Nonextensivity in magnetic nanoparticle ensembles

A superconducting quantum interference device and Faraday rotation technique are used to study dipolar interacting nanoparticles embedded in a polystyrene matrix. Magnetization isotherms are measured for three cylindrically shaped samples of constant diameter but various heights. Detailed analysis of the isotherms supports Tsallis’ conjecture of a magnetic equation of state that involves temperature and magnetic field variables scaled by the logarithm of the number of magnetic nanoparticles. This unusual scaling of thermodynamic variables, which are conventionally considered to be intensive, originates from the nonextensivity of the Gibbs free energy in three-dimensional dipolar interacting particle ensembles. Our experimental evidence for nonextensivity is based on the data collapse of various isotherms that require scaling of the field variable in accordance with Tsallis’ equation of state.

Figure 1

Transmission electron microscopy image of a 2D self-assembled array of $\gamma \text{−}{\mathrm{Fe}}_2{\mathrm{O}}_3$ nanoparticles. The arrows indicate a particle diameter of $12\mathrm{nm}$. Eye-guiding lines stress the hexagonal dot arrangement controlled by the length of the organic molecules attached to the dot surfaces.

Figure 2

$m$ vs $T$ of $\gamma \text{−}{\mathrm{Fe}}_2{\mathrm{O}}_3$ nanoparticles interacting magnetostatically in a random 3D array with an average particle spacing of $80\mathrm{nm}$. Solid triangles show the field heating curve (arrow labeled by ZFC/FH) in ${\mu }_0{H}_a=25\mathrm{mT}$ after zero field cooling. Open triangles show the subsequent field cooling curve (arrow labeled by FC), which separates from the FH branch at the blocking temperature $T_B\approx 127\mathrm{K}$ (arrow). The lines connecting the data points are eye-guiding splines.

Figure 3

(Color online) Isotherms $m$ vs. $H$ of sample $A$ (squares), $B$ (triangles), and $C$ (circles) measured at $T=300\mathrm{K}$, respectively. Note that only $1∕5$ of the measured data points are shown. Lines are best fits of the Langevin function given by Eq. (5).

Figure 4

Ratios $m_X∕m_Y$ vs $H$ of the isotherms of samples $A$, $B$, and $C$ measured at $T=280\mathrm{K}$ (a)–(c) and $T=300\mathrm{K}$ (d)–(f), respectively. Dashed and solid lines represent the ratios without and with field-scaling.

Figure 5

(Color online) Faraday rotation $\Theta$ vs $H_a$ for the laser beam diameters $d_1=3.37\mathrm{mm}$ (squares) and $d_2=1.97\mathrm{mm}$ (circles). Solid line is a best linear fit to the $\Theta \left(d_1\right)$ vs $H_a$ data for $0\le {\mu }_0{H}_a\le 0.02\mathrm{T}$. The slope determines the Verdet constant of our $\gamma \text{−}{\mathrm{Fe}}_2{\mathrm{O}}_3$ nanoparticles dispersed in polystyrene. The inset shows the ratio of the data sets $\Theta \left(d_1\right)$ vs $H_a$ and $\Theta \left(d_2\right)$ vs $H_a$. The straight line indicates the expected ratio $\Theta \left(d_1\right)∕\Theta \left(d_2\right)=1$ and visualizes the statistical fluctuations of the data.