Anomalous heat-capacity reduction driven by defects and phonon confinement in inverse spinel CoFe2O4 nanoparticles

Abstract This study explores the structural and thermal optimization of CoFe 2 O 4 spinel nanoparticles synthesized via the Pechini method and calcined at 450 $$^{\circ }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mmultiscripts> <mml:mrow/> <mml:mrow/> <mml:mo>∘</mml:mo> </mml:mmultiscripts> </mml:math> C and 650 $$^{\circ }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mmultiscripts> <mml:mrow/> <mml:mrow/> <mml:mo>∘</mml:mo> </mml:mmultiscripts> </mml:math> C. X-ray diffraction and Rietveld refinement confirm a single-phase cubic spinel (Fd $${\bar{3}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover> <mml:mrow> <mml:mn>3</mml:mn> </mml:mrow> <mml:mrow> <mml:mo>¯</mml:mo> </mml:mrow> </mml:mover> </mml:math> m) with high thermal stability and controlled lattice dynamics. Increasing the calcination temperature induces a slight lattice contraction (8.3877(16) Å to 8.3795(6) Å), while the microstrain remains relatively stable ( $$\epsilon$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ϵ</mml:mi> </mml:math> from 0.18 (6)% to 0.239 (23)%). Heat-capacity data (2–210 K) reveal dual phonon behavior: a low Debye temperature ( $$\Theta$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Θ</mml:mi> </mml:math> D =396.47 K) reflecting global phononic softening, and a high Einstein temperature ( $$\Theta$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Θ</mml:mi> </mml:math> E =683.34 K) indicating local rigidity from Co–O and Fe–O bonds. FTIR analysis suggests cation redistribution between A and B sites, increasing Fe–O stiffness (F CO from 124 to 128 N·m −1 ) and slightly relaxing tetrahedral bonds (F CT from 223 to 218 N·m −1 ), thus preserving the spinel framework. These phononic adjustments enhance lattice flexibility and yield a direct band gap of 1.92 eV suitable for visible-light absorption. CoFe 2 O 4 exhibits visible-light photocatalytic activity with negative Gibbs free energy (G 298.15 =−10,432 J mol −1 ), supporting effective electron–hole separation. The interplay of $$\Theta _D$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>Θ</mml:mi> <mml:mi>D</mml:mi> </mml:msub> </mml:math> and $$\Theta _E$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>Θ</mml:mi> <mml:mi>E</mml:mi> </mml:msub> </mml:math> governs structural stability and charge transport, offering guidance for the design of thermally robust and efficient catalysts.