A collider spectrometer proposed to Gesellschaft fur Schwerionenforschung mbH (GSI)


ELISe homepage

Proposal of high-resolution eA collider spectrometer
Proposed at "GSI-ELISe workshop", Oct. 24-28, 2005, GSI, Darmstadt, Germany

Possible Spectrometer for eA Collider
Presented to the workshop "Rare-Isotope Physics at Storage Rings", Feb. 3-8, 2002 at Hirschegg, Kleinwalsertal, Austria

背景 (Introduction)

2002年頃、理化学研究所の不安定核加速計画には不安定核ビームを冷却し電子ビームとの衝突実験を可能にするクーラーリングの構想がありました。その加速器と測定装置の配置図を下に示します。図の右上の地下1階部分に"e-RI collider"と表示された装置について説明します。これは左上の地下2階部分の上にあります。

There was a collider ring project in RIKEN RI-beam facility. The layout of the project is shown below. Pay attention to the upper-right part indicated as "e-RI collider".

図1 当時の理研RIビーム実験室配置図案 (Fig.1 Experimental hall layout of RIKEN RIBF project)

2期計画の部分で"e-RI Collider"と表示された実験室で不安定核と電子の衝突による不安定核からの電子散乱実験が計画されていました。その散乱電子の測定装置として図ではとりあえずMAMI研究所のスペクトロメータと同じものを描いています(赤色)。

In the "e-RI Collider" hall, electron scattering experiments from unstable nuclei are planned. In the figure, the electron spectrometers (shown in red) is copied from the MAMI-C spectrometer, tentatively.


Standard high-resolution spectrometers are not available for collider experiments because we cannot stop the beam in the scattering chamber. We also have to take the spreading of the scattering points into account which is caused by the length of the RI beam (10-50 cm). A new idea of the spectrometer is required.

コライダースペクトロメータの提案 (A proposal of a collider spectrometer)


We propose a new spectrometer dedicated to the collider experiments.

図2 理研に提案したスペクトロメータ (Fig.2 Collider spectrometer proposed to RIKEN)


In the above figure, the radioactive beam (RI beam) comes from upper end of the quadrupole magnet and the electron beam from the lower end. These beams collide along the symmetric axis of the quadrupole magnet. The scattering source spreads because the length of the RI beam is supposed to be 10-50 cm. Electrons with momentum spread of 20 % and horizontal angular range between 100 mr and 300 mr (6 deg and 11 deg) are traced.

光学的特長(Optical properties)


The magnetic field of the quadrupole magnet diverge the electrons horizontally while converging electrons vertically. Electrons scattered to extreme forward angles can be separated from the beam line. They are extracted from the side face of the quadrupole magnet where the magnetic field is dipole-like. By the following dipole magnet, electrons are converged both horizontally and verticall, focusing on the momentum-dependent focal point (i.e. on the focal line). By tripple measurement of the electron location (once in front of the dipole magnet and twice behind it), we can obtain the information on the initial conditions of the scattered electron. Up to now, the simulated momentum resolution and vertical angle resolution are satisfactory, but those of scattering position and horizontal angle are insufficient. The angular ambiguity caused by the multiple scattering at the front counter has to be minimized.


For measuring electrons at backward angles, the collision region is shfted downstream of electron beam and the strength of the quadrupole magnet is lowered so that the electrons enter the dipole magnet with the same location and angle as the case of the forward-angle measurements. Even if the shift of the collision region is difficult, we can lead electrons to the same location of the dipole magnet with a little different angle. We can measure electrons between 6 and 40 degrees by three settings of magnetic fields.

特異な四極電磁石の構造(Peculiar structure of the quadrupole magnet)

図3 四極電磁石の磁場分布 (Fig.3 Magnetic field distribution of the quadrupole magnet)


Field lines of a good quadrupole magnet are hyperbolae (leftt half). The strength of the field should be proportional to the distance from the symmetrical center (right half). A set of equi-distant co-centric circles shows that the quality of the quadrupole field is good. In the present magnet, we use not only the central part, but also all the horizontal region where the magnetic field is not quadrupole-like.

GSIへの提案(The proposal to GSI)


I proposed above idea to RIKEN at the Hirschegg workshop held by RIKEN and GSI in 2002. After the workshop, RIKEN gave up the collider facility because of the financial difficulty. On the other hand, GSI has been developping the similar project. Last year, I was asked to join the project.

図4 GSIの将来計画 (Fig.4 Future project of GSI)

上の図で青く表示されている部分が既存部分で赤く表示されているところが計画中の部分です。右下の"NESR" と表示されているのが不安定核のクーラーリングです。その部分を拡大すると下のようになります。

The existing parts are shown blue and future parts red. The cooler ring of radioactive nuclei is indicated as "NESR" at the lower-right corner. It is shown in detail in the following figure.

図5 蓄積リング (Fig.5 Accumulation ring)


The electron ring is located in the RI ring in contrast to Fig. 4. Along the common straight section of the both rings, electrons collide with RI beam. I proposed a new spectrometer for the electron scattering in 2005. In the following part of this page, we modified the proposal so that the spectrometer can be inserted within the given length of the straight section (4 m).

図6 スペクトロメータの平面図 (Fig. 6 Collider spectrometer)


Electron beam comes upwards along the vertical line in the left part. It collides with RI beam at the entrance of the quadrupole magnet. Scattered electrons are analyzed by the quadrupole and dipole magnets. Scattering angle of electrons can be changed by changing the strength of the quadrupole magnet, covering the angular range of 15-50 degrees. The spectrometer is unique that it can measure angular dependence of cross sections (angular distributions) without moving the spectrometer nor incident beams. If there is enough space for longer quadrupole magnet, we can extend the angular range for the forward angles. The animation suggests that we can measure the angular distributions by changing the current of the quadrupole magnet automatically.

図7 四極電磁石の磁場分布 (Fig.7 Field distribution of the quadrupole magnet)


Above figure shows the structure and the magnetic field distribution of the quadrupole magnet. At the center a magnetic shield is put in order to eliminate the magnetic field around the symmetry axis completely. We don't need to worry how to compensate the effect of the quadrupole magnet. Instead, we lost the angular range at forward region.


We set a reference system with its origin at the symmetrical center and with its length in unit of dipole-like gap. We parametrize y component of the magnetic filed distribution along x-axis.

式1 磁場分布をパラメータ化する式 (Eq.1 Equations for the parametrization of the field distribution)

ここで B0, a1, b1, c1, d1, p1, q1, a2, b2, c2, d2, p2, q2 がパラメータです。図7の磁場計算のx軸上の値を再現するようにパラメータの値を探します。ここで h1(x) が左側の立ち上がりを、h2(x) が右側の立下りを表します。

Here, B0, a1, b1, c1, d1, p1, q1, a2, b2, c2, d2, p2, and q1 are parameters. They are searched for so as to reproduce the magnetic field distribution along the x-axis. h1(x) reproduces the left rize-up and h2(x) the right decay.

図8 磁場分布のパラメータ化 (Fig.8 Parametrization of the field distribution)


The result of the parametrization is shown in the above figure. With this result, we can derive the field distribution in two-dimensional space.

式2 2次元磁場分布を与える式 (Eq.2 Two-dimensional field distribution)

この方法は複素変数法と呼ばれ、ラプラス方程式の解に適用できます。またラプラス方程式の解は恒等的にゼロでない場合には必ず特異点があることも知られています。一様磁場からゼロ磁場までの変化を表現するために、長い間 h(x) の関数形として分母にxの多項式の指数関数を含む式(エンゲ関数)が用いられ、級数展開を有限次数までしか加算しなかったために、特異点の存在に気づきませんでした。複素変数法を用いるとエンゲ関数には無限の数の特異点が現れることが分かります。式1の場合、特異点の位置は (a1, b1), (c1,d1), (a2,b2), (c2,d2) の4つだけであることが分かっていますので、パラメータを求めるときに特異点の位置をコントロールすることができます。詳しくは関連論文を参照してください。

This method is called "complex variable method". It is useful for solving Laplace's equations. The solution is known to have singularities unless it is equal to zero everywhere. In order to express the strength change from the uniform region to the zero-field region, a function, called Enge function, which contains an exponential function of polynomials of x in its denominator, has long bee used. No attention has been payed for the occurence of the singularities because the sum of a series expansion was terminated at a finite order without using the complex-variable method. We can see that the Enge function causes infinite number of singularities. On the other hand, we have only four singularities at (a1, b1), (c1,d1), (a2,b2) and (c2,d2) in the case of Eq. 1. We can restrict the location of the singularities when we search for the parameters. For detail, refer to the other report.

図9 2次元磁場分布の比較 (Fig.9 Comparizon of two-dimensional field distributions)

この図の下半分は図7の関心部分を拡大したもので、上半分は式2によって計算したものです。等高線は 0.1 テスラ間隔で表示しました。また2.5テスラ以上の部分は込み合うので等高線を省略しました。4つの特異点のうちの3つが見えています(4つ目は右上の場外)。左にある特異点が関心領域に入りかけて気になりますが、磁気シールド内径の半分以下には近づかないように制限しています。図6の軌道追跡計算はこのようにして与えられた磁場を用いて行われました。

The lower half shows the interesed region of figure 7 in detail. The upper one shows the field strength distribution calculated by equation 2. The contour curves are given with 0.1 tesla steps. Three of four singularities are seen. The location of the left one was regulated so that it does not aproach the region of interest. The rays in figure 6 were calculated with such magnetic field.