**Proton and Antiproton Injections into Tevatron with inserted liners in
the F0-Lambertson magnets.**

P.M. Ivanov

FNAL, BD/Tevatron

October 8-th, 2003

[Introduction]{.underline}

During this summer shutdown, the electro-dynamic liners were inserted
into four F0-Lambertson magnets to reduce the Tevatron transverse
impedances and, consequently, to improve stability of the coherent
head-tail modes for high intensity proton beam. The liners are
manufactured of copper-berillium bronze sheets with thickness of 0.4 mm.
Owing to liner excursions a real geometric aperture loss reaches up to 2
mm for individual magnet. The liner thickness is included in the
measured maxima of the horizontal excursions. Effective aperture loss is
smaller by a factor of $\sqrt{2}$ due to a specific shape of the
Lambertson notch area. Table 1 represents liner survey results for the
four injection magnets.

[Table 1. F0-Lambertson liner survey results**.**]{.underline}

[Magnet]{.underline} [Max Horizontal Excursion]{.underline} [Effective
Aperture Loss]{.underline}

L 003 ∆x= 1.9 mm ∆A≈ ∆x/√2≈ 1.3 mm

L 014 ∆x= 1.1 mm ∆A≈ ∆x/√2≈ 0.76 mm

L 009 ∆x= 1.4 mm ∆A≈ ∆x/√2≈ 1.0 mm

L 008 ∆x= 1.2 mm ∆A≈ ∆x/√2≈ 0.9 mm

It is necessary to rule out any chance connected with the liner
insertion and leading to degradation of machine performance such as
injection efficiency or beam lifetime.

In order to restore the lost aperture for the circulating beams there is
the most simple and reasonable solution such as displacement of the
F0-Lambertson magnets outside the ring by 2 mm. At the same time, we
have to make sure that a clear aperture inside the Lambertson field
region is enough to inject the proton and antiproton beams into
Tevatron. It would be optimal if we could keep without changes the
injection conditions including strength of the kicker-magnets. The
Lambertson field region horizontal aperture equals 50 mm that is
sufficient for optimization of the beam injection after the magnet
displacement. In this case, a space between the magnet pole and injected
beams is a critical parameter. Using the Tevatron actual lattice model
and working parameters of the kickers we can simulate the trajectories
of injected beams to define the aperture limitation at the current
injection conditions.

The main goal of this note is a detailed study of the aperture
restrictions at the F0-Lambertson magnets for circulating and injected
beams. Tevatron experimental data and actual parameters of the
kicker-magnets have been provided by J. Annala, D. Bollinger, A. Chen,
B. Hanna, V. Lebedev, V. Ranjbar, V. Shiltsev, and J. Volk.

**[Proton injection]{.underline}**

There are five kicker-magnets with total assembly design length
L~F17K~=243.75"=619.125 cm (interface to interface) according to the
drawing 1780.000-ME-119604. Length of each magnet equals to
L~1KMF17~=123.825 cm (from flange to flange). Effective magnetic length
of the magnet is defined by total length of the ferrite assemblies and
it can be estimated as L~kDF17PIK~=87.0 cm. Five proton kicker-magnets
are fed from two power supplies in configuration 3 + 2 as it shown in
Figure 1. In each set the magnets are connected in series. At standard
kicker setting U~0~=45 kV (a real voltage is
$\left\langle U_{0} \right\rangle = I_{\text{ave}} \cdot R = \text{1650}\ A \cdot \text{25}\ \Omega = \text{41}\text{.}\text{25}\ \text{kV}$)
the averaged value of the measured kicker currents is
$I_{\text{ave}} = \frac{\left( I_{1} + I_{2} \right)}{2}$=1650 Amps.

![](media/image1.png){width="6.284027777777778in" height="1.875in"}

Figure 1. Electrical Scheme of the Proton Kicker F17.

Vertical magnetic field in the kicker gap is proportional to the current
$I_{\text{ave}}$ and it can be estimated from Ampere's law
$\oint_{}^{}\overset{\rightarrow}{H} \cdot \overset{\rightarrow}{\text{dl}} = \ I$
in approximation of infinite ferrite permeability as:

$B_{y}\lbrack T\rbrack = \frac{4\ \pi \cdot \text{10}^{- 7}\ I\ \lbrack\text{Amp}\rbrack}{g\ \lbrack m\rbrack} = B_{0}$
, $B_{0} = 0\text{.}\text{0454}\ T$

where$g = 0\text{.}\text{0457}\ m$ is a vertical gap of the ferrite
poles.

In longitudinal direction of the magnet, all ferrite assemblies with
pole width $l_{0} = 2\text{.}2\ \text{cm}$ are separated from each other
by air gaps ${\Delta s}_{0} = 0\text{.}6\ \text{cm}$ that results in
variation of the vertical magnetic field $B_{y}(s)$ along a longitudinal
axis and effective reduction of the kick:

$${\Delta X}_{m}^{'} = \frac{\int_{0}^{L_{m}}{B_{y}(s)\text{ds}}}{\left\langle B \cdot \rho \right\rangle}\  \approx \frac{B_{0}5L_{m}}{\left\langle B \cdot \rho \right\rangle}F_{r}$$

It is easy to make it clear that the magnetic field integral reduction
factor can be estimated with reasonable accuracy according to the
equation:

$F_{r} \approx \frac{\int_{0}^{l_{0} + {\Delta s}_{0}}{\sqrt{f_{1}(s)\text{cos}^{2}\left( \frac{\pi\ s}{2l_{0}} \right) + f_{2}(s)\text{sin}^{2}\frac{\pi}{2l_{0}}\left( s - {\Delta s}_{0} \right)} \cdot \text{ds}}}{l_{0} + {\Delta s}_{0}}$,

where $s$ is a longitudinal coordinate with $s = 0$ in the center of
individual ferrite assembly having length $l_{0}$. The functions
$f_{1}(s)$,$f_{1}(s)$ are define as following:

$\begin{array}{r}
f_{1}(s) = 1,\ |s|\  < l_{0} \\
f_{1}(s) = 0,\ |s|\  \geq l_{0}
\end{array}$ and $\begin{array}{r}
f_{2}(s) = 0,\ |s|\  \leq {\Delta s}_{0} \\
f_{2}(s) = 1,\ |s|\  > {\Delta s}_{0}
\end{array}$

For the kicker design dimensions such as $l_{0} = 2\text{.}2\ \text{cm}$
and $\ {\Delta s}_{0} = 0\text{.}6\ \text{cm}$ we have
$F_{r} \approx 0\text{.}\text{85}$.

Full kick for the injected proton beam includes both the magnetic and
electrical component and it is directed to inside the machine:

$${\Delta X}_{P}^{'} = {\Delta X}_{m}^{'} + \ {\Delta X}_{e}^{'}$$

At the Tevatron injection energy E=150 GeV
($\left\langle B \cdot \rho \right\rangle \approx \text{500}\text{.}\text{35}\ \lbrack T \cdot m\rbrack$)
we have :

$${\Delta X}_{m}^{'} = 0\text{.}\text{336}\ \text{mrad}$$

Electric component of the kick can be estimated according to:

> ${\Delta X}_{e}^{'}\lbrack\text{rad}\rbrack = \frac{\text{10}^{- 3}}{3} \cdot \frac{U_{0}\lbrack\text{kV}\rbrack \cdot 5L_{\text{busbar}}}{w \cdot \left\langle B \cdot \rho \right\rangle\lbrack T \cdot m\rbrack} \approx 0\text{.}\text{018}\ \text{mrad}$
> at $U_{0} = \text{41}\text{.}3\ \text{kV}$

Here $w = 7\text{.}\text{01}\ \text{cm}$ is a horizontal space between
the kicker bus-bars and
$L_{\text{busbar}} \approx 0\text{.}\text{92}\ m$ is an effective
bus-bar length inside the magnet.

For a given kick angle
${\Delta X}_{P}^{'} = \left( 0\text{.}\text{336} + 0\text{.}\text{018} \right)\ \text{mrad} = 0\text{.}\text{354}\ \text{mrad}$
a trajectory offset for the injected beam inside the Lambertson magnet
can reach of $\Delta X = 2\text{.}5\ \text{cm}$ with respect to the
Tevatron axis but the separation of the injected beam and closed orbit
is 2.2 cm due to a local orbit offset
$\Delta X = 0\text{.}3\ \text{cm}$outside the ring during injection
pulse. According to a computation by the OPTIM code the horizontal
trajectory of the injected beam inside the Lambertson magnet is nearly
in parallel to the magnet poles. Result of this computation is presented
in Fig.2. Dots on the solid line correspond to the injected beam
trajectory measured by BPMs (measurements were performed by Jerry
Annala). There is a good agreement with the simulation. A vertical
trajectory inside the Lambertson field region aperture is shown in
Fig.3.

Kicker magnetic structure in the OPTIM code format is presented as
following:

oDF17PIK1 kF17PIK oDF17PIK2 kF17PIK oDF17PIK2 kF17PIK oDF17PIK2 kF17PIK
oDF17PIK2 kF17PIK oDF17PIK3

Effective lengths of the kicker lattice elements:

L~oDF17PIK1~=17.075 cm, L~oDF17PIK2~=36.825 cm, L~oDF17PIK3~=19.75 cm,
L~kDF17PIK~=87.0 cm

> ![](media/image2.png){width="5.282638888888889in"
> height="3.1069444444444443in"}

Figure 2. Horizontal trajectory of the injected proton beam from
downstream end of

the Lambertson magnet (L 008) towards the kicker at the F17 position.

> ![](media/image3.png){width="5.188888888888889in"
> height="3.0118055555555556in"}

Figure 3. Vertical trajectory of injected proton beam inside the
Lambertson field

region aperture.

**[Antiproton injection]{.underline}**

The two antiproton kicker-magnets are located at E48 but only one of
them is used as an injection kicker since a single magnet provides
sufficient kick for the injected beam from the Main Injector. The
antiproton kicker magnet has a similar structure like the proton kicker
but design specifications are different. For compensation of the
injection angle the kick should be directed outside the ring. As opposed
to the proton kicker the magnetic and electric components of the
antiproton kick are opposite directed as it is illustrated in Figure 4.

> ![](media/image4.png){width="4.625694444444444in" height="2.275in"}

Figure 4. Electrical Scheme of the Antiproton Kicker E48.

Assembly design length of each magnet equals to 255.52 cm (from flange
to flange). Effective magnetic length is defined by total length of the
ferrite assemblies and it is estimated as 245.0 cm according to the
drawing 2214-ME-291985.

In longitudinal direction the single ferrite assembly has a length of
$l_{0} = 2\text{.}5\ \text{cm}$. The separation air gap between neighbor
assemblies is ${\Delta s}_{0} = 1\text{.}0\ \text{cm}$. Using
corresponding equation from the section above, the magnetic field
integral reduction factor is computed as$F_{r} \approx 0\text{.}8$. At
standard setting for the antiproton injection
$U_{o} = \text{56}\text{.}5\ \text{kV}$ the kicker current reaches value
of $I_{k} = 4\text{.}\text{52}\ \text{kA}$.

Vertical peak magnetic field in the magnet gap is estimated as
$B_{0} = 0\text{.}\text{0993}\ T$ at$g = 0\text{.}\text{0572}\ m$.

Full antiproton kick consists of the difference between magnetic and
electric components:

${\Delta X}_{m}^{'} = 0\text{.}\text{389}\ \text{mrad}$ ;
${\Delta X}_{e}^{'} = 0\text{.}\text{014}\ \text{mrad}$

$${\Delta X}_{\text{Pbar}}^{'} = {\Delta X}_{m}^{'} - {\Delta X}_{e}^{'} \approx 0\text{.}\text{376}\ \text{mrad}$$

It is essential to draw attention to the significant difference in
strength of the kick for extraction of the reverse proton at nearly the
same E48-kicker setting (for proton
$U_{o} = \text{56}\text{.}0\ \text{kV}$):

$\frac{{\Delta X}_{P}^{'}}{{\Delta X}_{\text{Pbar}}^{'}} = 1\text{.}\text{075}$
($U_{o} = \text{56}\text{.}5\ \text{kV}$)

In order to obtain the same trajectories for the injected antiproton and
extracted proton beams one should decrease the E48-kicker voltage down
to $U_{o} = \text{52}\text{.}\text{56}\ \text{kV}$ for the reverse
protons.

The E48-kicker magnetic structure in the OPTIM code format is presented
as following (in proton direction):

oDE48SP1 kDPBINJK oDE48SP0 oDPBINJK oDE48SP2

Effective lengths of the kicker lattice elements:

L~oDE48SP1~=30.3425 cm, L~oDE48SP0~=35.6025 cm,

L~oDE48SP2~=34.2062 cm, L~kDPBINJK~=245.0 cm

Figure 5 presents the trajectory of the injected antiproton beam which
is perfectly matched with the closed antiproton helical orbit for the
given kick angle
${\Delta X}_{\text{Pbar}}^{'} = 0\text{.}\text{376}\ \text{mrad}$. A
beam separation at entrance of the upstream Lambertson magnet L 008
reaches of about 30 mm and it is decreased to 25.8 mm at the exit from
the L 003- magnet due to the initial horizontal injection angle of
${\Delta X}_{0}^{'} = - 0\text{.}\text{42}\ \text{mrad}$.

> ![](media/image5.png){width="5.566666666666666in"
> height="2.9298611111111112in"}

Figure 5. Trajectory of the injected antiproton beam.

**[Tevatron lattice functions at the F0-straight.]{.underline}**

Lattice functions are computed by OPTIM using Tevatron actual lattice
model that is simulated for machine settings at the injection energy.

> ![](media/image6.png){width="5.745833333333334in"
> height="3.004166666666667in"}

Figure 6. Tevatron lattice functions at the F0-straight section.

In order to estimate a reasonable aperture limitation in the
F0-Lambertson notch area, the horizontal and vertical beam sizes are
calculated for the proton beam with the transverse rms emittances
(ε~y~)~rms~≈ (ε~x~)~rms~≈ 2⋅10^-6^ cm and rms momentum spread σ~∆p/p~ ≈
5.5⋅10^-4^ at two points. Here the transverse beam sizes are given by

$$\left( \sigma_{x,y} \right)_{\text{tot}} = \sqrt{\sigma_{(x,y)_{\beta}}^{2} + \sigma_{(x,y)_{s}}^{2}} = \ \sqrt{\left( \varepsilon_{x,y} \right)_{\text{rms}} \cdot \beta_{x,y} + \left( D_{x,y} \cdot \sigma_{\Delta p/p} \right)^{2}}$$

Table 2. The beam sizes and lattice functions are given in millimeters
and in meters, correspondently.

> ![](media/image7.png){width="6.282638888888889in"
> height="0.9583333333333334in"}

As may be seen from Table 2, the variation of the beam sizes from
upstream of the first magnet L003 to downstream of L008-magnet is quite
small.

The elliptical beam profiles shown in Figure 7 represent proton beam
closed orbit positions with respect to the F0-Lamberston notch area at
different Tevatron operational modes and projection of the injection
trajectory inside the field region aperture. The beam profiles are
scaled in accordance with the boundary equation of form
$x^{2} + y^{2}\sigma_{\text{xtot}}^{2}/\sigma_{\text{ytot}}^{2} = \sigma_{\text{xtot}}^{2}$.

> ![](media/image8.png){width="4.875in" height="2.6868055555555554in"}

Figure 7. Proton beam positions at the F0-Lambertson magnets for the
different

Tevatron operational modes.

The smallest geometric aperture inside the Lambertson notch area takes
place during proton injection due to a horizontal local beam orbit bump
the amplitude of which reaches of about 8 mm with respect to the central
orbit. The steering errors lead to a residual coherent oscillation of
the injected beam with respect to closed orbit. According to
experimental observation the oscillation amplitude, usually does not
exceed 0.2 mm. In consequence, this effect can be ruled out any
consideration in the aperture limitation.

Green elliptical area in Figure 7 contains the particles with momentum
spread in vicinity of the equilibrium value
$\frac{\Delta P}{P} \simeq 0$ and transverse oscillation amplitudes of
which don't exceed the boundary of
$A_{x,y} \leq 4\left( \sigma_{x,y} \right)_{\text{rms}}$. Yellow
elliptical areas correspond to transverse spaces for protons with the
rms transverse emittances and momentum spread near the RF bucket
boundary ∆p/p= ± 2σ~s.~ As the aperture is decreased, these particles
can be lost in the first place. Visible beam scraping is experimentally
observed for high intensity coalesced proton bunch when the injection
beam orbit bump is increased up to ∆x=10 mm (by 2 mm larger than the
standard injection bump).

Figure 8 shows the antiproton beam closed orbit position and projections
of the injection trajectory. The shortest distance between the injected
beam and the magnet pole is about 11 mm at L003. After moving the
Lambertson magnets outside the machine this space will be decreased down
to 9 mm. For the injected proton beam, a minimum room inside the field
region aperture will be 8.3 mm.

![](media/image9.png){width="4.884722222222222in"
height="2.692361111111111in"}

Figure 8. Antiproton beam positions at the F0-Lambertson magnets.

**[Summary]{.underline}**

As a result of this analysis one can do the following conclusion. The
geometric aperture loss inside the F0-Lambertson notch area owing to
liner insertion can be restored to its original value by displacement of
the injection magnets outside the ring by 2 mm. It is reputed that
voltage margin allows increasing the strength of both the E48- and
F17-kickers to shift proportionally the injection trajectories inside
the Lambertson field region by the same distance but probably, it is not
necessary to do that. There is enough aperture room to avoid any
scraping effects for the injected proton and antiproton beams. All
injection conditions should be kept the same like before shutdown.

**[Specification for the beam orbit at Lambertsons]{.underline}**

The F0-Lambertson magnets are adjusted with accuracy of about ± 0.25 mm
with respect to each other and relative to the Tevatron axes of
reference at the same position like before summer shutdown. It is
reasonable to suggest that the nearest neighbor quadrupole doublets are
adjusted with accuracy that is not worse than it was early. It means
that old corrector settings for F0-local beam orbit bump creation can be
used. In accordance with computer simulation for the ideal proton beam
injection, both the closed orbit and injection trajectory should go in a
parallel way about the Lambertson magnet poles in horizontal plane. From
this point of view, it is reasonable to adjust the closed orbit inside
the F0-straight for proton beam at first.

The adjustment of the beam position inside the notch area should be done
at the injection local orbit bump with the horizontal amplitude ∆x=+5 mm
about Tevatron axis, applying sequential iterations in horizontal and
vertical planes. At these conditions, a hardly visible scraping can be
observed for uncoalecsed proton beam with standard parameters if the
closed orbit lies in the median plane of the magnet notch and it is in
parallel in the horizontal plane.

In order to put the proton orbit exactly into the notch along Lambertson
magnets four standard F0-local orbit bumps should be involved by
applying "T73 Mult:5". The following dipole correctors are needed to
create the F0-position and angle bumps:

vertical plane : **VE47, VE49, VF11, VF14**;

horizontal plane: **HE46, HE49, HF11, VF13.**

In accordance with computer simulation for the actual Tevatron lattice
model the multipliers for the corrector currents are:

[Horizontal position]{.underline} [Horizontal angle]{.underline}

> **T :HE46=- *I*\*0.1343 T :HE46=*I*\*0.0575**
>
> **T :HE49=*I*\*0.0056 T :HE49=-*I*\*0.09197**

**T :HF11=- *I*\*0.11464 T :HF11=*I*\*0.13177**

**T :HF13=-*I*\*0.08785 T :HF13=-*I*\*0.02574**

[Vertical position]{.underline} [Vertical angle]{.underline}

> **T :VE47=*I*\*0.0965 T :VE47=- *I*\*0.04183**
>
> **T :VE49=*I*\*0.0796 T :VE49=*I*\*0.07818**

**T :VF11=- *I*\*0.00565 T :VF11=- *I*\*0.07518**

**T :VF14=*I*\*0.12939 T :VF14=*I*\*0.02682**

![](media/image10.png){width="5.569444444444445in" height="2.9625in"}

> Figure 9. Configuration of the horizontal position orbit bump with
> parallel beam trajectory inside the F0-straight.

![](media/image11.png){width="5.496527777777778in"
height="3.338888888888889in"}

Figure10. Configuration of the F0-horizontal angle orbit bump with zero
displacement at the Lambertson center .

![](media/image12.png){width="5.496527777777778in"
height="3.063888888888889in"}

Figure11.Superposition of the two horizontal bumps:∆x=5 mm and ∆x′=0.117
mrad.

![](media/image13.png){width="5.9430555555555555in"
height="3.1284722222222223in"}

Figure12. Configuration of the vertical position orbit bump with
parallel beam trajectory inside the F0-straight.

![](media/image14.png){width="5.940972222222222in"
height="3.604861111111111in"}

Figure13. Configuration of the F0-vertical angle orbit bump with zero
displacement at the Lambertson center .