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11.3 Comparisons between dipole measurements and Astra modelization

11.3.1 Modelization of dipoles in Astra

In Astra, dipole magnetic fields are modelized as follow :

$$B_y = \displaystyle\frac{B_0}{1+\exp{ \frac{d(s)}{ 1.5 g}}}$$ (11.8)

where $d(s)$ is the distance to the dipole edge, and $g$ is the gap between the pole tips. The field map for the trapezoidal and parallelogram dipoles of A0 are plotted on figure 11.5.

Figure 11.5: Magnetic field map as modelized by Astra for the dipoles of A0. (a) trapezoidal dipole - (b) : Parallelogram dipole. The field plotted here is $B_n$, the transverse magnetic field $B_y$ normalised so that it's maximum value is one. On black solid line, the hard edges are plotted. The dashed line are at the distance 1.5$g$ from the hard edges. At first glance, one can see that important differences are present between the modelization and the experimental fields presented on figure 11.4

(a) (b)

.

11.3.2 Comparison of the different fields

We are still studying here the trapezoidal dipole of A0. For this dipole, the beam is suppose to enter at $x=-30$mm. We thus plot the different fields we have along this line. Of course, the beam will follow a curve inside the dipole, but for the fields comparison, we will be satisfied by studying the fields along the line $x=-30$mm. For experimental files, the data available are at $x=-25.4$mm. Fields from the three sets of measurements, and fields modelized by astra are presented on figure 11.6.

Figure 11.6: Comparison between the measurement fields and astra modelized fields. The three measurement data are one above the other, showing the consistency of the measurements that were done with several years of interval and with several dipoles (TDA006 and TDA011). With good eye, one can see, that only the data from exp3 goes far on each side. Nevertheless, no need to have good eye to notice that Astra fields are far from the measurement fields.

We wee here that the modelization of the field(fringe field) is bad in Astra. We know that the integral $\int B dl$, almost entirely determine the trajectory of the beam in the dipole. As a result of this, one can adjust the amplitude of the field in astra, or play with the gap paramter to have tp

11.3.3 Fit with Henge's field fall-off model

The measured fields from exp3 have been fitted with several Henge's fringe field model(see section 11.1.3). Henge model is function of a polynome. We fitted the measurement data for Henge fields with polynomes of degree one, three and five. In a sense, Astra corresponds to a Henge model with a polynome of degree 1, without any constant. Nevertheless, the Henge field for a polynome of degree one that fits best the measured field is far from Astra fields. The different fitted fields are plotted on figure 11.7. To compare the different fits, the integral of the field has been calculated, and compared to the integral for the measured field. If $B_$fn is a fitted field, and $B_m$ is the measured field, we define the ratio $R$ as:

$$R=\displaystyle\frac{\int B_\text{fn}(z) dz}{\int B_m(z)dz}$$ (11.9)

note that here we don't integrate on the trajectory of the particle, but on a straight line defined by $x=-0.03$m

Figure 11.7: Fitting the measured fields with a fringe field model. Note that the origin for the z axis is taken here at the edge of the dipole. For polynoms of degree above 3, the Henge fringe fields model can fit well any experimental fringe field. Indeed for $n=5$, the integral ratio $R$ show that the two integral differs from one percent. This figure clearly show us the need to implement a better field modelization in Astra.

From figure 11.7 we clearly see that one needs to implement a better field modelization in Astra. The other solution for the simulation, is to increase the field amplitude, of the current by a factor of 1.5(=1/R), or reduce the radius by a factor of 0.64 (=R). The underestimation of about 0.64 explains why the radius obtained by Astra when fitting the geometry of a0 or NML are smaller from a factor 1.5 compare to the real radius (see table 12.1 and 13.1).

Emmanuel Branlard
http://emmanuel.branlard.free.fr/