STUDIES OF ENERGY RECOVERY LINACS AT ... - CASA

tics could not be loaded. Because the quadrupoles were running at strengths
(Emax/Einj) = 51 times smaller than in standard CEBAF operations, scanning the
quadrupole by tens of Gauss has a negligible effect on the emittance measurement
- despite not producing compensatory optics. An expression for the relative change
in the beta functions due to a focusing error is given by [42]
δβ
βo
= ∓ βi
f
47
sin(2∆ψ) (2.15)
where βi is the initial, unperturbed beta function, ∆ψ is the phase advance for a
single revolution in the machine starting from the location of βi, f is the focal length
of the focusing error and the upper (lower) sign applies to the horizontal (vertical)
plane. To first order the betatron phase advances can be determined from the model
optics and are 60 ◦ and 145 ◦ for the vertical and horizontal planes, respectively.
Assuming a focusing error of 150 G (the extent to which each quadrupole was
scanned) gives
δβ
βo
δβ
βo
= −βi
= βi
150
sin(2 · 145
33.365 · 1020
◦ ) βi(0.004) (2.16)
150
sin(2 · 60
33.365 · 1020
◦ ) βi(0.004) (2.17)
where Eq. (2.16) corresponds to the horizontal plane and Eq. (2.17) corresponds
to the vertical plane. For the first pass beam with an initial unperturbed beta
function of 100 m, the result of the focusing perturbation caused by not producing
compensating optics leads to a contribution of 40 cm to the beta function on the
second pass. Because the beta functions are on the order of 35 m, the contribution
is approximately 1% and can safely be ignored.
Other sources of error include errors in the magnet-to-harp distance, magnet
power supplies, quadrupole excitation calibration, and beam energy. However, the