  Surface and size effect on the order parameter fluctuations and critical
phenomena in the intensively studied 3D-confined nanosized systems with
long-range order was not considered theoretically, while the calculations for
bulk samples and thick films were performed long ago. Since widely used
magnetic resonance, diffraction and scattering methods collect information
about both macro- and nanosystems via the structural factors, which are
directly related with fluctuations correlator, analytical expressions for the
correlation function of the order parameter fluctuations seem extremely
necessary for quantitative analyses of the experimental data broad spectrum. In
the letter we solve the vital problem within Landau-Ginzburg-Devonshire
phenomenological approach for the particles of arbitrary shape and consider
concrete examples of the spherical and cylindrical ferroic nanoparticles.
Allowing for the strong surface energy contribution, analytical expressions
derived for Ornstein-Zernike correlator of the long-range order parameter
fluctuations in 3D-confined system, dynamic generalized susceptibility,
relaxation times and correlation radii discrete spectrum are principally
different from those known for bulk system. Besides the great importance of the
fluctuations correlation function for the analyses of scattering and magnetic
resonance experimental spectra, proposed expression for fluctuations strength
defines the fundamental limit of phenomenological theory applicability for
3D-confined nanosystems.
