Transverse Beam Profile Measurement Using Scrape Scans Page: 3 of 9
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Pbar Note 665
Combining equations (2) and (7) gives:
Yi = a cos '
3. = --a sin n (8)
where 4n = p(s)- po +27nv . The trajectory of a particle in (q, ) coordinates is a circle of
radius a. The area of the phase space trajectory is 7ta2, as expected. If a95 denotes the phase
space circle containing 95% of the beam, then the emittance, 8 , is given by 8 =7r a95 .
C. Determination of the Transverse Distribution of the Beam
It is assumed in what follows that, prior to scraping, the transverse distribution of the beam
is static at the location of the scraper. The transverse beam distribution can be written as
N(q, ), where N(q,ir)did gives the number of beam particles in the phase space element
between (q, r) and (q + dry,i + dr). The q distribution of the beam is given by:
dN(ri) - N (9)
dN(i) gives the number of particles with q between q and q + dry.
It is more convenient to deal with the amplitude, a, than . Using rj2 +2 = a2, the
distribution function N(q, a)da is determined from N(q, r )dr as follows:
N(q, )di = N(q, a) a da
= N(q, a) 2a da (a >ig)
Va2 _ l2
where N(l, a) = N(,i(ri, a)). Equation (9) becomes:
dN(q) = 2 " N( za)a (11)
dl Il a2 _ 2
What remains is to relate N(rj,a) to what is measured in a scraper scan. The problem is
greatly simplified by the observation that N(il, a) depends only on the amplitude, a. This can be
seen as follows: Let n(q,a)- N(i,a)/NO, where No is the total number of beam particles.
n(q, a)dida is the joint probability of a beam particle with q between q and q + di and with a
between a and a + da. If g(a) is the probability distribution of a and F(q a) is the conditional
probability distribution of q for a given value of a, then
n(q, a)dida =[.F(il a)dri].[g(a)da] (12)
Since q = acos$, F(q a) depends on a and . They values of the particles in the beam, due
to filamentation, are uniformly distributed between 0 and 27G. Thus, q is uniformly distributed
between -a and a. Therefore, F(q a) is given by:
.F(q I a) a f a (13)
0 q > a
a Note: From equation (8): rj = -a cos 4 = - . This also follows from the application of Hamilton's equations to
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Werkema, Steven J. Transverse Beam Profile Measurement Using Scrape Scans, report, September 13, 2001; Batavia, Illinois. (https://digital.library.unt.edu/ark:/67531/metadc1014843/m1/3/: accessed June 19, 2019), University of North Texas Libraries, Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.