# Cylindric electron envelope for relativistic electron cooling Page: 4 of 6

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cos () 0 - sin 0 cos 2 0 sin 0 0

M(s s) = 0 cos ( ) 0 -sin (- cos 2 0 0

sin 0 cos 0 0 0 cos 2 20 sin

0 sin( 0 cos ()0 0 - cos

Let the optic functions at the first scraper be chosen for unknowns to be found. Together with the

emittance, this gives 7 unknown values. The optic functions at any positions are expressed through the

initial optic functions, see Eq. (5). From another side, the total number of equations is 3Ns : three ellipse

parameters are known at every scraper position, and these parameters relate to the optic functions according

to Eqs. (7). Number of explored scrapers has to be sufficient for the number of equations to exceed the

number of unknowns, 3Ns > 7 or N, > 3.

4 Optics Correction

After measuring the cross-sections and calculating the optic functions, they are seen as different from their

design values. The entire problem grows from the fact that the model of the beam optics is not good enough.

At best, it is only approximately correct, and the errors are expected to be higher than the tolerances. Let

it be assumed, first, that there is no reliable optic model at all. Could the designed cylindric envelope be

yet established in this case?

Let F = (//,a, /3y//3o, ay, u/0.5, 2v/7)T be a 6D vector of the found optic functions at the first

scraper, Fo = (1, 0, 1, 0, 1, 1)T be its design value, and dF - F - Fo. Let it be assumed now that there are

6 "linear independent" optic elements upstream the solenoid; the field magnitude in every one of them is

proportional to its supply current, so that the 6D vector H describes all of them. A term "linear independent"

means here that an optic functions response matrix

O - OF/BE (13)

is not degenerate, Rl 0.The response matrix R can be found in the same way, as the optic functions were:

it just requires to repeat the entire sequence of the envelope measurements and calculations 6 more times.

Let it be assumed now that this matrix is measured. If deviations of the optic functions from their design

values are small enough, a change of the currents

dE= -- dF (14)

establishes the design values of the optic functions. In practice, however, the deviations of the optic functions

might be not so small. If so, the linear correction (14) should be expected to bring us just closer to the design

values, not necessarily right on them. If so, then the functions have to be measured one more time, and

one more correction (14) applied. After some number of iterations, the design values should be established.

Note that during these iterations the response matrix R is not required to be measured as many times as

the functions F; normally, it should be sufficient to measure it once or twice.

The described procedure should work, but it has a significant practical drawback. Namely, the entire

set of cross-section measurements has to be multiply repeated here, which would take significant time. The

problem is exacerbated by the fact that these electron measurements are not compatible with the pbars: any

pbars circulating in the storage ring (Recycler) would die on the narrow aperture of the scrapers. Remember

now that this procedure does not use any model of the beam line optics: all that is needed is taken from the

measurements right away. Actually, this is not necessary and can be improved.

Indeed, envelope simulations require two things: first, the "initial conditions" of the basis vector (4),

or its value somewhere upstream of all the six tuning optical elements, and rather good knowledge of the4

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Burov, A.; Lebedev, V. & /Fermilab. Cylindric electron envelope for relativistic electron cooling, report, February 1, 2005; Batavia, Illinois. (https://digital.library.unt.edu/ark:/67531/metadc1409626/m1/4/: accessed April 23, 2019), University of North Texas Libraries, Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.