MATHEMATICS OF PHYSICAL PROPERTIES OF CRVSTALS 3 



electric field is half the product of the component of the induction in the 

 direction of the field and the electric field. This is, therefore: 



2W = ElDn + EiD^ + £^^3 + E^E.iD,, + D,.) + E,Ey{Dn + Du) + 



EiEiiDxo + Ai) 



Considering then a condenser made from a unit cube of crystal, the charge 

 is D and the energy content is IF. If there is no leakage loss, the charge 



that can be drawn from the condenser is D = - - . Whence Di — — = 



dE dEi 



DuEi + i(A2 + Ai) £2 + 1 (As + Ai) £3. If, therefore, the induction 



is derivable from a potential, A2 = h (A2 + Ai) or A2 = Ai- Similarly 



Di3 — Ai and A3 = A2- By a proper choice of axes the remaining six 



D's can be reduced to three. In the case of isotropic dielectrics Ai = A2 = 



As and 47rAi corresponds to k, the dielectric constant. 



SECTION 3 



The Symmetry of Crystals 



If a crystal has certain sorts of symmetry the number of constants re- 

 quired to describe each property is materially reduced. For this reason 

 we now turn our attention to a study of symmetry. 



In general, plotting a vector property of the medium for a crystal gives a 

 complicated surface which we shall call a property surface. Each property 

 surface of a homogeneous isotropic medium is a sphere. 



Because of the orderly arrangement of matter in a crystal, the property 

 surfaces of crystalline media are commonly symmetrical. If a casting of a 

 property surface were made it might fit into its mold in several positions. 

 A property surface for quartz for example, if lifted from its mold and rotated 

 through a third of a turn about the proper axis, would fit back into the mold. 

 That is, quartz has a three fold axis. The natural requirement that mole- 

 cules be laid down in a way economical of space limits the kinds of symmetry 

 possible for crystals to axes of two fold (binary) symmetry, of three fold 

 (trigonal), of four fold and of six fold symmetry, planes of reflection symme- 

 try- and combinations of axis-reflection symmetry, besides a simple sym- 

 metry through a center. From these elements it is possible to divide all 

 possible property surfaces into 32 classes. No other classes built from these 

 elements could be self-consistent. 



A diagram study will prove this point. On a sphere let us mark axes of 

 two fold symmetry by means of a solid boat shaped figure, three fold with a 

 solid triangle, four fold with a square, sLx fold with a hexagon, planes of 

 symmetry with a solid line (great circle) and combination axis reflection, 

 by means of similar hollow figures. Finally, we shall project the sphere 



