# A Large-APERTURE NARROW QUADRUPOLE FOR THE SNS ACCUMULATOR RING\*

N. Tsoupas, J. Brodowski, W. Meng, J. Wei, Y.Y. Lee, J. Tuozzolo, BNL,
Upton, NY 11973 USA

Abstract

The accumulator ring of the Spallation Neutron Source (SNS) \[1\] is
designed to accept high-intensity H^-^ beam of 1 GeV kinetic energy from
the injecting LINAC, and to accumulate, in a time interval of 1 msec,
2x10^14^ protons in a single bunch of 700 nsec. In order to optimize the
effective straight-section spaces for beam-injection\[2\],
extraction\[3\] and collimation\[4\], we have minimized the width of the
large aperture quadrupoles which are located in the same straight
sections of the accumulator ring with the injection and extraction
systems. By minimizing the width of the quadrupoles to ±40.4 cm, the
beam-injection and extraction angles are lowered to 8.75^o^ and 16.8^o^
respectively. Further optimization of the narrow quadrupole, minimizes
the strength of the dodecapole multipole component of the quadrupole,
thus reducing the width of the 12pole structure resonance and allowing a
larger tune space for stability of the circulating beam\[5\]. In this
paper we present results derived from magnetic field calculations of 2D
and 3D modeling, and discuss the method of optimizing the size of the
quadrupole and minimizing its dodecapole multipole component.

## 1 WHY USE NARROW QUADRUPOLE?

The SNS accumulator ring\[1\], schematically shown in Fig.1, is 248 m in
length and consists of four identical arcs of 90^o^ each. The arcs are
connected by four straight sections, each 31.5m in length. As part of
the ring's beam optics a set of two quadrupole-doublets are located
along each of the straight sections (see Fig. 1), and symmetrically
placed with respect to the center of each straight section. The presence
of the quadrupole doublets in the straight sections of the ring where
the beam-injection and beam-extraction systems are located (see Fig. 1),
suggests that the dimensions of the quadrupoles determine the various
parameters of the magnets associated with the beam-injection and
extraction systems. A more detailed view of the injection region in Fig.
2, shows the quadrupole doublets, the injection septum, and the four
chicane dipole magnets which generate the local orbit bump\[2\]
necessary to displace the closed orbit at the location of the stripping
foil. In order to minimize the magnetic-field-stripping of the H^-^ beam
and the H^0^ beam[^1] \[7\], the chicane magnets should be excited at
the lowest possible magnetic fields. Low excitation fields of the
chicane magnets requires smaller size of the local orbit bump (x~cod~)
shown in Fig. 1.

The size of (x~cod~) of the local bump can be reduced by moving the
injection septum magnet closer to the ring by reducing the transverse
dimensions (W~quad~) of the quadrupoles which otherwise would interfere.
Similarly the quadrupole doublets which are located at the straight
section where the beam-extraction system is located, interfere laterally
with the extracted beam as shown in Fig. 1. Reduction of the quadrupole
width (W~quad~) reduces the beam extraction angle and allows lower
B-field of the septum magnet, thus minimizing the effect of the septum
magnet's fringing fields on the circulating beam.

![](media/image1.png){width="2.4493055555555556in"
height="1.9673611111111111in"}Figure1:Schematic diagram of the SNS
accumulator ring.

![](media/image2.png){width="3.24375in" height="1.3777777777777778in"}

Figure 2: Schematic diagram of the injection region.

## 2 REQUIREMENTS OF THE NARROW QUAD

The cross-section of the narrow quadrupole is shown in Fig. 3.
(ironblue, coilsred). The dimensions of the narrow quadrupole magnets,
which appear in Table 1, were determined from a) the size of the beam at
the location of the quadrupoles, b) the allowed uncontrolled beam-losses
along the ring\[4\] and c) the magnetic-field requirements of the
quadrupoles. The length, the width, and the pole-width (columns 3,4,5
Table 1) of the quad have been chosen to minimize the effects of the
magnetic field saturation on the magnet and thus maintain the same field
quality for 1.0 and 1.3 GeV operations. The transverse width of ±40.4 cm
of the quadrupoles allows for 8.75^o^ and 16.8^o^ beam-injection and
extraction angles. The fringe field of the quadrupole at x\<-40.4 cm,
and x\>40.4 cm and \~5 cm away from the iron must be small enough[^2]
not to affect the injected and extracted beams that pass by the
quadrupoles.

Figure 3: Cross section of the narrow quadrupole.

The B-field gradient appears in column 6 and the field non-uniformity,
expressed as the ratio B~dod~/B~quad~ of the dodecapole field to the
quadrupole field at a r=10 cm should be ≤2x10^-4^ ,appears in culumn 7.
This limit keeps the width of the 12pole resonance\[5\] from infringing
into the tune spread of the beam thus allowing larger tune space for
beam stability of the circulating beam.

Table 1: Requirements of the narrow quadrupoles.

+----------+------+------+-------+-------+-------------+--------------+
| Name     | R cm | L cm | W cm  | PW cm | G           | B~           |
|          |      |      |       |       |             | dod~/B~quad~ |
|          |      |      |       |       | \[T/m\]     |              |
+----------+------+------+-------+-------+-------------+--------------+
| 30Q44    | 15   | 44   | 80.8  | 17.8  | 4.2 , 5.1   | ≤2x10^-4^    |
+----------+------+------+-------+-------+-------------+--------------+
| 30Q58    | 15   | 58   | 80.8  | 17.8  | 4.2 , 5.1   | ≤2x10^-4^    |
+----------+------+------+-------+-------+-------------+--------------+

### **3 MODELLING OF THE NARROW QUAD**

The magnetic modeling of the narrow-Quad was performed to determine a)
the contour of the pole face, and b) the minimum dimensions of the width
and pole-width of the magnet, that satisfy the field requirements. The
commercially available computer code "opera" of Vector-Fields Inc. \[6\]
was used for the modeling of the magnet and the calculations were
performed in two steps; first 2D-modeling and second 3D modeling.

### 3.1 2D Modeling

The 2D modeling establishes the contour of the pole-face

![](media/image3.png){width="2.9902777777777776in"
height="3.3569444444444443in"}that minimizes the dodecapole
multipole[^3]. The contour of the pole face that creates the multipoles
shown in Table 2 is very simple and is described in \[8\]. The 2D
modeling establishes also the minimum allowed width of the iron-pole
(17.8 cm) that minimizes the magnetic saturation effects in the iron.
Indeed Table 2 shows that the multipoles do not vary linearly with the
quadrupole strength because of the nonlinear permeability of the iron.
However the strength of the multipoles shown in Table 2, for the
excitation energies of 1.0 and 1.3 GeV operations is too small to affect
the field quality of the magnet. The ratio B~dod~/B~quad~ at r=10 cm is
≤2x10^-4^ as required.

Table 2. The strength of the allowed multipoles[^4] b~n~.

+--------+----------+------------+----------+-----------+------------+
| T GeV  | b~       | b~3\ oct~  | b        | b~7       | b~         |
|        | 1\ quad~ |            | ~5\ dod~ | \ 16pole~ | 9\ 20pole~ |
|        |          | Tm^-3^     |          |           |            |
|        | Tm^-1^   |            | Tm^-5^   | Tm^-7^    | T\*m^-9^   |
+--------+----------+------------+----------+-----------+------------+
| 1.0    | 4.236    | -          | 2.20     | -0.30     | -2.6       |
|        |          | 1.1x10^-4^ |          |           | x10^5^     |
+--------+----------+------------+----------+-----------+------------+
| 1.3    | 5.084    | -          | -0.22    | -0.67     | -3.1x10^5^ |
|        |          | 2.2x10^-4^ |          |           |            |
+--------+----------+------------+----------+-----------+------------+

The absolute value of the magnetic field B~mod~ in the iron, shown in
Fig.4, is below the value of 16 kGauss.

![](media/image4.png){width="2.6173611111111112in"
height="2.3673611111111112in"}

Figure 4: Contour plot of the B~mod~ in the iron.

3.2 3D Modeling

The 3D modeling provides complete information about the magnetic field
of the magnet in 3D space and helps further in the optimization of the
shape of the magnet which should satisfy all the magnetic field
requirements. The 3D model of a 44 cm long quadrupole was generated by
using the same cross section as the cross section of an optimized
quadrupole from the 2D calculations described earlier. The 3D magnetic
field calculations have been performed using the TOSCA module of the
opera computer code\[6\]. The aim of the 3D calculations is to minimize
the relative strength[^5] of the allowed multipoles of the magnet by
shaping the ends of the pole-pieces. The results from the 3D
calculations of the 3D models were analyzed by calculating the radial
field B~r~(r,z) on a radius r=10 cm, and expanding it as in the
expansion (1) below:

B~r~(r,z) =∑b~n~(r,z)cos\[(n+1)θ\] {n=1 quad n=3 oct ...} (1)

Subsequently the amplitude of the expansion coefficients that correspond
to the various multipoles were compared among the various 3D models.

In particular, the calculated coefficient of expansion b~5~(r,z) which
is a measure of the strength of the dodecapole multipole is plotted as a
function of the distance z from the centre of the quadrupole in Fig. 5
for two 3D models a) No "pole-piece chamfer"[^6] and b) with 30.25^o^
pole-piece chamfer[^7] (see Fig. 6).

Figure 5: Plots of the expansion coefficient
b~5~(r=10,z![](media/image5.pct){width="2.9930555555555554in"
height="2.463888888888889in"}).

The variation of the b~5~ coefficient plotted in Fig. 5 is due to the
end effects of the magnet. In order to set a basis for comparison with
the experimentally measured quantities, we calculated the values of the
ratios

R~n~= ∫b~n~(r,z)dz/∫b~1~(r,z)dz where ∫b~n~(r,z)dz is the amplitude of
the radial field b~n~(r,z) of the n^th^ multipole, integrated along the
z-direction at r=10 cm.

Figure 6. An isometric view of the chamfered end of one of the
pole-piece of the quadrupole. A chamfer of 30.25^o^ minimizes the value
of ∫b~5~(r,z)dz.

Table 3 contains the values of the calculated ratios R~n~ for the
various multipoles. All the value of the ratios R~n~ are well below the
limit of 2x10^-4^ except the value of R~5~ ratio[^8] at 1.3 GeV. The
value of the effective quadrupole length L~eff~ of the magnet is shown
in column 5 for 1.0 and 1.3 GeV excitations.

Table 3: The Ratio R~n~ of few allowed multipoles.

  -------------- ------------ -------------- -------------- -----------------
  T GeV          R~3\ oct~    R~5\ dod~      R~7\ 16pole~   L~eff~ \[cm\]

  1.0            -8x10^-6^    6x10^-5^       -2x10^-7^      53.3

  1.3            -4x10^-5^    5x10^-4^       -2x10^-6^      53.1
  -------------- ------------ -------------- -------------- -----------------

## 4 MECHANICAL TOLERANCES

The 3D modeling also provides an upper limit in the mechanical
tolerances within which the magnet still satisfies the field
requirements, thus avoiding "over or under"-design of the magnet. The
mechanical tolerances were studied in a full 3D model by introducing
possible mechanical errors in the optimized 3D model.

Among the possible mechanical errors studied we report:

a)  An error of 0.5 mm in the longitudinal length of one of the poles of
    the magnet.

b)  Rotation of a single pole-face by 1.5 mrad about an axis parallel to
    the symmetry axis of the quadupole and passing through the pole tip.

In both cases (a) and (b) above the variation of the ratio R~5~ was
\~10% of the value shown in Table 3, with the rest of the ratios
maintaining still small values as those in Table 3.

## 5 CONCLUSIONS

![](media/image6.png){width="2.9583333333333335in"
height="2.359722222222222in"}Magnetic field calculations on 2D and 3D
models of a large aperture narrow quadrupole provided optimum pole shape
and dimensions for an SNS ring quadrupole which satisfies the field
requirements of the SNS ring.

Upper limits on the mechanical tolerances were obtained by introducing
mechanical errors on a full 3D-model.

## 6 References

\[1\] J. Wei, et. al. Phys. Rev. ST AB **3**, 080101 (2000)

\[2\] J. Wei, et. al. EPAC2000, p. 2560 Vienna, Austria

\[3\] N. Tsoupas et al. EPAC2000 p. 2270 Vienna, Austria

\[4\] Catalan-Lasheras et al. Phys. Rev. ST-AB **4**, 010101

\[5\] A. Fedotov et al. EPAC2000 p. 1492 Vienna, Austria

\[6\] Vector-Fields-Inc.

\[7\] D. Abell, et al. EPAC2000 p. 2107 Vienna, Austria

\[8\] N. Tsoupas et al. BNL SNS Tech. Note 110

[^1]: The H^0^ beam is produced by the incomplete stripping of the H^-^
    beam by the stripping foil.

[^2]: For 1.0 GeV operation the fringe field is \~5 Gauss

[^3]: The symmetry of this "narrow-quad" is not four-fold symmetric
    therefore the dodecapole multipole is not the first allowed
    multipole (Table 2). However the strength of the allowed multipoles
    introduced due to the break of the four fold symmetry of the quad
    are negligible as shown in Table 2.

[^4]: The multipoles b~n~ have been calculated from the expansion of the
    radial field B~r~= Σb~n~r^n^cos\[(n+1)θ\] {n=0 Dip, n=1 Quad...} at
    r=15 cm.

[^5]: Relative to the strength of the quadrupole multipole.

[^6]: The cross section of the pole piece of the 3D model is identical
    to the cross section of the 2D model.

[^7]: The ends of the pole pieces have been chamfered by 30.25^o^. The
    chamfer starts at 6 cm from the edge of the pole face.

[^8]: Calculations show that an alternative model with a ratio
    R~5~\~2x10^-4^ at 1.0 GeV generates a ratio R~5~\~3x10^-4^ at 1.3
    GeV excitation. However we have chosen to use the model that
    generates the R~n~ shown in Table 3.