MATHEMATICS OF PHYSICAL PROPERTIES OF CRYSTALS 



33 



Isothermal and Adiahatic Elastic Modulii 



Let us take a unit crystal cube at temperature d, apply any stress X 

 adiabatically, heat it to bring the temperature back to 6 at constant stress 

 then release it iso thermally. The cycle is analyzed in the table: 



Summing the strains to zero: 



(S'" - S'')X = t(A° + HX) 



If we equate the total entropy change to zero we obtain an expression 

 for Q that can be substituted in the relation "work in = Heat out." This 

 gives us: 



-^XXS'" - S'')X + tX{A° ^ HX) =h ^-^ 



6 



and from these two expressions we derive, writing for S'" —5": 



= A {A° + HX){A° -f HX)c (10.8) 



pa-P 



which is, to the first order of the small quantities X: 



4> = —^ A°A° + 2HXA° (10.9) 



and since .Y = Ce we have also 



paP 



{A° + 2HCe)A° (10.10) 



Whence we see that as the stress approaches zero as a limit </> approaches 



a - - 



0° = A° Ac . If we write similarly C'° — C = i/' we have multiplying 



S^" = S" -\- (f) by C'° = C + ^ and dropping higher orders of small quan- 

 tities: 



tA = - &"<!>€"' (10.11) 



