Development and testing of a three-period, subsize 2G AIMI MgB₂ planar undulator

The FEM simulations were performed using a COMSOL multiphysics software package. The three-period subsize undulator coil was modeled in a 2D configuration with length along the coil bore and width along coil crosssection, using the exact dimensions of the actual fabricated and tested sub-size undulator coil. This sub-size undulator coil has a period length of 14.4 mm, and the coil winding was 5 mm deep and 4.8 mm wide. All material properties were taken from the COMSOL library, the iron yoke was chosen as to be low carbon steel—1018 (using the ASTM standard for magnetic permeability), and we used unity for the relative permeability of the superconductor windings. For the longer (1 meter) undulator simulation, all parameters were kept the same, except for the length, which was set to 1 m.

We performed magnetic field studies using COMSOL's physics module, which employs Maxwell's equations to solve for the results. The control equations used are as follows:

$$\begin{align*}\nabla {\text{ }} \times {{H}} = {{J}}\end{align*}$$

$$\begin{align*}B = \nabla \times A\end{align*}$$

$$\begin{align*}{{J}} = \sigma {{E}} + {\text{ }}{{{J}}_{\text{e}}}\end{align*}$$

where H is the Magnetic field strength (A m⁻¹), B is Magnetic flux density (T), A is the magnetic vector potential, and E is the electric field. Here also J_e is the superconducting critical current density normalized to the whole of the winding (i.e. J_e is the 'superconducting' component of the current density in the winding, while J is the total average current density in the coil winding's cross-section, and thus the σ E term in the third equation is the contribution of the normal electrons to the overall current density in the winding). The winding is considered as an averaged continuum, and the winding is considered to have a nearly infinite conductivity below J_e, and a conductivity of σ (taken as the stabilizer resistivity × the area fraction of stabilizer in the winding cross section) for the fraction of current above J_e.

We then solved the control equations using the finite element method, which discretizes the problem domain into a mesh of elements and then solves the equations at each node in the mesh. This results in a system of linear equations that can be solved using a variety of numerical methods. We used a stationary solver, with an extremely fine mesh density (with a minimum mesh element size of 4.2 × 10⁻⁶ m and a maximum mesh element size of 0.0021 m). The COMSOL multiphysics simulations provided us with detailed information about the magnetic field distribution in the undulator. This information is essential for optimizing the design of the undulator and ensuring that it meets the required performance specifications.

The undulator was wound with 6-filament 2 G advanced internal Mg infiltration (AIMI) MgB₂ strands 0.61 mm in diameter, fabricated by Hyper-Tech Research (0.74 mm OD with 0.065 mm thick S-glass insulation). The 2 G multifilamentary AIMI strand was made by restacking and drawing a bundle of monocore elements, figure 1(a). The MgB₂ wires are comprised of several subelements, each consisting of an axially positioned Mg rod embedded in B powder, clad in a Nb sheath, and an outer Monel tube. After drawing down the monocore, its outer monel sheath was etched away. The remaining Nb sheathed element was cut, and the pieces were stacked into a monel-clad Cu tube, figures 1(b) and (c), in preparation for further drawing.

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The undulator was three periods long (the period, λ, was 14.4 mm) and consisted of two side-by-side stacks of four (racetrack-like) coils each, designed for the the undulator bore field to be produced between them in operation. The magnetic gap was 6.35 mm, and the individual racetrack coils were 100 mm long by 52 mm wide. Each racetrack coil had an edge slot for conductor winding. The racetrack winding groove was 5 mm deep and 4.8 mm wide. All of the coil specifications are also listed in table 1. Figure 2(a) is a longitudinal cross-section of planar undulator coils. Figure 2(b) shows the 6–5–6 structure of the winding. The material used for the yoke was 1018 steel, whose properties were fed into the FEM models.

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The coils were wound one by one with a single wire, which was S-glass insulated, and in order to change the winding direction, a groove bypass in the edge plates was used. The assembled coil was wrapped with Nb foil, put in an argon flow oven, and heat-treated at 685 °C for 2 h. The Nb foil worked as an oxygen getter. After taking out the coil, it was soldered with the wire ends in the lead connections. After this, an epoxy coating (CTD101) was applied to the undulator via vacuum impregnation in a glove box. Finally, the coil set was attached to a G10 frame for testing. Figure 3 displays the engineering critical current density of the conductor as a function of the field at 4.2 K. The measured values (higher field data points shown) were extrapolated to lower fields using the following expression [25]:

$$\begin{align*}{J_c}\left( {B,t} \right) = {J_{c0}}\left( t \right){\text{Exp}}\left( { - {\raise0.7ex\hbox{${{B^m}}$} \!\mathord{\left/ \right.} \!\lower0.7ex\hbox{${{B_0}\left( t \right)}$}}} \right)\end{align*}$$

$$\begin{align*}{J_{c0}}\left( t \right) = {\text{ }}{J_{c00}}{\text{ }}\left( {1 - {\text{ }}\alpha {t^2}} \right)\end{align*}$$

$$\begin{align*}{B_0}\left( t \right) = {\text{ }}{B_{00}}{\text{ }}\left( {1 - {\text{ }}\beta t} \right)\end{align*}$$

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where, B is the external applied magnetic flux density, α is 1, β is 1.1, J_c00 = 7.09 × 10⁵ A m⁻², which at 4.2 K corresponds to B₀₀ = 5.5 T^m, and the m-value = 1.2.

4.1.1. Short (3 period) undulator FEM.

FEM was used to model the fields in a short (3 period) undulator and a 1-m long (69 period) undulator as well as their transport properties, and the results of load line analysis enabled FEM-derived field maps to be generated. Figure 6(a) shows the magnetic field map in the cross-section of the undulator, and figure 6(b) shows the field (along the y-axis) as a function of distance along the center of the undulator beam tube. Note that since the subsize undulator consists of only twelve coils, the central field minimum is more profound than the outer maxima; the field amplitude is also asymmetric, with 1.19 T in the positive domain and 0.24 T in the negative domain.

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4.1.2. Long (1-m, 69 period) undulator FEM.

After a temperature of 4.2 K was reached, a current of 14.8 A was applied to the coil. The field was then measured as a function of position along the coil bore using a transverse hall sensor moving along the coil bore (figure 7). The maximum field measured was 0.116 T in the positive domain and 0.056 T in the negative domain. The difference in field values is due to an end effect. After this, we measured the coil's critical current (I_c). The voltage taps were applied to the current leads, and the gauge length was the total length of the conductor in the magnet (131 m).

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Measurements of B vs I were performed with the hall sensor placed at a point of maximum field (B_max) at 4.2 K by increasing the current in steps and pausing at intervals to record the voltage on the hall sensor. An I_c of 325.7 A was achieved, corresponding to B_max = 1.19 T. Figure 8 illustrates the correlation between the hall probe measurements and the FEM field calculations.

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After this, we measured the I_c of the coil at various temperatures. For these measurements, the LHe level was allowed to drop below the bottom of the coil, and Cu cooling straps were attached to various places on the coil and draped into the LHe, providing conduction cooling. Resistive heaters attached to the coil enable its temperature to be adjusted. The I_c runs took about 5 min to execute, during which the coil temperature would increase slightly due to ohmic heating at the leads (typically 1.5 K); the temperature recorded in figure 9 is the last before transition. Table 2 and figure 9 present I_c as a function of temperature measured on the upper Cernox sensor (the warmest point on the conductor) just before the transition by quench.

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The transport properties (I_c vs B) of a short sample of the MgB₂ conductor (figure 3) were converted to B vs I_c and plotted along with the load lines of the magnet both in the winding (calculated by FEM) and in the bore (hall probe measured) as shown in figure 10. It can be seen that the performance of the coil reached 94% that of the short sample.

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