64 BELL SYSTEM TECHNICAL JOURNAL 



closer its approach to perfect order. Perfect order would consist in every 

 atom located at rest in exactly its proper place in the crystalline lattice. 

 If such is the state of affairs at the absolute zero, then a crystalline phase 

 at the absolute zero exhibits perfect order. Now as I have implied without 

 precisely saying, entropy is taken to be a measure of disorder. If this is 

 proper, then it is sensible and correct to say that entropy vanishes for the 

 perfect crystal at the absolute zero. This is a reason, it is in fact the reason, 

 for assigning the value zero to the additive constant Sq. 



If I were to stop my exposition at this point, it would certainly be an 

 impressive conclusion, but hardly a just one. We cannot solve the mysteries 

 of entropy in any so simple a way; in fact they have never been completely 

 solved. Many additions should be made to the foregoing paragraph: 

 some would help out its evident meaning and some would not. Thus, as it 

 stands, it certainly implies that a liquid at the absolute zero — helium 

 offering such a case — must exhibit disorder and therefore have a greater 

 value of entropy than does the crystalline solid. However, it turns out that 

 liquid helium near the absolute zero has the same entropy as solid helium 

 does, and therefore if one value is zero so also must the other be. If instead 

 of liquid helium I had taken the supercooled and toughened liquid which is 

 called a "glass" — glycerol or alcohol in the glassy state, for instance — the 

 result would have been more agreeable. Several of these glasses have been 

 studied with great care, and the right-hand member of equation (16) 

 evaluated for them; it turns out that the entropy near the absolute zero i^ 

 markedly and indubitably greater for the glass than for the crystal "" 

 order" must therefore prevail in the glass, but the result with liquid helium 

 impedes the physicist from proclaiming that it is simply the obvious disorder 

 of the irregular placing of the atoms in the glass. A mixture of two sub- 

 stances of a single kind, even when this mixture forms a beautiful crystal, 

 may nevertheless present disorder; for the two kinds of atoms may be 

 sprinkled in a thoroughly chaotic manner over the available points of the 

 crystalline lattice. Such a mixture, it has been found, does have an entropy 

 near the absolute zero which is definitely greater than the sum of the en- 

 tropies of the two substances when unmixed. But it is also possible for the 

 mixture of two substances to be orderly, in the usual sense of the word! 

 Thus in a mixture — "alloy" is the customary word — of equal numbers of 

 atoms of copper and zinc, the copper atoms may all be found upon one 

 lattice and the zinc atoms upon another, the two lattices interpenetrating 

 and interlocking with each other so as to form the lattice of the crystal of 

 the alloy. This is one of the extreme possibilities: the other is, the en- 

 tirely chaotic besprinkling of the points of the two lattices with atoms of the 

 two kinds. Both can be realized, the transition between the two being 



