Some effects of the transverse-stability requirement on the design of a grating linac Page: 6 of 8
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Eqs (14) and (IS) are linear differential equations with periodic
coefficients, and can be solved by the standard matrix method^. The
discission of the stability of the motion involves two parts: First, Eq (14)
is inhomogenious. Therefore, we have to find a periodic solution to define
the equilibrium orbit. If the equilibrium orbit deviates too much from x » 0,
the particle will either hit the grating surface, while if it deviates too
much from x « 0 the acceleration becomes small. Second, we have to consider
the stability of the homogeneous betatron motion around the equilibrium
orbit. We consider the homogeneous part first.
Tne betatron motion is stable if
| cos ux | . < 1, | cos py | < 1, (17)
where
cos ux - cosh eg cos 6] - j (^1_ - ^£) sinh e2 sin e^,
^2 ql t18)
cos u = cosh e1 cos e2 + j (fl - ^2) sinh e1 sin e2 .
q2 d]
Here we have defined
6i = qi L/2 (19)
in the above, q-j and could be imaginary if a < 0. The above formulae
still apply with the replacement sin i e - i sinh e, etc. In addition to
(18), we must require
-x/2 < ± A/2 + <t>0 < x/2, (20)
since otherwise the particle will be decelerated.
The inequalities (18) and (20) are analyzed numerically to find the
stable region of the parameter space. It is convenient to present the results
in terms of a, <t>0 and
KL=/l JL sin (a/2) (21)
We wish to determine the range of <}>0 corresponding to stable motion for
given and a. We find
(i) for all values of a, the maximum phase acceptance (range of d>0) is
obtained near 41* x,
(ii) setting^ x, the value a = .7 x corresponds to the maximum phase
acceptance | lt,0 | $ . 11 x,
(iii) for comparison, the phase acceptance for the case 'i = x, a = .5 *
is |<t 0 | « .06 x.
If the injected beam is uniformly distributed in phase, case (ii) accepts
11 percent of the beam while case (iii) accepts only 6 percent. The ratio of
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Kim, K. J. & Kroll, N. M. Some effects of the transverse-stability requirement on the design of a grating linac, article, April 1, 1982; [Berkeley,] California. (https://digital.library.unt.edu/ark:/67531/metadc1066578/m1/6/: accessed July 16, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.