Requirement for the D field mapping and Hall probe calibration

The D magnet's bending power dominates $\smallint\!B dl$ in the HKS magnets over Q1 and Q2, therefore the field map of D is the most crucial for the momentum resolution of the HKS. In order to estimate how good a field map is required for the D magnet, we performed a Monte Carlo simulation with artificially deteriorated field maps. The condition of the simulation can be summarized as follows:

• The kaons were uniformly generated in the momentum range of 1.05 GeV/$c$ to 1.35 GeV/$c$ and in the angle to the central ray of 0 to 6 degrees.
• The TOSCA calculated field map was used for the splitter magnet.
• HKS dipole field is given by the following empirical formula.
  $$\begin{eqnarray*} B_z &=& \frac{B_0}{h^n},\\ B_y &=& -B_0 \frac{Z N}{a G} \exp(\frac{s - X}{h^{n+1}}),\\ h &=& 1 + \exp(\frac{s - X}{a}). \end{eqnarray*}$$
  
  
  Here, $B_0$ is the maximum magnetic field (1.44 T), $G$ the gap length (20 cm), $s$ the distance from the iron boundary divided by $G$. Other parameters were given as $X = -0.1720, a = 0.1750, n = 0.2690$.
• In order to see the effect only due to a field map inaccuracy, we assumed the kaon beam transports in vacuum and the position resolution of the drift chamber is 20 $\mu$m.
• The function which associates the kaon momentum and chamber hit position was constructed by a fitting code based on the principal component analysis (ERIKA, see section3.2.4) of the standard events (without any artificial skewing of the magnetic field).
• To estimate mesh size effect, two mesh sizes (10 mm and 40 mm) were used.

We set the error caused by inaccuracy of the field map to be less than 50 keV/$c$ which is smaller than the other factors. The results of the simulation are: 1) In order to achieve 50 keV/$c$ momentum resolution, the position resolutions of 700 $\mu$m (rms) and 300$\mu$m (rms) are required respectively for the 10 mm mesh and 40 mm mesh. 2) the accuracy of the measured field is qualitatively expected to affect directly the accuracy of the momentum resolution, since it affects directly $BL$; the simulation shows this trend. In order to achieve 50 keV/$c$ momentum resolution , a field accuracy of $<1 \times 10^{-3}$ and $<2 \times 10^{-3}$ are required for the 10 mm mesh and 40 mm mesh, respectively.

The requirements for the field map can be summarized as:

• Position resolution is $<300 \mu$m,
• Field accuracy is $<1 \times 10^{-3}$,
• measuring mesh size is $<40$ mm.

In the dipole magnet, the vertical component of the field is much larger than others, and thus it is essential to know the direction of the probe and planar Hall effect if we use a Hall probe. Furthermore, the D magnet is so large compared to Q1 and Q2 that the mapper used for Qs cannot be used for D. Therefore, we developed a tri-axial Hall probe system with a 3D field mapper.

Three GMW Group3 Hall probes (model DTM-151, accuracy $1 \times 10^{-4}$, temperature stability $\pm10$ppm) were mounted on an aluminum cube which has three mirrors (parallelism 5''). The directions of the facets were measured by a laser beam reflected by those mirrors.

As shown in figure 33, the tri-axial Hall probe was calibrated and planar Hall effect was measured using a well-known dipole field (KEK 8D320 magnet). Two Hall probes' (Bx, By) responses are naturally described by trigonometric functions of rotation angle ($\phi$). The remain probe (Bz) is expected to have no dependence on $\phi$, but a few percent effect with a $2\phi$ component is observed. This is planar Hall effect and it is essential to calibrate this effect for our purpose. The stability of the magnetic field was monitored by a NMR probe and the absolute readout values of the Hall probes were calibrated with this NMR probe.