What is average particle size expected. Generally, FESEM provides for such measurements reasonably well.
One simple technique to estimate an average grain size is the intercept technique. A random straight line is drawn though the micrograph. The number of grain boundaries intersecting the line are counted. The average grain size is found by dividing the number of intersections by the actual line length.
No one has mentioned the taking of a representative sample and evaluating statistically valid numbers of particles (e.g. to estimate the mean to a required standard error of 1% will require the measurement of 10000 random particles).
All the mentioned techniques take tiny amounts of sample. How many tons of material do you have?
No one has mentioned the need to separate particles from aggregates and agglomerates and the difference between crystallite and particle size. All the preceding are far more important (and apply to all techniques) than a discussion on the precision of aforesaid techniques. All the mentioned techniques have international standards associated with them and take precise and accurate measurements on what is presented to them.
Plus, all images are 2D representations of 3D particles and shape is a 3D issue… We have equivalents to deal with - circular equivalent usually in the case of imaging.
The Scherrer equation calculates crystallite size, not particle size. It also is complicated by the fact that the XRD peak broadening is caused by the instrument and strain, as well as crystallite size.
We still have no knowledge as to the approximate size and range of your system. If it’s greater than around 20 microns, then manual microscopy combined with some form of image analysis will be far more appropriate than any form of electron microscopy for size determination.
Do you want to measure the size of primary particles (crystallites) or their aggregates?
Hydroxyapatite is an HCP crystal (not cubic) and it is very likely that primary particles show a significantly big aspect ratio.
Aggregates might randomly orient themselves and (thermodynamically) there is the tendency in them to form spherical shapes but they might not have the chance to do so.
Try SEM images and post-analyze them using ImageJ.
FESEM is the same thing as an SEM in terms of the image they produce.