Corresponding author: K. Fabian, Geological Survey of Norway, Leiv Eirikssons vei 39, Postboks 6315 Sluppen, 7491 Trondheim, Norway. E-mail: email@example.com
 Curie point temperatures (TC) of natural and synthetic magnetic materials are commonly determined in rock magnetism by several measurement methods that can be mutually incompatible and may lead to inconsistent results. Here the common evaluation routines for high-temperature magnetization and magnetic initial susceptibility curves are analyzed and revised based on Landau's theory of second-order phase transitions. It is confirmed that in high-field magnetization curves TC corresponds to the inflection point, below the temperature of maximum curvature or the double-tangent intersection point. At least four different physical processes contribute to the initial magnetic susceptibility near the ordering temperature. They include variation of saturation magnetization, superparamagnetic behavior, magnetization rotation, and magnetic domain wall motion. Because each of these processes may influence the apparent position of TC, initial susceptibility and high-field curves can yield deviating estimates of TC. A new procedure is proposed to efficiently determine the temperature variation of several magnetic parameters on a vibrating-sample magnetometer, by repeatedly measuring quarter-hysteresis loops during a single heating cycle. This procedure takes measurements during the inevitable waiting time necessary for thermal equilibration of the sample, whereby it is not slower than the commonly performed measurements on a Curie balance. However, it returns saturation magnetization, saturation remanence, high-field and low-field slopes, and other parameters as a function of temperature, which provide independent information about TC and other sample properties.
 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  and use the term Curie point temperature (TC) to denote all cases. The measurement of TC 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:
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].
 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 TC 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 TC 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 TC 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 TC using a high-temperature vibrating-sample magnetometer (VSM).
1.2 Common Methods for Curie Point Determination
 There are several types of instruments in use to determine approximate Curie point temperatures. An overview of the different methods to determine Ms(T)-curves can be found in Collinson . In a Curie balance, a field—typically between μ0Hex=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 Hex. Ferromagnets (and also ferrimagnets) are paramagnetic above TC and still carry an induced magnetization. Most natural samples also contain other paramagnetic minerals contributing a magnetization χp(T)Hex 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.
Ade-Hall et al.  used a Chevallier torsion balance for measuring the induced magnetization in 0.1 T. They introduced the “practical definition” of the Curie point TC, 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 TC in rock magnetism [e.g., Tauxe, 1998; Leonhardt, 2006; Lagroix et al., 2004].
 The other very common method is the two-tangent method, initially described by Grommé et al. : “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 Hex 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.
 To devise a physically based method to determine Curie points for irreversible heating curves of titanomaghemites, Moskowitz  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 TC by using only data acquired below TC. 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.
 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  (section 5.1.3).
 Curie point temperatures have also been determined from curves of temperature-dependent magnetic initial susceptibility χ0(T). The importance of the difference between determining TC from Ms(T) and χ0(T) is pointed out by Petrovský and Kapička , where methods to determine TC from measurements of the initial susceptibility are analyzed. They conclude that the two-tangent method is not suitable for χ0(T) and can considerably overestimate TC.
 The physical origin of χ0(T) close to TC is more challenging than that of Ms(T), because a number of low-field effects are important for χ0(T), but become negligible in the higher fields used to infer Ms(T). The variation of m depends not only on the variation of Ms(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 χ0(T) even slightly below TC. In nanoparticles the inhomogeneity of MS, 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
 In this article, we focus on the behavior of Ms(T) and χ0(T) of bulk material, where magnetic long-range order can develop without significant influence from particle boundaries and interfaces. Variation of TC and Ms(T) with particle size is relevant for nanoparticles only [Shcherbakov et al., 2012].
2 The Curie Point Temperature in Landau Theory With Field Term
 The study of the magnetic phase transition at the Curie point is commonly formulated in terms of the Landau theory based on the approximate fourth-order expansion of the zero-field free energy near TC in terms of the order parameter m = Ms(T)/Ms(0), which yields
 Here τ = (T − TC)/TC, F0 is constant, and a, b > 0. Due to the time inversion symmetry of the Maxwell equations, F contains no odd terms in m. At the minima of F one has dF/dm = 0, d2F/dm2 > 0 and therefore
where the Iverson bracket [X] is defined to be one, if the logical statement X is true, and zero otherwise. This characteristic square root behavior of m(T) apparently sharply defines TC. Unfortunately, this result is based on the absence of external fields, even though it has traditionally been used to interpret data from measurements performed within external fields of considerable strength. In the full Landau theory [e.g., Landau and Lifshitz, 1980, chap. 144], the field energy − mH is explicitly taken into account and the free energy becomes
 The condition dF/dm = 0 leads to
 By rescaling m and H to mscaled = σm with σ2 = 2b/a, and h = σ/(2a)H the equation becomes
 After rescaling, the derivatives m′, m″ of m with respect to τ fulfill the equations
 At an inflection point with m″ = 0 and m′ ≠ 0 the last equation yields 3 mm′ = − 1, which if substituted into (6) yields τ = 0.
 This implies that within an external field, the Curie point corresponds to the temperature where the magnetization curve has an inflection point with m″ = 0. This also is the minimum of the derivative m′. The theoretical predictions from Landau theory are plotted for different normalized fields h in Figure 2. Note that in case of h ≠ 0 there is no second-order phase transition from a disordered to an ordered state anymore, because only for h = 0 the magnetization m has a discontinuity of its first derivative at τ = 0. Within an external field, the condition m″ = 0 rather marks a crossover from a field-ordered to an exchange-ordered state, which develops into a phase transition when the field is removed.
 The methods of linear extrapolation or maximum curvature, which are commonly used in geophysics, are inspired by the zero-field analysis (2) and result in too high estimates of TC when applied to in-field measurements. At the Curie point τ = 0 one obtains from (5) the scaling relation
 This relation can be used for a precise determination of TC by either plotting m3 versus h in the Arrott plot [Arrott, 1957], which is widely used in physics to determine TC for pure materials, or by plotting m3/h as a function of h for different temperatures, and choosing the temperature where for high h the plot best approaches a constant value m3/h. A method equivalent to the Arrott plot was used in Moskowitz  to determine TC for synthetic titanomagnetites. The main disadvantage of this method is that admixtures of other magnetic minerals destroy the scaling relation (8).
 Calculating the derivative d/dh in (5) yields for the inverse of the susceptibility χ = dm/dh the relation
 This relation predicts a singularity of the initial susceptibility exactly at TC. Generalizations of this equation, e.g., for ferrimagnets [Néel, 1948], predict a similar peak. Yet, experimental data show that low-field, or initial susceptibility χ0(T) typically has a step-like shape with a sharp decrease near or at TC on heating. A remnant of the above theoretical peak, which here will be denoted as Landau peak, is often only observed for the high-field slope χhf. To understand this behavior, it is necessary to extend the free-energy expression (1) by an anisotropy-energy term that describes an additional degree of freedom for the ordered magnetic moment, namely a rotation away from its easy axis. This and other additional susceptibility mechanisms are discussed in the next section.
3 Initial Susceptibility Near the Curie Point
 Close to the ordering temperature, the magnetic initial susceptibility χ0(T) is influenced by several independent mechanisms by which a small applied field generates a magnetization along the field axis. To classify these, it is assumed that a thermally stable magnetic particle with total volume v, and saturation magnetization Ms contains a number of magnetic phases with volumes vi and magnetization directions mi. At zero field almost all phases correspond to magnetic domains aligned with easy anisotropy axes. The total magnetization then is
whereby the susceptibility contains the three terms
 The first term represents the change of Ms with field (para-effect), the second term represents domain wall motion and the corresponding change in domain volumes. The third term represents rotation of magnetization directions within the domains against magnetic anisotropy. At high temperatures above TC, exchange couplings between atoms are small in comparison to thermal energy, and the individual atoms react to an external field through the mechanisms of diamagnetism and paramagnetism. Near, but below TC, the whole particle volume is only one magnetic domain, and because thermal energy dominates, its direction behaves paramagnetically, and magnetic anisotropy is less effective. This is the superparamagnetic state with its associated susceptibility. These main physical mechanisms of magnetic susceptibility are sketched in Figure 3, and below will be discussed separately with respect to their influence upon the determination of TC. Their relative contributions at a specific temperature depend on the grain-size distribution within the sample. Accordingly, the shape of the magnetic susceptibility curve strongly depends on grain size.
3.1 Susceptibility Due to Variation of Ms(H)
 The first term on the right-hand side of (11) describes the so-called para-effect due to the variation dMs /dH at a fixed temperature [Holstein and Primakoff, 1940]. Physically, it results from the fact that below TC the saturation magnetization in zero field represents an equilibrium between exchange coupling and thermal activation. The average spin contributed to Ms by a magnetic ion like Fe3+ is lower than its theoretical value at T = 0 K. By applying a field, the coupling between neighboring spins is effectively enhanced which shifts the equilibrium to increased spin alignment and in turn also increases Ms. In Landau theory the para-effect is implicitly described by equation (5), and Figure 2b shows that only close to TC it contributes noticeably to χ0(T). According to (9), χ0 has a singularity at TC, where τ and m are zero. This singularity is effectively suppressed in any finite applied field, even in those used for measuring χ0. The para-effect therefore is not necessarily recognized in real measurements of χ0(T), where other susceptibility mechanisms are more prominent. However, the increase of Ms(H) with field still occurs at high fields, where all other mechanisms are saturated, whereby the para-effect can generate a clear Landau peak of high-field susceptibility χhf(T) slightly above TC, which is in accordance with Figure 2b. Note that within a field, Ms is nonzero even above TC, because the field cooperates with the exchange coupling. Inserting (8) into (9) yields exactly at TC the relation χhf(h) ∝ h− 2/3. The general analysis by Holstein and Primakoff  otherwise gives χhf(h) ∝ h− 1/2.
3.2 Susceptibility Due to Anisotropy
 A significant contribution to χ0(T) just below TC is rotation of a single-domain magnetic moment against magnetic anisotropy (Figure 3c). By this mechanism χ0(T) maintains a high value on cooling. Magnetic anisotropy defines preferred magnetization directions (easy axes) for the ordered spins in zero field, and can be due to either crystallographic anisotropy, or shape anisotropy, whereby the latter arises from the self-demagnetizing effect in elongated single-domain particles. Applying a field rotates the spins away from its easy axis, thereby providing a susceptibility mechanism even when there is no change in Ms.
 For a theoretical treatment, one can extend the Ginzburg-Landau theory from the previous section. The mean magnetic moment per atom is M = mSμB, where S is the spin quantum number, and μB the Bohr magneton. The uniaxial anisotropy energy from shape anisotropy per atom contributes an energy K(m) = km2sin2(ϕ − θ), where ϕ denotes the angle between easy axis and field direction, and θ ≤ ϕ denotes the angle between field and mean magnetization vector m. The modified Landau free energy per atom thereby becomes
 To estimate the relative size of these terms, Shcherbakov et al.  use a mean-field approximation of the Heisenberg Hamiltonian to obtain an approximation Fmf for the free energy F, which results in
 The normalized parameters a and b are therefore
 Equation (12), after division by kBTC therefore yields
 Here the normalized material and field parameters are
 A realistic order of magnitude for the rescaled anisotropy constant k and field h can be estimated. For magnetite the shape anisotropy constant at room temperature is approximately J/m3 and kBTC ≈ 1.2 ⋅ 10− 20 J/atom. The magnetite unit-cell contains 3Z/2 = 12 pairs of Fe atoms—or ferrimagnetic moments—and has volume V ≈ 5.91 ⋅ 10− 28 m3. In combination, this results in k ≈ 2 ⋅ 10− 5. On the other hand, an external field of B = 1 mT corresponds to h ≈ 2 ⋅ 10− 7.
 For a given angle ϕ ∈ [0, π/2], minimization of (16) with respect to θ and m results in the equations
 Clearly, for h = 0, m = 0, θ = ϕ is a solution, which becomes unstable at τ = 0, where ∂ 2F/∂ m2 changes sign. For the region, where τ < 0, these equations cannot be solved analytically, but numerical solutions can be readily found. The resulting magnetization curve depends on the relative values of h and k (Figure 4). If km ≪ h, which for constant h applies somewhere in the vicinity of TC, then θ ≈ 0, and the magnetization is directed along the field direction. In this case anisotropy is negligible, and (19) and (20) reduce to the pure Landau theory, where , such that h > km is equivalent to τ < β/α(h/k)2. Because k(τ) vanishes rapidly near TC, for constant h there always is a temperature range close to TC, where anisotropy is negligible and m(τ,h), and accordingly the susceptibility measured with a constant amplitude, follows pure Landau theory. For lower temperature and small field the situation is more interesting. For example, at h/k ≈ 0.01, and sufficiently low temperature, m is almost constant; however, its direction changes. Its component along the field follows from (19), and for a small field with h ≪ km one obtains
 The observed magnetization projection on the field direction, mobs, can then be estimated by
 By integrating this over a constant random distribution of easy axes, the first term averages to zero. For the second term the fact that ϕ − θ is small implies by (21) that
and the susceptibility due to rotation is obtained as
where the dependence of Ms on h due to the para-effect has been neglected. Otherwise, the approximation agrees with the exact calculation in the appendix, that yields for T < TC or τ < 0
 This result shows that the rotational initial susceptibility near TC creates the characteristic step function, which is often observed as a prominent feature in measurements of initial susceptibility as a function of temperature.
3.3 Susceptibility Due to Domain Wall Motion
 In case of large crystals of a soft magnetic material, domain-wall motion could contribute to the initial susceptibility in the vicinity of TC. Some models show that the initial susceptibility χdw due to domain wall motion is proportional to, where K(T) denotes the magnetocrystalline anisotropy constant [Träuble, 1966]. At high temperatures magnetostrictive effects and magnetization rotation constrain this susceptibility. For large magnetite particles of size d one can assume that classical domain wall movement can occur only if the wall thickness . Because for magnetite A(T) ∝ Ms(T)2, and K(T) ∝ Ms(T)12 [Fletcher and O'Reilly, 1974] the total effect of domain wall motion can be written as
where ms(T) = Ms(T)/Ms(0), k(T) = K(T)/K(0), δ0 is the domain wall width, χdw,0 the domain wall susceptibility at T = 0 K, and […] again is the Iverson bracket. As long as there are domain walls in the particles, the demagnetizing effect further reduces the externally measured apparent susceptibility χa with respect to the internal susceptibility χdw through
where Nd ≈ 1/3 is an average demagnetizing factor [Dunlop and Özdemir, 1997, equation (9.4)]. The fact that χdw is set to zero above a critical nucleation temperature does not necessarily mean that the experimentally determined susceptibility changes substantially. It only means that domain wall motion is no longer the predominant mechanism by which the particle responds to field variations. When the domain-wall width δ becomes comparable to the grain size, the grain can no longer host multiple domains and different magnetization structures form. One of the possible configurations is a vortex structure [Williams and Dunlop, 1989] positioned approximately in the center of the particle, such that the net magnetization is about zero for h = 0. Micromagnetic calculations indicate that the susceptibility due to spin motions in a vortex state with fixed central axis is comparable to that from other susceptibility mechanisms for a field applied along the vortex axis [Heider et al., 1996], and that the exchange energy for platelets is practically insensitive to the movement of the vortex axis during the initial stage of the magnetization process under the action of the external field [Scholz et al., 2003; Shcherbakov and Winklhofer, 2010], which means that the apparent susceptibility of the vortex structure essentially follows equation (27).
 Figure 5a shows that the width of a 180° Bloch wall for magnetite increases very strongly as a function of temperature. Consequently, only very large particles may show a contribution of χdw near TC, as indicated in Figure 5b. Therefore, domain wall motion is of little importance near TC, but for large particles it leads to a noticeable susceptibility below TC that decays toward lower temperatures as the anisotropy |K(T)| increases.
3.4 Susceptibility Due to Superparamagnetism
 A third influence upon χ0(T) is due to the superparamagnetic behavior of small particles. This effect has two components, both of which can be extremely important. If the particle size of the ferrimagnetic crystals is so small that they contain only a small number of atoms in at least one dimension, this can lead to a considerable reduction of the particles’ ordering temperature TC(particle) as compared to the TC of the bulk material. A detailed theoretical evaluation of this effect is given in Shcherbakov et al. . In this case, the magnetic susceptibility reflects the lowered TC due to the particle size distribution. For magnetite this becomes a noticeable effect when at least one of the particle dimensions is less than ≈15 nm. The reduction in TC, e.g., for spherical magnetite grains of 10 nm diameter is about 30 K, and for smaller grains becomes rapidly larger [Sadeh et al., 2000].
 If the particles are large enough to essentially order at the bulk TC, they still are superparamagnetic in the range between TC and their respective blocking temperatures TB. In this temperature interval the initial magnetic susceptibility is dominated by thermal fluctuations of the relatively large ordered particles. This superparamagnetic susceptibility is considerably more prominent than the para-effect due to the variation of Ms with field. Thereby the peak of the total magnetic initial susceptibility will be lowered to TB, clearly below TC.
 A calculation of the peak position for magnetite can be based on a single-domain particle of volume v, and shape anisotropy with demagnetizing factor 1/3 ≤ N ≤ 2/3. Its energy barrier to magnetization changes then is
leading to a relaxation time
where τ0 ≈ 10− 9 s is the spin relaxation time, and kB is the Boltzmann constant [Shcherbakov and Fabian, 2005]. For susceptibility measurements a particle is considered blocked with χSP = 0 if the critical relaxation time is more than τ = 0.1 − 1 s, or
 With Ms(T) ≈ Ms,0(1 − T/TC)1/2, and by defining a particle temperature , the blocking temperature TB from (30) after substituting (29) and (28) yields
TB corresponds to the Hopkinson peak temperature T∗ of magnetic susceptibility above which the volume specific superparamagnetic susceptibility
is the dominant contribution. In Figure 6 this peak temperature is plotted as a function of diameter for spherical magnetite particles.
4 A New Procedure to Improve Curie Point Measurements
 In this section an improved technique for fast high-temperature measurements, using a VSM, is described. The common measurement scheme, as, e.g., implemented in the PMC Inc. HT-VSM and most Curie balances, is to apply a constant field H during heating and cooling, and to measure the values M(H,T). A typical result of this procedure is shown in Figure 7a for a calibration sample containing magnetite and hematite (TTC is the temperature at the thermocouple). To minimize thermal hysteresis, and to achieve an accurate value for TC, the temperature changes must be slow enough to permit thermal equilibration of the sample at each temperature step. Hence, even for relatively small samples, a full heating curve up to 600–700°C requires a measurement time of 30–60 min.
 The proposed new method uses the inevitable waiting time at each temperature step to measure not only a single value, but a full magnetization curve at different field values. For a high-temperature VSM (PMC Inc.), it is convenient to measure a sequence of saturation-initial curves Msi(H,T) [Fabian and von Dobeneck, 1997; Fabian, 2003].
 At each temperature step Ti to Ti+1, the sample is heated in zero field to Ti+1. To minimize measuring time, the field is then increased in relatively large steps (e.g., ΔH = 50 mT) to the maximum field (e.g., 1.5 T). Besides the initial saturation, which took place at Ti, this procedure essentially measures Msi(H,Ti + 1) at temperature Ti+1. The field is switched off before further heating. Instead of the single curve shown in Figure 7a, this procedure returns the two-dimensional data set as in Figure 7b.
 If the temperature step is small (e.g., ΔT = 1 – 3°C), Mrs(Ti) can be determined to good approximation by the first measurement of Msi(H,Ti) at H = 0. Otherwise, the average (Msi(0,Ti + 1) + Msi(0,Ti − 1))/2 could provide a better estimate of Mrs(Ti). When heating rate, temperature step ΔT, and measurement time (through ΔH) for each Ms(T)-curve are appropriately chosen, this Msi measurement requires the same total measurement time as a common M(H, T) measurement with the same overall heating rate. However, by acquiring many data points in different fields, a more sophisticated data analysis becomes possible.
 Figure 7 visualizes how the measured Msi(T) curves for several temperatures increase the information obtained during one heating cycle. A linear fit of the high-field region of the Msi(H,Ti) curve at temperature Ti has the equation Ms(Ti) + χhf(Ti)H, where χhf(Ti) is the high-field susceptibility, determined by paramagnetism and diamagnetism, the para-effect, or field-induced spin canting. These fits are represented by gray lines in Figure 7c and yield estimates for Ms(Ti) and χhf(Ti), respectively. Due to the small temperature step ΔT, nearly continuous curves of Ms(T), Mrs(T) and χhf(T) can be derived. Figure 7d shows that for the magnetite-hematite calibration sample a clear peak in χhf(T) for magnetite indicates the para-effect close to the Curie point, while the antiferromagnetic ordering transition for hematite shows no discernible peak.
4.1 Field-dependent M(T) Curves
 Using temperature-dependent measurements of quarter hysteresis loops it is possible to construct induced magnetization M(H,T) curves for any value of H. In Figure 8a this is done for an Icelandic basalt containing fine-grained magnetite. Using this data set, it is also possible to construct a plot of m3/h, which according to equation (8) in case of a pure magnetic mineralogy at TC should be constant as a function of h. Figure 8b shows that this version of the Arrott plot gives a precise estimate of TC ≈ 574°C for this sample. To compare this with estimates from other determinations, Figure 8c plots m′, m″, high-field susceptibility, Ms(T), and Mrs(T). In agreement with Landau theory, the minimum of m′, or the zero of m″, appears to be independent of the applied field within measurement precision. However, the position of this minimum is closer to 571°C, and thus slightly lower than the estimate from the Arrott plot. In any case, the maximum of m″ varies with field intensity and always lies about 10–15°C above TC. The lower panel shows that the two-tangent method of Grommé et al.  essentially coincides with the peak determination of m″ and correspondingly overestimates TC. It should be noted that the measured Mrs(T) curve does not fully vanish at TC ≈ 574°C. This residual remanence is probably due to mineralogical inhomogeneity within this natural sample, leading to a distribution of Curie points as sketched in Figure 1. The vanishing of remanent magnetization sensitively traces the upper end of the TC distribution, while the other methods record a lower average TC value. The peak of χhf(T) in Figure 8c lies some 5°C above TC, in agreement with Landau theory. The advantage of this Landau peak is that it also can be found in samples containing several magnetic minerals, as in Figure 9. Here it very accurately delimitates the Curie point of the magnetite phase, which lies 30–40°C above the maximum of the initial susceptibility.
5 Discussion and Conclusions
 The aim of this article is twofold. It gives an overview of the relevant physical mechanisms that influence the classic methods of Curie point temperature determination, and introduces an improved measurement method to infer several magnetization parameters in one single high-temperature run.
 By evaluating previous work, including our own, we became aware that for the last decades the methods used for determining Curie points in rock and paleomagnetism were not based on a solid physical foundation, and mostly systematically overestimate TC. The reason why this has not been improved previously is that for most practical purposes the errors are negligible, and that the calibration of the instrumentation is based on similarly overestimated Curie point temperatures. Thereby, the numerical values reported were not too far off from the correct values. However, in cases where the difference between the calibration point and the measured TC is large, the literature certainly contains Curie point temperatures off by tens of degrees. Serious problems occur in studies which compare Curie point temperatures obtained by different physical measurements, like susceptibility and in-field measurements. Many of these results should be reevaluated.
 The new measurement scheme presented here has several advantages over a usual high-field measurement. It efficiently uses the waiting time for thermal equilibration to determine Ms(T), Mrs(T), and χhf(T) in one run. By interpolation one can construct all curves M(H,T), and generate Arrott plots. Especially the measurement of χhf(T) has not been used previously to determine TC, but proves useful to distinguish ferrimagnetic from (weak) antiferromagnetic phases.
5.1.1 The Role of Paramagnetic Susceptibility
 For rock magnetic applications it is of interest to estimate at which concentration of an admixed paramagnetic material the determination of TC for the ferrimagnetic phase becomes problematic. The magnetization of the paramagnetic material near the Curie point of the ferrimagnetic phase is
where h is defined by (18). Using the ferrimagnetic magnetization given by (8) with a concentration c of Fe ions yields the total magnetization
 The two contributions from paramagnetic and ferrimagnetic materials are approximately of the same size, when c ≈ h2/3, which for high fields (h = 10− 3) means c < 1%. This result explains why the determination of TC from a single high-field M(H,T) curve can be seriously influenced or made impossible by a coexisting paramagnetic mineral that contains 100–1000 times more Fe than the ferrimagnetic mineral, especially if the latter is inhomogeneous, with Curie point temperatures distributed over a few degrees Celsius.
 In this case again the new method has the advantage that it determines an Ms(T) curve that, due to extrapolation to H = 0, is less affected by the paramagnetic phase.
 When looking at the initial susceptibility, a relative concentration of c ≈ 0.01 − 0.1 % of ferrimagnetic minerals marks a practical detection limit, because at this point the paramagnetic susceptibility exceeds the ferrimagnetic, and the onset of ordering cannot be identified reliably.
 The outcome of the Curie point determination depends on grain size and on the measurement method. The approximate grain sizes given below are values for magnetite, but their order of magnitude applies similarly to many magnetic minerals.
For extremely fine grains with sizes of 1–20 nm, the ordering temperature Tord lies below the bulk TC by about 1–100°C, depending on size [Shcherbakov et al., 2012]. In this case, any measurement method yields an incorrect result, if the measured ordering temperature is interpreted as a bulk TC for a particular mineral.
Particles with size up to some 100 nm behave superparamagnetically above their blocking temperature, which can be several tens to hundreds degrees below TC. For these grains, the peak of weak field susceptibility (Hopkinson peak) corresponds to this average blocking temperature. Only strong field measurements provide the bulk Ms(T) curve indicating TC by the minimum of dMs/dT. They may also show the Landau peak of dMs/dH slightly above TC (Figure 2b).
Grains with single-domain blocking temperature close to TC. In these grains, the Hopkinson peak lies within a few degrees of TC. In a strong field the minimum of dMs/dT corresponds to TC and the Landau peak of dMs/dH lies only 5–10° above TC. This peak theoretically occurs also in weak-field susceptibility, but might be completely buried under the step-like increase of the induced magnetization generated by the rotation of vectors m from the easy axes directions. The contributions of superparamagnetic and rotational susceptibilities are
Their ratio χSP/χrot = 2KV/(kBT) increases on cooling, and by definition assumes a value of about 50 at the blocking temperature Tb. Hence, χSP dominates over χrot until more than 98% of the particle volume is thermally blocked.
 Multidomain grains. In a strong field again the minimum of dMsdT indicates TC, and the Landau peak of dMs/dH lies only 5–10° above TC. In weak fields the dominant magnetization change is due to domain wall movement. Close below TC, domain walls become very large due to rapidly vanishing magnetocrystalline anisotropy. Other inhomogeneous magnetization structures, like multivortex structures, may emerge. These structures are also easily moved by an external field and produce a susceptibility similar to that of domain wall movement.
 In summary, strong field measurements always produce a minimum of dMs/dT at TC, and for ferrimagnetic minerals a Landau peak of dMs/dH slightly above TC, while weak-field magnetic susceptibility rather yields a step-like increase with maximum below TC. For superparamagnetic grain sizes, the downward shift of the maximum can be several tens of degrees. If the particles contain only few atomic layers, surface effects decrease the average exchange coupling and can dramatically lower the ordering temperature.
Appendix A: Initial Susceptibility Due to Rotation Against Anisotropy Near the Curie Point
 We start from equations ((12)), ((19)), and ((20)). The measured magnetization in a field h is the projection on the field direction averaged over all directions of anisotropy axes.
 The susceptibility χ0 = (dmavg/dh)(0) is then obtained from
 From the derivatives of (19) and (20) with respect to h one obtains for h = 0 with θ(ϕ,0) = ϕ
 In the first expression one can insert the relations m2 = − ατ/β, cos θ = cos ϕ, and in the second one can use sin θ = sin ϕ to obtain
 Inserting this into ((A2)) and again replacing θ by ϕ the equation simplifies to
 This integral is now readily evaluated and yields
 The research was supported by grant 189721 from the Research Council of Norway (Nanomat Program) in the EU Matera Program. VSM measurements have been performed using visiting research fellowships to the Institute of Rock Magnetism (IRM), University of Minnesota, U.S.A., which is supported by an NSF Instruments and Facilities Grant. We gratefully acknowledge thoughtful comments by Mike Jackson, Bruce Moskowitz, and an anonymous reviewer.