# Notes on tune drift measurements at 150 Gev

### Mike Martens, Jerry Annala

### 8/21/02

## Introduction

During Collider Run II operations it has been noticed that the tunes in
the Tevatron drift with time by as much as 0.015 units while the
Tevatron is at its front porch energy of 150 Gev. The time scale of the
tune drifts and the b2 chromaticity correction algorithm are the same
suggesting that the drift might be caused by orbit offsets and sextupole
feeddown effects in the Tevatron. In order to investigate this
possibility the tunes in the Tevatron were measured as a function of
time at 150 Gev, RF frequency, and current in the chromaticity
correction sextupoles T:SF and T:SD. The data were collected on 5/15/02
while the Tevatron was at 150 Gev after at 30 minutes store at 980 Gev.
The data from these measurements and analysis of the sextupole feeddown
hypothesis are presented in this note.

In the Tevatron the main sources of time varying sextupole fields are
the b2 component of the Tevatron dipoles and the chromaticity correction
sextupole circuits T:SF and T:SD. If the observed tune drifts are caused
by time varying sextupole fields then there must be an average
horizontal orbit offset in these magnets (since a vertical orbit offset
creates a coupling effect.) To determine the average orbit offset in the
T:SF and T:SD magnets the horizontal orbit positions were changed by
varying the RF frequency and then measurements were made of the tune
changes as a function of current in the T:SF and T:SD circuits. For
instance we determined when the horizontal orbit, averaged over all the
T:SF magnets, was centered in the T:SF magnets by finding the RF
frequency that produced no tune change when the current in the T:SF
circuit was varied. Once the average orbit offset was determined then
the magnitude of the tune drift could be calculated.

An analysis of these data shows that the time varying strength of the
chromaticity sextupoles does not explain the observed tune drift. We
also conclude that an average horizontal orbit offset of about 1 mm in
the dipoles is required to explain the tune drift if the sextupole
feeddown hypothesis is correct. During this particular study we did not
measure coupling changes. However, we note that previous studies also
show a time varying coupling change that, if left uncorrected, results
in a minimum tune split of about 0.02 tune units after about 2 hours. If
this coupling change is due to feeddown effects then a vertical orbit
offset in the sextupole fields is required but this was not examined in
this study.

Another possible explanation for the time varying coupling and tune
shifts is time varying quadrupole fields from persistent currents in the
Tevatron dipoles or quadrupoles. This possibility is not examined in
detail in this note.

## Analysis

The tune change from horizontal orbit offsets in sextupole magnetic
fields is given by the formula

$${\Delta\nu}_{x} = \frac{1}{4\pi}\sum_{}^{}{(\beta_{x}\ {\Delta K}_{2}\ L\ x})$$

where ~x~ is the horizontal beta function, K~2~ is the change in
sextupole field, L is the length of the magnet, and x is the horizontal
orbit position. The sum is over all sextupole fields which, in the case
of the Tevatron, consists of the main dipoles and the magnets in the
T:SF and T:SD circuits.

For convenience we express the change in the T:SF and T:SD sextupole
fields in two parts. The first part, K~2~, is the time varying part
which is used to compensate for the time varying b2 component in the
dipoles and keeps the chromaticity at a constant value. The second part,
K~2~, is the amount of sextupole field added to T:SF and T:SD as part of
the tune drift measurement study. We can also express the orbit offset,
x, in two parts. The first, x~0~, is the offset in the magnet with the
nominal orbit and nominal RF frequency and the second, D~x~ (p/p), is
the orbit offset due to the dispersion and a beam momentum offset. If we
include these effects then the tune change is given by

${\Delta\nu}_{x} = \frac{1}{4\pi}\sum_{}^{}\left( \beta_{x}\ L\ \left( {\Delta K}_{2} + {\delta K}_{2} \right)\ \left( \ \left( \frac{\Delta p}{p} \right)\ D_{x} + x_{0} \right)\  \right)$.

The strength of the T:SF and T:SD circuits are adjusted as a function of
time to keep the chromaticity constant while the Tevatron is at 150 Gev.
Since we know the current in the T:SF and T:SD circuits and we know that
the measured chromaticity does not change, ~x~ = 0, we can calculate the
average value of the sextupole field in the dipoles from the expression
for the change in chromaticity

$${\Delta\xi}_{x} = \frac{1}{4\pi}\sum_{\text{Dipole}}^{}\left( \beta_{x}{\Delta K}_{2}\text{LD}_{x} \right) + \frac{1}{4\pi}\sum_{\text{T:SF}}^{}\left( \beta_{x}{\Delta K}_{2}\text{LD}_{x} \right) + \frac{1}{4\pi}\sum_{\text{T:SD}}^{}\left( \beta_{x}{\Delta K}_{2}\text{LD}_{x} \right) = 0$$

With this expression for the change in chromaticity we can express the
change in tune as

$${\Delta\nu}_{x} = \frac{1}{4\pi}\sum_{}^{}\left( \beta_{x}\ {\Delta K}_{2}\ L\ x_{0} \right) + \left( \xi_{x} + {\Delta\xi}_{x} \right)\left( \frac{\Delta p}{p} \right)\  + \frac{1}{4\pi}\sum_{}^{}\left( \beta_{x}\ L\ {\delta K}_{2}\ \left( \ \left( \frac{\Delta p}{p} \right)\ D_{x} + x_{0} \right)\  \right)$$

where ~x~ is the horizontal chromaticity and ~x~ is the change in
horizontal chromaticity as a function of time.

The first term in this equation is the tune drift at 150 Gev due to the
sextupole feeddown effects. This is the term that we are interested in
calculating to determine if the time varying sextupoles are responsible
for the drifting tune. We assume we know the lattice functions, that the
changing sextupole fields in the T:SF and T:SD circuits are known, and
that we can calculate the average time varying sextupole field in the
dipoles from the expression ~x~ = 0. This leaves the orbit offsets,
x~0~, in the dipoles and T:SF and T:SD magnets as the unknown variables
in the first term of the equation. The second term is the change in tune
due to the change in momentum and the chromaticity and does not change
as a function of time since the chromaticity is kept constant.

The third term is useful because we can adjust the sextupole field,
K~2~, in the T:SF or T:SD magnets and then measure the change in tune as
a function of p/p. This allows us to determine the average value of the
horizontal orbit offset in T:SF and T:SD circuit. Using the relationship
between the RF frequency and the beam momentum,

(f/f) = -(p/p) where is the slip factor, we can write the third term as

$${\Delta\nu}_{x} = \left( {\delta K}_{2}L \right)\frac{N}{4\pi}\left( - \frac{1}{\eta}\left\langle \beta_{x}D_{x} \right\rangle\left( \frac{\Delta f}{f} \right) + \left\langle \beta_{x}x_{0} \right\rangle \right)$$

where N is the number of magnets in the circuit and denotes the average
value.

The value ~x~x~0~ is then determined from

$$\left\langle \beta_{x}x_{0} \right\rangle = \frac{1}{\eta}\left\langle \beta_{x}D_{x} \right\rangle \ast \left( \frac{\Delta f}{f} \right)_{{\Delta\nu}_{x} = 0}$$

where $\left( \frac{\Delta f}{f} \right)_{{\Delta\nu}_{x} = 0}$ is the
value of the RF frequency shift that gives zero tune shift when the
current is changed in the T:SF or T:SD circuits.

## Tune measurements

The tunes of the Tevatron were measured as a function of time at 150
Gev, a function of RF frequency, and current in the T:SF and T:SD
sextupoles. The data were collected on 5/15/02 while the Tevatron was at
150 Gev after a 30 minutes store at 980 Gev. At each RF frequency offset
(-40 Hz, -20 Hz, 0 Hz, +20 Hz, +40 Hz) the tune was measured with 1) no
change in T:SF and T:SD, 2) +0.5 Amps added to T:SD, and 3) +0.5 Amps
added to T:SF. The complete set of data is plotted in Appendix A.

For this section of the note we focus on the measurements of the tune
drifts as a function of time at 150 Gev and the measurements of the tune
difference resulting from a 0.5 Amp change in the chromaticity sextupole
circuits. For each RF frequency offset the change in tune was measured
and the results are shown in the table and figure below. These data were
extracted from the tune measurements taken when the Tevatron was at 150
Gev for 60 minutes.

+-------------+-------------+-------------+-------------+-------------+
| RF          | Vert tune   | Vert tune   | Horz tune   | Horz tune   |
| frequency   | (SF+.5      | (SD+.5      | (SF+.5      | (SD+.5      |
|             | Amps)       | Amps)       | Amps)       | Amps)       |
| Offset (Hz) |             |             |             |             |
+-------------+-------------+-------------+-------------+-------------+
| -40         | -0.0014     | -0.0058     | 0.0073      | 0.002       |
+-------------+-------------+-------------+-------------+-------------+
| -20         | -0.0006     | -0.0037     | 0.0064      | 0.0015      |
+-------------+-------------+-------------+-------------+-------------+
| 0           | 0           | -0.0018     | 0.0015      | 0           |
+-------------+-------------+-------------+-------------+-------------+
| 20          | 0.0011      | -0.0002     | -0.0035     | -3E-04      |
+-------------+-------------+-------------+-------------+-------------+
| 40          | 0.0023      | 0.0015      | -0.0076     | 0           |
+-------------+-------------+-------------+-------------+-------------+

![](media/image1.wmf)

Linear fits to this data are used to determine the RF frequency that
results in no net change in the tune when the currents in T:SF and T:SD
are varied. The results of these fits are shown in the table below.

+-----------------+-------+-------+-------+-------+-------+-------+
|                 | S     | Inte  | $(\D  | Mom   | x~0~  | x~0~  |
|                 | lope, | rcept | elta  | entum |       |       |
|                 | /RF   |       | f)_{{ | o     | (     | \     |
|                 | (Hz   | (tune | \Delt | ffset | m^2)^ | (mm\) |
|                 | ^-1^) | u     | a\nu} | p/p   |       |       |
|                 |       | nits) | _{x}  |       |       |       |
|                 |       |       | = 0}$ |       |       |       |
|                 |       |       | (Hz)  |       |       |       |
+-----------------+-------+-------+-------+-------+-------+-------+
| Vert tune       | 4.5   | 0.    | -6.15 | 4.1   | -0.4  | -     |
| (SF+.5 Amps)    | 5E-05 | 00028 |       | 6E-05 | 85E-2 | 0.158 |
+-----------------+-------+-------+-------+-------+-------+-------+
| Horz tune       | -19.  | 0.    | 4.13  | -2.   | 3.    | +     |
| (SF+.5 Amps)    | 9E-05 | 00082 |       | 8E-05 | 20E-2 | 0.106 |
+-----------------+-------+-------+-------+-------+-------+-------+
| Vert tune       | 9.0   | -     | 22.1  | -15   | 1.    | +     |
| (SD+.5 Amps)    | 5E-05 | 0.002 |       | .0E-5 | 00E-2 | 0.343 |
+-----------------+-------+-------+-------+-------+-------+-------+
| Horz tune       | -2.9  | 0.    | 22.1  | -15   | 1.    | +     |
| (SD+.5 Amps)    | 0E-05 | 00064 |       | .0E-5 | 06E-2 | 0.343 |
+-----------------+-------+-------+-------+-------+-------+-------+

From the results of the fit we determine the intercept on the frequency
axis, $(\Delta f)_{{\Delta\nu}_{x} = 0}$, and the value of ~x~x~0~.
Using the approximation x~0~ = ~x~x~0~/~x~ we calculate the average
horizontal offset in the T:SF and T:SD circuits. The data suggests that
the average orbit offset in the T:SD sextupoles is +0.34 mm horizontally
and the average orbit offset in the T:SF sextupoles is -.03 mm although
the vertical tune and horizontal tune measurements for T:SF give
different average orbit offsets. (No attempt to estimate the errors on
these measurements has been made.)

We also show, in the table below, the measured slope and the theoretical
slope of /*f* for a 0.5 Amp change assuming the design values for the
lattice functions and the known gradient fields in the T:SF and T:SD
magnets. There is good agreement in the calculated and measured values
for the T:SD circuit, but the T:SF circuit shows a disagreement of about
20-30%

+--------------------------+---------------------+---------------------+
|                          | Calculated Slope    | Measured Slope,     |
|                          |                     |                     |
|                          | /f (Hz^-1^)         | /f (Hz^-1^)         |
+--------------------------+---------------------+---------------------+
| Vert tune (SF+.5 Amps)   | 5.85E-5             | 4.55E-05            |
+--------------------------+---------------------+---------------------+
| Horz tune (SF+.5 Amps)   | -15.0E-5            | -19.9E-05           |
+--------------------------+---------------------+---------------------+
| Vert tune (SD+.5 Amps)   | 9.08E-5             | 9.05E-05            |
+--------------------------+---------------------+---------------------+
| Horz tune (SD+.5 Amps)   | -2.93E-5            | -2.90E-05           |
+--------------------------+---------------------+---------------------+

Another part of the data is the measurement of the horizontal and
vertical tune as a function of time at 150 Gev with the nominal RF
frequency and nominal current in T:SF and T:SD. This is shown in the
plot below.

![](media/image2.wmf)

A fit to the horizontal and vertical tune drift data using a logarithmic
function gives

$$\begin{array}{r}
{\Delta\nu}_{x} = \text{ -2}\text{.}\text{7x10}^{\text{-3}}\text{*ln}(t) + 0\text{.}\text{5962} \\
{\Delta\nu}_{y} = + 4\text{.}\text{1x10}^{\text{-3}}\text{*ln}(t) + 0\text{.}\text{5562}
\end{array}$$

where t is the time at 150 Gev in minutes.

We also have the fits to the time varying sextupole gradients in the
Tevatron dipoles and T:SF and T:SD circuits that were calculated in
Appendix B,

$$\begin{array}{r}
\Delta(K_{2}L)_{\text{SF}}(m^{\text{-2}}) = \text{ -3}\text{.}\text{12x10}^{\text{-3}}\text{*ln}(t)\text{ -}1\text{.}\text{13x10}^{\text{-3}} \\
\Delta(K_{2}L)_{\text{SD}}(m^{\text{-2}}) = \text{  -4}\text{.}\text{79x10}^{\text{-3}}\text{*ln}(t)\text{ -}4\text{.}\text{79x10}^{\text{-3}} \\
\Delta\left( K_{2}L \right)_{\text{Dipole}}(m^{\text{-2}}) = 9\text{.}\text{51x10}^{\text{-4}}\text{*ln}(t) + 2\text{.}\text{94x10}^{\text{-3}}
\end{array}$$

Combining the measured orbit offsets, the tune drift measurement, and
the time varying sextupole fields we can calculate the average orbit
offset in the Tevatron dipoles, ~x~x~0Dipole~. We substitute the results
of the measurements into the equation for the tune drift due to the
orbit offsets in the sextupole fields,

$${\Delta\nu}_{x} = \frac{1}{4\pi}\sum_{}^{}\left( \beta_{x}\ {\Delta K}_{2}\ L\ x_{0} \right)$$

which can be written as

$$\begin{array}{r}
 \\
{\Delta\nu}_{x} = \frac{1}{4\pi}\begin{array}{r}
\left( N_{\text{Dipole}}\Delta\left( K_{2}L \right)_{\text{Dipole}}\left\langle \beta_{x}x_{0} \right\rangle_{\text{Dipole}} + \middle| \right)\left( N_{\text{T:SF}}\Delta\left( K_{2}L \right)_{\text{T:SF}}\left\langle \beta_{x}x_{0} \right\rangle_{\text{T:SF}} + \middle| \right) \\

\end{array} \\

\end{array}$$

where all the terms are known except the average orbit offset in the
dipoles, ~x~x~0Dipole~.

Concentrating on the logarithmic variation we get

$$\begin{array}{r}
{\Delta\nu}_{x} = \text{-2}\text{.}\text{7x10}^{\text{-3}} \ast \text{ln}(t) = \frac{1}{4\pi}\begin{array}{r}
\left( \text{774*}(9\text{.}\text{74x10}^{\text{-4}}) \ast \left\langle \beta_{x}x_{0} \right\rangle_{\text{Dipole}} + \middle| \right)\left( \text{88*}\left( \text{-3}\text{.}\text{12x10}^{\text{-3}} \right) \ast \left( 1\text{.}\text{00x10}^{\text{-2}} \right) + \middle| \right) \\

\end{array} \\

\end{array}$$

which gives

$$\left\langle \beta_{x}x_{0} \right\rangle_{\text{Dipole}} = \text{ -3}\text{.}\text{54x10}^{\text{-2}}\text{ m}^{2}$$

or

$$\left\langle x_{0} \right\rangle_{\text{Dipole}} = \frac{\left\langle \beta_{x}x_{0} \right\rangle_{\text{Dipole}}}{\left\langle \beta_{x} \right\rangle_{\text{Dipole}}} = \text{ -0}\text{.}\text{632 mm}$$

This means that there must be an average horizontal offset of --0.632 mm
in the Tevatron dipoles if the time varying sextupole fields are to
explain the horizontal tune drift at 150 Gev. If the feeddown hypothesis
explaining the tune drift is correct then the T:SF and T:SD circuits
contribute 8% and 13% to the horizontal tune drift respectively while
the dipoles contribute 79% to the horizontal tune drift.

The same analysis is done using the vertical tune drift measurement and
the equation for the vertical tune drift

$${\Delta\nu}_{y} = - \frac{1}{4\pi}\sum_{}^{}\left( \beta_{y}\ {\Delta K}_{2}\ L\ x_{0} \right)$$

which can be written as

$${\Delta\nu}_{y} = - \frac{1}{4\pi}\begin{array}{r}
\left( N_{\text{Dipole}}\Delta\left( K_{2}L \right)_{\text{Dipole}}\left\langle \beta_{y}x_{0} \right\rangle_{\text{Dipole}} + \middle| \right)\left( N_{\text{T:SF}}\Delta\left( K_{2}L \right)_{\text{T:SF}}\left\langle \beta_{y}x_{0} \right\rangle_{\text{T:SF}} + \middle| \right) \\

\end{array}$$

where all the terms are known except the average orbit offset in the
dipoles, ~y~x~0Dipole~.

Concentrating on the logarithmic variation we get

$$\begin{array}{r}
{\Delta\nu}_{y} = 4\text{.}\text{1x10}^{\text{-3}} \ast \text{ln}(t) = - \frac{1}{4\pi}\begin{array}{r}
\left( \text{774*9}\text{.}\text{51x10}^{\text{-4}} \ast \left\langle \beta_{y}x_{0} \right\rangle_{\text{Dipole}} + \middle| \right)\left( \text{88*}\left( \text{-3}\text{.}\text{12x10}^{\text{-3}} \right) \ast \left( \text{-4}\text{.}\text{85x10}^{\text{-3}} \right) + \middle| \right) \\

\end{array} \\

\end{array}$$

which gives

$$\left\langle \beta_{y}x_{0} \right\rangle_{\text{Dipole}} = \text{ -5}\text{.}\text{35x10}^{\text{-2}}\text{ m}^{2}$$

or

$$\left\langle x_{0} \right\rangle_{\text{Dipole}} = \frac{\left\langle \beta_{y}x_{0} \right\rangle_{\text{Dipole}}}{\left\langle \beta_{y} \right\rangle_{\text{Dipole}}} = \text{ -0}\text{.}\text{923 mm}$$

This means that there must be an average horizontal offset of --0.923 mm
in the Tevatron dipoles if the time varying sextupole fields are to
explain the vertical tune drift at 150 Gev. If the feeddown hypothesis
explaining the vertical tune drift is correct then the T:SF and T:SD
circuits contribute --2.6% and 26% to the vertical tune drift
respectively while the dipoles contribute 76% to the vertical tune
drift.

## Orbits during tune drift

The horizontal orbit in the Tevatron arcs is shown in the plot below.
The BPMs in the straight sections, from the 48 locations through the 11
locations, have been removed from this plot. Also shown is the
calculated orbit for a beam momentum offset of

p/p = -1.1E-4 which is equivalent to a frequency offset of f = 16 Hz.
The measured orbit and the calculated orbit with the beam momentum
offset both have an average horizontal BPM reading of -0.41 mm in the
BPMs in the Tevatron arcs. As is clear from the plot the orbit
distortions in the horizontal plan cannot be explained by the dispersion
function and a momentum offset.

Because the magnets in the T:SF circuit are very close to the horizontal
BPMs we can determine that the horizontal position in the T:SF magnets
is -0.45 mm according to the BPM measurements. This data is not
consistent with the measurement of the average orbit offset determined
from the tune change measurements made by varying the current in T:SF.
(The tune versus T:SF current measurements gave results suggesting that
the average horizontal orbit offset was -0.158 mm if the vertical tune
was used to determine the orbit offset and +0.106 mm if the horizontal
tune was used to determine the orbit offset in the T:SF sextupoles.)

The vertical orbits in the arcs of the Tevatron are shown in the plot
below. The BPMs in the straight sections, from the 49 locations to the
11 locations, have been removed from the plot. With these BPMs removed
the average vertical offset is 0.03 mm which is small relative to the
average horizontal orbit offset.

![](media/image3.wmf)

![](media/image4.wmf)

##  Appendix A: Tune measurement as function of RF frequency, time at 150 Gev, and current in T:SF and T:SD.

### Tune measurement

These are plots of the horizontal and vertical tunes measured as a
function of time while the Tevatron was at 150 Gev after at 30 minutes
store at 980 Gev. These data were collected on 5/15/02 and are recorded
in the Tevatron E-log. In these plots the horizontal axis is the time at
150 Gev in minutes and the vertical axis is the measured tune. At each
RF frequency (-40 Hz, -20 Hz, 0 Hz, +20 Hz, +40 Hz) the tune was
measured with 1) no change in T:SF and T:SD, 2) +0.5 Amps added to T:SD,
and 3) +0.5 Amps added to T:SF.

![](media/image5.wmf)

![](media/image6.wmf)

![](media/image7.wmf)![](media/image8.wmf)![](media/image9.wmf)![](media/image10.wmf)![](media/image11.wmf)

![](media/image12.wmf)

![](media/image13.wmf)

![](media/image14.wmf)

### Tune difference with current in T:SF and T:SD.

These plots show the difference between the tune with and without T:SF
and T:SD with an additional 0.5 Amps. In each case the data at the same
RF frequency and with zero current in T:SF and T:SD were fitted to a
logarithmic function (t) = a ln(t~150\ Gev~) + b. This fit was then
subtracted from the measured data for all three cases 1) no current in
T:SF and T:SD, 2) an additional 0.5 amps in T:SD, and 3) an additional
0.5 amps in T:SF. If the average orbit in the sextupole circuits were
zero then we would expect to see no change in the tune when the current
in the sextupoles was changed.

![](media/image15.wmf)

![](media/image16.wmf)

# ![](media/image17.wmf)![](media/image18.wmf)![](media/image19.wmf)![](media/image20.wmf)![](media/image21.wmf)![](media/image22.wmf)![](media/image23.wmf)![](media/image24.wmf) 

## Appendix B: Sextupole fields in the Tevatron

This appendix gives formulas, calculations, and data concerning the time
varying sextupole fields in the Tevatron at 150 Gev. The main sources of
these fields are the b2 component of the Tevatron dipoles and the
chromaticity correction sextupole circuits T:SF and T:SD.

### Chromaticity sextupole magnet strength

First we define the sextupole magnetic field as

$$\begin{array}{r}
B_{x}(x,y) = B_{2}\text{xy} \\
B_{y}(x,y) = \frac{1}{2}B_{2}(x^{2} - y^{2})
\end{array}$$

and want to relate this to the magnetic field in a Tevatron sextupole
magnet. From \"The Tevatron Energy Doubler: A superconducting
Accelerator\", H. Edwards, (*Ann. Rev. Nucl. Part Sci.* 1985, 35:605-60)
we have that the integrated strength of a sextupole magnet is 57 kG-in
at 1 inch at 50 Amps for the chromaticity sextupoles and 44 kG-in at 1
inch at 50 Amps for the sextupole magnets in the downstream packages.
The chromaticity sextupoles are located in the upstream packages so
their strength is

$$B_{2}L/I = 8\text{.}\text{98}\ \text{T/m/A}$$

where L is the length of the sextupole and I is the current in the
magnet.

$$\int_{}^{}{}B_{y}(\text{at}\ 1\ \text{inch}) \cdot \text{dl} = \text{57}\ \text{kG} - \text{in}@\text{50}\text{Amp}$$

$$(0\text{.}5)B_{2}L(1\ \text{inch})^{2}/I = 5\text{.}7\ T - \frac{\text{inch}}{\text{50}}A$$

$$B_{2}L/I = 2\text{.}0 \ast 5\text{.}7\ \frac{T}{\text{50}}\frac{A}{\text{inch}} \ast (\frac{\text{inch}}{0}\text{.}\text{0254}\text{meter}) = 8\text{.}\text{98}\frac{\frac{T}{m}}{A}$$

If the sextupoles are treated as a thin sextupole magnet, then in the
MAD convention the thin sextupole strength is given by

$$K_{2}L = (B_{2}\frac{L}{I})\frac{1}{|B\rho|}I$$

where I is the current in the sextupole, and $|B\rho|$ is the magnetic
rigidity and we can also relate the sextupole field, K~2~, to the
magnetic field

$K_{2} = \frac{1}{|B\rho|}\frac{\partial^{2}B_{y}}{\partial x^{2}}$.

With 1 Amp in the T:SF circuit the gradient strength at 150 Gev is,

$$\begin{array}{r}
K_{2}L = (B_{2}\frac{L}{I})\frac{1}{|B\rho|}I = (8\text{.}\text{98}\frac{\frac{T}{m}}{A}\frac{)}{(}\text{3335}\text{.}\text{64}T - \frac{m}{\text{Tev}} \ast 0\text{.}\text{15}\text{Tev}\ )(1\text{.}0\ A) \\
K_{2}\text{L } = 0\text{.}\text{01795}\ \frac{m^{\text{-2}}}{\text{Amp}}
\end{array}$$

In the Tevatron there are about 88 elements in each of the T:SF and T:SD
circuits. (Some individual magnets may be jumpered out of the circuit
due to magnet failures.) The average beta functions, ~x~, ~y~, average
dispersion, D~x~, and average product, ~x~\*D~x~, are listed in the
table below as calculated using MAD with the design Tevatron lattice.

  ---------- ---------- ---------- ---------- ---------- ----------- -----------
  Circuit    Number of  Average    Average    Average    Average     Average
             elements   ~x~ (m)    ~y~ (m)    D~x~ (m)   ~x~\*D~x~   ~y~\*D~x~
                                                         (m^2^)      (m)

  T:SF       88         93.81      30.09      3.810      358.7       115.8

  T:SD       88         30.30      93.29      2.301      70.84       214.6
  ---------- ---------- ---------- ---------- ---------- ----------- -----------

### Tune shift and coupling from feeddown effect

Next we look at the effective quadrupole field that the sextupole field
creates for an off-center closed orbit. If the closed orbit through a
thin sextupole has the coordinates x~0~ and y~0~, then a thin sextupole
with strength K~2~L and a tilt angle will give a normal quadrupole field
with strength

$$(K_{1}L)_{\text{no}} = \text{ K}_{2}\text{L }\left( x_{0}\text{ cos 3}\psi\text{ + }y_{0}\text{sin}3\psi \right)$$

and a skew quadrupole field with strength

$$\left( K_{1}L \right)_{\text{ sq}} = \text{ K}_{2}\text{L }\left( x_{0}\text{ sin 3}\psi - y_{0}\text{cos}3\psi \right)$$

For the Tevatron the chromaticity sextupoles have zero tilt angle, = 0,
and then

$$\begin{array}{r}
(K_{1}L)_{\text{no}} = \text{ K}_{2}\text{L x}_{0} \\
\left( K_{1}L \right)_{\text{ sq}} = \text{ -K}_{2}\text{Ly}_{0}
\end{array}$$

Once we know the strengths and feeddown effects of the sextupoles we can
calculate the effect they have on the tune and coupling. The change in
horizontal and vertical tune is given by the simple equations

$${\Delta\nu}_{x} = \frac{1}{4\pi}\sum_{}^{}\beta_{\text{x,i}}(K_{1}L)_{\text{no,i}}$$

$${\Delta\nu}_{y} = - \frac{1}{4\pi}\sum_{}^{}\beta_{y,i}(K_{1}L)_{\text{no,i}}$$

where $\beta_{x},\beta_{y}$ are the beta functions at the locations of
the feeddown sextupoles, and $(K_{1}L)_{\text{no,i}}$ is the normal
component of the quadrupole field created by the off-center closed orbit
and the sextupole field.

To quantify the effect that the feeddowns have on the coupling we
calculate a sine and cosine term of the coupling as

$${\Delta C}_{\text{sq}} = \ \frac{1}{2\pi}\sum_{}^{}{(K_{1}L)_{\text{sq,i }}\sqrt{\beta_{x,i}\beta_{y,i}}\text{ cos }(\varphi_{y}\text{ - }\varphi_{x})}$$

$${\Delta S}_{\text{sq}} = \text{  }\frac{1}{2\pi}\sum_{}^{}{(K_{1}L)_{\text{sq,i }}\sqrt{\beta_{x,i}\beta_{y,i}}\text{ sin }(\varphi_{y}\text{ - }\varphi_{x})}$$

where $(K_{1}L)_{\text{sq,i }}$ is the skew component of the quadrupole
field and $(\varphi_{y}\text{ - }\varphi_{x})$is the difference between
the vertical and horizontal betatron phase advances.

To better understand the coupling constants we express the minimum tune
split in terms of ${\Delta C}_{\text{sq}},{\Delta S}_{\text{sq}}$. If
the Tevatron is tuned up such that there is no coupling on the proton
helix then in principle the horizontal and vertical tunes could be set
equal to one another. However, if we add a small amount of coupling then
the minimum tune split is
${\Delta\nu}_{\text{min}} = \text{  }\sqrt{{\Delta C}_{\text{sq}}^{2} + {\Delta S}_{\text{sq}}^{2}}$

### Chromaticity correction from T:SF and T:SD

The chromaticity is defined as

$$\Delta\nu = \xi\ \left( \frac{\Delta p}{p} \right)$$

from which we can calculate the expected change in chromaticity from the
T:SF and T:SD circuits:

$$\begin{array}{r}
{\Delta\nu}_{x} = \frac{1}{4\pi}\sum_{}^{}\beta_{\text{x,i}}(K_{1}L)_{\text{no,i}} \\
{\Delta\nu}_{x} = \frac{1}{4\pi}\sum_{}^{}\beta_{\text{x,i}}(K_{2}\text{Lx}) \\
{\Delta\nu}_{x} = \frac{1}{4\pi}\sum_{}^{}\beta_{\text{x,i}}(K_{2}\text{LD}_{x})\ \left( \frac{\Delta p}{p} \right) \\
\xi_{x} = \frac{1}{4\pi}\sum_{}^{}\beta_{\text{x,i}}(K_{2}\text{LD}_{x}) \\
\xi_{y} = - \frac{1}{4\pi}\sum_{}^{}\beta_{\text{y,i}}(K_{2}\text{LD}_{x})
\end{array}$$

With the chromaticity sextupole strength of K~2~L = 0.01795 m^-2^ and
using the average beta functions and dispersion functions at the
locations of the chromaticity sextupole magnets we can calculate the
change in chromaticity from the T:SF and T:SD circuits at 150 Gev in the
Tevatron.

There are 88 SF magnets at an average horizontal beta of about 93.8
meters, an average vertical beta of 30.1 meters, and an average
dispersion of 3.81 meters. There are 88 SD magnets at an average
horizontal beta of about 30.3 meters, an average vertical beta of 93.3
meters, and an average dispersion of 2.3 meters. This gives

$\begin{array}{r}
\xi_{x} = \frac{\text{88}}{4\pi} \ast \text{93}\text{.}\text{8 m*0}\text{.}\text{01795 m}^{\text{-2}}\text{*3}\text{.}\text{81}\ m = \text{44}\text{.}\text{9 units/}(\text{Amp in T:SF}) \\
\xi_{y} = - \frac{\text{88}}{4\pi} \ast \text{30}\text{.}\text{1 m*0}\text{.}\text{01795  m}^{\text{-2}}\text{*3}\text{.}\text{81 m} = - \text{14}\text{.}\text{4 units/}(\text{Amp in T:SF}) \\
 \\
\xi_{x} = \frac{\text{88}}{4\pi} \ast \text{30}\text{.}\text{3 m*0}\text{.}\text{01795 m}^{\text{-2}}\text{*2}\text{.}\text{3 m} = 8\text{.}\text{76 units/}(\text{Amp in T:SD}) \\
\xi_{y} = - \frac{\text{88}}{4\pi} \ast \text{93}\text{.}\text{3 m*0}\text{.}\text{01795 m}^{\text{-2}}\text{*2}\text{.}\text{3 m} = - \text{27}\text{.}\text{0 units/}(\text{Amp in T:SD})
\end{array}$

A MAD calculation with K~2~L increased by 0.01795 m^-2^ (corresponding
to 1.0 Amps at 150 Gev) gives the following changes to the chromaticity

$$\begin{pmatrix}
\text{dCh} \\
\text{dCv}
\end{pmatrix} = \begin{pmatrix}
\text{45} & 8\text{.}9 \\
\text{-14}\text{.}\text{36} & \text{-26}\text{.}\text{96}
\end{pmatrix}\begin{pmatrix}
\text{T:SF }(\text{amps}) \\
\text{T:SD}(\text{amps})
\end{pmatrix}$$

which is consistent with the above calculations.

As a comparison between actual Tevatron and the theoretical calculations
based on the design lattice we give the calculational constants from C49
that are used to relate the change in current to the change in
chromaticity

$$\begin{pmatrix}
\text{dCh} \\
\text{dCv}
\end{pmatrix} = \begin{pmatrix}
\text{43}\text{.}\text{14} & 6\text{.}\text{99} \\
\text{-12}\text{.}\text{51} & \text{-24}\text{.}\text{25}
\end{pmatrix}\begin{pmatrix}
\text{T:SF }(\text{amps}) \\
\text{T:SD}(\text{amps})
\end{pmatrix}$$

## which agrees well with the theoretical calculations.

### Chromaticity from b2 component of dipoles

There are 772 dipoles in the Tevatron. The average beta functions, ~x~,
~y~, average dispersion, D~x~, and average product, ~x~\*D~x~, are
listed in the table below as calculated using MAD with the design
Tevatron lattice.

+---------+---------+---------+---------+---------+---------+---------+
| Circuit | Number  | Average | Average | Average | Average | Average |
|         | of      | ~x~ (m) | ~y~ (m) | D~x~    | ~x      | ~y      |
|         | e       |         |         | (m)     | ~\*D~x~ | ~\*D~x~ |
|         | lements |         |         |         | (m^2^)  | (m^2^)  |
+---------+---------+---------+---------+---------+---------+---------+
| Dipoles | 772     | 57.41   | 57.94   | 2.827   | 170.2   |         |
|         |         |         |         |         |         | ------- |
|         |         |         |         |         |         |   155.6 |
|         |         |         |         |         |         |         |
|         |         |         |         |         |         |         |
|         |         |         |         |         |         | ------- |
+---------+---------+---------+---------+---------+---------+---------+

Thus if K2 = 1E-4 m^-3^ then

$$\begin{array}{r}
\xi_{x} = \frac{1}{4\pi}\sum_{}^{}\beta_{\text{x,i}}(K_{2}\text{LD}_{x}) \\
\xi_{x} = \frac{\text{772}}{4\pi} \ast \text{57}\text{.}\text{41 m*1E} - \text{4 m}^{\text{-3}}\text{*6}\text{.}\text{1214 m*2}\text{.}\text{827 m} = 6\text{.}\text{10 units/}(\text{1E-4 m}^{3}\text{of K2}) \\
\xi_{y} = - \frac{1}{4\pi}\sum_{}^{}\beta_{\text{y,i}}(K_{2}\text{LD}_{x}) \\
\xi_{y} = - \frac{\text{772}}{4\pi} \ast \text{57}\text{.}\text{94 m*1E} - \text{4 m}^{\text{-3}}\text{*6}\text{.}\text{1214 m*2}\text{.}\text{827 m} = - 6\text{.}\text{16 units/}(\text{1E-4 m}^{3}\text{of K2})
\end{array}$$

Changing K2 in the BEND elements in the MAD file by 1E-4 changes the
horizontal chromaticity from 10 units to 16.44 units and the vertical
chromaticity from 10 units to 4.14 units which confirms the above
calculations

### Estimate of b2 component of dipoles

The average, over all the dipoles in the Tevatron, of the time varying
sextupole field, K~2~, can be estimated from the time variation in the
T:SF and T:SD sextupoles and the fact that these circuits are tuned up
to keep the chromaticity constant.

During the studies on tune drift measurements taken on 5/15/02 the T:SF
and T:SD b2 windup curves played out the following current as a function
of time

![](media/image25.wmf)

and fits to a logarithmic function gives

$$\begin{array}{r}
{\Delta I}_{\text{SF}}(\text{amps}) = \text{ -0}\text{.}\text{174*ln}(t)\text{ -0}\text{.}\text{0627} \\
{\Delta I}_{\text{SD}}(\text{amps}) = \text{ -0}\text{.}\text{267*ln}(t)\text{ -0}\text{.}\text{0974}
\end{array}$$

or, with the transfer constant K~2~L = 0.01795 m^-2^/Amp, in terms of
the sextupole gradient strength

$$\begin{array}{r}
\Delta(K_{2}L)_{\text{SF}}(m^{\text{-2}}) = \text{ -3}\text{.}\text{12x10}^{\text{-3}}\text{*ln}(t)\text{ -1}\text{.}\text{13x10}^{\text{-3}} \\
\Delta(K_{2}L)_{\text{SD}}(m^{\text{-2}}) = \text{  -4}\text{.}\text{79x10}^{\text{-3}}\text{*ln}(t)\text{ -1}\text{.}\text{21x10}^{\text{-3}}
\end{array}$$

Using this data we can calculate the change in chromaticity due to the
change in the T:SF and T:SD circuits and then estimate the change in the
K~2~ component of the Tevatron dipoles. Recent measurements showed that
the b2 windup curves were tuned up well at 150 Gev and the measured
chromaticity changed by only one or two units (out of about 40 to 50
units of chromaticity change over two hours if the b2 were not
compensated.) So we take ~x~ = 0 and use

$${\Delta\xi}_{x} = \frac{1}{4\pi}\sum_{\text{Dipole}}^{}\left( \beta_{x}{\Delta K}_{2}\text{LD}_{x} \right) + \frac{1}{4\pi}\sum_{\text{T:SF}}^{}\left( \beta_{x}{\Delta K}_{2}\text{LD}_{x} \right) + \frac{1}{4\pi}\sum_{\text{T:SD}}^{}\left( \beta_{x}{\Delta K}_{2}\text{LD}_{x} \right) = 0$$

to calculate (K~2~L)~Dipole~. From a MAD calculation of the design
lattice at 150 Gev

$$\sum_{\text{Dipole}}^{}\left( \beta_{x}D_{x} \right) = 1\text{.}\text{32x10}^{5}\ m^{2},\ \sum_{\text{T:SF}}^{}\left( \beta_{x}D_{x} \right) = 3\text{.}\text{16x10}^{4}\ m^{2},\ \sum_{\text{T:SD}}^{}\left( \beta_{x}D_{x} \right) = 6\text{.}\text{23x10}^{3}\ m^{2}$$

which gives

$$\begin{array}{r}
\left( {\Delta K}_{2}L \right)_{\text{Dipole}}\sum_{\text{Dipole}}^{}\left( \beta_{x}D_{x} \right) = - \Delta(K_{2}L)_{\text{SF}}\sum_{\text{T:SF}}^{}\left( \beta_{x}D_{x} \right) - \Delta(K_{2}L)_{\text{SD}}\sum_{\text{T:SD}}^{}\left( \beta_{x}D_{x} \right) \\
\left( {\Delta K}_{2}L \right)_{\text{Dipole}}\text{*1}\text{.}\text{32x10}^{5} = \begin{array}{r}
\left( (3\text{.}\text{12x10}^{\text{-3}}\text{*ln}(t) + 1\text{.}\text{13x10}^{\text{-3}})\text{*3}\text{.}\text{16x10}^{4} + \middle| \right) \\

\end{array} \\

\end{array}$$

If we use the vertical chromaticity to calculate the K~2~ component of
the dipole we get

$$\begin{array}{r}
\left( {\Delta K}_{2}L \right)_{\text{Dipole}}\sum_{\text{Dipole}}^{}\left( \beta_{y}D_{x} \right) = - \Delta(K_{2}L)_{\text{SF}}\sum_{\text{T:SF}}^{}\left( \beta_{y}D_{x} \right) - \Delta(K_{2}L)_{\text{SD}}\sum_{\text{T:SD}}^{}\left( \beta_{y}D_{x} \right) \\
\left( {\Delta K}_{2}L \right)_{\text{Dipole}}\text{*1}\text{.}\text{33x10}^{5} = \begin{array}{r}
\left( (3\text{.}\text{12x10}^{\text{-3}}\text{*ln}(t) + 1\text{.}\text{13x10}^{\text{-3}})\text{*1}\text{.}\text{01x10}^{4} + \middle| \right) \\

\end{array} \\

\end{array}$$

The factors multiplying the ln(t) term when using the horizontal and
vertical chromaticity agree to within 5%. The difference in the constant
term is only 4% of the value of (K~2~at time t=5 minutes and even less
for later times. Therefore we quote the average value of these two
determinations K~2~ and get

$$\left( {\Delta K}_{2}L \right)_{\text{Dipole}}(m^{\text{-3}}) = 9\text{.}\text{51x10}^{\text{-4}}\text{*ln}(t) + 2\text{.}\text{94x10}^{\text{-4}}$$

for the average time variation of the sextupole field in the Tevatron
dipoles.

## Appendix C: Momentum, circumference, and RF 

For a constant bend field the change in revolution period is

$$\begin{array}{r}
\frac{\Delta\tau}{\tau} = \frac{\Delta L}{L} - \frac{\Delta v}{v} \\
 \\
 \\
\frac{\Delta L}{L} = \frac{1}{\gamma_{t}^{2}}\left( \frac{\Delta p}{p} \right) \\
\frac{\Delta v}{v} = \frac{1}{\gamma^{2}}\left( \frac{\Delta p}{p} \right) \\
\eta = \frac{1}{\gamma_{t}^{2}} - \frac{1}{\gamma^{2}} \\
\frac{\Delta\tau}{\tau} = \eta\ \left( \frac{\Delta p}{p} \right) \\
\frac{\Delta f}{f} = - \eta\ \left( \frac{\Delta p}{p} \right)
\end{array}$$

For the Tevatron at 150 Gev we have $\alpha = \frac{1}{\gamma_{t}^{2}}$
from a MAD calculation

$$\begin{array}{r}
\alpha = \frac{1}{\gamma_{t}^{2}} = 0\text{.}\text{0028266} = \frac{1}{\text{18}\text{.}\text{809}^{2}} \\
\eta = \frac{1}{\text{18}\text{.}\text{809}^{2}} - \frac{1}{(\frac{\text{150}}{0}\text{.}\text{938})^{2}} = 0\text{.}\text{002787} \\
\left( \frac{\Delta p}{p} \right) = - \text{358}\text{.}\text{74}\left( \frac{\Delta f}{f} \right) \\
\left( \frac{\Delta p}{p} \right)@ + \text{40}\text{Hz} = - \text{358}\text{.}\text{74} \ast \left( \frac{\text{40}\text{Hz}}{\text{53}\text{.}\text{103703}\text{MHz}} \right) = - 0\text{.}\text{0002702} \\

\end{array}$$

## Appendix D: Tevatron tune quadrupoles

### Quadrupole magnet strength

First we define the quadrupole magnetic field as

$$\begin{array}{r}
B_{x}(x,y) = B_{1}y \\
B_{y}(x,y) = B_{1}x
\end{array}$$

and want to relate this to the magnetic field in a Tevatron quadrupole
trim magnet. From \"The Tevatron Energy Doubler: A superconducting
Accelerator\", H. Edwards, (*Ann. Rev. Nucl. Part Sci.* 1985, 35:605-60)
we have that the integrated strength of a trim quadrupole magnet is 75
kG-in at 1 inch at 50 Amps giving a strength of

$$B_{1}L/I = 0\text{.}\text{15}\ \text{T/A}$$

where L is the length of the sextupole and I is the current in the
magnet.

$$\int_{}^{}{}B_{y}(\text{at}\ 1\ \text{inch}) \cdot \text{dl} = \text{75}\ \text{kG} - \text{in}@\text{50}\text{Amp}$$

$$B_{1}L(1\ \text{inch})/I = 7\text{.}5\ T - \frac{\text{inch}}{\text{50}}A$$

$$B_{1}L/I = 7\text{.}5\ \frac{T}{\text{50}}A = 0\text{.}\text{15}\frac{T}{A}$$

If the quadrupoles are treated as a thin quadrupole magnet, then in the
MAD convention the thin quadrupole strength is given by

$$K_{1}L = (B_{1}\frac{L}{I})\frac{1}{|B\rho|}I$$

where I is the current in the quadrupole, and $|B\rho|$ is the magnetic
rigidity. With 1 Amp in the trim quadrupole circuit we can calculate the
gradient strength at 150 Gev,

$$\begin{array}{r}
K_{1}L = (B_{1}\frac{L}{I})\frac{1}{|B\rho|}I = (0\text{.}\text{15}\frac{T}{A}\frac{)}{(}\text{3335}\text{.}\text{64}T - \frac{m}{\text{Tev}} \ast 0\text{.}\text{15}\text{Tev}\ )(1\text{.}0\ A) \\
K_{1}\text{L } = 0\text{.}\text{00030}\ \frac{m^{\text{-1}}}{\text{Amp}}
\end{array}$$

This can then be related to the change in tune that it produces

$${\Delta\nu}_{x} = \frac{1}{4\pi}\sum_{}^{}\beta_{\text{x,i}}(K_{1}L)_{\text{no,i}}$$

$${\Delta\nu}_{y} = - \frac{1}{4\pi}\sum_{}^{}\beta_{y,i}(K_{1}L)_{\text{no,i}}$$

As an order of magnitude we use 1 Amp in the tune quad at a location
where the beta function is 93 meters. Then the tune change is

$${\Delta\nu}_{x} = \frac{1}{4\pi}\sum_{}^{}{\text{93}m}\text{*0}\text{.}\text{00030 m}^{\text{-1}}\text{/Amp } = \text{ 0}\text{.}\text{00222/Amp}$$