# . Collider Accelerator Physics and Design

## 3.1 Two Colliders in One Tunnel {#two-colliders-in-one-tunnel .unnumbered}

### 3.1.1 Footprint Issues {#footprint-issues .unnumbered}

For the design study it has been envisioned that the Low Field (LF) ring
will be comprised of long, combined function magnets---the bending of
the central trajectory and the focusing of the particle beam will be
performed using gradient magnets with central field strength on the
order of 2 T. The High Field (HF) ring will use separated function
magnets---dipole magnets for bending the central trajectory and
quadrupole magnets for focusing. Since the LF and HF designs have
different focusing characteristics, the dispersion suppressors must also
be de­signed simultaneously to ensure that the LF and HF orbits line up
appropriately when entering the long straight sections for the
Interaction Regions and Utility Regions.

As stated earlier, the size of the arcs is determined primarily by the
low magnetic field of the first stage collider, while the lengths of the
major straight sections are determined by the high magnetic rigidity of
the beam in the second stage collider. The interaction regions must be
made long enough to accommodate trajectory and optics manipulations of
high energy proton beams using reasonable guesses for sizes of future
high energy, high luminosity detectors. The utility straight section
regions must be made long enough to provide space for extracting a very
high energy proton beam toward a beam dump and for beam scraping and
beam instrumenta­tion at high energy. All of these functions will be
required for the LF collider as well, of course, but the technologies
used for injection, extraction, acceleration, and so forth will be
similar for the two rings. Thus, the requirements of the HF design will
set the scale for the straight sections.

### 3.1.2 Optics Issues {#optics-issues .unnumbered}

In addition to the geometric layout of the two rings, many optical
properties must be decided simultaneously as well. The Low Field ring
uses gradient magnets in the arcs of the collider, while the High Field
ring uses quadrupole FODO cells. For simplicity of instrumentation,
power distribution, cable runs, etc., it is desirable for the maximum
beam size locations in the HF arc to correspond with those in the LF
arc. Dispersion suppressor regions typically are made by reducing the
total bending in two consecutive cells by half. The proportion of
bending to leave out in each of the cells depends upon the phase advance
per cell. In combined function lattices, such as the LF design, the
bending magnets also provide focusing, so this tactic must be modified.
While combined function designs of dispersion suppressors have been
performed in the past (e.g., Fermilab Recycler), the choice of cell
phase advance and cell lengths must be settled upon for both rings to
ensure a consistent dispersion suppressor design. (Obviously, this is an
issue for both optics and geometry as well.) This is another reason that
it was chosen to give the two colliders equal cell lengths and equal
betatron phase advances per cell. Each collider, LF and HF, is made of
two separate, but linked, accelerators. The choice must be made as to
whether adjacent magnets have the same or opposite focusing
characteristic with respect to each other for the beam traveling through
them. This affects the matching of sections going into and out of
straight sections, as well as the design of the beam transfer system
joining the LF and HF rings. The decision must be made in coordination
with the magnet design effort. Finally, the magnet choice for the LF
collider is a horizontally separated two-in-one iron dominated magnet.
To ensure that the two beam trajectories have the same path length, the
beams will need to be crossed several times about the circumference.
Space and optical designs must be incorporated into the layout to
accommodate this feature. The two beams of the high field collider are
naturally separated vertically in the common coil arc magnets.

### 3.1.3 Approach to Design {#approach-to-design .unnumbered}

As indicated in Figure 3.1 each collider is made up of two major arcs
that connect two clusters of straight sections. At the outset, only one
cluster region---located at or near the Fermilab site---will be equipped
with full interaction region and utility region optics. The opposite
cluster will have the same straight section lengths, but optically will
consist of simpler, FODO-type modules where applicable, with the
exception of a special straight section designs to accommodate beam
scraping and future beam transfers between the LF and HF accelerators.

![](media/image1.wmf)

Figure 3.1. Schematic layout of collider modules. The figure is not to
scale, emphasizing the straight section functions.

Dispersion suppressor modules are adopted following the SSC design,
which was also later used in the Fermilab Main Injector. A cell phase
advance of 90 degrees has been chosen for which a dispersion suppressor
can be made using two special cells, each 3/4 the length of a standard
cell and each containing 2/3 the bending of a standard cell. These
modules reduce the periodic arc dispersion function to zero at the
output of two such cells. The short bending regions between Interaction
Regions and Utility Regions will be composed of back-to-back dispersion
suppressor modules. Any additional bending required in these regions is
generated by inserting standard half-cells between the DS modules.

### 3.1.4 Footprint Parameters {#footprint-parameters .unnumbered}

Table 3.1 lists the fundamental parameters of the footprint. In the
transition from Stage-1 to Stage-2 operation, the experiments will
remain centered on the same interaction points. At the same time it is
envisaged that they will be upgraded to take up more space along the
beam line, increasing the distance from the IP to the first magnet,
*L^\*^,* from 20 to 30 m. It is therefore necessary to allow for a
bypass to keep the low field ring beams well clear of the experiments.
This must be done without changing the total low field ring
circumference.

One advantage of modular construction is that low and high field
lattices are guaranteed to have almost identical footprints, and
therefore to fit in the same tunnel, so long as correspond­ing low and
high field modules are placed on top of each other. The maximum
deviation is only a few millimeters, easily allowing one ring to be
placed on top of the other at all locations in the VLHC tunnel.

Table 3.1. Fundamental lattice parameters common to both low and high
field rings.

  -------------------------------------------- -------------- ------------
  Circumference, *C*                           233.037        km

  Average arc radius, *R*                      35.0           km

  Number of interaction points                 2              

  Half cell length, *L~hc~*                    135.486        m

  Half cell bend angle, *θ*~hc~                3.875          mrad

  Half cell count                              1720           

  Half cell harmonic, *n~λ~*                   24             

  Bunch spacing (53.1 MHz), *S~B~*             5.645          m

  Time between bunches                         18.8           ns

  Number of buckets                            41280          

  Number of bunches, *M*                       37152          

  Phase advance per cell                       90.0           deg

  Revolution frequency                         1.286          kHz

  Revolution period, *T*                       0.778          ms

  Harmonic number, *h*                         371520         

  RF frequency (9 *x* 53.1)                    478.0          MHz
  -------------------------------------------- -------------- ------------

### 3.1.5 Half-cell Length {#half-cell-length .unnumbered}

The average bending radius of the arcs is made as close as possible to
the goal value of *R* = 35 km by adjusting the total equivalent number
of (bending) arc cells in each ring. Each of the 4 short half-cells in
each dispersion suppressor has 2/3 the bending of an arc half-cell.
Thus, the 20 dispersion suppressors (on either side of 10 straights)
have a total bending equivalent to 160/3 arc half-cells. If there are
*N* = 784 arc half cells in each half of the VLHC, then the bend per
half cell is

$q_{\text{hc}} = \frac{2p}{2N + \frac{\text{160}}{3}} = 3\text{.}\text{875}\text{mrad}$
Eq. 3.1

and the average arc bend radius is

$R = \frac{L_{\text{hc}}}{q_{\text{hc}}} = \text{34}\text{.}\text{961}\text{km}$
Eq. 3.2

The (minimum) bunch spacing *S~B~* is taken to be one 53.1 MHz RF
wavelength in the Tevatron

$S_{B} = \lambda_{\text{53}} = \frac{(\text{2000}m)p}{\text{1113}} = 5\text{.}\text{645}m$
Eq. 3.3

This guarantees that the filling schemes from the Tevatron will be
relatively simple. The bunch time spacing of about 18.8 ns is not
unreasonably challenging to the experiments.

The half-cell length is

$L_{\text{hc}} = n_{\lambda}S_{B}$ Eq. 3.4

The bunch spacing can trivially be increased (for example during
commissioning) by any of the common factors of the "half cell harmonic
number" *n~λ~.* The value of *n~λ~* = 24 was chosen for both low and
high field rings, corresponding to a half-cell length of *L~hc~* =
135.4865 meters.

This half cell length is short enough to keep the natural horizontal
emittance due to syn­chrotron radiation in the high field ring, *ε~nat~*
= 0.04 μm, much smaller than the horizontal emittance *ε~x~* ≈ 0.20 μm
which must be maintained (by heating) to avoid beam-beam limits in the
store. A stronger limit to the maximum arc half-cell length is the need
to limit the vulner­ability of the low and high field rings to systematic
field harmonic errors in the arc bending magnets. This is more critical
for the low field ring, simply because the injection energy is an order
of magnitude lower (1 TeV) than for the high field ring (10 TeV).

The current value of *L~hc~* = 135.4865 m is by no means fully
optimized, not least because we do not accurately understand what values
of systematic magnet harmonics might be achieved in industrial
production of the arc bending magnets.

## 3.2 Stage 1 -- the Low Field Ring {#stage-1-the-low-field-ring .unnumbered}

The first stage low-field collider will have a top energy of 20 TeV and
peak (initial) luminosity of 10^34^ cm^-2^sec^-1^. The collider will use
the Tevatron as its injector operating at a transfer energy of 900 GeV.
For this study a transverse emittance of 1.5 π mm-mrad (rms, normalized)
is used, which is typical for the Fermilab Booster, though about half
the value at the Tevatron under recent normal operations. It is
anticipated that with further Run II experience, the efficiency of
emittance preservation will increase. Using the present acceleration
systems of the Tevatron injector chain, the collider has a bunch spacing
of 5.645 m (53.1 MHz), which sets the number of available RF buckets.
With a 90% filling fraction to allow for gaps in the beam for various
kicker rise times, the number of protons per bunch required in the
Stage-1 collider is approxi­mately 2.6 × 10^10^, similar to Tevatron
bunch intensities during previous Fixed Target opera­tions. Using an
interaction region design with a *β*^\*^ = 0.3 m, we arrive at the
desired initial luminosity. Table 3.2 lists the general parameters of
the Stage-1 collider.

The typical store parameter evolution is depicted in Figure 3.2. The
figure shows the proton bunch intensity and luminosity decreasing due to
collisions. The vertical emittance also de­creases slightly due to
synchrotron radiation damping which occurs with a damping time of about
100 hr. Due to the use of gradient magnets in the Stage-1 collider the
horizontal emit­tance actually will be *anti-damped*, and will increase
at approximately the same rate. This should not be a problem since store
times will be much less than 100 hours.

Table 3.2. Nominal store parameters for the low field collider.

  ------------------------------------------------- ---------- ---------------
  Storage energy                                    20         TeV

  Peak luminosity                                   10^34^     cm^-2^sec^-1^

  Packing fraction                                  89         \%

  Injection energy                                  0.9        TeV

                                                               

  Transverse normalized emittance, rms (H&V,        1.5 π      μm
  inject)                                                      

  Initial bunch intensity                           2.6 *x*    
                                                    10^10^     

  Average beam current                              195        mA

  Stored energy per beam at collision               3.0        GJ

                                                               

  Bend field at storage                             2.0        T

  Bend magnet gradient                              9.0        T/m

  Phase advance per cell                            90.0       deg

  Max rms arc beam size (inject)                    1.2        mm

  Bunch spacing (53.1 MHz)                          5.645      m

  Time between bunches                              18.8       nsec

  Bunch length                                      30         mm

  Longitudinal emittance, rms (inject)              0.4        eV-s

  RF voltage at storage                             50         MV

  Fill time                                         60         min

  Acceleration time                                 1000       sec

  Beam size (rms) at IP (storage)                   4.6        μm

  Total crossing angle (10*σ* separation in drift   153        μrad
  space)                                                       

  Distance from IP to first magnet                  21         m

  *β^\*^* at IP (H & V)                             0.3        m

  Maximum interactions per crossing                 20         

  Debris power at IP (each direction)               3          kW

                                                               

  SYNCHROTRON RADIATION AT STORE                               

  Energy loss per turn per particle                 38         keV

  Radiation damping time, *τ*~0~                    100        hr

  ***(anti-damping in H plane)***                              
  ------------------------------------------------- ---------- ---------------

### 3.2.1 Lattice {#lattice .unnumbered}

The bulk of the lattice is characterized by its construction from just
three optical building blocks---arc, dispersion suppressing, and
straight-section cells. The arc and suppressor cells consist of combined
function gradient magnets and have space allocated at maximum-*β* sites
for 4 correction elements. Additional corrector space is available at
the quarter-cell locations (*β*~x~ = *β*~y~) for lumped correction of
systematic multipoles \[[^1]\]. Two utility straights, each of which
must accommodate 900 GeV injection from the Tevatron plus abort of 20
TeV beams are needed. The beams collide with a horizontal crossing angle
at two interaction points where *β*^\*^ = 30 cm. In a special cell,
midway between the IP\'s, the beams are made to cross over between the
inner and outer horizontal apertures. Diametrically opposed in the ring,
a mirror of this cell is installed to ensure identical path lengths for
the two circulating beams.

![](media/image2.wmf){width="2.953472222222222in"
height="2.472916666666667in"}
![](media/image3.wmf){width="2.9743055555555555in"
height="2.472916666666667in"}

Figure 3.2. Evolution of collider parameters during a typical store.

#### 3.2.1.1 Interaction Regions {#interaction-regions .unnumbered}

The final foci in the IRs are anti-symmetric triplets, formed from
single-bore, 300 T/m mag­nets. Four additional quadrupole circuits,
comprising double-bore, 70 T/m magnets, are also used in optical
matching. With a total of 6 independently-tunable quadrupole circuits
available it is possible to match the four β\'s and α\'s from the IP to
regular FODO cells, and hold the phase advance ∆μ constant across the IR
through the squeeze from β^\*^ = 6.00 m → 0.30 m. Fixing ∆μ eliminates
the necessity of a special phase trombone to maintain the nominal
oper­ating point. Figure 3.3 shows the lattice functions through the IR
during collisions.

The circulating beams are separated horizontally by 15 cm throughout the
arcs and 70 T/m straight-section quads. Four dipoles, 13.6 m each at
1.97 T, situated between the Q3 and Q4 quadrupoles bring the beams
together at the entrance to the triplet for collisions at the IP.
Dipoles downstream of the IP separate the beams again and channel them
back into the inner and outer rings. A half-crossing angle of 77 μr at
the IP gives 10σ separation between the beams at the first parasitic
crossing.

#### 3.2.1.2 Injection and Abort Straight Sections {#injection-and-abort-straight-sections .unnumbered}

To accommodate 900 GeV injection from the Tevatron, as well as 20 TeV
proton abort, 5 straight-section cells are modified as shown in Figure
3.4. The long drift space (377 m) created by triplet focusing produces
sufficient room for abort Lambertsons (see Section 5.1.5.4) and beam
extraction.

Dipoles separate the circulating beams through the utility straight to
make room for RF cavities downstream of the second triplet. Three 21.54
m dipoles at 2 T bend the beams apart by 40 cm, followed immediately by
3 more dipoles that flatten out the trajectories. This pattern is
reversed downstream of the RF to bring the beams back into standard
double-bore magnets.

![](media/image4.wmf){width="5.763194444444444in"
height="3.8520833333333333in"}

Figure 3.3. IR lattice functions in the collision configuration.

![](media/image5.wmf){width="5.502777777777778in"
height="3.670138888888889in"}

Figure 3.4. Layout of a utility straight supporting beam transfer from
the Tevatron and 20 TeV abort. (See Section 5.2.5.4.)

#### 3.2.1.3 Beam Crossovers {#beam-crossovers .unnumbered}

With the beams separated horizontally in the arcs it is necessary to
move the beams between the inner and outer rings to maintain identical
path lengths. In the LF ring 2 crossover cells are inserted on opposite
sides of the ring. Four rolled dipoles move the beams the 15 cm from one
channel to the other. Each dipole is 36 m at 1.944 T. The first dipole
is rolled by 0.12 rad to start separating the beams vertically, while
bringing them closer together horizontally. The second bend is rolled by
−0.12 rad to flatten the vertical trajectory. At the crossing point the
beams are separated vertically by 9 mm. This is 17.8σ at the injection
energy of 900 GeV. Rolled dipoles after the crossing point remove the
vertical offsets and complete the 15 cm transverse transfer of beams to
the opposite channel.

#### 3.2.1.4 Miscellaneous Insertions {#miscellaneous-insertions .unnumbered}

For momentum collimation a localized dispersion wave is generated in a
5-cell straight section (Figure 3.5). Four 45 m dipoles (2 T @ 20 TeV)
at the upstream end of the straight move the beams transversely by 0.36
m, creating ±70 cm of dispersion. Four more dipoles downstream bring the
beams back onto their nominal trajectory and cancel the dispersion.

Free space for damping systems, beam instrumentation, etc. (see, for
example, Figure 5.53) can be generated by replacing standard arc
gradient magnets with a shorter, modified version plus a small
quadrupole. Through this insertion, the β\'s, α\'s, and phase advance of
an arc cell are exactly reproduced by a gradient magnet with *B\'* =
9.185 T/m (1% weaker than the regular arc magnets) plus a 2.0 m
quadrupole with *B\'* = 26.14 T/m. The low field of the quadrupole makes
it just a longer version of the normal arc trim quads. The dispersion
wave created by the holes is cancelled locally with 90° of phase advance
per cell.

![](media/image7.wmf){width="5.5159722222222225in"
height="3.678472222222222in"}

Figure 3.5. Dispersion creation in a 5-cell straight for collimation.

### 3.2.2 Magnet Apertures and Field Quality {#magnet-apertures-and-field-quality .unnumbered}

The available mechanical aperture determines the required dynamic
aperture. Collimators define the limiting aperture in most cases and
they are usually placed at around 6σ from the center of the beam pipe.
This allows 3σ for the beam distribution, and another 3σ for injection
errors and orbit drifts over time. The real dynamic aperture therefore
must be greater than 6σ in order to avoid significant losses of
particles at places in the ring outside the beam cleaning section.

Errors, time dependent fields and other imperfections in a real machine
can never be mod­eled accurately enough in a simulation, so the dynamic
aperture calculated by a simulation always exceeds the measured dynamic
aperture. Observations at the Tevatron, HERA, and SPS have shown that
the agreement between the measured and calculated dynamic aperture
varies between 20-100%. As a consequence the LHC, for example, requires
that the calculated dynamic aperture after 10^5^ turns at both injection
and collision (with only magnetic non-linearities) equal 12σ. Since the
errors listed above are just as likely for the VLHC, it seems reasonable
to adopt a dynamic aperture requirement of 12σ after 10^5^ turns.

#### 3.2.2.1 Scaling the Main Injector Errors {#scaling-the-main-injector-errors .unnumbered}

As a starting hypothesis, the errors in the Main Injector are used as a
scaling basis, assuming that the mechanical errors scale with the pole
tip gap. Thus we assume that the field errors in the VLHC low field
transmission line magnets at a pole tip radius of 10 mm are the same as
the errors in the Main Injector magnets at a pole tip radius of 1 inch.
The bend fields in the Main Injector at 8 GeV and in the VLHC at an
injection energy of 1 TeV are nearly the same: *B~0~^MI\ ^*= 0.101 T,
*B~0~^VLHC^* = 0.098 T.

We make the somewhat stronger assumption that the equality is satisfied
for every har­monic *n* so that we obtain the following scaling relation
between the multipole coefficients in the Main Injector and those in the
VLHC,

$\left\lbrack \frac{b_{n}}{R_{\text{ref}}^{n}},\frac{a_{n}}{R_{\text{ref}}^{n}} \right\rbrack^{\text{VLHC}} = (2\text{.}\text{54})^{n}´\left\lbrack \frac{b_{n}}{R_{\text{ref}}^{n}},\frac{a_{n}}{R_{\text{ref}}^{n}} \right\rbrack^{\text{MI}}$
Eq. 3.5

Table 3.3. Systematic \<\> and random σ of multipole coefficients used
for the VLHC, scaled from Main Injector magnets using Equation 3.5. The
coefficients are shown in units of 10^-4^ at a reference radius of 10
mm.

  ------------------ -------------- -------------- --------------- ---------------
  **Order, *n***     **\<b~n~\>**   **\<a~n~\>**   ***σ*(b~n~)**   ***σ*(a~n~)**

  2                  -0.600         0.000          0.600           0.200

  3                  0.000          0.000          0.149           0.300

  4                  0.300          0.000          0.300           0.150

  5                  0.000          0.000          0.100           0.500

  6                  0.000          0.000          0.250           0.250
  ------------------ -------------- -------------- --------------- ---------------

Table 3.3 shows the derived harmonics for the low field magnets. We have
assumed that the measured systematic skew harmonics in the Main Injector
magnets are the result of the top down asymmetry in the magnet bus work
\[[^2]\]. This should not be an issue for the low-field magnets since
the bus work is symmetric, and so the nonlinear skew harmonics have been
set to zero.

#### 3.2.2.2 Dynamic Aperture at Injection {#dynamic-aperture-at-injection .unnumbered}

The main issue is to determine whether the field quality is good enough
to meet the target set for the acceptable dynamic aperture. If not, a
nonlinear corrector system that increases the dynamic aperture to the
target value must be developed. The results shown below will be refined
at a later stage when more information on the field quality is
available. For most calculations reported here, particles are tracked
for 1024 turns without synchrotron oscillations in a perfectly aligned
lattice. The only nonlinearities are the chromaticity sextupoles and the
field errors in the arc magnets. Sources of coupling are not introduced
so correction with skew quadrupoles is not necessary. The fractional
tunes are set to the Tevatron values ν~x~ =0.581, ν~y~ = 0.575.

The impact of linear coupling is reduced when there is a difference in
horizontal and verti­cal integer tunes. In the perfect aligned lattice
there are no sources of linear coupling. Never­theless the influence of
tune splits on the dynamic aperture for the cases *ν~x~- ν~y~* = ±2, ±4,
±6 were studied. In no case did the dynamic aperture improve and in some
cases the aperture was reduced significantly, presumably due to the beta
beating.

Figure 3.6 shows the dynamic aperture with only chromaticity sextupoles,
only random er­rors, only systematic errors and all the errors. The
dynamic aperture with all errors is nearly the same as that with only
the random errors. Tracking with all the field errors at injection shows
that the dynamic aperture is about 20σ as shown in Table 3.4. Increasing
the number of turns to 100,000 typically results in a decrease of the
dynamic aperture by about 2-3σ. This suggests that the target dynamic
aperture of 12σ at 100,000 turns should be met with this set of assumed
errors.

#### 3.2.2.3 Dynamic Aperture at Collision {#dynamic-aperture-at-collision .unnumbered}

The dynamic aperture at collision optics will be dominated by the field
errors of the IR quad­rupoles due to the large beam sizes in these
magnets. The IR quadrupoles with their high gradients will be built with
"conventional" superconducting magnets. The nonlinear harmonics in these
magnets are well understood following decades of development and also
through the Fermilab program of building these quadrupoles for the LHC.
The errors in the arc magnets built with conventional technology usually
are not significant in determining the dynamic aperture at top energy.
For example, the target dynamic aperture in the LHC at top energy is set
at 12σ with only the IR quadrupole field errors. We will adopt the same
target dynamic aper­ture of 12σ in the simulations.

![](media/image8.wmf){width="6.233333333333333in"
height="4.641666666666667in"}

Figure 3.6. Dynamic aperture at injection with only chromaticity
sextupoles(CS), CS and random errors only, CS and systematic errors
only, and CS with all errors. The target dynamic aperture is 12σ.

Table 3.4. Dynamic aperture at injection energy with various sets of
errors in the arc magnets. These results show that the random errors
together have a somewhat larger impact on the dynamic aperture than the
two systematic components \< b~2~\>, \< b~4~\>. The dynamic aperture
with all the errors is about the same (20σ) as without the systematic
errors.

  ------------------------------------------------ ------------ -----------
  **Errors**                                       **Dynamic    
                                                   Aperture**   

                                                   Average      Minimum

  Only chromaticity sextupoles (CS)                30.9         25.4

  CS and only random errors                        20.0         18.4

  CS and only systematic errors                    23.4         21.1

  CS and all errors                                19.7         18.4

  CS and only random errors and *σ(b~2~)= σ(a~2~)* 20.4         18.6
  = 0                                                           

  CS and only random errors and *σ(b~3~)= σ(a~3~)  20.2         17.6
  =* 0                                                          

  CS and only \<*b~2~*\> ≠ 0                       31.5         25.2

  CS and only \<*b~4~*\> ≠ 0                       24.5         21.1
  ------------------------------------------------ ------------ -----------

The arc magnets in the Stage-1 collider, however, are built with
transmission line magnets, which have different characteristics from
conventional superconducting magnets. At high current, the field in the
iron saturates leading to a drop in the gradient and a large negative
sextupole component (*b~2~*). While quadrupole correctors in every cell
to maintain the correct phase advance per cell will compensate the
change in gradient, the compensation of the sex­tupole component is a
more serious issue. If the nonlinear fields due to magnet saturation
have a significant impact on the dynamic aperture, then this could
potentially limit the maximum energy reach of the low field collider.

The straightforward approach to compensating the saturation
*b~2\ ~*component is to use the chromaticity sextupoles in the cells to
correct the linear chromaticity. At large \| *b~2~*\| this leads to
strong chromaticity sextupoles lumped at two locations in the cell, one
SF next to the focusing magnet (F), the other SD next to the defocusing
magnet (D). A calculation of the dynamic aperture at 20 TeV without the
IR errors but with the *b~2\ ~*component in the arc magnets due to
saturation shows that the dynamic aperture is quite large even up to 10
units of *b~2.\ ~*At *b~2~=0,* integrated strengths required for the F
and D type sextupoles are (946, 1761) T/m respectively. At positive
*b~2~ ≤ 3,* the F sextupoles decrease and D sextupoles increase in
magnitude and vice versa at negative *b~2~ ≥ -3.* At larger absolute
values of *b~2~,* both sextupole strengths increase in magnitude. If
required, additional sextupoles could be placed in the 1 m gap between
the F and D and the D and F magnets to correct the chromaticity
generated by the iron saturation. This would lower the integrated
sextupole strengths and allow for larger values of *b~2\ ~*to be
compensated.

The impact of magnet saturation should, however, be determined in the
presence of the dominant field errors of the IR quadrupoles. We have
based our dynamic aperture calculations on the expected field errors
calculated by G. Ambrosio et al. \[[^3]\].

The dynamic aperture was calculated for three seeds for the random
errors with the signs of the uncertainties in the systematics
alternating between positive and negative. The random errors were
truncated at 3σ. For each seed the dynamic aperture was calculated with
-10 ≤ *b~2~* ≤ 10. At *b~2~ = 0*, the dynamic aperture drops to around
20σ in the presence of the IR errors, compared to a value of around 130σ
without these errors but only chromaticity sextupoles. The average
dynamic aperture varies over a small range of 20-23 σ when *b~2~* ≤ 10.
Clearly the saturation sextupoles do not have a significant impact on
the dynamic aperture at 20 TeV. The maximum tolerable value of
*b~2\ ~*may instead be limited by the sextupole strengths required to
correct the chromaticity.

#### 3.2.2.4 Dynamic Aperture with Synchrotron Oscillations {#dynamic-aperture-with-synchrotron-oscillations .unnumbered}

After correction of the orbit due to misaligned magnets, the rms closed
orbit deviations were reduced to \<*x*\>~rms~ = 0.22 mm, \<*y*\>~rms~ =
0.21 mm. The fractional tunes were *ν~x~* = 0.585, *ν~y~* =0.573. Figure
3.7 shows the average and minimum dynamic aperture with synchrotron
oscil­lations and closed orbit errors as a function of the momentum
deviation amplitude. The average and minimum are taken over three random
seeds. The average dynamic aperture at *δp/p* = 0 is 19*σ* while at
*δp/p* = 3 × 10^-4^ ≈ 3*σ~p~*, the average dynamic aperture drops to
about 11*σ* which is only slightly under the target value.

![](media/image9.wmf){width="3.865972222222222in"
height="3.342361111111111in"}

Figure 3.7. Dynamic aperture with synchrotron oscillations and closed
orbit errors. Particles were tracked for 10,000 turns and with three
random seeds.

### 3.2.3 Tolerances and Corrections {#tolerances-and-corrections .unnumbered}

#### 3.2.3.1 Closed Orbit {#closed-orbit .unnumbered}

Distortions of the ideal closed trajectory in the collider will be
primarily generated by trans­verse alignment errors. The trajectory in
the horizontal plane will also be influenced by errors in the main bend
field of the magnets, and the vertical trajectory will be further
distorted due to roll errors in the main bending field.

For a bending magnet of length *L* with gradient *B\'* which is
displaced transversely from its ideal position by an amount *d*, a
particle with magnetic rigidity *Bρ = p/e* will have its trajectory
deflected through an angle *θ = (B\'L/Bρ) d*. For the low field
collider, the gradient magnets have lengths roughly equal to the
half-cell length and will be supported along its length by several
adjustable magnet stands. If the arc magnets are divided into *n*
equal-length sections for support and alignment, then the rms expected
closed orbit deviations in the horizontal (*x*) and vertical (*y*)
degrees of freedom at maximum β locations in the arcs are given by

$Dx_{\text{co}} = \frac{\sqrt{\beta\left\langle \beta \right\rangle}}{2\left| \text{sin}\text{pν} \right|}\sqrt{\frac{N}{2}}\left( \frac{B_{0}L}{B\rho} \right)\sqrt{\frac{1}{n}\left( \frac{B'd_{\text{rms}}}{B_{0}} \right)^{2} + \left( \frac{DB}{B_{0}} \right)_{\text{rms}}^{2}}$
Eq. 3.6

and

$Dy_{\text{co}} = \frac{\sqrt{\beta\left\langle \beta \right\rangle}}{2\left| \text{sin}\text{pν} \right|}\sqrt{\frac{N}{2}}\left( \frac{B_{0}L}{B\rho} \right)\sqrt{\frac{1}{n}\left\lbrack \left( \frac{B'd_{\text{rms}}}{B_{0}} \right)^{2} + f_{\text{rms}}^{2} \right\rbrack}$
Eq. 3.7

where *(∆B/B~0~)~rms~* is the rms field strength deviation of the full
magnet, *d~rms~* is the rms displace­ment of the magnet sections, and
*φ~rms~* is the rms roll angle (with respect to vertical) at the *n*
locations within a half-cell. *N* is the number of half-cells in the
collider.

For typical values of the above variables we use *n* = 5 over a half
cell length of *L* = 67 m, *β*~max~= 411 m, \<*β*\> = 250 m, *B~0~* = 2
T, *B\'* = 9 T/m, *d~rms~* = 250 μm, *φ~rms~* = 0.5 mrad, and
*(∆B/B~0~)~rms~* = 10^-4^. From a distribution of errors over the 780
arc cells in the collider, we obtain expected rms orbit errors on the
scale of

$Dx_{\text{co}}»9\text{.}5\text{mm}$ Eq. 3.8

$Dy_{\text{co}}»\text{10}\text{.}3\text{mm}$ Eq. 3.9

The maximum uncorrected closed orbit excursions clearly venture outside
any reasonable beam pipe radius.

To correct the orbit distortions, we assume dipole correctors are placed
at maximum *β* lo­cations in the arcs (H & V), and a standard 3-bump
algorithm is performed. The corrector strength necessary to perform the
correction using the parameters introduced above would be

$Q_{x} = \frac{\sqrt{2}}{\text{sin}m}\sqrt{\frac{\left\langle \beta \right\rangle}{\beta}}\left( \frac{B_{0}L}{B\rho} \right)\sqrt{\frac{1}{n}\left( \frac{B'd_{\text{rms}}}{B_{0}} \right)^{2} + \left( \frac{DB}{B_{0}} \right)_{\text{rms}}^{2}}$
Eq. 3.10

and

$Q_{y} = \frac{\sqrt{2}}{\text{sin}m}\sqrt{\frac{\left\langle \beta \right\rangle}{\beta}}\left( \frac{B_{0}L}{B\rho} \right)\sqrt{\frac{1}{n}\left\lbrack \left( \frac{B'd_{\text{rms}}}{B_{0}} \right)^{2} + f_{\text{rms}}^{2} \right\rbrack}$
Eq. 3.11

where *μ* is the cell phase advance. For *μ* = 90°, and the same
parameter values as before, the rms corrector strengths are

$Q_{x}»2\text{.}3m\text{rad}$ Eq. 3.12

$Qy»3\text{.}2m\text{rad}$ Eq. 3.13

At 20 TeV, a 3 μrad bend angle requires an integrated field strength of
0.2 T-m. To be able to correct at the 2.5-σ level, steering correctors
should have strengths of approximately 0.5 T-m. The maximum corrector
strengths will be dominated by displacement errors caused by diffu­sive
ground movement and ground settlement over time.

#### 3.2.3.2 Quadrupole Effects {#quadrupole-effects .unnumbered}

The quadrupole gradient in the transmission line magnets drops steeply
at high fields due to saturation of the iron. Quadrupole correctors, one
in each plane, will be placed in every arc cell to maintain the required
phase advance of 90° per cell. At present these correctors are 0.5 m
long and placed at the beginning of each half-cell. The correctors,
independently powered, have been sized to correct the gradient shift due
to magnet saturation at 2 Tesla.

Sources of errors in the tune include gradient errors and closed orbit
distortions in the non­linear magnets. Quadrupole correctors in the arcs
will be used to control the global tunes and to split the integer tunes
to reduce the sensitivity to systematic linear coupling sources.

It will be important to restrict the beta beating to a low value
everywhere, both for on-momentum and off-momentum particles. It will be
especially important to isolate the interac­tion regions from
beta-beating errors generated elsewhere in the ring. Quadrupoles at the
entrances and exits of the arcs and at the entrances and exits of the
straight sections can be used to control both the amplitude and phase of
∆β/β.

Normal quadrupole gradient errors and skew quadrupole gradient errors
will contribute to horizontal and vertical dispersion beating
respectively. The errors generated in the arcs and utility straight
sections will be corrected either by quadrupole correctors or by the
closed orbit correctors. It will be important to keep the dispersion at
the IPs very small. Local dispersion correction within the IRs will be
necessary to correct the dispersion generated by the crossing angle.

#### 3.2.3.3 Coupling {#coupling .unnumbered}

Sources of coupling include skew quadrupole errors in the main magnets,
rolls of the gradient magnets and vertical orbit distortion through the
nonlinear magnets. Requiring that the mini­mum tune split be less than
0.05 before correction may set the tolerance for the allowed cou­pling.
Larger values make coupling correction with skew quadrupoles difficult.

Skew quadrupoles placed in regions of horizontal dispersion will
generate vertical disper­sion, which will need additional correction. It
is therefore preferable to place these quadrupoles in straight sections.
Two families of skew quadrupoles in the utility straight sections and
the IRs will be sufficient to control the effects due to the difference
resonance *ν~x~ - ν~y~ = p*. These families will be placed so that the
difference in phases *ψ~x~ -- ψ~y~* differs by *π*/2 at the families.
The coupling sum resonance *ν~x~ + ν~y~ = p* may also need to be
corrected (depending on the strength of the skew quadrupole errors). In
this case an additional two families will be required, placed so that
*ψ~x~ + ψ~y~* differs by *π*/2 at these families.

#### 3.2.3.4 Chromaticity Effects {#chromaticity-effects .unnumbered}

The linear chromaticity is controlled by two families of sextupoles, one
in each plane, placed in every arc cell. These families can be defined
in software, since all correctors are independently powered, as
described in Section 5.2.3. This will suffice provided the nonlinear
change of tune with momentum is small over the required momentum
aperture. If the global nonlinear chromaticity requires correction, then
two families of sextupoles in each plane for the 90°cells will be
required. It would there­fore be desirable to place sextupoles (of say
the F type) in every other cell on one bus and the other family of
sextupoles in the other cells on another bus.

In addition to the chromaticity due to the quadrupoles, there is a
contribution due to sex­tupole fields caused by iron saturation at fields
close to 2 Tesla. This chromaticity must also be compensated by the
chromaticity sextupoles. This has been discussed in the section on the
dynamic aperture at collision.

If the beta function is strongly dependent on the momentum, then it can
lead to an increased spot size if not corrected at the IPs and to
emittance growth due to injection errors if not corrected at the
injection point. Correcting the momentum dependence of the beta
functions will require additional sextupole families.

For the IRs, a local chromaticity correction system was proposed for the
SSC \[[^4]\], which increased the momentum aperture from about 2.5σ with
linear chromaticity correction to about 8σ with correction of the
nonlinear chromaticity generated by the IRs. A similar system may be
required for the IRs if the momentum aperture needs to be increased.

#### 3.2.3.5 Alignment Tolerances in the IRs {#alignment-tolerances-in-the-irs .unnumbered}

These tolerances are more stringent for the IR quadrupoles than for any
other magnets in the ring. Several criteria have to be met including:

-   Dispersion at the IP should be small, both to attain maximum
    luminosity and to avoid synchro-betatron resonances.

-   Beam excursions in the triplets should be limited to preserve the
    dynamic aperture.

-   Separations between the beams at the parasitic collision points
    should not decrease.

In the IR quadrupoles the beta functions change rapidly and are not
symmetric about the quadrupole center. As a consequence, pitch and yaw
misalignments will introduce substantial beam separations at the IPs if
not corrected. Due to the crossing angle a roll misalignment of an IR
quadrupole will introduce orbit shifts in the plane orthogonal to the
crossing plane. This is in addition to the extra coupling generated by
the roll angle. Thus misalignments in all six degrees of freedom of
these IR quadrupoles need to be well controlled and corrected.

##### **Orbit offsets at the IPs** {#orbit-offsets-at-the-ips .unnumbered}

The orbit offset at the IP *z~offset~(IP)* due to a transverse
displacement *∆z~Q~* of an IR quadrupole (assumed to be at 90 degrees
from the IP) is

$\frac{z_{\text{offset}}(\text{IP})}{{Dz}_{Q}}µ\frac{\sqrt{\beta^{}\beta_{Q}}}{F}$
Eq. 3.14

Here *β*~Q~ is the beta function at the quadrupole and *F* is the focal
length of the displaced quadrupole. The maximum offset of the quadrupole
that can be corrected with orbit correctors is proportional to this
ratio *z~offset~(IP)/ ∆z~Q~*. In Table 3.5 we compare the parameters for
the quadrupole where the beta function is the largest.

Tolerances for the LHC quadrupoles were derived in \[[^5]\]. The rms
tolerance for the offset was obtained to be 0.27 mm. Scaling this number
by the ratio 2.3/3.1, we obtain a tolerance of 0.2 mm for the VLHC IR
quadrupoles. This assumes that corrector strengths similar to those in
the LHC (in kick angle) will be used in the VLHC.

Table 3.5. Parameters of the strongest IR quadrupole in the LHC and the
VLHC.

  ------------------------------ --------------- -------------------------
  **Parameter**                  **LHC IR**      **VLHC (LF) IR**

  *β*^\*^ \[m\]                  0.5             0.3

  *β*~max~ \[m\]                 4567            11263

  Focal Length, *F* \[m\]        21.1            18.7

  *z*~offset~(IP)/*∆z*~Q~        \~2.3           ∼3.1
  ------------------------------ --------------- -------------------------

Correctors to correct the orbit at the IP are best placed at 90 degrees
from the IP. As in the LHC, correction packages with dipole and skew
dipole correction coils should be placed at several high beta locations
next to the IR quadrupoles. Spaces between Q2 and Q3 and after Q3 would
be ideal locations for these correctors.

##### **IR quadrupole roll angles** {#ir-quadrupole-roll-angles .unnumbered}

Tolerances on the allowed roll angles of the IR quadrupoles may be set
by specifying the minimum tune split tolerable before correction. We
will assume that this tune split due to roll angles in all sixteen
quadrupoles from the two IRs is less than 0.05 before correction. RMS
roll angles of 0.5 mrad can be tolerated for the IR quads, assuming that
all the IR quadrupole assemblies include a skew quadrupole corrector.

Normally two families of skew quadrupoles are required to correct both
the real and imagi­nary parts of the coupling resonance term. However the
phase advance through the IRs is very small and so the imaginary part of
the resonance term is negligibly small. Thus only a single skew
quadrupole on each side of the IP will suffice to locally correct the
coupling in the IR.

Table 3.6 summarizes the misalignment tolerances for the IR quadrupoles.
These are based on scaling from tolerances derived for the LHC and
should be replaced at a later stage with more exact calculations.

Table 3.6. Misalignment tolerances for the VLHC IR quadrupoles derived
by scaling from the LHC IR quadrupole tolerances \[Error: Reference
source not found\].

  ------------------------------------------------- ---------------------
  **Misalignment**                                  **Tolerance**

  Offsets \[mm\]                                    0.2

  Longitudinal placement \[mm\]                     1

  Roll \[mrad\]                                     0.5
  ------------------------------------------------- ---------------------

#### 3.2.3.6 Crossing Angle {#crossing-angle .unnumbered}

A crossing angle is required to separate the beams at the parasitic
collisions. As is well known, the crossing angle has several effects on
the linear and nonlinear optics and beam dynamics, including generation
of dispersion due to orbit offsets in the quadrupoles, coupling due to
skew multipole errors, and feed down of higher order multipoles. Local
dispersion control is required to prevent dispersion waves from
propagating into the arcs. This can be performed adjusting pairs of
quadrupoles in the cells adjacent to the IRs where there is natural
dispersion. A scheme such as this was proposed for the SSC \[[^6]\].
Correction of local coupling from this and other sources will require
skew quadrupoles in the IR as already mentioned. The linear effects from
feed-down of higher order multipoles will be corrected by orbit and tune
correctors. Nonlinear effects will require nonlinear correction packages
in the IRs similar to those in the LHC and RHIC.

### 3.2.4 Ground Motion and Emittance Growth {#ground-motion-and-emittance-growth .unnumbered}

We have measured the slow ground motion in the 300-ft deep dolomite
Conco mine in Aurora, IL, about 3 miles west of Fermilab. The goal of
the studies is to record and analyze vertical ground motion in 8 points
separated by 30 meters over a time interval of about 1 year. The
experiment in the Aurora mine started in October 2000. The first data
reveal that dolomite mine motion is rather small and can be described by
the ATL law with coefficient of *A* = (6.8 ± 2.7) × 10^-7^ μm^2^/m/s
\[[^7]\]. The resulting maximum orbit distortion in the VLHC which will
be caused by this coefficient is of the order of 9 mm/(year)^1/2^.

The only drawback of the Aurora mine experiment is that there are too
few sensors, which limits the statistics for studies of spatial
correlations. Therefore, a system with about 20 sensors is needed, and
long-term measurements should be repeated on a larger scale. Analytical/
computer studies of the slow ground motion effects and orbit correction
system and procedures must be performed.

Turn-to-turn dipole magnetic field fluctuations and vibration of
quadrupole magnets are of concern, because they can excite coherent beam
motion. If the motion is not corrected over the *decoherence time* of
about 1/*ξ*~beam-beam~ ≈ 1000 turns, then the coherent motion will be
converted into transverse emittance increase and can cause substantial
emittance growth over about 10 hours of the store. So far,
experimentally measured high-frequency ground or magnet vibra­tions do
not pose a big problem for VLHC \[[^8]\]. Nevertheless, more detailed
analysis on how the lattice may affect the tolerances is needed. For
example reference \[[^9]\] claims that tolerances on vibration amplitude
in a lattice with combined function magnets is about 3 times the
tolerances in a separated function FODO lattices.

In 1999 experimental studies of high-frequency magnetic field
fluctuations were carried out in a Tevatron superconducting dipole
magnet. An effective rms dB/B ≈ 10^-10^ which is about 10 times the
tolerance for the VLHC, was measured. This amplitude could cause
emittance doubling after about 6 minutes. The measurements have to be
repeated in the VLHC dipole prototypes. Again, we need to analyze and
perform computer simulations for lattices with separated and combined
function magnets.

Suppression of the emittance growth can be provided by low noise
feedback. The first analysis of the FB is presented in \[[^10]\]. The
system has to have sub-micron equivalent input noise. Design and test of
such a system at the Tevatron is an important VLHC beam physics R&D
task.

### 3.2.5 Collective Effects {#collective-effects .unnumbered}

A VLHC Instability Workshop was held at SLAC, March 21-23, 2001
\[[^11]\]. Most instability issues examined at this workshop are not
expected to be serious concerns for the VLHC. Two exceptions are the
transverse mode-coupling instability (TMCI) and the transverse resistive
wall instability, which are both most pronounced for the Stage-1
collider at injection. In par­ticular, the design intensity of 2.5 ×
10^10^ particles per bunch is a factor of two higher than the expected
TMCI threshold. The resistive wall instability will require an extension
of the state-of-the-art feedback system. To compensate for the very
large incoherent and coherent tune shifts, as well as tune variation
along the bunch train due to resistive wall effects, the ring will need
to be filled in an appropriate bunch pattern to mitigate this issue.

The rings of both stages have the same size, *2πR* = 233.037 km in
circumference, and the same RF frequency. In order to have shorter
bunches, the RF frequencies in both phases are chosen 9 times higher
than that of the Tevatron, so that the bunches will be separated by 9 RF
buckets. The RF harmonic is *h* = 371520, which is the Tevatron harmonic
number scaled to the VLHC circumference.

In the low-field rings, the RF voltage has been chosen to be *V~rf~* =
50 MV. The rms bunch area of bunches inside the buckets is chosen to be
0.4 eV-sec, the rms bunch area \[[^12]\] of the Tevatron bunches at 900
GeV. With *V~rf~* = 50 MV, the maximum energy spread is 2 × 10^-3^, and
the bucket area is 5 eV-sec. Note that the present Tevatron RF system
has a maximum voltage of 1 MV only, thus the bunch length would be too
long and the momentum spread too small to fit the higher harmonic
buckets at injection. Thus, the Tevatron bunches must be shortened by
either a bunch rotation or through the installation of a higher RF
voltage before the transfer to the 20 TeV ring is possible.

A bunch excited by forces in the vacuum chamber oscillates in the
longitudinal phase space in modes describable by the radial mode
parameter *n~r~* designating *n~r~* radial nodes and the azimuthal mode
parameter *m* designating *\|m\|* azimuthal nodes. For a given *m,* the
most easily excited radial mode is *n~r~ = \|m\|*. Almost distribution
independent, the spectrum for the mode designated by *m* ≠ 0 peaks at
frequency *f(m) ≈ (\|m\|+1)/2τ~L~*, where *τ~L~* is the *total* length
of the bunch in time. For a Gaussian distribution, we approximate it by
the 95% length *τ~L~* = 2√6*σ*~τ~,, where *σ*~τ~ is the rms bunch
length. We note that the frequencies for the various VLHC bunch modes of
interest are in the GHz range.

#### 3.2.5.1 Transverse Mode Coupling Instability {#transverse-mode-coupling-instability .unnumbered}

The TMCI threshold for the resistive wall has been calculated in various
ways, including analytical formulas, a matrix approach for up to 25
modes, and through particle tracking simulations \[Error: Reference
source not found\]. The analytic formulas estimate a threshold
approximately 40% larger than the matrix approach. The difference
between the matrix approach and particle tracking was about 30%. These
discrepancies were discussed at the workshop, and future calculations
and checks have been planned.

The problem is very important for the low field VLHC. These calculations
show that the threshold number for the bunch population at injection
with the nominal set of parameters is equal to 1.4 × 10^10^ protons,
about half the nominal intensity. The above estimate assumes a round
beam pipe of 9 mm radius. The threshold estimate needs to be redone for
the nominal oval-shaped beam pipe.

Assuming that TMCI will exist in the VLHC, the following measures could
be a cure:

-   Inject low intensity bunches and coalesce them to make high
    intensity bunches at high field where the TMCI threshold is
    increased.

-   Provide active feedback (resistive damping) at a few low order modes
    *l* = ±1, ±2,· · ·

-   Introduce an RF quadrupole to provide a tune shift between the head
    and tail of the bunch.

For this reason, a bunch coalescing scheme has been adopted, which is
described in a subsec­tion below. The other two measures will continue to
be studied.

The TMCI theory for proton beams is technically challenging because the
bunches are longer and other forces may be present. Experimental data
for comparison would be most useful. For instance, it may be possible to
study the TMCI using the Tevatron Electron Lens (TEL).

#### 3.2.5.2 Resistive Wall Effects {#resistive-wall-effects .unnumbered}

For the low field ring parameter range the various formulae for the
resistive wall impedance agree within a factor of two. The e-folding
time for the low frequency resistive wall instability is less than 1
turn. Additionally, the very low revolution frequency leads to a
variation in the magnetic image Laslett tune shift when the ring is
partially filled during the injection process. The variation is due to
the fact that the revolution period is comparable to the magnetic
diffu­sion time through the beam pipe. Initial estimates show a DC tune
shift on the order of 0.5 units, with an AC component of about 0.1 units
due to the time dependence of the magnetic diffusion into the beam pipe.
Increasing the vacuum pipe thickness and/or radius could reduce both of
these problems. Increasing the dipole magnet gap can reduce the DC
Laslett tune shift, while the variation in tune along the bunch train
could be reduced by quadrupoles running at multiples of the revolution
frequency. It is envisioned, however, that injecting beam in an
appropriately symmetric manner, with bunches entering the ring far apart
from each other and gradually filling in between the already injected
bunches, the tune variations can be reduced to a tolerable level.

The resistive wall instability needs to be damped. Realistic studies of
the feedback system are needed. To fight the transverse resistive wall
instability, a fancy feedback system is envi­sioned \[Error: Reference
source not found\]. The system would have high-gain, low bandwidth
(perhaps 100 kHz) and the pickup signal would arrive at the kicker
slightly behind the beam bunch that produced the signal. This feedback
system indeed is capable of suppressing the dipole multi-bunch resistive
wall growth time of one turn. It is not 100% clear exactly how fast the
growth rate is because the theory of resistive wall impedance in this
complicated environment (cold, thin coating, magnets outside, etc.) is
not easy to calculate. Whether it can suppress beyond one turn is yet to
be demonstrated.

#### 3.2.5.3 Bunch Coalescing in the Stage-1 Collider {#bunch-coalescing-in-the-stage-1-collider .unnumbered}

To alleviate beam instabilities at its injection energy, the Stage-1
collider will have a second RF system for performing bunch coalescing.
Five bunches each containing approximately 0.5 × 10^10^ protons will be
coalesced into one bunch of 2.5 × 10^10^ using a 53.1 MHz system in the
Stage-1 collider \[[^13]\]. If this is performed at 20 TeV, then the 478
MHz system voltage must be reduced from 50 MV to about 6.5 MV and the
bunches allowed to rotate 1/4 of a synchrotron period. The main RF
system is then turned off, and the second RF system, operating at 53.1
MHz, is snapped on with voltage of 2-3 MV. After 1/4 period in this
system, the 53.1 MHz RF is turned back off and the 478 MHz turned on
again at approximately 20 MV. The RF voltage is then raised
adiabatically to 50 MV completing the process.

This system naturally implies that a 478 MHz RF system be employed at
some early stage in the Tevatron injector system. The details of this
injector upgrade are yet to be worked out.

## 3.3 Stage 2 -- the High Field Ring {#stage-2-the-high-field-ring .unnumbered}

### 3.3.1 Luminosity versus Energy {#luminosity-versus-energy .unnumbered}

The High Field VLHC ring will be the first cryogenic collider to operate
in the synchrotron radiation dominated regime, in which the radiation
damping time is shorter than the storage time. In this regime there are
practical and economic limits to the cryogenic system that can be
installed. There is a maximum value *P~cryo~* for the synchrotron
radiation, which can be absorbed in each ring. It is necessary to adjust
the beam parameters to stay within the installed power limit, resulting
in an average luminosity that depends on the beam energy *E* according
to a "maximum power law,"

$L_{\text{ave}} < \frac{P_{\text{cryo}}}{E}\left( \frac{1}{N_{\text{IP}}s_{\text{tot}}} \right)\left( \frac{T_{0}}{T_{\text{store}}} \right)$
Eq. 3.15

Although there is a clear advantage in reducing the storage time,
*T~store~* must remain signifi­cantly larger than the damping time *T~0~*
in order to take advantage of radiatively damped beam sizes. Other
factors (such as the refill time) will also play a role in determining
the optimum value of *T~store~* , but it is safe to estimate that

$\frac{T_{0}}{T_{\text{store}}}»0\text{.}2$ Eq. 3.16

Except for some uncertainty in this factor, the maximum power law
clearly states that the installed capacity to absorb synchrotron
radiation at cryogenic temperatures directly limits the attainable
product of average luminosity and energy. This trade-off between higher
luminosity and higher energy was emphasized in Table 1.2 in Chapter 1.

Similarly, if the beam stored energy per ring *U = NME* must be kept
below a maximum value \-- for example if the beam dump has a limited
capacity \-- then

$L_{\text{ave}} < \frac{U_{\text{max}}}{E}\left( \frac{1}{N_{\text{IP}}s_{\text{tot}}} \right)\left( \frac{1}{T_{\text{store}}} \right)$
Eq. 3.17

Insofar as the stored energy is a practical limit to high field
performance \-- to the product *L~ave~E* \-- then there is pressure to
reduce *T~store~* , and hence to increase the dipole field, to reduce
the circumference, and to reduce the refill time.

### 3.3.2 Operational Performance {#operational-performance .unnumbered}

For the Design Study, the most straightforward approach is to use as a
baseline standard triplet optics and round beams, as in previous hadron
colliders. It should be noted, however, that the High Field ring will be
the first hadron collider with the potential to take advantage of
naturally flat beams. While still under discussion, flat beams are
sufficiently new and interesting for hadron colliders that most of the
Stage-2 Design Study effort was devoted to analyzing this case up to
now. The parameters described below have assumed that the beam
emittances evolve into an asymmetric state characterized by *κ =
ε~y~/ε~x~.* The high field ring should be able to achieve *κ* \<\< 0.1
during a store, consistent with conventional electron storage rings.

Table 3.7. Nominal parameters for stores in the high field ring, using
flat beams and doublet IR optics.

  ----------------------------------------------- ----------- ---------------
  Storage energy, *E*                             87.5        TeV

  Peak luminosity, *L*~max~                       2 *x*       cm^-2^sec^-1^
                                                  10^34^      

  Collision debris power, per IP                  73          kW

  Dipole field at storage                         9.765       T

  Distance from IP to first magnet                30          M

  Injection energy                                10          TeV

  Fill time                                       30          sec

  Acceleration time                               2000        sec

  Synchrotron radiation damping time, *T*~0~      2.48        hours

  Energy loss per turn, *U*~0~                    15.3        MeV

  Natural transverse emittance (H)                0.0397      μm

  Natural RMS momentum width                      5.5         10^-6^

  Collision beta horz, *β^\*^~x~*                 3.7         m

  Collision beta vert, *β^\*^~y~*                 0.37        m

  Equilibrium emittance ratio, *κ*                0.1         

  Initial bunch intensity, *N*                    7.5         10^9^

  Beam current                                    57.4        mA

  Synch. rad. power, per beam, *P*                .88         MW

  Dipole linear heat load                         4.7         W/m

  Stored energy, per beam, *U*                    3.9         GJ
  ----------------------------------------------- ----------- ---------------

The instantaneous luminosity is

$L = \frac{M}{4pT}\frac{N^{2}g}{\sqrt{e_{x}e_{y}\beta_{x}^{}\beta_{y}^{}}}$
Eq. 3.18

where *T* is the revolution period. The total number of protons in the
ring, *MN*, is approximately set by the need to provide enough for
"luminosity burn-off." Since the number of bunches *M* is fixed, the
need for a given peak luminosity then sets the single bunch population
*N*. Nominal values for these and other parameters, including a
conservative value of *κ* = 0.1, are given in Table 3.7.

It is the head on beam-beam interaction that sets the minimum horizontal
emittance, whether the beams are flat or round. Figure 3.8 shows the
horizontal and vertical emittances decreasing to plateau values of
*ε~x~* and *ε~y~* that are consistent with the beam-beam limit, and
which are maintained by transverse beam heating. Figure 3.9 shows the
corresponding evolu­tion of instantaneous luminosity, and its average,
during the store.

#### 3.3.2.1 The Head-on Beam-beam Interaction {#the-head-on-beam-beam-interaction .unnumbered}

The horizontal and vertical tune shift parameters for bi-Gaussian round
beams *(β^\*^~x~ = β^\*^~y~, ε~x~ = ε~y~*) are

$x_{x} = x_{y} = \frac{r}{4p}\frac{N}{e_{x}}$ Eq. 3.19

![](media/image10.wmf){width="4.7243055555555555in"
height="3.5104166666666665in"}

Figure 3.8. Evolution of the transverse emittances and the rms momentum
spread during a store.

![](media/image11.wmf){width="4.079166666666667in"
height="3.029861111111111in"}

Figure 3.9. Evolution of the instantaneous and average luminosities,
during a store.

The beam-beam limit is expected to be approximately the same for round
or flat beams, given by

> *ξ* ≤ 0.008.

This value is justified by practical experience at the SPS and at the
Tevatron. The SPS operated at *ξ* ≈ 0.004 (or slightly higher) with 6
collisions per turn. More recently, in Run Ib the Teva­tron operated with
*ξ* ≈ 0.0075 with only 2 head on collisions per turn (as in the VLHC).
Simu­lations predict that radiation damping might give the high field
ring a slight additional advan­tage, which is by no means as strong as
that commonly observed in electron storage rings. The numerical value of
0.008 is illustrated in Figure 3.10, which displays empirical data
compiled by Keil and Talman for electron storage rings \[[^14]\]. The
"damping decrement" for the high field VLHC---the fraction of a damping
period per head on collision---is approximately 10^-7^.

Then, re-writing Eqn. 3.18 as

$L = \frac{M}{4pT}\frac{N^{2}g}{\text{ke}_{x}\beta_{x}^{}}$ Eq. 3.22

it is explicitly clear that with flat beams the value of *β*^\*^~x~ can
be increased by a factor of about 1/*κ*, significantly simplifying the
optics. Equation 3.21 then shows that the value of *β*^\*^~y~ is about
the same for flat or round beams.

![](media/image12.wmf){width="4.8694444444444445in"
height="3.626388888888889in"}

Figure 3.10. Maximum beam-beam parameter vs. damping decrement.

#### 3.3.2.2 Crossing Angles and Parasitic Beam-beam Collisions {#crossing-angles-and-parasitic-beam-beam-collisions .unnumbered}

The horizontal and vertical tune shifts due to a single parasitic
collision are

${Dq}_{x,y}» \pm \frac{\text{rN}}{2\text{pg}}\frac{\beta_{x,y}}{D^{2}}$
Eq. 3.23

where the approximation is valid if the full beam separation *∆* is much
greater than both horizontal and vertical beam sizes, so that the beam
acts like a moving line charge.

If the beams are thoroughly separated (into separate beam pipes, or with
very large separa­tions) at a distance *L~sep~* from the IP, then there
are *4L~sep~/S~B~* parasitic collisions around each IP, where *S~B~* is
the longitudinal bunch separation. The total tune shifts from all
parasitic collisions with crossing angle *α* in one interaction region
are

${DQ}_{x,y}» \pm \frac{\text{rN}}{2\text{pg}}\frac{4}{S_{B}}\frac{L_{\text{sep}}}{\beta_{x,y}^{}\alpha^{2}}$
Eq. 3.24

where the approximation is most valid if the beams are fully separated
before the first IR quadrupole.

Again, assuming one exercises the flat beam option, the horizontal beta
function at each collision is much less than the vertical

$\frac{\beta_{x}}{\beta_{y}}»\frac{\beta_{y}^{}}{\beta_{x}^{}} = k$ Eq.
3.25

so that, taking Eqns 3.24 and 3.25 together gives

${DQ}_{x}\text{»-}{kDQ}_{y}$ Eq. 3.26

The horizontal tune shift is greatly suppressed with flat beams, to be
much smaller than it is with round beams \-- if the values of *L~sep~*
and *α* compare favorably between the two cases.

### 3.3.3 Advantages and Disadvantages of Flat Beams {#advantages-and-disadvantages-of-flat-beams .unnumbered}

All electron colliders, whether circular or straight, take advantage of
flat beams and use doublet optics. However, it has never before been
possible to use flat beams in a hadron collider. Flat beams have the
advantages and disadvantages discussed below, which continue to be
investigated.

Flat beams require the first quadrupole on both sides of the IP to be
vertically focusing to both counter-rotating beams (whether the crossing
angle is vertical or horizontal). Thus, the optics must be symmetric
across the IP, and the first quadrupole must be a 2-in-1 magnet. As seen
in the next Section, both optical solutions have been developed.

Table 3.8 compares the performance of flat and round beam in the high
field ring. In both cases the beam-beam limit of *ξ* = 0.008 is reached
5 or 6 hours into the store, when the lumi­nosity is at its peak. The
horizontal and vertical emittances values recorded in Table 3.8 are
those that are initially maintained (preventing further emittance
shrinkage) when the horizontal and vertical beam-beam parameters first
saturate. The same peak luminosity is achieved in both flat and round
beam cases by adjusting the horizontal and vertical *β^\*^* values.

The value of the total crossing angle α is set to be identical with flat
or round beams---and is independent of the plane of the crossing angle.
The major advantages and disadvantages of flat beams are outlined in the
list below:

##### **Flat Beam Advantages** {#flat-beam-advantages .unnumbered}

1.  Order of magnitude increase of *β^\*^~x\ ~*, and subsequent
    reduction of horizontal *β~max~* in the doublet quadrupoles. While
    early separation pushes the first quadrupole further from the IP,
    the values of *β~max\ ~*, both horizontal and vertical, are still
    reduced, as indicated in Table 3.8.

2.  There are far fewer parasitic collisions per interaction region with
    flat beams than with round beams, greatly reducing the long-range
    (especially horizontal) tune shift.

##### **Flat Beam Disadvantages** {#flat-beam-disadvantages .unnumbered}

1.  The design of the first quadrupole and splitting dipole are
    difficult. A 2-in-1 quadrupole with relatively close separation will
    not have as good field quality a single bore element. But, with
    lower *β~max~* in the doublet design, the tolerable field errors are
    likely to be larger.

```{=html}
<!-- -->
```
3.  Neutral particles generated at the IP will aim head-on for the
    conductor located at the center of the first 2-in-1 quadrupole.

4.  Lack of energy flexibility may be a disadvantage to flat beams. At
    lower energies, damping times are longer and thus flat beams are no
    longer viable. (However, a lattice has been demonstrated in which 4
    IR quadrupoles can act as a doublet or as a triplet \[[^15]\].)

5.  An operational disadvantage is the need for careful tuning to keep
    the vertical emittance small. Electron rings, however, routinely
    achieve beam flatness in the range 0.001 \< *κ* \< 0.01 by
    controlling dispersion and global betatron coupling. Flatness values
    of *κ* ≈ 0.01 should be achievable.

Table 3.8. Flat and round beam performance parameters after about 6
hours into the store, just after peak luminosity, when both horizontal
and vertical beam-beam parameters are saturated.

  ----------------------------------------------- ---------- -------------
                                                  **FLAT**   **ROUND**

  Flatness parameter, *κ*                         0.1        1

  Beam-beam parameter *ξ~x~ = ξ~y~*               0.008      0.008

  Peak luminosity *L* (10^34^ cm^-2^sec^-1^)      2.0        2.0

                                                             

  Initial bunch intensity *N*(10^9^)              7.5        7.5

  Collision beta horz *β^\*^*~x~ (m)              3.7        0.71

  Collision beta vert *β^\*^*~y~ (m)              0.37       0.71

  Maximum beta horz *β*^\*^~max~ (km)             7.84       14.58

  Maximum beta vert *β^\*^*~max~ (km)             10.75      14.58

  Horizontal emittance *ε~x~* (μm)                0.161      0.082

  Vertical emittance *ε~y~* (μm)                  0.016      0.082

  Collision beam size horz *σ^\*^~x~* (μm)        2.53       0.79

  Collision beam size vert *σ^\*^~y~* (μm)        0.25       0.79

  Maximum beam size horz *σ^\*^~x-max~* (μm)      116        113

  Maximum beam size vert *σ^\*^~y-max~* (μm)      43         113

                                                             

  Total crossing angle *α* (μrad)                 10.0       10.0

  Separation distance, *L*~sep~ (m)               30         120

  Number of long range collisions per IR          20         84

  Long range tune shift per IR, horz *∆Q~x~*      0.0008     0.0166

  Long range tune shift per IR, vert *∆Q~y~*      0.0081     0.0166
  ----------------------------------------------- ---------- -------------

A round beam solution for the VLHC is in hand, including IR magnet
designs and optical layouts. Because of the difficult design of a 2-in-1
quadrupole magnet for the doublet optics, the flat beam scenario is at
this time a promising upgrade path. The flat beam case with its possible
advantages merits further work and discussion and will continue to be
studied over the coming months.

### 3.3.4 Lattice {#lattice-1 .unnumbered}

The lattice geometrically matches the low field (LF) ring lattice so
that they both fit in the same tunnel. See Table 3.9 for a selection of
HF lattice parameters. The arcs are composed of 90 degree FODO cells
using separated function magnets, as are the dispersion suppressors.

#### 3.3.4.1 Triplet IR Optics Approach {#triplet-ir-optics-approach .unnumbered}

Using interaction region optics standard in modern hadron colliders,
collisions are created using anti-symmetric triplets for the final
focusing. The "round beam" model discussed here is qualitatively similar
to the low-field IR design. The triplet quadrupoles are 400 T/m
single-bore magnets. Four additional circuits, comprising double-bore
400 T/m magnets are also used for optical matching. With a total of 6
independently-tunable quadrupole circuits available it is possible to
match the four *β*\'s and *α*\'s from the IP into the regular FODO
cells, plus hold the phase advance *∆μ* constant across the IR through
the squeeze from *β^\*\ ^*= 12.0 → 0.50 m. Fixing *∆μ* eliminates the
need for a special phase trombone somewhere in the ring to maintain the
nominal operating point. Figure 3.11 shows the lattice functions through
the insertion region at collision.

Table 3.9. Short list of some high field lattice parameters.

  --------------------------------------------- --------------- ---------
  Horizontal tune                               218.19          

  Vertical tune                                 212.18          

  Transition gamma                              194.13          

  Maximum arc beta                              459             m

  Maximum arc dispersion                        1.42            m

  Vertical *β* (store)                          0.37            m

  Horizontal *β* (store)                        3.7             m

  Maximum *β,* injection                        0.61            km

  Maximum Horz *β,* store                       7.84            km

  Maximum Vert *β,* store                       10.75           km
  --------------------------------------------- --------------- ---------

The circulating beams are separated vertically everywhere in the ring,
except in the triplet quadrupoles. Four 10 T dipoles between the Q3 and
Q4 quadrupoles bring the beams together at the entrance to the triplet
for collisions at the IP. Dipoles downstream of the IP separate the
beams again vertically and channel them back into the upper and lower
rings. A half-crossing angle of 28.8 μr gives 10*σ* separation between
the beams at the first parasitic crossing 2.823 m downstream of the IP
(*ε~N~* = 1.5 π μm at 87.5 TeV/c).

#### 3.3.4.2 Doublet IR Optics Approach {#doublet-ir-optics-approach .unnumbered}

Doublet optics are more naturally suited to the flat beams of the
high-field ring. The IR gives 30 m free space from the interaction point
to the first magnetic element. For the doublet optics, the first magnet
encountered is a high field small bore magnet of 16 T field. This can
handle the beams until the total separation becomes 8 mm. Then the beams
enter a lower field, 12 T, magnet with a larger bore. These beam
separation dipoles bend the beam vertically. This crossing region
requires that the beam separation at the first quadrupole be 30 mm. With
these dimensions the first quadrupole is limited to 400 T/m gradient. As
the beams separate, a higher quadrupole field can be attained with a
limit of 600 T/m.

![](media/image13.wmf){width="6.166666666666667in" height="4.1125in"}

Figure 3.11 Lattice functions near the IP in collision optics, using
triplet focusing.

With this design of the crossing region, two interaction regions can fit
in the on-site cluster region. The *β^\*^* can be varied from 0.37 m to
7.12 m (vertically), *β^\*\ ^* horizontally is 10 times larger). The
maximum beta is 10.6 km with the collision optics and only about 760 m
at injection. Figure 3.12 shows the collision optics of the interaction
region.

### 3.3.5 Magnet Apertures and Field Quality {#magnet-apertures-and-field-quality-1 .unnumbered}

The effect of the magnetic field quality on the dynamic aperture was
investigated at the injec­tion energy of 10 TeV. At injection the beam
size has its maximum and errors in the arc magnets dominate the dynamic
aperture. With the current one-turn injection scenario, a storage time
on the order of seconds is sufficient. To estimate the effect of the
magnetic field errors on the single particle dynamic, the nonlinear
chromaticity, the tune spread from nonlinear fields and the 1000-turn
dynamic aperture were computed. 1000 turns is equivalent to 0.7 seconds
real time in the VLHC. Experiments have shown that the dynamic aperture
can be computed for storage rings with nonlinear field errors within a
30% error when the field errors are well known \[[^16],[^17]\]. For a
future machine, however, a larger safety margin is required.

![](media/image14.wmf){width="4.9527777777777775in" height="3.68125in"}

Figure 3.12. Lattice functions near the IP in collision optics, using
doublet focusing.

In the lattice that is used for the evaluation, no interaction regions
were inserted since the arcs dominate the dynamic aperture at injection.
Arc magnet errors at injection were estimated for a 2.0 T field in the
dipoles. The harmonic multipole content used in the simulation is given
in Table 3.10. The systematic errors were computed as the maximum
possible absolute value with given mean and uncertainties. Random errors
were created from a Gaussian distribution, cut at two sigma.

Table 3.10. Systematic \<\> and random σ of multipole coefficients used
for Stage-2 tracking at injection. The coefficients are shown in units
of 10^-4^ at a reference radius of 10 mm.

  ------------------ -------------- -------------- --------------- ---------------
  **Order, *n***     **\<b~n~\>**   **\<a~n~\>**   ***σ*(b~n~)**   ***σ*(a~n~)**

  1                  0.000          0.000          1.20            1.20

  2                  -2.000         0.000          0.85            0.85

  3                  0.000          0.000          0.35            0.35

  4                  0.500          0.000          0.12            0.12

  5                  -0.006         0.000          0.05            0.05

  6                  -0.004         0.000          0.02            0.02
  ------------------ -------------- -------------- --------------- ---------------

Quadrupoles were horizontally and vertically displaced randomly with a
Gaussian distribu­tion with a 0.3 mm rms value, cut at three sigma. In
addition, beam position monitors were displaced with a 0.2 mm rms value.
The resulting non-zero closed orbit was corrected to zero in the beam
position monitors with horizontal orbit correctors at focusing
quadrupoles and vertical orbit correctors at defocusing quadrupoles.
Skew quadrupole errors in the arcs were disre­garded and no coupling
correction was necessary. The transverse tunes were set to (218.190,
212.180). Both horizontal and vertical chromaticities were set to 2.

No synchrotron motion was included but particles had a relative momentum
deviation of *∆p/p* = 7.2 × 10^-4^, corresponding to three times the rms
of the momentum distribution. The physical aperture of 10 mm in the arcs
corresponds to 35*σ* of the transverse beam size in the quadrupoles.
Figure 3.13 shows dynamic aperture averaged over 10 seeds along with the
minimum of the 10 seeds. The error bars of the average dynamic aperture
are the rms value of 10 seeds.

Although linear coupling and synchrotron motion were disregarded in the
determination of the dynamic aperture, the field quality with the magnet
field errors used is sufficient for injec­tion. At collision, we expect
the dynamic aperture to be determined by the errors in the IR
quadrupoles. Detailed modeling remains to be performed.

![](media/image15.wmf){width="6.004166666666666in"
height="4.463888888888889in"}

Figure 3.13. 1000-turn dynamic aperture at injection. Shown are the
average and minimum over 10 seeds, and the physical aperture, all in
units of the transverse rms beam size.

### 3.3.6 Tolerances and Corrections {#tolerances-and-corrections-1 .unnumbered}

In this section we consider the effect of magnet misalignments as well
as the effect of main dipole and quadrupole field errors on the beam
orbit and betatron tunes. We discuss the toler­ances for misalignment and
field errors listed in Table 3.11. To this purpose we use the high-field
storage optics with *β^\*^~x~* = 3.7 m and *β^\*^~y~* = 0.37 m.

Table 3.11. Tolerance summary for magnet alignment.

  -------------------------------------- ------------ -------- -----------
  **RMS ERROR**                          **ARC**      **IR**   

  QUADRUPOLES                                                  

  Transverse offset                      0.25         0.2      mm

  BPM to quad offset                     0.15         0.1      mm

  Roll                                   0.5          0.5      mrad

  Field Error *∆K/K*                     0.5          0.5      10^-3^

  DIPOLES                                                      

  Roll                                   0.5          0.5      mrad

  Field Error *∆B/B*                     0.5          0.5      10^-3^
  -------------------------------------- ------------ -------- -----------

#### 3.3.6.1 Orbit Errors {#orbit-errors .unnumbered}

Possible sources of orbit error include transverse quadrupole
misalignments (in both the horizontal and vertical directions), main
dipole field integral errors (for the horizontal orbit), and roll of the
main dipole magnets (for the vertical orbit). In order to estimate and
compare the value of the orbit distortion coming from these different
types of error sources, the rms closed orbit has been calculated. The
calculations assumed the rms magnet misalignments listed in Table 3.11.
Each of five different individual perturbation sources were considered
in turn. In some cases the errors are introduced in all magnets in the
accelerator arcs, while in others the errors are applied to the
quadrupoles of one of the interaction region doublets. In the
interaction region doublets there is one corrector per plane. The
biggest effect comes from quadrupole offsets. The dipole integral field
error and dipole roll errors quoted in Table 3.11 have little influence
on the required corrector strength.

#### 3.3.6.2 Tune Error, Coupling and IR Chromatic Effects {#tune-error-coupling-and-ir-chromatic-effects .unnumbered}

The betatron tune shift caused by an rms quadrupole field error of
10^-4^ and the betatron cou­pling effect produced by rms quadrupole rolls
of 1 mrad have been calculated. The tolerance for arc quadrupole error
can be put at 5 × 10^-4^ providing a 2.5 rms beta-beating value of about
9% is acceptable. A small number of quadrupole correctors placed in the
arcs will easily compensate for this beta-beating. Field errors in
interaction region quadrupoles can be cor­rected by adjusting the
currents of power supply for these quads. After the current adjustment
is done the current fluctuations should be kept well below the 10^-4^
level. Both global correction in the arcs and local skew quadrupole
correction in the interaction regions are required for the compensation
of betatron coupling effects. With these correction systems in place it
is possible to relax the quad roll tolerances to about 0.5 mrad.

Because of the very large maximum *β* values in the interaction region
quadrupoles, the chromatic effects produced by these quadrupoles have
been evaluated. For the design value of rms momentum spread of 5 ×
10^-5^ the amplitude of the chromatic vertical beta-wave was found to be
just a few percent. Thus the effect is not large, mainly due to the
small rms momentum spread. The vertical chromaticity produced by the
quadrupoles from one interaction region has been found to be about -42.
The horizontal chromatic effects are smaller than the vertical.

#### 3.3.6.3 Corrector Strengths {#corrector-strengths .unnumbered}

The maximum dipole corrector strengths displayed in Table 3.12 are 2.5
times the rms value when all error sources are combined in quadrature.
Assuming a maximum achievable field of 3.5 T for the IR dipole
correctors, and 2.2 T for the arc dipole correctors, magnetic lengths of
1.05 m in the arcs and 2.5 m in the IRs are required. If the maximum
corrector strengths are increased to higher values than those cited in
Table 3.12 then the corresponding tolerances in Table 3.11 can be
relaxed accordingly.

Even the maximum dipole corrector strengths listed in Table 3.12 cannot
make up for the magnet alignment diffusion caused by ground motion.
Ground motion might put magnet offsets well beyond the defined
tolerances after a few years of movement. Thus, periodic magnet
realignment will be necessary. Some means to simplify the realignment
procedure, such as using stepping motors, should be evaluated.

Table 3.12. Corrector magnetic lengths and strengths. The maximum
strengths are 2.5 times the rms value required when all contributions
are added in quadrature.

  --------------------------------------------------- ---------- ---------
  **ARC CORRECTORS**                                             

  Dipole magnetic length                              1.05       m

  Dipole max strength                                 2.3        T-m

  Skew quad length                                    0.95       m

  Skew quad max strength                              95         T

  Sextupole magnetic length                           0.95       m

  Sextupole max strength, *L(d^2^B/dx^2^)*            7400       T/m

  **IR CORRECTORS**                                              

  Magnetic length                                     2.5        m

  Dipole max strength                                 8.6        T-m

  Skew quad max strength                              95         T
  --------------------------------------------------- ---------- ---------

### 3.3.7 Collective Effects {#collective-effects-1 .unnumbered}

The reduced bunch population and higher energy of the Stage-2 collider
greatly diminish the instability issues found in the Stage-1 collider.
While uncertainties of the transverse impedance are larger for the high
field ring, worst case estimates lead to resistive wall e-folding times
of order 3 turns, significantly slower than in the low field case. Any
damper technology required should easily follow from that developed for
the low field case. Future studies may indicate that increasing the
thickness of the copper layer is beneficial for reducing any resistive
wall effects. The effect of dipole magnetic fields on the resistivity of
the various materials used in the high field design needs to be well
understood.

#### **References** {#references .Reference-heading .unnumbered}

[^1]: \[?\] David Neuffer, *Particle Accelerators* **23** (1988), 21.

[^2]: \[?\] F. Ostiguy, personal communication, March 19, 2001.

[^3]: \[?\] G. Ambrosio, personal communication, March 22, 2001.

[^4]: \[?\] T. Sen et al, "Chromaticity Correction for the SSC Collider
    Rings," *Proc. 1993 IEEE Part. Accel. Conf*, 143 (1993).

[^5]: \[?\] T. Sen, "Alignment tolerances of IR quadrupoles in the LHC,"
    FNAL-Conf-99-304 (October 1999).

[^6]: \[?\] Y. Nosochkov and D. Ritson, "Provision of IP crossing angles
    for the SSC," SSCL-Preprint-367 (1993).

[^7]: \[?\] V. Shiltsev, *Proc. ICFA Workshop on Ground Motion*, Nov.
    2000, SLAC;\
    http://www-project.slac.stanford.edu/ls/wkshp/GM2000.

[^8]: \[?\] B. Baklakov et al., *PRST Accel. Beams*, 031001, 1998.

[^9]: \[?\] V. Shiltsev, FNAL TM-1987, 1996.

[^10]: \[?\] A. Burov et al., "Beam Stability Issues in a Very Large
    Hadron Collider," *NIMA* 450, p. 194, 2000.

[^11]: \[?\] Workshop information and presentations can be found at
    <http://www.slac.stanford.edu/~achao/VLHCworkshop.html>.

[^12]: \[?\] See for example, K.Y. Ng, "Impedances and Collective
    instabilities of the Tevatron at Run II," Fermilab Report TM-2055,
    1998.

[^13]: \[?\] K.Y. Ng, "Bunch Coalescence in Low-Field VLHC Rings,"
    report in progress (April 2001).

[^14]: \[?\] E. Keil, R. Talman, "Scaling of Luminosity Data Between
    e^+^e^-^ Storage Rings," *Particle Accelerators,* Vol. 14 pp.
    109-118, 1983.

[^15]: \[?\] S. Peggs, M. Harrison, F. Pilat, M. Syphers, "Flat beams in
    a 50 TeV hadron collider," Vancouver PAC 97, 1997.

[^16]: \[?\] W. Fischer, *An Experimental Study on the Long-term
    Stability of Particle Motion in Hadron Storage Rings*, Ph.D. thesis,
    Hamburg University, DESY 95-235 (1995) and CERN SL/96-10 (AP), 1996.

[^17]: \[?\] W. Fischer, M. Giovannozzi and F. Schmidt, "Dynamic
    Aperture Experiment at a Synchrotron," *Phys. Rev. E*, Vol. 55,
    Number 3, p. 3507, 1997.