You do not need to determine the optimal immersion depth of the disperser rod. You need to determine the optimal nanoparticle size distribution after the dispersion that you need.
The shape of synthesized nano particles is not affected by sonication but size is affected by sonication because it disperse the nano particles which also prevent agglomeration. The suspension of nano particles is dispersed in ultrasonicator under normal condition.
To obtain the desired shape of nanoparticles, you must precisely fulfill all the conditions for their synthesis. Shape control of nanocrystals is achieved by accurately tuning the nucleation and growth processes, which may be under both thermodynamic and kinetic control. May be seed-mediated , template-assisted, control of growth kinetic, surfactant-directed, supramolecular-directed, complex-template... synthesis.
I guess if the nanoparticles are small enough, with no aggregation, their size and shape would not change on sonication. I think the change in size and shape are highly dependent on the forces by which the nanoparticles attached to each other.
Particle is the smallest entity which cant be separated further by the application of ultrasonic energy. Hence we can expect the shape and size to remain the same.
first of all, result may depend on actual chemical composition. E.g. nanotubes or even needle like crystals may break. If there are several crystal modifications, provided sufficient solubility, theoretically there might be also a transformation from one to another modification. Take nothing for granted. Do the experiment.
Bhaskaranand Bhatt I agree with @Titus, you have to study your case. You also need to have in consideration both time and sonication potency. Keep in mind that in synthesis of some polymeric nanoparticles, final size obtained depend on these factors.