The heating elements in thermal demagnetizers are typically constructed of non-inductive windings in order to minimize the AC demagnetization effect of the heating current (Chamalaun and Porath 1968). There are generally two types of non-inductive heater winding, one is a bifilar solenoid wrapped on a cylindrical form to heat specimens within (Fig. 1a). This style of winding is widely used in small furnaces such as the Sogo Fine-TD furnace in Japan (Zheng et al. 2010a, b). It is also used in some Kappabridges and variable field translation balances (VFTBs) to heat specimens. In this configuration, the opposing currents of adjacent wires reduce the AC magnetic field. In addition, the wire spacing should be small enough to efficiently reduce the residual magnetic field. A disadvantage of this system, however, is that it will increase the risk of short circuit, and often lead to limited sample capacity in order to maintain an acceptable resistance and workable power consumption.

An alternative method is to use solenoids arranged in opposite direction (Fig. 1b) or wound into a helix as in many commercial furnaces. It is usually assumed that the magnetic field generated by a uniform solenoid is only in the interior and both ends of the solenoid, but this neglects the fact that for a real solenoid the wire loops (and current loops) are not perfectly circular, the current must go from one end to the other, which results in non-zero external fields. Therefore, it is necessary to consider the magnetic field distribution outside a helical solenoid.

### Simulated calculation of solenoid magnetic field

The magnetic field distribution outside a solenoid can be calculated using the Biot–Savart law (Hagel et al. 1994; Budnik and Machczynski 2011). Assuming a current, *I*, flowing in a finite solenoid with a radius of *R* and a pitch of *p*, the vectors of magnetic field at point P (Fig. 2) produce by a differential current element *Idl* at source point N(*Xs*, *Ys*, *Zs*) can be written as:

$$ dB = \frac{{\mu_{0} I}}{4\pi }\frac{{d\mathop{l}\limits^{\rightharpoonup} \times \mathop{r}\limits^{\rightharpoonup} }}{{r^{3} }}, $$

(1)

where *r* is distance from the point N on the filament to the field point P. *Xs* = *RcosƟ*, *Ys* = *RsinƟ* and *Zs* = *aƟ* since the point N is on the solenoid, where *a* is the pitch factor.

From cylindrical coordinates to Cartesian coordinates, *dl* and *r* can be expressed as:

$$ d\mathop{l}\limits^{\rightharpoonup} = \mathop{i}\limits^{\rightharpoonup} dx + \mathop{j}\limits^{\rightharpoonup} dy + \mathop{k}\limits^{\rightharpoonup} dz = - \mathop{i}\limits^{\rightharpoonup} R\sin \theta d\theta + \mathop{j}\limits^{\rightharpoonup} R\cos \theta d\theta + \mathop{k}\limits^{\rightharpoonup} ad\theta , $$

(2)

$$ \mathop{r}\limits^{\rightharpoonup} = \mathop{i}\limits^{\rightharpoonup} (x_{0} - R\cos \theta ) + \mathop{j}\limits^{\rightharpoonup} (y_{0} - R\sin \theta ) + \mathop{k}\limits^{\rightharpoonup} (z_{0} - a\theta ). $$

(3)

Therefore:

$$ d\mathop{l}\limits^{\rightharpoonup} \times \mathop{r}\limits^{\rightharpoonup} = \left| {\begin{array}{*{20}c} {\mathop{i}\limits^{\rightharpoonup} } & {\mathop{j}\limits^{\rightharpoonup} } & {\mathop{k}\limits^{\rightharpoonup} } \\ { - R\sin \theta d\theta } & {R\cos \theta d\theta } & {ad\theta } \\ {x_{0} - R\cos \theta } & {y_{0} - R\sin \theta } & {z_{0} - a\theta } \\ \end{array} } \right|, $$

(4)

and we have

$$ dB_{x} = \frac{{\mu_{0} I}}{4\pi }\frac{{\left[ {(z_{0} - a\theta )R\cos \theta - (y_{0} - R\sin \theta )a} \right]d\theta }}{{\left[ {(x_{0} - R\cos \theta )^{2} + (y_{0} - R\sin \theta )^{2} + (z_{0} - a\theta )^{2} } \right]^{{{\raise0.7ex\hbox{$3$} \!\mathord{\left/ {\vphantom {3 2}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$2$}}}} }}, $$

(5)

$$ dB_{y} = \frac{{\mu_{0} I}}{4\pi }\frac{{\left[ {(z_{0} - a\theta )R\sin \theta + (x_{0} - R\cos \theta )a} \right]d\theta }}{{\left[ {(x_{0} - R\cos \theta )^{2} + (y_{0} - R\sin \theta )^{2} + (z_{0} - a\theta )^{2} } \right]^{{{\raise0.7ex\hbox{$3$} \!\mathord{\left/ {\vphantom {3 2}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$2$}}}} }}, $$

(6)

$$ dB_{z} = - \frac{{\mu_{0} I}}{4\pi }\frac{{\left[ {(y_{0} - R\sin \theta )R\sin \theta + (x_{0} - R\cos \theta )R\cos \theta } \right]d\theta }}{{\left[ {(x_{0} - R\cos \theta )^{2} + (y_{0} - R\sin \theta )^{2} + (z_{0} - a\theta )^{2} } \right]^{{{\raise0.7ex\hbox{$3$} \!\mathord{\left/ {\vphantom {3 2}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$2$}}}} }}. $$

(7)

As the radius of the solenoid decreases, or when the distance between the point P and solenoid is much larger than the radius, R can be approximately ignored. The point N on a straight wire is \((0,0,a\theta )\), and filament became as \(d\mathop{l}\limits^{\rightharpoonup} = \mathop{k}\limits^{\rightharpoonup} ad\theta\), and \(\mathop{r}\limits^{\rightharpoonup} = \mathop{i}\limits^{\rightharpoonup} x_{0} + \mathop{j}\limits^{\rightharpoonup} y_{0} + \mathop{k}\limits^{\rightharpoonup} (z_{0} - a\theta )\), and therefore, the formula of magnetic field can be simplified as:

$$ dB_{x} = - \frac{{\mu_{0} I}}{4\pi }\frac{{y_{0} ad\theta }}{{\left[ {x_{0}^{2} + y_{0}^{2} + (z_{0} - a\theta )^{2} } \right]^{{{\raise0.7ex\hbox{$3$} \!\mathord{\left/ {\vphantom {3 2}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$2$}}}} }}, $$

(8)

$$ dB_{y} = \frac{{\mu_{0} I}}{4\pi }\frac{{x_{0} ad\theta }}{{\left[ {x_{0}^{2} + y_{0}^{2} + (z_{0} - a\theta )^{2} } \right]^{{{\raise0.7ex\hbox{$3$} \!\mathord{\left/ {\vphantom {3 2}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$2$}}}} }}, $$

(9)

It can be seen that when the solenoid diameter is small, the magnetic field generated by the solenoid is roughly the same as that of a straight wire. Therefore, to reduce the stray external fields of a solenoid, a straight wire can be inserted through its core and connected with solenoid at one end, such that the current in the straight wire is flowing opposite to the current in the solenoid (Fig. 1c). A new heating element is designed based on this configuration which we call it straight core solenoid, and will be used to manufacture our new thermal demagnetizer.

To simulate a heating element solenoid, we assume that the solenoid length is 1 m, the radius is 4 mm, and the pitch is also 4 mm. And the current is set to 1 A. We simulate the fields in YZ plane (Fig. 2) along the direction of solenoid axis, at a distance of 20 mm and 10 mm from central axis. The results are shown in Fig. 3. Only absolute values of field are shown in Fig. 3a because of the logarithmic scale. It is found that the perpendicular component (X) is higher than the radial component (Y) and parallel components (Z), and the X component can be reduced by two orders of magnitude if a straight wire with reverse current is passed through the center of the solenoid. The Y and Z components remain unchanged, since the magnetic field produced by the straight wire wraps around the wire, which is shown here in the X direction only. The fluctuations of field value were also observed at very close to the solenoid and the period is consistent with the pitch (Fig. 3b). As the distance increases, the periodic fluctuation becomes insignificant (Hagel et al. 1994). The magnetic field distribution in other cases can be obtained by changing parameters of the solenoid, such as radius, pitch. It is useful to optimize the structure of heating wire before practical manufacture.

In fact, the whole heater group consists of a series of parallel or anti-parallel heating element solenoid and its configuration determined the magnetic field inside the furnace. We calculate the magnetic field distribution in the furnace. The heater composed of 12 conventional solenoids or new heating elements with opposite current direction between adjacent is simulated. Contour maps of different components are shown in Fig. 4. The parameters of each solenoid are the same as those used in Fig. 3, and the current is still set to 1 A. X, Y and Z represent the magnetic field in horizontal, vertical and along the axis direction in the furnace, respectively. It can be seen that the magnetic field in the center of the furnace is extremely low for both configurations, but the low magnetic field area is considerably enlarged in the furnace with new heating elements. The magnetic field inside the solenoid is relatively complex, but does not affect the thermal demagnetization samples. Different configurations which are consistent with some practical applications can also be simulated. A large magnetic field may be produced by parallel connection of heating wire groups (Fig. 5). However, the magnetic field is still very low inside the furnace with same configuration using the new straight core solenoid.

### Measurement of solenoid magnetic field

In order to verify the effectiveness and determine the magnetic field of the new heating element, the respective magnetic fields produced by the conventional solenoid heating wire and the new heating element were measured with a triaxial fluxgate magnetometer (APS520). One solenoid heating element with length of 1 m and diameter of 8 mm was placed inside a magnetically shielding cylinder with 1 A DC current applied in the wire (Fig. 6a). The fluxgate sensor was placed to the side of solenoid at vertical distance of about 20 mm and moved along the long axis direction. The magnetic field at different positions was observed with the current on then off. In the same way, we measured the magnetic field of our new straight core solenoid heating element, which is composed of the same solenoid with a straight wire passing through it (Fig. 6b). The solenoid and the straight wire were isolated by a ceramic sheath. The results are shown in Fig. 6c. The magnetic field of conventional solenoid is about 6000–7000 nT and field direction is almost perpendicular to the long axis of solenoid and is consistent with the simulation, which is almost equivalent to that of an axial line current. The stray magnetic field of the straight core solenoid is much lower than the standard solenoid, with a total field intensity of 26–535 nT.

In our prototype, thermal demagnetizer adjacent elements are arranged in opposite directions (Fig. 7a). The magnetic field at different positions along the axis of the heating chamber were measured when a DC current of 1 A is applied to the heating wire. We measured the magnetic field four times at different vertical distances from the center axis. The peak magnetic field in the sample region is about 130 nT (Fig. 7b). In order to verify the effect, we also arranged the conventional solenoids following the same structure, and the current of adjacent solenoids is opposite. It is found that the magnetic field in the center is relatively low (115–285 nT in sample zone), but gradually increases toward the edge of the furnace (Fig. 7c). However, if the wire connection is changed, parallel connection of two groups are adopted, the magnetic field will become higher and mainly in the vertical direction (Fig. 7d). The measured results are consistent with the numerical simulation.

### Degaussing the shield cylinder and the TD-PGL-100

In order to shield the influence of external magnetic field, the chamber of thermal demagnetization furnace is usually installed in a cylindrical magnetic shield cylinder. The residual fields in sample zone typically vary from several nT to 150 nT in different furnaces (Paterson et al. 2012), which is mainly determined by the remanent magnetization of magnetic materials within the furnace, especially the shields (Freake and Thorp 1971; Thiel et al. 2007). A lower residual field can be achieved by AF demagnetization of the shield cylinder. Early experiments found that the best shield demagnetization results were obtained by passing the AC current through conductors within the shield cylinders or by passing the current longitudinally through the shielding material itself (Herrmannsfeldt and Salsburg 1964). A circular magnetic field perpendicular to the central longitudinal axis will be produced in the cylindrical shield if a current conducting straight wire is passed though the longitudinal center axis of the shield cylinder, however, this requires a large current to reach peak demagnetization. In practice, we use a total of eight 4 mm^{2} copper lines wound from inside to outside of the shield cylinder to form a large coil (Fig. 8a), and connected to a transformer with a twisted-pair cable. The transformer has separated primary and secondary windings which can be pulled apart. Separating the windings will generate a nonlinear attenuation of the alternating current in the coil which performs AF demagnetization of the shield cylinder. The residual magnetic field in the sample region can be as low as 1 nT (Fig. 8b), which makes it possible to further reduce spurious fields during thermal demagnetization.

A new furnace named TD-PGL-100 had been built based on above-mentioned techniques. The furnace has a chamber of 120 cm length and 9 cm diameter, and the sample zone is designed as 60 cm in length, which allows up to 100 standard paleomagnetic specimens be treated together. Temperature gradient in the sample zone is within 10 °C of target temperature below 600 °C and 12 °C at 700 °C. The maximum power consumption of the furnace is 4.0 KVA. Employing an adjustable voltage silicon-controlled rectifier over a solid-state relay allows the output power to be decreased as the temperature approaches the target point, which helps to avoid temperature overshoots. With careful tuning, the oven temperature has a 1 °C accuracy with a resolution of 0.1 °C. As the examples of heating and cooling curves shown in Fig. 9, the temperature rises rapidly at the beginning of heating, and gradually approaches the target temperature after ~ 30 min and then the temperature is maintained for a hold time to ensure the samples are thoroughly heated. The cooling fan can be turned on manually or automatically to cool the samples down. The temperature drops rapidly and slows when approaching room temperature. The heating and cooling data can be logged into USB memory automatically, which is essential for the thermal treatment and related experiments particularly for absolute paleointensity studies, where temperature reproducibility is important.