Determination of Blocking Temperature in Magnetization and

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Determination of Blocking Temperature in Magnetization and Mössbauer Time Scale: A Functional Form Approach Giorgio Concas, Francesco Congiu, Giuseppe Muscas, and Davide Peddis J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.7b01748 • Publication Date (Web): 29 Jun 2017 Downloaded from http://pubs.acs.org on July 4, 2017

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The Journal of Physical Chemistry

Determination of Blocking Temperature in Magnetization and Mössbauer Time Scale: a Functional Form Approach

G. Concas a*, F. Congiu a, G. Muscas b, D. Peddis c,d

a

b

Dipartimento di Fisica, Università di Cagliari, S. P. Monserrato-Sestu km 0,700, 09042 Monserrato (CA), Italy Department of Physics and Astronomy, Materials Physics, Uppsala University, Box 516, SE-751 204, Uppsala,

Sweden c

Vinča Institute of Nuclear Sciences, PO Box 522, 11001 Belgrade, Serbia

d

Istituto di Struttura della Materia-CNR, 00015 Monterotondo Scalo (RM), Italy

*

Corresponding author.

E-mail address: [email protected]

ABSTRACT We studied the temperature dependence of the magnetization in an ensemble of monodomain nanoparticles both with DC magnetometry and Mössbauer spectroscopy. The analytical form of the temperature dependence is given by the complementary cumulative distribution function. This allows to determine the magnetization blocking temperatures of the sample by a fitting procedure. It is possible to calculate the Mössbauer blocking temperature by a single spectrum and the DC magnetization blocking temperature by two points of the thermo-remanent magnetization curve, thus with a large reduction of the experimental work. The method may be used for particles with not too strong interactions, such happens in the Fe28 sample and not for samples with strong interactions as N30; it may be used for interparticle interaction energies up to 2 yJ and not for energies larger than 60 yJ. This method of analysis of the data should be used in the future work concerning the thermoremanent magnetization and Mössbauer spectra of magnetic nanoparticles.

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1. Introduction

Magnetic nanomaterials are intensively studied from both a fundamental and a technological point of view (e.g. Magnetic Resonance Imaging, hyperthermia, drug delivery, catalysis, microwaves applications).1-5 As magnetic materials reduce in size to nanometer scale, multidomain organization is energetically unfavorable and single magnetic domain particles are formed, each one with a large collective magnetic moment. For isolated spherical particles with uniaxial anisotropy, the lowest energy configuration corresponds to the magnetization aligned in one of the two opposite directions of the easy anisotropy axis. These two minima are separated by an energy barrier, which is an intrinsic property of individual magnetic material. The magnetic behavior of a random ensemble of nanoparticles (NPs is characterized by the instability of the magnetization due to thermal energy, which can overcome the barrier inducing the switching of the magnetization between the two minima. This behavior is analogous to paramagnetism, but with different time and magnetization scale and, for this reason it is called superparamagnetism.6-13 For non-interacting particles, the rate of magnetization flipping (i.e. superparamagnetic relaxation) is characterized by a relaxation time governed by an Arrhenius-like Néel-Brown equation:14-15

∆

= ( )

(1)

where T is the temperature, τ0 the characteristic relaxation time and kB is the Boltzmann constant. ∆Ea is the energy barrier between two energy minima; for NPs with uniaxial anisotropy, it can be considered to be proportional to the magnetic anisotropy constant Ka and particle volume V (∆Ea = Ka V). An ensemble of magnetic NPs show superparamagnetic behavior above a certain temperature (blocking temperature denoted by Tb in the equations), while below this temperature, the magnetic properties are similar to the bulk material. The blocking temperature (BT) can be


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defined as the temperature at which the relaxation time (t) is equal to the experimental measuring time of the technique used (τm), i. e.:

=

( )

(2)

Equation (2) shows that BT depends on the experimental measuring time. Since Mössbauer Spectroscopy (MS) and DC magnetization measurements have significantly different time scales, the BT estimated by the two techniques differ considerably. Inserting the time scales of DC magnetometry (τm ~ 100 s) and MS (τm ~ 5×10-9 s) and values of τ0 (10-11-10-12 s) for ferrimagnetic materials in Eq. (2), one finds that the ratio between the BT measured by Mössbauer and DC magnetometry should be in the range 3-7.16-20 The first aim of this work is to investigate the temperature dependence of the superparamagnetic relaxation based on the analytical form. The application of this analysis on experimental data collected by means of Mössbauer and DC magnetometry allows to estimate the blocking temperature of monodomain nanoparticles ensembles by an easy fitting procedure, which is capable to provide a BT value with a minimum of only two values of magnetization at different temperatures from TRM curves. Moreover, it is possible to determine the Mössbauer BT by a single spectrum saving a considerable amount of time. In real samples, there is a distribution of blocking temperatures that depends mainly on the volume distribution. A common approach in order to study the superparamagnetic relaxation by DC magnetometry, is to measure the thermo-remanent magnetization (TRM). In this measurement the sample is cooled from a temperature well higher than BT to low temperature (5 K in the present work) in a small external magnetic field (5 mT), and then the field is turned off and the magnetization is measured on warming up.



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The final aim of this work is to determine the BT in Mössbauer (TbM) and DC magnetometry (Tbm) time scale of the sample by a fitting procedure based on the analytical form of the evolution of the unblocked fraction versus temperature, or the evolution of MTRM vs. temperature. The BT is obtained as a free parameter of the fit. As a collateral result, it is possible to determine the Mössbauer BT by a single spectrum, even if with less precision. In addition, it is also possible to determine the magnetization BT by using only two values of TRM curve. As model systems to verify our method, a set of samples of spine iron oxides (γ-Fe2O3 and CoFe2O4), has been chosen. All these samples are well known and well characterized , appearing ideal systems to test this new method.

2. State of the art

As we already discuss, BT can be defined as the temperature for which the superparamagnetic relaxation time is equal to the measuring time of the experimental technique. Anyway, in practice samples of small particles exhibit particle size distribution and BT is defined as the temperature at which 50% of the sample is in the superparamagnetic state. When an assembly of magnetic monodomain particles is observed by means of Mössbauer spectroscopy, it is found that the superparamagnetic relaxation phenomena result in a broadening of the absorption Zeeman lines for relaxation times of the order of 10-8 s.19 For 5×10-9 s, the magnetic hyperfine splitting disappears.19 The total spectrum is thus a superposition of a magnetic and a quadrupolar pattern, the relative weight of the latter increasing with temperature as the number rapidly fluctuacting particles. In this framework, and using the previous definitions, we can define TbM the temperature at which 50% of the spectral area is magnetically split. This corresponds to an average relaxation time of about 5×10-9 s.


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The common procedure to determine the TbM is to record spectra at different temperatures, considering two spectra with unblocked fraction >50% and

Determination of Blocking Temperature in Magnetization and

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