## 2 Materials and Methods
### Magnet Holder  
Magnet holders (Figure 4a) are one of the most important components of the undulator. Such magnet holders are used for mounting permanent magnets enabling precise calibration of the magnet position. The magnet blocks (Figure 4b) with dimensions 100 \(\times\) 100 \(\times\) 39.9 mm (height \(\times\) width \(\times\) thickness) are made of a neodymium magnet (NdFeB) with properties given in Table 1. To fulfil the tolerance requirements, the nominal thickness of the magnets is reduced from 40 mm to 39.9 mm. Each magnet block is subjected to magnetic, mounting, and gravity forces. These magnetic and mounting forces are the main loading acting on the magnet holders.  
Due to the large dimensions of the magnets (100 \(\times\) 100 \(\times\) 39.9 mm), large magnetic forces (see Section 3.1), and magnet positioning requirements (\(\pm\)0.150 mm), the magnet holders (Figure 4a) are made of two separated parts called the outer frame and the inner U-shaped frame. The outer frame must be sufficiently rigid due to large magnetic forces acting on the magnet, as well as forces caused by mounting and calibration. The geometry and the stiffness of the inner frame are designed to keep the magnet in a certain position (ensure its stability) and to enable its positioning. The above-mentioned demands are contradictory and require a proper balance of the inner frame geometry. The crucial dimensions of the inner frame influencing its stiffness are the thickness of the frame and the distances between the bolts and the magnet. The final geometry of the inner frame was set based on the FEM analyses carried out in ANSYS Workbench 2023 R2 [39]. The remaining parts of the proposed holders are (Figure 4a) as follows:  
* brass shims;
* magnet mounting clamps;
* clamp mounting screws;
* inner frame fixing screws;
* positioning (alignment) screws.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Component**} & \multicolumn{5}{c}{**Mechanical Properties of Material**} \\  & **Material** & **Remanence** & **Intrinsic Coercivity** & **Yield Stress** & **Tensile Strength** \\  & & **(T)** & **(kA/m)** & **(MPa)** & **(MPa)** \\ \hline Magnets & N42 UH & 1.28 & 1.91 & - & - \\ Magnet holders & 7075 T6 & - & - & 434 & 506 \\ Bolts & BUMAX 109 & - & - & 900 & 1000 \\ Shims & CuZn37 & - & - & 130 & 330 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Mechanical material properties.  
Figure 4: Geometry of investigated structure: (**a**) assembly drawing of magnet holder; (**b**) geometry of magnet block.  
Brass shims are used to provide the required (certain) offset between the inner and outer frame. This offset depends on the magnitudes of magnetic forces acting on the individual magnets, and it enables additional compensation for manufacturing inaccuracies. The thickness of brass shims is assumed to be adjustable to allow for additional calibration of the magnet position. The thickness will be evaluated separately for each magnet after its measurements (i.e., magnet height and magnetization) and shimming-sorting calculations.  
The clamps (Figure 4a) are used to keep the magnet inside the inner frame. They are mounted through the use of M8 bolts made of non-magnetic steel BUMAX 109. The remaining bolts are also made of the same steel with the same dimensions but with a different pitch. The fixing screws are used to assemble and fix the inner frame to the outer frame. The positioning screws play a crucial role in positioning the magnet relative to the outer frame (and jaw). By tightening or loosening these screws, it is possible to change the deformation of the inner frame and as a consequence to change the position (vertical movement and angle of inclination) of the magnet (calibration, positioning, and alignment).