Magnetic relaxation and correlating effective magnetic moment with particle size distribution in maghemite nanoparticles

The role of particle size distribution inherently present in magnetic nanoparticles (NPs) is examined in considerable detail in relation to the measured magnetic properties of oleic acid-coated maghemite ({\gamma}-Fe$_2$O$_3$) NPs. Transmission electron microscopy (TEM) of the sol-gel synthesized $\gamma$-$Fe$$_2$$O$$_3$ NPs showed a log-normal distribution of sizes with average diameter $<D>$= 7.04 nm and standard deviation $\sigma$= 0.78 nm. Magnetization, $M$, vs. temperature (2 K to 350 K) of the NPs was measured in an applied magnetic field $H$ up to 90 kOe along with the temperature dependence of the ac susceptibilities, $\chi$$'$ and $\chi$$"$, at various frequencies, $f$$_m$, from 10 Hz to 10 kHz. From the shift of the blocking temperature from $T$$_B$ =35 K at 10 Hz to $T$$_B$ = 48 K at 10 kHz, the absence of any significant interparticle interaction is inferred and the relaxation frequency $f$$_o$= 2.6 x 10$^{10}$ Hz and anisotropy constant $K$$_a$= 5.48 x 10$^5$ ergs/cm$^3$ are determined. For $T$ < $T$$_B$, the coercivity $H$$_C$ is practically negligible. For $T$ > $T$$_B$, the data of $M$ vs. $H$ up to 90 kOe at several temperatures are analyzed two different ways: (i) in terms of the modified Langevin function yielding an average magnetic moment per particle $\mu$$_p$=7300 (500) $\mu$$_B$; and (ii) in terms of log-normal distribution of moments yielding $<$$\mu$$>$= 6670 $\mu$$_B$ at 150 K decreasing to $<$$\mu$$>$= 6100 $\mu$$_B$ at 300 K with standard deviations $\sigma$ $\approx$ $<$$\mu$$>$$/2$. The above two approaches yield consistent and physically meaningful results as long as the width parameter, $s$, of the log-normal distribution is less than 0.83.