# Beam Physics at Tevatron COMPLEX^\*^

Valeri Lebedev^\#^, FNAL, Batavia, IL 60510, USA

Abstract

The challenge of achieving the Tevatron Run II luminosity goal of
310^32^ cm^-2^s^-1^ requires high level of engineering and machine
operation, good and reliable diagnostics, and clear understanding of the
underlying accelerator physics. Recent history demonstrated steady
increase of the Tevatron luminosity, which was supported by each of the
three listed above items. This report reviews major developments in the
accelerator physics, which contributed in the Run II luminosity growth.
Present limitations of the luminosity and projections of further
luminosity growth are also discussed.

## Introduction

The commissioning of Tevatron Run II began in the spring of 2001 with
the first luminosity seen in June. By the year end the luminosity was in
the range of (5-10)10^30^ cm^-2^s^-1^. Although the luminosity growth
was significantly slower than expected, steady growth of luminosity has
been demonstrated during last two years with the peak luminosity of
4210^30^ cm^-2^s^-1^ achieved in April 2003. This luminosity growth
would not be possible without deep insight into the accelerator physics
problems, which have restricted the machine operation. Important
contributions came from (1) optics correction in transfer lines, (2)
improvements of helical beam separation in Tevatron^\[1\]^, (3)
introduction of dual lattice operation in Accumulator to suppress
intrabeam scattering (IBS), (4) feedforward compensation of beam loading
in Main Injector, (5) correction of injection errors for antiproton
transfers from MI to Tevatron, (6) understanding transverse instability
in Tevatron with subsequent transverse impedance reduction^\[2\]^, and
(7) active damping of instabilities in Tevatron^\[3,4\]^. Although
understanding of longitudinal undamped oscillations in Tevatron^\[5\]^
not contribute directly to the luminosity growth it has been critical
for planning Run II upgrades. Several interesting accelerator physics
problems encountered in Run II commissioning are considered in this
report.

## 1. IBS in Accumulator

Accumulator^\[6\]^ stacks antiprotons with an average accumulation rate
of \~10^11^ per hour, so that after 20 hours the stack of \~210^12^
antiprotons can be accumulated. The total number of antiprotons along
with the longitudinal and transverse emittances are the major parameters
which determine the collider luminosity. At the beginning of 2002, once
the antiproton source was successfully commissioned, it became clear
that there is a strong heating of horizontal degree of freedom in
Accumulator. That caused the horizontal emittance to be significantly
higher than in Run I (see Figure 1).

The source of the problem was found to be the upgrade of machine optics,
which caused an increase in IBS heating. The optics upgrade was aimed to
increase the antiproton production^\[7\]^. That required an increase of
frequency band for stochastic cooling systems with subsequent decrease
of the slip factor to prevent the bad mixing. Although the betatron
tunes were changed insignificantly the IBS heating rate was strongly
increased.

![](media/image1.wmf)

Figure 1. Dependences of antiproton normalized effective emittance,
(*~x~* + *~y~*)/2, on beam current (100 mA = 10^12^) before upgrades (),
after the stochastic cooling upgrade (), and after both the cooling and
optics upgrades were implemented (). Each dot corresponds to one
collider shot. Lines present emittance boundaries for Run Ib.

Typically, in Accumulator, the longitudinal velocity spread in the beam
frame is much smaller than the transverse ones, and the transverse
emittance growth is dominated by excitation of betatron motion due to
energy changes at collisions. That significantly simplifies formulas. If
*k*~\|\|~=2.2*~p~*/()0.15 the IBS growth rates are:

$\frac{d}{\text{dt}}\begin{bmatrix}
\sigma_{p}^{2} \\
\varepsilon_{x} \\
\varepsilon_{y}
\end{bmatrix} \approx \frac{r_{p^{2}}\text{cN}}{4\gamma^{3}\beta^{3}C}\left\langle \frac{\sqrt{2\pi}L_{C}}{\sigma_{x}\sigma_{y}\theta_{}}\begin{bmatrix}
2\left( 1 - k_{||} \right) \\
A_{x}(1 - \kappa) \\
A_{x}\kappa
\end{bmatrix} \right\rangle_{s}\ \text{.}$ (1)

Here *r~p~* and *c* are the proton classical radius and the speed of
light, and are the relativistic factors, *C* is the ring circumference,
*N* is the number of particles, *~p~* is the rms relative momentum
spread, *~x~*, *~y~*, *~x~*, and *~y~* are the rms sizes and the local
angular spreads, is the *x*-*y* coupling parameter,
$\theta_{} = \sqrt{\theta_{x^{2}} + \theta_{y^{2}}}$,
$\left\langle \right\rangle_{s}$ denotes averaging over the ring, *L~C~*
is the Coulomb logarithm (*L~C~*20),

$A_{x} = \frac{D_{x^{2}} + \left( D_{x}^{'}\beta_{x} + \alpha_{x}D_{x} \right)^{2}}{\beta_{x}}$
, (2)

*~x~* and *~x~* are the horizontal beta- and alpha-functions, and *D~x~*
and $D_{x}^{'}$ are the dispersion and its derivative.

Usually ring optics is sufficiently smooth, and the second term in Eq.
(2) can be neglected. However, in the case of Accumulator optics
upgrade, average value of *A~x~* is well above the value corresponding
to the smooth lattice approximation and 2.5 times higher the value
before the upgrade. That caused strong amplification of horizontal
emittance growth. To reduce the IBS we introduced dual lattice
operation. The stacking is performed at the stacking lattice, which has
been designed for fast stacking but has strong heating due to IBS. After
stacking is completed the machine optics is retuned to the \"shot\"
lattice (similar to the Run I lattice), which has the same tunes but
smaller IBS heating, and therefore is more suitable for the final
cooling. After about 20 minutes cooling, the beam is ready for
extraction.

![](media/image2.wmf)

![](media/image3.wmf)

Figure 2. Measured (solid lines) and calculated (dashed lines)
normalized 95% emittances (top) and energy spread (bottom) for
antiproton current of 105 mA in Accumulator, = 0.1 is used in the
simulations.

Figure 2 presents computed and measured emittances of antiproton beam
for stacking lattice with cooling turned off. There is good agreement
between theory and measurements for the energy spread and the vertical
emittance but it is not as good for horizontal emittance. However good
agreement was found in the case of proton beam. The difference is caused
by additional heating from the self-stabilized two-beam instability due
to small amount of ions stored in the beam. The instability appears at
antiproton currents above \~30 mA. The emittance growth related to
multiple collisions with the residual gas is sufficiently large and was
taken into account in the simulations.

The described optics manipulations together with the core cooling
upgrade were introduced after the 2002 summer shutdown and allowed us to
achieve the antiproton beam emittances required for Run II.

## 2. single and multiple scattering 

There are a few applications in the Tevatron complex for which
simultaneous consideration of multiple and single Coulomb scattering is
important.

The first one is associated with separation of different heating
mechanisms contributing to transverse emittance growth in Tevatron. The
three basic sources of the beam heating are IBS, multiple scattering on
the residual gas and the noise in magnets, kickers, *etc*. While the
first one can be easily separated due to strong dependence on the beam
parameters, the effects of the other two look very similar.
Nevertheless, there is a fundamental difference between them. In
distinguish from the noise the beam interaction with residual gas,
additionally to diffusion, creates non-gaussian tails due to single
scattering. Approach developed in Ref. \[8\] uses an
integro-differential equation to describe the evolution of the
distribution in a linear focusing field. The equation correctly treats
both single and multiple scattering and can be written in the following
form

$\frac{\partial f}{\partial t} - \lambda\frac{\partial}{\partial I}\left( \text{If} \right) = \int_{0}^{\infty}{W(I,I^{'})\left( f(I^{'},t) - f(I,t) \right)dI^{'}}$
, (3)

where the kernel is

$W(I,I^{'}) = \frac{D\ \left( I + I^{'} + \frac{I_{\text{min}}}{2} \right)}{L_{C}\left( (I - I^{'})^{2} + (I + I^{'})I_{\text{min}} + \frac{I_{\text{min}^{2}}}{4} \right)^{\ \frac{3}{2}}}\ ,$(4)

is the damping decrement (if cooling is present), *D* is the diffusion
coefficient,
$L_{C} = \text{ln}\left( \sqrt{\frac{I_{\text{max}}}{I_{\text{min}}}} \right)$
is the Coulomb logarithm, *I*~min~ and *I*~max~ are the minimum and
maximum actions.

The following experiment was performed to measure the emittance growth
rate and to separate contributions of noise and gas scattering. A low
intensity beam was injected into Tevatron. The beam was debunched (to
reduce IBS) and scraped in both planes to a known size. The scrapers
were then removed, and the beam was left alone for 1 hour. To measure
resulting distribution we scraped the beam vertically while measuring
the beam current as function of scraper position. Good agreement between
the measurements and numerical solution of Eq. (3) (=0) has been found.
Both the emittance growth in the core (5 mm mrad/hour) and the tail
population were described well by the model. That means that in the case
of small intensity beam, when IBS is negligible, gas scattering is a
major source of the beam heating and there is no visible heating could
be associated with noise. Last year vacuum improvements yielded some
reduction of the beam heating due to gas scattering and reduced
background in CDF and D0 detectors.

The second example is related to the beam lifetime computations in
Recycler where the beam emittances and machine acceptances are quite
close. Solving Eq. (3) with zero boundary condition at the machine
acceptance we found the dependence of the rms beam size and the beam
lifetime as functions of time for different cooling decrements. After
some time the system comes into equilibrium, where the shape of
distribution function does not depend on time, and the intensity decays
exponentially with the decay time $\tau_{\infty}$. We define the
lifetime correction factor, *K*, as a ratio of $\tau_{\infty}$at given
equilibrium emittance (determined by cooling decrement) to the lifetime
of zero-emittance beam (determined by the single scattering only).
Figure 3 presents *K* as a function of ratio of the rms equilibrium
emittance to the machine acceptance. Without cooling the rms beam
emittance reaches its maximum of 0.391, where the lifetime is 40 times
worse than for a point-like beam. Although, the model described above
was developed for a one-dimensional case, when the aperture is limited
in one plane only, the results presented in Figure 3 can be used for the
lifetime correction factor of two-dimensional case if the emittances and
the acceptances in both planes are equal.

![](media/image4.wmf)

Figure 3. Dependence of the lifetime correction factor, , on the ratio
of the rms emittance to the machine acceptance.

The last example of this section is related to the evolution of
longitudinal distribution in Tevatron governed by IBS. In this case the
non-linearity of potential well changes the kernel in Eq. (3) to^\[9\]^:

$\overset{\sim}{W}(E,E^{'}) = \frac{{B\omega}_{0}\omega\omega^{'}}{(E - E^{'})^{2}}\left\{ \begin{matrix}
\frac{1}{2\omega} + \frac{I}{E^{'} - E}\ ,\ E^{'} \geq E + \delta E\ , \\
 \\
\frac{1}{2\omega^{'}} + \frac{I^{'}}{E - E^{'}}\ ,\ E^{'} \leq E - \delta E\ \text{.}
\end{matrix} \right.\ $ (5)

Here ~0~ is the frequency of small amplitude motion, *E* is the energy,
$I = (1/2\pi)\oint_{}^{}\text{pdx}$ and
$\omega = \partial\frac{E}{\partial}I$ are the action and the frequency.
Similar to Eq. (4) the divergence in Eq. (5) at $E \approx E^{'}$ need
to be confined for the energy difference below *E*\~~0~*I~min~*. Eq. (5)
reduces to Eq. (4) for linear motion. For the sinusoidal potential of
longitudinal motion the energy is,

$E = p^{2}/2 + \omega_{0^{2}}\left( 1 - \text{cos}(x) \right)$ . (6)

Evolution of the longitudinal bunch profile in time obtained by
numerical solution of Eq. (3) with kernel of Eq. (5) is presented in
Figure 4. The initial distribution function has no tails, because before
acceleration the particle distribution fits into 4 eV s bucket size,
while after acceleration the bucket size is 10 eV s. Therefore the
initial particle loss occurs due to the single scattering only, later,
however, the tail population grows and diffusion loss begins to
dominate. In distinguish from the standard (local) diffusion the large
non-gaussian tails are created from the very beginning. For a point-like
beam the lifetime is determined by single scattering and is equal to
$\tau_{0} = 4\frac{L_{C}}{D}$. The lifetime decreases with beam
expansion and, when the beam size achieves its maximum rms size of 0.931
rad, the lifetime reaches its asymptotic value of
$\tau_{0} \approx 0\text{.}\frac{\text{741}}{D}$.

![](media/image5.wmf)

Figure 4. Numerical simulation of the longitudinal bunch profile
evolution during the store in Tevatron.

## 3. luminosity evolution 

Numerous factors affect the Tevatron luminosity and its evolution in
time. Each store is different and because of finite instrumentation
accuracy it is practically impossible to state what was different or
what came wrong for every particular store. Nevertheless the luminosity
evolution is similar for most of the stores. It is driven by some basic
processes, which are not very sensitive to the details of distribution
function, and therefore the luminosity evolution can be described by
comparatively simple parametric model. The model takes into account the
major beam heating and particle loss mechanisms. They are (1) the
emittance growth and the particle loss due to scattering on the residual
gas, (2) the particle loss and the emittance growth due to scattering in
IPs, (3) the transverse and longitudinal emittance growth due to IBS,
(4) the bunch lengthening due to RF noise, and (5) the particle loss
from the bucket due to heating of longitudinal degree of
freedom^\[10\]^. If the collider tunes are correctly set and the beam
intensity is not too high the beam-beam effects are not very important
and the model describes the observed dynamics of beam parameters and the
luminosity comparatively well.

Figure 5 presents measured and computed luminosity for the Store 2138
(Jan.05.2003). The only free parameters used in the model were the
residual gas pressure of 1.210^-9^ Torr of molecular nitrogen equivalent
at room temperature, the *x*-*y* coupling parameter = 0.45, and the
spectral density of RF phase noise of 50 rad^2^/Hz. They correspond to
the gas scattering lifetime of 380 hours, and the bunch lengthening due
to RF noise of 2.210^-3^ rad^2^/hour. RF phase noise was measured
directly^\[11\]^ and agrees with the fit to the model within the
measurement accuracy (factor of 2). The computed proton and antiproton
intensities are close to the measured ones as can be seen in Figure 6.
It has been critical to use the described above model for non-local
diffusion in longitudinal direction to achieve such a good agreement

![](media/image6.wmf)

Figure 5. Dependence of luminosity on time for Store 2138; solid and
dotted lines - luminosity measured by CDF and D0 detectors, dashed line
-- model prediction.

![](media/image7.wmf)

Figure 6. Dependence of intensities for proton and antiproton beams on
time for Store 2138; solid lines -- measurements, dashed lines - model
prediction.

![](media/image8.wmf)

Figure 7. Dependence of bunch lengths for proton and antiproton beams on
time for Store 2138; solid lines -- measurements, dashed lines - model
prediction.

The major mechanism for loss of antiprotons is the luminosity loss. The
cross-section of 70 mbarn is used for proton-antiproton scattering in
the IP. The proton bunch lengthening (see Figure 7) is mainly driven by
IBS, while for low intensity antiproton beam the RF phase noise usually
dominates. Figure 8 presents a comparison of measured and computed
antiproton emittances versus time. The emittances were measured with
synchrotron light monitors^\[12\]^. The obtained values were corrected
for diffraction to match them with the effective emittance computed from
the luminosity and the emittance measurements performed with flying
wires at the beginning and the end of the store. At the beginning of the
store, the vertical emittance grows significantly faster than the model
prediction. Our present belief is that it is related to an amplification
of diffusion by the beam-beam effects.

![](media/image9.wmf)

Figure 8. Dependence of horizontal and vertical antiproton beam
emittances on time for Store 2138; solid lines -- measurements, dashed
lines - model prediction.

The store 2138 discussed above has moderate discrepancies with the
model, and can be considered a regular store. Most of our stores are
influenced more by the beam-beam interaction, but fortunately it weakly
affects the luminosity decay and the luminosity integral. Figures 9 and
10 present measured and computed parameters for Store 2328
(Mar.20.2003). The same vacuum and RF phase noise were used in the
model. Unlike Store 2138, both the proton and antiproton beam
intensities decay faster and the proton bunch length grows significantly
slower than the model predicts^\[12,13\]^. The most probable cause of
such misbehavior is small, uncontrolled store-to-store tune variation
causing a loss of dynamic stability for particles at large synchrotron
amplitudes with subsequent particle loss and reduction of bunch
lengthening.

The results presented above show that the beam-beam interactions
certainly affect the luminosity decay. However this effect is
sufficiently small. Thus, the developed model, with some reservations,
can be used to analyze the luminosity dynamics for the final Run II
parameters. Table II presents parameters for one of our best stores
(Store 2328), typical collider parameters in April 2003 and projections
for the final Run II parameters. Evidently, in order to increase the
luminosity by a factor of 7.2 times we need to quadruple the number of
antiprotons extracted from the stack. The remaining factor of 1.8 should
result from the improvements in the antiproton transport and Tevatron.
Three major contributions are an increase of the proton intensity by
\~30%, an improvement of coalescing in MI, and improvements of
antiproton transport efficiency (from the antiproton stack to the
collisions in Tevatron). Two last items are expected to increase the
transfer efficiency from \~60% to \~80%. The chosen proton intensity,
2.710^11^ per bunch, corresponds to the linear head-on tune shift of
0.01 for each of two IPs. This is the maximum tune shift achieved in Run
Ib with 66 bunch operation. We choose the maximum antiproton intensity
to be half of the proton intensity. Further increase of antiproton
intensity is limited by antiproton production.

![](media/image10.wmf)

Figure 9. Dependence of luminosity on time for Store 2328; solid and
dotted lines - luminosity measured by CDF and D0 detectors, dashed line
-- model prediction.

![](media/image11.wmf)

![](media/image12.wmf)

Figure 10. Dependence of intensities (top) and bunch lengths (bottom)
for proton and antiproton beams on time for Store 2328; solid lines --
measurements, dashed lines - model prediction.

Due to reduction of luminosity lifetime with growth of peak luminosity,
the average luminosity grows slower than the peak luminosity. After all
Run II upgrades are introduced the luminosity integral is estimated to
be \~2.5 fb^-1^/year.

Table 2. Present and final Run II parameters

+---------------------------------------+---------+---------+--------+
|                                       | Store   | Typical | Final  |
|                                       | 2328    | Apr.03  |        |
|                                       |         |         | Run II |
+---------------------------------------+---------+---------+--------+
| Protons, 10^10^ / bunch               | 20.7    | 20      | 27     |
+---------------------------------------+---------+---------+--------+
| Pbars, 10^10^/ bunch                  | 2.54    | 2.2     | 13.5   |
+---------------------------------------+---------+---------+--------+
| Norm. 95% proton emittances,          | \~14/24 | \~15/25 | 20/20  |
| *~x~*/*~y~*, mm mrad                  |         |         |        |
+---------------------------------------+---------+---------+--------+
| Norm. 95% pbar emittances,            | \~15/24 | \~16/25 | 20/20  |
|                                       |         |         |        |
| *~x~* /*~y~*, mm mrad                 |         |         |        |
+---------------------------------------+---------+---------+--------+
| Proton bunch length, cm               | 65      | 62      | 50     |
+---------------------------------------+---------+---------+--------+
| Pbar bunch length, cm                 | 59      | 58      | 50     |
+---------------------------------------+---------+---------+--------+
| Initial luminosity, 10^30^            | 40.5    | 35      | 290    |
| cm^-2^s^-1^                           |         |         |        |
+---------------------------------------+---------+---------+--------+
| Initial luminosity lifetime, hour     | 11      | 12      | 7.1    |
+---------------------------------------+---------+---------+--------+
| Store duration, hour                  | 19      | 20      | 15.2   |
+---------------------------------------+---------+---------+--------+
| Lumin. integral per store, pb^-1^     | 1.71    | 1.2     | 8.65   |
+---------------------------------------+---------+---------+--------+

The work reported in this paper is the result of expertise, ingenuity
and dedication of a large number of people in Fermilab. The author
wishes to acknowledge their excellent results, and is grateful for the
contribution received.

## References

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\[9\] A. Burov, Private communications, 2003.

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