Relationship between stress and d—spacing

Consider an isotropic film on a substrate for a case in which the equi-biaxial mismatch stress cTm and the corresponding mismatch strain Cm are to be 

Based on the discussion in Sections 2.1 and 3.5, the nonzero components of stress in the film, with reference to the a ia 2a 3—coordinate system 

Note that calculation of using this equation presumes a knowledge of the reference d-spacing value do of the film in the unstressed state. Equations (3.53) and (3.54) together yield the mismatch stress in the film in terms of the measured d-spacing as 

A tensile mismatch stress in the film implies that cTm 0 and, therefore, 

The same line of reasoning applies for the case of an anisotropic material supporting and equi-biaxial extensional mismatch strain and zero shear strain. Consider a cubic single crystal thin film with a (001) film-substrate interface plane. The mismatch strain and mismatch stress in this film can be estimated using x-ray diffraction measurements along with (3.43) and (3.45) by noting that 

---