An Exact Method To Determine the Complete Orientation Distribution Function of the Chain Axis from an Arbitrary (hkl) Reflection

A pole inversion formula is derived for textures with an axis of symmetry, for example, planar and fiber orientation obtained by uniaxial and biaxial deformation processes, respectively. The measured pole density distribution function of an (hkl) reflection is expressed as an integral transform of the (required) pole density distribution function (which is usually the chain axis in polymers). Unlike other methods for pole inversion, this formulation does not involve any series expansion of the orientation functions. The integral equation is analytically solved for the special case when the (hkl) reflection is perpendicular to the chain axis, a common feature to most of the semicrystalline and liquid-crystalline polymers.