Measurements of Transverse Beam Diffusion Rates in the Fermilab Tevatron Collider Page: 2 of 3
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Action [pm]Figure 3: Calculated evolution of the distribution function
during an inward collimator step. The vertical lines rep-
resent the positions of the collimator vs. time. Collimator
action varies between Ji= 0.05 pm and Jcf 0.04 pm in
a time At 1 s. The initial and final slopes of the tails are
Ai 0.8 pm-2 and Af = 1 pm-2. The diffusion coefficient
is D 10-5 pm2/s.
function is tens of meters. It is assumed that the collima-
tor steps are small enough so that the diffusion coefficient
can be treated as a constant in that region. This hypothesis
is justified by the fact that the fractional change in action
is of the order of AJC/J ~ (2)(25 pm)/(2 mm) - 2.5%.
Because the diffusion coefficient is a strong function of ac-
tion (D ~ J4), this translates into a variation of 10% in the
diffusion rate, an acceptable systematic in a quantity that
varies by orders of magnitude. If D is constant, the dif-
fusion equation becomes 9 f D djf. With these defini-
tions, the particle loss rate at the collimator is equal to the
flux at that location: L -D - [dif]> y,. Particle showers
caused by the loss of beam are measured with scintillator
counters placed close to the collimator jaw. The observed
shower rate is parameterized as S= kL + B, where k is a
normalization constant including detector acceptance and
efficiency and B is a background term which includes, for
instance, the effect of residual activation. Both k and B are
assumed to be independent of collimator position and time
during the scan.
Under the hypotheses described above, the diffusion
equation can be solved analytically using the method of
Green's functions, subject to the boundary condition of
vanishing density at the collimator and beyond. Details
are given in Ref. [11]. An example of the evolution of
the phase-space density according to this model is shown
in Figure 3. A few representative snapshots in time are
chosen: during collimator movement (0 < t At); a
short time after the step, with a time scale determined by0 50 100
Time [s]150
Figure 4: Example of least-squares fit of the model to the
observed loss rates during an inward collimator step.
Jci - cf 2 /D = 10 s; and a long time after the step, with a
characteristic time J2/D = 160 s.
Local losses are proportional to the gradient of the dis-
tribution function at the collimator. The gradients differ in
the two cases of inward and outward step, denoted by the I
and 0 subscripts, respectively:
- 1
ajff(Jc,t) r -Ai+2(Ai -Ac)P JC + 2 -a
- {2Ai(Jci -Jc)+2(AiJe -AcJc) exp - ()2]}difo(J, t)
-2AiP (JccJc) +2(Ai -Ac)P (JO +
A -AJJ
+2 exp 2 (a})The parameters Ai and Af are the slopes of the distribu-
tion function before and after the step, whereas Ac varies
linearly between Ai and Af as the collimator moves. The
parameter a is defined as a _ 2Dt; its effect is to ex-
pose the dependence of losses on the inverse square root
of time, as is typical for diffusion processes. The func-
tion P(x) is the S-shaped cumulative Gaussian distribution
function: P(-oo) = 0, P(0) = 1/2, and P(oo) = 1.
The above expressions are used to model the measured
shower rates. Parameters are estimated from a least-squares
fit to the experimental data. An example is shown in Fig-
ure 4, where the best-fit function from the model is super-
imposed on the data points. The inset shows a detail of
the first few seconds after the collimator step. The oscilla-
tions in the data are due to coherent beam jitter. The back-
ground B is measured before and after the scan when thecn
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50
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Stancari, G.; Annala, G.; Johnson, T. R.; Still, D. A. & Valishev, A. Measurements of Transverse Beam Diffusion Rates in the Fermilab Tevatron Collider, article, August 1, 2011; Batavia, Illinois. (https://digital.library.unt.edu/ark:/67531/metadc830229/m1/2/: accessed April 24, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.