of work has been done in recent years 

 on the specific heats of metals, but, so far as I know, 



NO. 2409, VOL. 96] 



no effect has been found which can be traced to elec- 

 trons, nor is there any trace of energy being absorbed 

 in dissociating the atoms of the metal into electrons 

 and positively charged atoms. 



Variation of Electrical Resistance with Temperature. 



The fact that to a considerable degree of approxi- 

 mation the electrical resistance of all pure metals is at 

 all but the lowest temperatures proportional to the 

 absolute temperature, is a result of such generality 

 that we should expect it to be a direct consequence of 

 any adequate theory of electrical resistance. On the 

 theory that the conductivity is due to free electrons, 

 this result, so far from being an obvious consequence 

 of the theory, is very difficult to reconcile with it. 

 The conductivity of the metal is, as we have seen, 

 the product of two factors, n the number of free elec- 

 trons, and k the mobility of an electron, and since 

 the conductivity is approximately inversely propor- 

 tional to the absolute temperature, nk must be very 

 much greater at low temperatures than at high. Con- 

 sider the factors separately, we certainly should not 

 a priori expect n to be larger at low temperatures 

 than at high ones, for we regard the free ion« as due 

 to the dissociation of the atoms of the metal, and thus 

 to have much the same connection with temperature 

 as the products of dissociation in such a case as the 

 dissociation of molecules of iodine. In such cases, 



however, n is proportional to e'^'^, where T is the 

 absolute temperature, R the gas constant, and w 

 proportional to the work required to dissociate the 

 system. Now this factor, instead of increasing as the 

 temperature diminishes, decreases, and does so very 

 rapidly indeed, when RT is small compared with w. 

 If, then, the product nk is much greater at low tem- 

 peratures than at higher ones, the increase must be 

 due to the factor k — that is, the mobility of the system 

 must increase very rapidly as the temperature 

 diminishes. Now the theory of the motion of an 

 electron through a number of centres of force, each 

 of which acts on the electron with a force inversely 

 proportional to the />*'» power of the distance, leads to 



the result that k is proportional to ^^—i /m, where m 

 is the number of attracting centres per unit volume 

 and 6 the temperature. The product nk varies approxi- 

 mately as 6-^, and since n diminishes as 6 diminishes, 

 k must increase more rapidly as 6 diminishes than 

 6-^; for this to be the case p must be less than one, 

 an extremely improbable result. If we suppose that 

 the force exerted by a positively charged atom on the 

 negative electron is so much greater than that exerted 

 by a charged one that it is the collisions with the 

 charged atoms that determine the mobility, m = n and 

 p = 2, then nk varies 6i—i.e. the conductivity should 

 increase with the temperature instead of diminishing. 



The very remarkable results got within the last few 

 years by Kamerlingh Onnes, in his experiments on 

 the electrical resistance of metals at very low tem- 

 peratures, seem to me to tell against the theory that 

 the conduction is due to free electrons, while they 

 receive a simple explanation on the theory which I am 

 about to suggest. Kamerlingh Onnes finds that the 

 resistance of some pure metals, such as mercury or 

 lead at the temperature of liquid helium or there- 

 abouts, becomes too small to be measured, and is cer- 

 tainly less than one-thousandth-millionth of its value 

 at 0° C, whereas, if it diminished in proportion to 

 the temperature it would only be diminished by one- 

 seventieth. He has found, too, the very remarkable 

 result that a current once started by moving a magnet 

 in the neighbourhood of a lead ring at this tempera- 

 ture, lasted with little diminution more than two hours 



