## 3 Linear Accelerators at GSI
### UNILAC as a Heavy Ion High Current Linac
#### 3.1.1 Beam Matching to the RFQ  
Beam matching to the HSI-RFQ acceptance was performed by a Quadrupole Quartet (QQ), comprising four magnetic quadrupole lenses. Due to the short distance between the lenses and the relatively high aperture, an overlapping of the magnetic field from neighboring quadrupoles has a strong influence on the beam dynamics. Additionally, high magnetic gradients, being applied to a large beam spot, provide for a significant non-linear deformation of the 6D beam phase space. The absence of the beam diagnostics between QQ and RFQ allows for the optimization procedure only for the whole frontend. The used set of gradients was found experimentally in order to provide for the maximum beam current behind the HSI-RFQ.  
The standard envelope codes for beam line optimization (with separately described quadrupoles) do not take an overlapping of magnetic field from neighboring quadrupoles into account. Also, the use of the paraxial approximation and matrix approach leads to a simplified description of the 3D particle motion in the magnetic field. Therefore, such codes can only provide for roughly estimated matching settings for the quadrupole lenses.  
Usually only one four-parametric matched solution for the beamline settings is obtained by the use of a standard envelope software, while general theory predicts four solutions for a common case of four variable elements. As the polarity of the quadrupoles is technically fixed, the four focusing-defocusing elements might provide also two independent matching solutions.  
Generally, the DYNAMION code is not foreseen for a typical many-elements beam line optimization. Nevertheless, with a slight adaptation DYNAMION can solve such specific problems by means of massive beam dynamics simulations (in automatic mode) with the Monte Carlo method for the magnetic gradients of four quadrupoles. A dedicated set of beam dynamics simulations was performed, randomly varying four quadrupole gradients in the full range (0-12 T/m). The distribution of the magnetic field has been measured for each quadrupole separately and was introduced into the DYNAMION code as input data. An overlapping of the field for the whole quadrupole quartet has been calculated automatically inside the code, in accordance with the random quadrupole settings for each run. The input particle distribution, generated from the emittance measurement data in front of the QQ, has been used as well.  
As expected, two different QQ settings, leading to a high particle transmission for the system, have been found. One solution represents well the gradients for all four quadrupoles, which have been found experimentally during HSI-RFQ recommissioning in 2009. However, newly found QQ settings have been experimentally confirmed for different ions and beam intensities. For the low current argon beam, a 100% particle transmission for the entire HSI (LEBT, RFQ, MEBT and two IH-DTL) has been reached for the first time since commissioning in 1999, while previously achieved particle transmission did not exceed 85%. Also, a remarkable improvement for the HSI transmission was demonstrated experimentally for medium current heavy ion beams, in particular for a 4 mA Ta\({}^{4+}\) ion beam [30]. Moreover, the corresponding DYNAMION beam dynamics simulations withthe new QQ gradients showed after HSI-RFQ an about twofold increase of the transverse beam brilliance and a 60% higher longitudinal one. Mostly this result is explained by a significantly lower nonlinear deformation of the beam phase shape inside the QQ with newly found lower quadrupole gradients. Therefore, a better beam matching to the HSI-RFQ in turn leads to the correspondingly lower emittance growth along the RFQ channel.