## 1 Introduction

### 1.1 Curie Point Determination in Rock Magnetism

[2] The Curie, or Néel temperature of a ferromagnetic or ferrimagnetic, or antiferromagnetic material, is the temperature where its uncompensated spins in zero-field undergo a second-order phase transition from a thermally disordered high-temperature to a magnetically ordered low-temperature state. For simplicity, we here follow the practice of *Petrovský and Kapička* [2006] and use the term *Curie point temperature* (*T _{C}*) to denote all cases. The measurement of

*T*is one of the central techniques in rock magnetism. In most applications it is a fast, reliable, and well-proven method to determine the predominant magnetic minerals in natural or synthetic samples, even if their concentrations are small. This is so for several reasons:

_{C}- Magnetic measurements are very sensitive, and the presence or absence of magnetic signals are very easily and precisely detectable.
- The two most common natural magnetic oxides have clearly distinct Curie points: magnetite at 580°C, and hematite at 680°C [
*Petersen and Bleil*, 1982]. - Within the common natural solid-solution series of magnetic minerals, titanomagnetites, titanomaghemites, titanohematites, and ferri-ilmenites, the Curie point temperature varies over a wide temperature range and can be used to tightly constrain the magnetic-mineral composition or oxidation state [
*Petersen and Bleil*, 1982].

[3] Because magnetic measurements are very sensitive, the onset of magnetic ordering can be observed using different magnetic measurement parameters, like static in-field magnetization, or AC-susceptibility. In both cases the transition from a less magnetized to a more magnetized state within a relatively small temperature interval is observed and correctly interpreted as resulting from the magnetic ordering that takes place when crossing the Curie point during cooling. However, there is no clear practical agreement about the accurate position of *T _{C}* within this temperature interval that for magnetite easily covers from 10°C up to 30°C or more. This usually is of minor importance, e.g., when the value of

*T*is used to distinguish between magnetite and hematite, or when the measurement should yield an average estimate for the composition of a titanomagnetite. However, when a precise composition determination is needed, or when two phases coexist with very close Curie point temperatures, inconsistent measurement methods for

_{C}*T*can lead to severely misleading conclusions. The aim of this article is to clarify the theoretical background of different approaches presently used to measure the Curie point temperature, and in addition to propose a new experimental protocol to reliably determine an accurate value of

_{C}*T*using a high-temperature vibrating-sample magnetometer (VSM).

_{C}### 1.2 Common Methods for Curie Point Determination

[4] There are several types of instruments in use to determine approximate Curie point temperatures. An overview of the different methods to determine *M*_{s}(*T*)-curves can be found in *Collinson* [1993]. In a Curie balance, a field—typically between *μ*_{0}*H*_{ex}=0.1 T and 1 T—is generated in a small area, leading to a strong field gradient that draws ferromagnetic and paramagnetic substances toward the stronger magnetic field, while diamagnetic substances are repulsed. This force is then precisely compensated by an additional coil. The current required for this compensation is proportional to the magnetization of the sample in the external field *H*_{ex}. Ferromagnets (and also ferrimagnets) are paramagnetic above *T _{C}* and still carry an induced magnetization. Most natural samples also contain other paramagnetic minerals contributing a magnetization

*χ*

_{p}(

*T*)

*H*

_{ex}that is considered as a major source of smoothing of the transition [e.g.,

*Tauxe*, 1998]. Two other possible smoothing mechanisms are inhomogeneity of the ferrimagnetic material and a temperature gradient inside the sample. Both effects lead to a smoothing of the transition due to an apparent or real distribution of Curie point temperatures as sketched in Figure 1. To obtain the correct Curie points from such smoothed measurements, a number of different methods have been suggested and used in the rock magnetic literature.

[5] *Ade-Hall et al*. [1965] used a Chevallier torsion balance for measuring the induced magnetization in 0.1 T. They introduced the “practical definition” of the Curie point *T _{C}*, as the “temperature at which the curvature of the concave part of the heating curve is a maximum”. In view of our theoretical considerations below, their justification for this definition is interesting: “The point of inflection was far too poorly defined to provide a practical Curie point” [

*Ade-Hall et al*., 1965]. This method of maximum curvature is now one of the two most common methods used to determine

*T*in rock magnetism [e.g.,

_{C}*Tauxe*, 1998;

*Leonhardt*, 2006;

*Lagroix et al*., 2004].

[6] The other very common method is the two-tangent method, initially described by *Grommé et al*. [1969]: “Curie points were determined by drawing straight lines approximately coinciding with the *J–T*-curve respectively above and below the estimated Curie point, and projecting their intersection to the temperature axis.” They do not give a physical argument for this method, but point out that they individually chose the external field *H*_{ex} between 140 and 400 mT to coincide with the “knee” of the *J–H*-curve of the sample—probably measured at room temperature—in order to avoid unnecessary large induced magnetization from paramagnetic substances in the sample.

[7] To devise a physically based method to determine Curie points for irreversible heating curves of titanomaghemites, *Moskowitz* [1981] analyzed the graphical two-tangent method and proposed a new extrapolation method that is based on the Landau theory of second-order phase transitions, and provides an estimate of *T _{C}* by using only data acquired below

*T*. A drawback of this approach is that it essentially uses the absolute value of the magnetization, and therefore can only be used for the highest Curie point temperature occurring in the sample.

_{C}[8] This disadvantage also applies to the *Arrott plot*, which is based on a molecular-field approximation of the free energy as a function of spontaneous magnetization, and widely used in physics to determine the Curie point of pure substances [*Arrott*, 1957; *Arrott and Noakes*, 1967]. A general introduction is found in *Bertotti* [1998] (section 5.1.3).

[9] Curie point temperatures have also been determined from curves of temperature-dependent magnetic initial susceptibility *χ*_{0}(*T*). The importance of the difference between determining *T _{C}* from

*M*

_{s}(

*T*) and

*χ*

_{0}(

*T*) is pointed out by

*Petrovský and Kapička*[2006], where methods to determine

*T*from measurements of the initial susceptibility are analyzed. They conclude that the two-tangent method is not suitable for

_{C}*χ*

_{0}(

*T*) and can considerably overestimate

*T*.

_{C}[10] The physical origin of *χ*_{0}(*T*) close to *T _{C}* is more challenging than that of

*M*

_{s}(

*T*), because a number of low-field effects are important for

*χ*

_{0}(

*T*), but become negligible in the higher fields used to infer

*M*

_{s}(

*T*). The variation of

*m*depends not only on the variation of

*M*

_{s}(

*H*,

*T*) with field

*H*, it also contains a contribution from a rotation of the ordered moment with respect to an easy magnetization axis, and contributions from thermally activated switching of small independent – but already magnetically ordered – regions (e.g., SP particles). In large bulk material domain-wall movement contributes to

*χ*

_{0}(

*T*) even slightly below

*T*. In nanoparticles the inhomogeneity of

_{C}*M*

_{S}, due to the different exchange coupling of inner and surface atoms, is of additional importance.

### 1.3 Toward Unifying Theory and Experiment in Curie Point Determination

[11] In this article, we focus on the behavior of *M*_{s}(*T*) and *χ*_{0}(*T*) of bulk material, where magnetic long-range order can develop without significant influence from particle boundaries and interfaces. Variation of *T _{C}* and

*M*

_{s}(

*T*) with particle size is relevant for nanoparticles only [

*Shcherbakov et al*., 2012].