Beam Based Alignment (BBA) is widely used in synchrotron light sources and other accelerators. By correlating the magnetic centre of a quadrupole or sextupole with a nearby beam position monitor via the beam, the true centre of the BPM can be physically established to high precision. It is the only technique available with such precision, and is therefore important for accelerators needing very accurate beam alignment, such as light sources and colliders.
To apply a BBA system, the current through a given magnet is perturbed and any resulting shift of the beam orbit is observed. An orbit shift implies that the beam does not pass through the centre of the perturbed magnet. The beam can be aligned through both the centre of the magnet and an adjacent BPM by adjusting the position of the orbit in the magnet until no orbit shift is observed. A BBA system can also be used to measure the amplitude of the lattice functions by measuring the tune shifts resulting from magnet perturbation.
Theory:
In storage ring, if there is a kick variation ∆θ at the position s the position variation at s¢ can be expressed as
(1)
Where z denotes the horizontal or vertical position and β, ψ, and ν are corresponding beta function, phase, and tune. If the beam passes a quadrupole with an offset, the kick to the beam is changed when the quadrupole current is changed. If we denote the offset by ξ the kick variation at the quadrupole can be expressed as
∆θ = ξ∆kL (2)
Here, ∆k and L are the field gradient change and the length of the quadrupole. Different from the case when the field strength of a corrector magnet is changed, when we change that of the quadrupole the Twiss parameters also change and the Equation 1 does not hold exactly. However, if we reasonably assume that the offsets are small enough, we can combine Equations 1 and 2 to get
(3)
Here we use the subscripts i and j instead of s and s¢. The i represents the ith quadrupole and j represents the jth BPM.
Measurements:
If we express Equation 3 as
∆zj = Mijξi (4)
We can find the offsets; ξ’s, by measuring the position changes ∆z’s and matrix Mij, and using a non-square matrix inversion algorithm. However, this process will give the reliable result only under the assumption that all BPMs are working equally well. Even though we carefully eliminated a number of unbelievable BPMs in the measuring process, still there is a possibility that some BPMs could work abnormally at some specific measurements. So, we obtained the offsets by directly comparing the measured orbit distortions with the simulated ones using the assumed offsets. The measured orbit distortions are direct BPM reading variations due to the quadrupole field strength variations.
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