Calibration of ac and dc magnetometers with a Dy 2 O 3 standard

The ac susceptibility and magnetization curves of a glued Dy2O3 powder sample are measured by an ac susceptometer and a dc superconducting quantum interference device magnetometer, both of which have been calibrated previously. It is shown that the magnetic moment of the paramagnetic sample as a function of field and temperature may be accurately expressed by a combination of the Curie–Weiss law and the Langevin function at T > 45 K with three adjusting parameters, so that the dc magnetization curves and the magnitude and phase of ac susceptibility at different values of dc bias field measured by any magnetometer can be calibrated by using Dy2O3 as a standard. The expressions are empirical and cannot be justified in the entire field and temperature range by existing theories of paramagnetism. Below 10 K, indication of approaching a possible phase transition is found. It is shown that pure Dy2O3 powder may be used as a primary standard, with susceptibility [13.28(T + 17)]−1 emu/Oe/g at T > 50 K and H < 10 kOe, in consistency with the Curie–Weiss law and the quantum mechanical theory of paramagnetism. © 2011 American Institute of Physics. [doi:10.1063/1.3581224]


I. INTRODUCTION
Instruments for magnetic measurements may be named as magnetometers if the magnetic moment of the sample is directly measured by them.2][3] According to the Quantum Design manuals, a paramagnetic Pd cylinder of mass about 0.27 g is provided by the factory to calibrate the SQUID dc magnetometer and the extraction dc magnetometer in PPMS at a high field, whose mass susceptibility is 5.25 × 10 −6 emu/Oe/g at T = 298 K with uncertainty less than 0.5%, 4 and a glued Dy 2 O 3 powder (i.e., Dy 2 O 3 powder embedded in epoxy) cylinder is provided by the factory with 1 order of magnitude higher moment susceptibility (magnetic moment per unit field) to calibrate the ac susceptometer.The susceptometer is calibrated using the Dy 2 O 3 standard sample at T = 298 K, field amplitude H m = 800 A/m, and frequency f = 1000 Hz, whereas the moment susceptibility of the Dy 2 O 3 standard sample itself is obtained by measurement at T = 298 K and H = 800 kA/m in the extraction dc magnetometer, calibrated by using the Pd standard sample at T = 298 K and H = 1600 kA/m.
All such recommended calibrations are carried out by measuring the magnetic moment of standard samples at fixed temperature and field, assuming the error of magnetic field to be negligible.This assumption is basically valid in instruments commercially available, whose magnetizing field coils are made accurately with a stable ratio of field to current, so that the field is obtained by measuring the current according to the Ampère law.However, if the coil is a superconducting magnet, the hysteretic magnetization of superconductor itself will give a complicated demagnetizing field in the bore of the magnet, so that the field cannot be measured by the current accurately.This may be a reason why dc magnetometers with superconducting magnet are calibrated at a very high field, at which the demagnetizing effects are negligible.In all such calibrations, temperature and frequency are also assumed to be accurate.
In order to know the performance of the studied ac susceptometer in the entire range of T , H m , and f , we have calibrated it with a copper cylinder standard of 5 mm in diameter and length. 5Different from conventional calibration techniques mentioned above, the electromagnetic properties of the copper cylinder are unknown and are determined during the calibration of the ac susceptometer.In other words, the calibration is directly based on the Maxwell equations and the Ohm law but is not transmitted from another primary standard sample.Thus, it can be widely adapted by users of any kind of commercial or home-made ac susceptometers.By comparing the magnitude of ac susceptibility measured at T = 10 K and a number of values of f with the calculated eddy-current susceptibility, we have obtained an averaged magnitude correction factor k = 1.081 for that particular ac susceptometer.In the present paper, we will use the same ac susceptometer and a dc SQUID magnetometer to measure a glued Dy 2 O 3 powder cylinder of diameter 5 mm and length 6 mm provided by the factory (with unknown Dy 2 O 3 mass) to show how both magnetometers may be further calibrated by this standard sample.We will also show that a pure Dy 2 O 3 powder sample with known mass can serve as a primary standard.
Such a calibration will also be made in a large range of T , H m , and f , for which the actual functional properties of the Dy 2 O 3 sample have to be expressed accurately.In Sec.II, the Curie-Weiss law is used for the low-field moment susceptibility, with the Curie and the Weiss constants determined accurately by ac susceptibility measurements at T > 45 K.The resulting expression is used for the magnitude and phase calibration of the ac susceptometer at different T and f and the low-field susceptibility calibration of the dc magnetometer at different T .In Sec.III, the Curie-Weiss law is combined with the Langevin function to describe the T and H dependence of magnetic moment with an additional adjusting parameter, the elementary moment, determined by dc magnetic measurements.The resulting expression for dc field-dependent differential susceptibility is used for calibrating the magnitude and phase of the incremental ac susceptibility at different dc field.
It is found that complex behaviors occur at low temperatures and that the resulting elementary moment is illogically T dependent and larger than that predicted by the quantum mechanical theory, as explained in Sec.IV.However, as described in Sec.V, the measurements of a pure Dy 2 O 3 powder sample of known mass validate the quantum mechanical theory at low fields so that pure Dy 2 O 3 may serve as a primary standard, whose low-field susceptibility per unit mass is a function of T only.In Sec.VI, the influence of finite sample dimensions on the sensitivity of moment measurements is calculated to make additional error analysis.Further discussion and the final conclusion are presented in Secs.VII and VIII.
We will use electromagnetic units for the data, since they are widely used in the literature and the data files of both instruments.However, as generally recommended, SI units will be used in basic physical equations.Which units system to be used will be noted in the text.In electromagnetic system, the units of magnetic moment and field are erg/G and Oe, respectively, but the former is written emu customarily in the present paper.In SI, the units of moment and field are Am 2 and A/m, respectively.The conversion between both systems may be made by 1 Am 2 = 10 3 emu and 1 A/m = 4π × 10 −3 Oe.

A. Curie-Weiss moment susceptibility
The amplitude of complex ac moment m m = m m − jm m of the glued Dy 2 O 3 sample was, using the incorporated PPMS software, measured at zero dc field, ac field amplitude H m = 10 Oe, f between 11 and 3333 Hz, and T between 2 and 300 K, following a standard procedure after the sample was automatically centered at a certain temperature. 1The results of complex ac moment susceptibility χ = m m /H m = χ − jχ , after a magnitude correction with k = 1.081 was applied to m m , are shown in Fig. 1 for f = 11, 33, 111, 333, 1111, and 3333 Hz and T ≤ 40 K.We observe that positive χ is weakly f dependent and it decreases with increasing T , whereas χ is 2 orders of magnitude smaller than χ and may be positive or negative with a remarkable f and T dependence.
As a paramagnetic material, the low-field susceptibility of Dy 2 O 3 sample should follow the Curie-Weiss law, 6 expressed in SI units by where μ 0 = 4π × 10 −7 H/m is the permeability of free space, m 0 (Am 2 ) and n are the elementary moment of the material and its number per unit volume, k B = 1.381 × 10 −23 J/K is the Boltzmann constant, and T 0 (K) is the Weiss constant.Since all these quantities and the volume are constant for the sample, Eq. (1) may be conveniently written for the moment susceptibility χ CW in electromagnetic units as where C is a constant.For comparing experimental data with Eq. ( 2), we calculate the magnitude and phase of the measured χ as and plot 1/|χ | as a function of T in Fig. 2(a) for f = 333 and 1111 Hz, at which χ measurements may be performed with higher resolution and smaller T -dependent magnitude error. 5The straight line shows the best Curie-Weiss fit to the data, with C −1 = 138.it may have about 1% difference at 300 K between f = 333 and 1111 Hz.The fitting 1/χ CW line is located in between with a difference about 0.1% from the average 1/|χ | at both values of frequency when T ≥ 45 K, but the line is above the data points for 0.6% and 1% at T = 40 and 35 K, respectively.The fitting error is remarkably increased at even lower T , as seen from Fig. 2(a).
Thus, the Curie-Weiss moment susceptibility of the particular glued Dy 2 O 3 sample is finally obtained as which can be used for low-field calibration of magnetometers at T > 45 K.

B. Calibration of ac susceptometer at H dc = 0
Using χ CW in Eq. ( 4) as a standard, we calculate the relative error of the measured |χ |, |χ |/χ CW − 1, as a function of frequency and temperature, as shown in Fig. 3(a).This error is between −0.8% and 0.6% at all values of frequency and temperature, which is consistent with ±0.7% calibrated by the copper cylinder stated in Ref. 5. Since the standard φ is zero for paramagnetism at audio frequencies, the error in φ is actually the measured φ itself, as shown in Fig. 3(b).It is within ±0.3 • and also comparable with the same error at f = 111-1111 stated in Ref. 5.However, there are anomalies occurring at T < 10 K, where φ rises sharply.

C. Calibration of SQUID magnetometer
The glued Dy 2 O 3 sample was measured by the SQUID magnetometer at stepwise changed field from −70 kOe to 70 kOe to −70 kOe at T = 10, 20, 50, 100, 200, and 300 K after centering at each temperature.The field was set in noovershoot mode.The magnetic moment at each field was measured with one scan of distance 4 cm and 64 points 2 applying the "iterative regression" algorithm incorporated in the software.The SQUID magnetometer had been calibrated independently by using a Pd standard sample but a data correction will be made later in Sec.VI after the error due to sample dimensions is analyzed.The results are shown in Fig. 4(a) for the complete field range and in Fig. 4(b) for a low field portion, where hysteresis can be seen.Arrows in Fig. 4(b) indicate the directions of magnetic state evolution, which has an opposite sense to that of a normal hysteresis loop.
The hysteresis is a consequence of the error in field produced by superconducting magnet.The instrument reported field is proportional to the current flowing in the superconducting solenoid calculated from the Ampère law but the actual applied field at the sample is the sum of this field produced by the current and the field produced by the magnetic poles in the magnetized superconductor itself.According to the critical-state model, [7][8][9] a positive magnetic moment is induced in a hard superconductor and is proportional to the critical-current density J c , when the field decreases from a large positive value.Such a moment will produce a negative demagnetizing field in the bore of the magnet.When the field increases from a large negative value, the moment becomes negative and the demagnetizing field becomes positive.This is the reason for the hysteresis shown in Fig. 4(b), as already explained in Ref. 3. Since J c decreases with increasing the magnitude of the field |H |, 8,9 this error decreases with increasing |H |.However, we may add +/−18.2Oe to the reported field values on the ascending/descending curve to effectively reduce the relative error in the entire range, since a major portion of the error is corrected at low |H | and the over correction is much smaller than the field value at high |H |.This will be further discussed when Fig. 12 is shown.After such a correction, the comparison between the m/H measured by the SQUID magnetometer and χ CW calculated from Eq. ( 4) is shown in Fig. 5.We see that for T = 50, 100, 200, and 300 K, the measured ascending and descending m/H are quite stable and lower than χ CW of Eq. ( 4) by about 1% in a large field interval.Thus, the SQUID magnetometer can be calibrated by using the Dy 2 O 3 sample, which has 1 order of magnitude larger χ than the Pd sample so a higher resolution at low fields can be achieved.

A. Field and temperature dependence of magnetic moment
The measured m vs H curves after the field correction (i.e., the shift of +/ − 18.2 Oe) are shown in Fig. 6.All the experimental curves at different T may be well fitted by the classical theory of paramagnetism in terms of the Langevin function, plotted by the red solid lines.In this theory, 6 the magnetization of a paramagnetic material is expressed, after combining with the Curie-Weiss law, by where M 0 = nm 0 and the Langevin function with x expressed in SI units as It can be derived from Eqs. ( 5)-( 7) that the differential susceptibility at any values of H and T is whose limit at H = 0 is Eq.(1).
To compare with the measured data in electromagnetic units, these equations are rewritten as where As Eq. ( 4), all these equations are valid for the data at T > 45 K, where the only adjustable parameter is m 0 /k B , which should be obtained by fitting the measured m(H ) curves.However, the measured m(H ) curves at T ≥ 100 K are basically linear, for which Eq. ( 9) is reduced to Eq. ( 2), where m 0 /k B is absent.Thus, we use data for T = 20 K, where nonlinear m vs H is clearly shown, to start the fit.Unfortunately, the Curie-Weiss law is not valid accurately for T = 20 K, and we choose to assume the same C in Eq. ( 2) and to change T 0 to an effective one −15.5K to get the value of χ measured by ac susceptibility at 20 K.The obtained fitting parameter is m 0 /k B = 9.8 × 10 −4 K/Oe.With these parameter, the results for ≥ 50 K are well fitted, as seen in Fig. 6.Even so, the results for T = 10 K cannot be well fitted by such a value of m 0 /k B after the effective T 0 is properly set as −13.5 K; a good fit requires m 0 /k B = 1.1 × 10 −3 K/Oe at T = 10 K.

B. Calibration of incremental χ
After the m(H ) fits by Eqs. ( 9) and (11), Eqs. ( 10) and ( 11) may be used for the calibration of incremental χ at different values of H dc , f , and T .
The 1/|χ | and φ of the incremental ac susceptibility measured at H m = 10 Oe, f = 1111 Hz, eight values of H dc between 0 and 80 kOe, and T = 10, 100, 200, and 298 K are shown in Figs.7(a) and 7(b), respectively.We see in Figs.7(a) and 7(b) that 1/|χ | increases monotonically and φ increases and then decreases with increasing H dc at T = 10 K; similar but smaller variation occurs at T = 100 K for 1/|χ | but not for φ.The behavior at T = 20 K is similar to T = 10 K but less pronounced (data not shown).
We plot the measured χ and χ in Figs.8(a) and 8(b), respectively, as functions of H dc at T = 10 and 100 K.We see that χ decreases with increasing H dc at both values of T , and χ shows a peak at T = 10 K but equals zero for T = 100 K. Regarding χ as dm/d H at H = H dc , we compare it with the results calculated from Eqs. (10) and (11), as plotted by red lines in Fig. 8(a).We see that the line matches well the data points for T = 100 K, but the line is obviously above the points at T = 10 K.It seems that the properties of the Dy 2 O 3 are anomalous at T = 10 K for paramagnetism, and the over-low incremental χ is accompanied by the χ peak.10) and (11).

A. Low-T anomalies in susceptibility
Anomalies in phase similar to those shown in Fig. 3(b) are not reported in Ref. 5, where a high-quality superconducting film is measured at a low field in the Meissner state so that φ should be 0. Comparing the measured φ vs T of the superconductor and Dy 2 O 3 in Fig. 9, we see that the φ of the former is stable at T < 20 K, but for the latter, it is larger than the former and increases quickly with decreasing T to below 5 K.This suggests that the low-T anomalous φ in Fig. 3(b) should not be a consequence of the error in ac susceptibility measurements alone, but be also from a possible low-T phase transition of Dy 2 O 3 itself.Using the results of the superconducting film as a standard, the phase corrected χ as a function of T is plotted in Fig. 1(b), showing χ > 0 till T = 40 K with a systematic f dependence.
Such an anomaly may be related to the over-low incremental χ and the χ peak occurring at low T .
It is worth mentioning the early work 10 reporting that Dy 2 O 3 undergoes a transition into antiferromagnetic state below the Neel temperature T N = 1.2 K.This temperature is much smaller in magnitude than the Weiss constant T 0 = −16.8K obtained in the present work, indicating a high degree of frustration between the magnetic moments in the ordered state. 11The much higher Weiss constant T 0 = −3.2K quoted in Ref. 10 is probably due to the very lim- ited temperature range (up to only 4.2 K) from where the data were extrapolated.A phase transition of Dy 2 O 3 at T ≈ 1 K was also mentioned in Ref. 12.
It is important to realize that the true Curie-Weiss behavior sets in well above T N , when any short-range order effects would have died out and all the levels of the crystal-field-split ground multiplet of Dy 3+ ion are significantly populated. 13e argue that this is valid at temperatures above 45 K, where a good fit of the experimental data is found.

B. Over-large m 0 at low fields
According to the quantum mechanical theory of paramagnetism, the Langevin function L(x) should be replaced by the Brillouin function B J (y) as 6 M = M 0 B J (y), (12)   where and the Brillouin function is with y expressed in SI units as The corresponding low-field susceptibility may be derived as Since the elementary moment of Dy 2 O 3 is carried by Dy 3+ , whose spin, orbital, and total angular momenta are S = 5/2, L = 5, and J = 15/2, so that the g factor The elementary moment m 0 = gμ B √ J (J + 1) = 10.6μB may be calculated from Eqs. ( 16) and (17).
Since m 0 /k B is fixed in this case, the measured m(H ) curve cannot be well fitted.It is interesting to note that the classical theory fits lead to m 0 = 14.6μB and m 0 = 16.4μB for T ≥ 20 K and T = 10 K, respectively.
This T -dependent over-large m 0 detected from the glued Dy 2 O 3 powder is an anomalous behavior with respect to paramagnetism.However, we have validated Eq. ( 16) with m 0 = gμ B √ J (J + 1) = 10.6μB by measuring a pure Dy 2 O 3 powder sample of known mass, as described below.Therefore, the discovered over-large m 0 at low fields is actually a consequence of over-small saturation M 0 .A value of M 0 = 10nμ B is calculated from Eqs. ( 13) and ( 17) for Dy 3+ , but M 0 ≈ 5.3nμ B is obtained from the magnetization of the Dy 2 O 3 powder sample at T = 1.8K and μ 0 H = 7 T. We note that even smaller M 0 ≈ 2.8nμ B was reported previously for Dy 2 O 3 powder. 14Since the actual magnetic saturation is lower than that appearing in the quantum mechanical theory of paramagnetism, the measured m(H ) curves cannot be well fitted by this theory, but they still can be well fitted by the classical theory of paramagnetism, if m 0 may be adjusted arbitrarily.The over-large values of m 0 are thus obtained by Langevin fitting to the measured m(H ) curves.

V. PURE DY 2 O 3 POWDER AS A PRIMARY STANDARD
Regarding n as the number of Dy 3+ per unit mass, calculated by the molar mass 373.0 g/mol of Dy 2 O 3 and the Avogadro constant N A = 6.022 × 10 23 mol −1 , and assuming m 0 = 10.6μB , the Curie-Weiss susceptibility per unit mass, written as χ m,CW , may be calculated from Eq. (1) in both unit systems as There is a T -independent diamagnetic contribution from the core electrons of the atoms.From the Pascal constant of Dy 3+ and O −2 to be −19 × 10 −6 and −12 × 10 −6 emu/mol, 15 the diamagnetic partial susceptibility of Dy 2 O 3 is calculated to be −2 × 10 −7 emu/Oe/g, which is 10 3 times in magnitude smaller than χ m,CW at 300 K and so negligible at T < 300 K. Thus, pure Dy 2 O 3 powder with known mass can be used as a primary standard if Eq. ( 18) is valid.The validity of Eq. ( 18) derived from quantum mechanical theory is tested experimentally.We have prepared a spherical sample of Dy 2 O 3 powder of purity 99.9% (Alfa Aesar, Germany, provided by Quantum Design) covered with a small piece of food wrap foil to avoid extra diamagnetic contribution, and measured its m vs T by the SQUID magnetometer at H = 1000 Oe.In order to get accurate H , the superconducting magnet was quenched before measurements.The sample was automatically centered at each value of T .The results are given in Fig. 10, from which we see that the measured data may be well fitted by The constant factor is practically the same as that in Eq. ( 18), but T 0 = −17.5K is 0.7 K lower than that for the glued sample determined by ac measurements.We suspect that the dif- Linear vertical scale is used in (a), so that the Curie-Weiss relation is linear, but logarithmic vertical scale is used in (b) and (c), in order to show clearly the relative differences among the data at each value of T .
ference in T 0 comes from different errors in T between the ac and dc magnetometers, since T 0 = −17.5K is also obtained by fitting the data in Fig. 5 for the glued sample and T 0 = −16.9K has been obtained by ac measurements of another pure powder sample.Without direct calibration of T for the magnetometers, T 0 = −17 K may be chosen in Eq. ( 18) for simplicity.We should mention that previous measurements in Ref. 16 gave m 0 = 9.14μ B for Dy 3+ , which is 14% smaller than 10.6μ B .
The equal validity of the glued and pure Dy 2 O 3 powder samples in the phase calibration of the ac susceptometer may be seen from Figs. 3(b) and 9, where symbols and lines are the results of the glued and pure samples, respectively.

VI. ERROR DUE TO FINITE SAMPLE DIMENSIONS
Until now, we have overlooked an error in moment measurements due to finite sample dimensions.The moment sensitivity of magnetometers is defined ideally for a sample of negligible dimensions compared with the size of sensing coil or for a spherical sample of finite diameter.The calibration samples provided by Quantum Design are cylinders and their moment sensitivity should depend on their dimensions.Since a uniformly magnetized cylinder is equivalent to a current-flowing uniformly wound thin solenoid, 17 the flux linkage in the sensing coil produced by the magnetic moment of the sample can be calculated numerically based on the formula for the mutual inductance L m between two coaxial coils of radius ρ 1 and ρ 2 and distance δ, 18 where To estimate the error, we consider one sensing coil (thin solenoid) of diameter 9 mm and length 19 mm for the ac susceptometer and one single-turn coil of diameter 19.4 mm for the dc SQUID magnetometer. 19Assuming the sample diameter to be d s = 5 or 2.9 mm (corresponding to the copper and the glued Dy 2 O 3 cylinders or the Pd cylinder), we calculate the moment sensitivity with respect to the sensed flux linkage for a cylinder sample of length l s and diameter d s , S mφ (l s , d s ).The resulting S mφ (l s , d s )/S mφ (0, 0) − 1 as a function of l s /d s is shown in Fig. 11.
We see from Fig. 11 that the error for the ac moment measurements of the Cu and the glued powder is −0.2% and −0.5%, respectively, and the error for the dc moment measurements of the Pd and the glued powder is −0.7% and −2.4%, respectively.Since the pure powder sample is spherical, such an error is zero for its measurements.
Before making additional error analysis, we should mention that a magnitude correction was already made for the data of ac measurements with a factor k = 1.081 obtained by measuring the copper standard, but a similar correction was not made yet for the data of dc measurements with another factor k = 0.980 obtained by measuring the Pd standard.Combining the −0.7% error, this factor should be replaced by k = 0.987.
Multiplying 0.987 to 13.30 in Eq. ( 19) results in 13.13, which is 1.1% lower than the theoretical 13.28 in Eq. ( 18).The over-high susceptibility of the powder sample may partially result from a negative error in H about −0.3%, estimated by measuring the initial and hysteresis loop after quenching the magnet.Considering this, the measured susceptibility is 0.8% higher than its theoretical value.
The ac susceptibility of the glued Dy 2 O 3 cylinder measured by the ac susceptometer, which was calibrated and corrected by measuring the copper cylinder, has a −0.3% error owing finite dimensions of both cylinders, so that 138.6 in Eq. ( 4) should be corrected to 138.2.The dc susceptibility of the glued sample measured by the dc SQUID magnetometer before correction has a 0.3% error owing to the finite dimensions of the Pd and Dy 2 O 3 cylinders and k = 0.980.As a result, the low-field dc susceptibility becomes about 1.6% lower than those calculated from the corrected Eq. ( 4).
We have calculated the moment sensitivity with respect to a single coil to calculate the error for the SQUID magnetometer rather than that with respect to the coil assembly with iterative regression as calculated in Ref. 19.However, both results are similar; for a cylinder of length and diameter 5 mm, the error is −0.97% and −1.04% for the former and the latter calculations, respectively.
From the results shown in Fig 11, we may conclude that when cylindrical samples are used for calibrating ac and dc magnetometers, their optimum l s /d s = 0.85, at which the error due to finite dimensions is less than 0.1%.Standard samples provided by the factory have l s /d s ≈ 1.2, which may result in an error about −0.5%.

VII. FURTHER DISCUSSION
The difference about 1.6% in low-field susceptibility of the glued sample between the ac and dc measurements could be due to temperature, whose accuracy is 1% according to the manuals.We make the following comparison in m(H ) between the results obtained by direct SQUID magnetometer measurements and those calculated from Eqs. ( 9) and (11).The difference m(H ) between the measured and calculated m at T = 200 K is plotted in Fig. 12 by red open squares.We see that the difference between the ascending and descending values remains constant at H < 1000 Oe and it decreases with further increasing H and that the average m increases in magnitude with increasing H remarkably at H > 1000 Oe.If 200 K is replaced by 202 K in the calculation, then such an increase becomes a small oscillation as shown by black open circles.The constant m at H < 1000 Oe indicates the applicability of the constant field shift used for correcting the field in quite a large range of field.
We have calibrated dc and/or ac magnetometers by using a copper cylinder, a high-temperature superconducting film, a glued Dy 2 O 3 powder sample, and a pure Dy 2 O 3 powder sample.It is necessary to compare the different standards.
Based on its ac susceptibility calculated from the eddy-current effects, the copper cylinder can calibrate ac susceptometers only.It may be regarded as a quasiprimary standard sample since its properties and the performance of the ac susceptometer can be mutually calibrated. 5The calibration may be performed in the entire T , H m , and f range, but owing to a remarkable magnetoresistance, the incremental susceptibility at H dc > 0 cannot be calibrated by it.Based on the calculated susceptibility in the Meissner state, the superconducting film may also be regarded as a primary standard sample.It can be used at enough low temperatures and very low fields, satisfying the calibration of ac susceptometer at T < 70 K.The glued Dy 2 O 3 powder sample is not a primary standard, but as soon as C and T 0 are obtained by using a well calibrated magnetometer, the performance of any dc and ac magnetometer may be calibrated by it at any fields and T > 45 K.The standard χ for the latter two is simple with φ = 0, whereas both |χ | and φ are functions of frequency and temperature in a complex way when copper cylinder is used.The pure Dy 2 O 3 powder sample can serve as a primary standard with accuracy on the order of 1% used at fields below 10 kOe and temperature above 50 K (see Fig. 5).However, as a standard, a glued powder sample may still be a better choice, since it is easy to handle and to be maintained in a good condition.
Although an overall agreement has been observed between different calibrations, there are remarkable differences found, which leaves the topic of calibration still open.

VIII. CONCLUSION
The performance of ac and dc magnetometers may be calibrated with a glued Dy 2 O 3 powder sample using its temperature and field dependence of magnetic moment expressed empirically by a combination of the Curie-Weiss law and the Langevin function at T > 45 K.The expressions for the studied sample are Eqs.( 2) and ( 9)- (11), where the Weiss constant T 0 = −16.8K, the normalized elementary magnetic moment m 0 /k B = 9.8 × 10 −4 K/Oe, and the inverse moment Curie constant C −1 = 138.6Oe emu −1 K −1 .When calibrating ac susceptometers, the measured |χ | should be compared with Eq. (2) or Eqs.(10) and (11) and the measured φ should be compared with zero; when calibrating dc magnetometers, the measured m(H ) should be compared with Eqs. ( 9) and (11).Nonzero φ may occur at lower T , indicating a possible phase transition.Without knowing the mass of Dy 2 O 3 , the properties of such glued samples vary between samples.However, pure Dy 2 O 3 powder samples of known mass may serve as a primary standard for low-field susceptibility, calculated by Eq. ( 18) with T 0 = −17 K, which is a combination of the Curie-Weiss law and the quantum mechanical theory of paramagnetism.The optimum aspect ratio of cylindrical standard samples is 0.85.
FIG. 1. (Color online) Real and imaginary ac moment susceptibilities, χ (a) and χ (b), of the glued Dy 2 O 3 standard sample measured at H m = 10 Oe and f from 11 to 3333 Hz as a function of temperature T .The lines in (b) show the phase corrected χ (see Fig. 9) with the arrow indicating the direction of increasing frequency.

FIG. 2 .
FIG. 2. (Color online) 1/|χ | of the glued Dy 2 O 3 sample at f = 333 and 1111 Hz (symbols) and 1/χ CW (line) as functions of T .Linear vertical scale is used in (a), so that the Curie-Weiss relation is linear, but logarithmic vertical scale is used in (b) and (c), in order to show clearly the relative differences among the data at each value of T .

3 FIG. 3 .
FIG. 3. (Color online) (a) Relative magnitude error (|χ |/χ CW − 1) × 100 (%) at T > 39 K and (b) phase error of ac susceptibility measurements at frequencies between 11 and 3333 Hz, determined by measuring the glued Dy 2 O 3 sample, as functions of T (symbols).The phase error determined by measuring a pure Dy 2 O 3 sample is also shown in (b) by lines.

FIG. 4 .
FIG. 4. Hysteresis m vs H curves of the glued Dy 2 O 3 sample measured by the dc SQUID magnetometer.The arrow in (a) indicates the direction of decreasing T .The arrows in (b) indicate the sense of state evolution.

FIG. 5 .
FIG.5.The m/H calculated from the m vs H curves measured by the dc SQUID magnetometer at T = 50, 100, 200, and 300 K after a field shift of ±18.2 Oe compared with the ac moment susceptibility of the glued Dy 2 O 3 sample obtained by calibration of the ac susceptometer, Eq. (4).

FIG. 6 .FIG. 7 .
FIG. 6. (Color online) m vs H loops of the glued Dy 2 O 3 sample measured by the dc SQUID magnetometer at T = 10, 20, 50, 100, 200, and 300 K, increasing in the arrow direction, after a field shift of ±18.2 Oe.Open and solid circles show the ascending and descending branches, respectively.Red solid lines show the Curie-Weiss and Langevin function fitting.

FIG. 8 .
FIG. 8. (Color online) Real and imaginary incremental ac moment susceptibilities, χ (a) and χ (b), of the glued Dy 2 O 3 sample measured at 10 and 100 K as a function of H dc .In (a), red solid lines show results of differential susceptibilities calculated from Eqs. (10) and(11).

3 (
FIG. 9. (Color online) Comparison of the phase of the ac susceptibility of the glued (solid symbols) and a pure (lines) Dy 2 O 3 samples with a superconducting film in the Meissner state studied in Ref. 5 (open symbols).

FIG. 10
FIG.10.1/χ m of a pure Dy 2 O 3 powder sample measured by the dc SQUID magnetometer at H = 1000 Oe as a function of T and its Curie-Weiss fit.Linear vertical scale is used in (a), so that the Curie-Weiss relation is linear, but logarithmic vertical scale is used in (b) and (c), in order to show clearly the relative differences among the data at each value of T .

FIG. 12 .
FIG. 12. (Color online) The difference between the m vs H loop of the glued sample directly measured by the dc SQUID magnetometer (without field shift) at T = 200 K and its Curie-Weiss and Langevin function fitting with T = 200 and 202 K.
FIG. 11.The relative difference between cylindrical samples of length l s and diameter d s and a point moment in moment sensitivity with respect to the sensed flux linkage, S mφ (l s , d s )/S mφ (0, 0) − 1, as a function of l s /d s .The one-turn coil diameter is 19.4 mm for the dc SQUID magnetometer, and the coil diameter and length are 9 and 19 mm, respectively, for the ac susceptometer.The sample diameter and length are 5 and 5 mm for copper, 5 and 6 mm for the glued Dy 2 O 3 , and 2.9 and 3.4 mm for Pd.