# Self-Consistent Computation of Electromagnetic Fields and Phase Space Densities for Particles on Curved Planar Orbits Page: 3 of 5

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Figure 2: Plan for B integration.

s tsecond step the variables s and v are interchanged making

s the new independent variable and in the final step z :=

s - 3ru replaces v as a dependent variable and pz := ( -

7,)/7, replaces ps. (11) defines s = s(R) and x = x(R)

so that z = z(R, u) = s(R) - ,/u and we have the identity

R - RT(z(R, u) + ),u) + x(R)n(z(R, u) + ),u). Now z

is small in the bunch and expanding for small z gives R

RrO(rv) + M(j3Tu)r + O(r(z2 + xz) and we obtain ob-

tain the approximate inverse r = MT (,u) (R -RO,u)).

Here M(s) = (t(s), n(s)) and r = (z, x)T.

The equations of motion in (z, x, pz, p; s) have the

fields 8(R, u) evaluated at R = R,(s)) + xn(s) and

u = (s - z)/),. We have the following approximations

8(R,(s) + xn(s), (s - z)/),) 8 S(R,(s + z) + xn(s +

z), s//h) E(R(O,3u) + M( u)r, s). At the first ap-

proximation we use the fact that the fields are slowly vary-

ing in s for fixed r. The second approximation uses the fact

that ), j 1 and we are only interested in the fields in the

bunch for r small. We obtainz -n(s)x pz Fzi(R, s) +pxFz2(R, s)

X' - pX p' _= (s)pz + Fx(Rs),(13)

where R := Rn(s) + M(s)r and' = d/ds. The self-forces

are given approximately byFzi = P Eli * t(s),

Fz2 = -EI n(s)

P c ()Y

Figure 3: Beam frame coordinates.

The beam frame is defined in terms of the reference or-

bit using the Frenet-Serret coordinates (s, x), where s is the

arc length along the reference orbit and x is the perpendic-

ular distance along n from the orbit at RT (s) as shown in

Fig. 3. Recall the reference orbit for a bunch compressor

in Fig. 1.

The transformation (Z, X) to (s, x) is

R = R, (s) + xn(s), (11)

where R,(s) (Z,(s), X,(s)) and the unit normal vec-

tor n(s) :- (-X'(s), Z'(s)). The corresponding tangent

vector is t(s) R,(s) (Z'(s), X'(s)). In addition,

we define ps and px by P : PT(pst(s) + pyn(s)), where

PT- miyr#c is the momentum of the reference particle.

Finally, we define the curvature K(s) by n'(s) n(s)t(s)

and it follows that t'(s) -n(s)n(s). In terms of Fig. 1

this makes K negative in the first magnet, positive in the

second and so on.

Our lab to beam transformation has three steps:

(Z, X, Pz, Px; u) - (s, x, ps, pX; U) (12)

- (U, x, Ps, P; s) - (z, x, pz,pr; s).

The first step is the transformation just discussed, in theFx = q (-EzX'(s) + ExZ'(s) - cBy), (14)

P4c

where (Ez, Ex, By) are evaluated at (R, s). We have ex-

panded Fx in order to point out that each of the last two

terms are large whereas their difference is small.

The equations of motion (13), without the self fields,

have been linearized. They can be solved and the solution

written ( 4)(s)(o, where ( (z, x, pz, px)T. Here 4) is

the principal solution matrix which is defined in terms of

the dispersion function, D(s), and R56(s) (see [2]). The

equations of motion in the interaction picture become(o 4)-1(s)(0, F, 0, F) T (s, (o).

(15)

4)(s) varies more slowly than the self-forces and so we nu-

merically solve these rather than (13). A larger time step,

which is controlled entirely by the self-forces, can be used.

Our field formula is in the lab frame so the lab charge

and current densities must be determined from the beam

frame PSD. The relation between lab and beam PSDs is4'L(Z,X,PZ,PX;u) =#fB(zxpz,p2; s).

r2

This leads to

pL (R; u) f dpzdp fB =: PB(r; s),(16)

(17)

JL(R; u) 0, c[pB(z, x; s)t(s) + T(z, x; s)n(s)], (18)

where r(z, x; s) = f p~fB(z, x, pZ, pX; s)dpzdpz. Us-

ing the fact that fB(z, x, pz, pX; -) is slowly varyingx

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Ellison, J.A.; Bassi, G.; Heinemann, K.A.; U., /New Mexico; Venturini, M.; /LBL, Berkeley et al. Self-Consistent Computation of Electromagnetic Fields and Phase Space Densities for Particles on Curved Planar Orbits, article, November 2, 2007; [Menlo Park, California]. (digital.library.unt.edu/ark:/67531/metadc888527/m1/3/: accessed December 11, 2018), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.