Critical evaluation of the Lotgering degree of orientation texture indicator
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Preferred orientation in textured ceramics is often assessed by comparing the relative intensities of x-ray diffraction reflections to those of a randomly oriented ceramic using the Lotgering degree of orientation (f ). However, this paper provides evidence that indiscriminate assessments of f can be misleading. Using measured intensities of a modestly textured tape cast bismuth titanate (Na0.5Bi4.5Ti4O15) ceramic, calculated f values vary from 7.4 to 73.2% depending on the reflections included in the calculation. The texture is also quantified by calculating the orientation distribution function (ODF) using measured pole figures. A model is then presented that demonstrates f is nonlinear with the multiple of preferred (00l)-orientations, the standard unit of the 00l pole figure.
I. INTRODUCTION
Piezoelectricity is the coupling of electrical and mechanical energy and is apparent in ferroelectric ceramics because the spontaneous polarization vectors can be preferentially reoriented from their initial random configuration.1 These materials translate a stress into a dielectric displacement and vice versa, making them ideal for sensors, actuators, and transducers in industrial, medical, and military applications. Bismuth titanate ceramics are preferred for high-temperature sensor applications because the structure is piezoelectric at higher temperatures than other ferroelectric ceramics.2 The linear coefficients that relate the electrical and mechanical signals are called the piezoelectric constants. Because these constants are often modest in high-temperature piezoelectrics, use of crystallographic texturing has been successfully adopted as a way to increase the piezoelectric constants through alignment of the spontaneous polarization vectors. Some processes that have been used to induce a crystallographic texture in bismuth titanate ceramics are hot forging,3–8 tape casting,9–14 and extrusion.15 Quantifying the degree of crystallographic texture in these systems is critical to optimizing processing parameters for enhanced property anisotropy. The most complete descriptions of texture quantify the probability of given crystallite orientations with respect to the reference sample coordinate axes, a so-called orientation distribution function (ODF).16,17 The value of the ODF for a given orientation is the probability of finding a crystallite of similar orientation, in the unit called multiples of a random distribution (MRD). Despite the availability of
DOI: 10.1557/JMR.2004.0440 3414
J. Mater. Res., Vol. 19, No. 11, Nov 2004
these more complete approaches, a less demanding estimate of texture in bismuth titanate has persisted.3–15,18,19 This estimate is given by the Lotgering degree of orientation (f ), which is a comparison of the relative intensities of (00l) reflections to all observed reflections in a coupled -2 powder x-ray diffraction (XRD) spectrum20
f=
p − po ; 1 − po
p=
冱I 冱I +冱I 00l
00l
,
(1)
non−00l
where f is considered a measure of the “degree of orientation” and ranges from 0% to 100%, Ihkl
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