# **Optic Measurements and Optics Correction in Tevatron**

#### Valeri Lebedev

FNAL, September 18, 2003

## [Contents]{.underline}

1.  Differential optics measurements

2.  Formalism for X-Y coupled betatron motion

3.  On-line optics measurements and correction

4.  Emittance growth due to coupling and optics mismatch

5.  Conclusions

## [1. Differential optics measurements]{.underline}

-   Presently, the only optics measurements for the entire machine have
    been performed with the differential orbits measurements

-   [Counsel application]{.underline} (P163)

    -   Tevatron, MI, Accumulator and all transfer lines with exception
        of MI-8, and MI-to-Recycler

        -   Four correctors (two horizontal and two vertical) and energy
            change

    -   For Tevatron

        -   Correctors: HE42, HE44, VE45, VE47

            1.  50 μrad excites \~ ±3 mm betatron wave

        -   Energy change

            1.  80 Hz in T:VFKNOB corresponds to ∆p/p ≈ --4.8⋅10^-4^

        -   The measurements are fast (\~5 minutes) and can be easily
            acquired during shot setup or whenever it is necessary.

-   [Tedious analysis]{.underline}

    -   Manual fitting of differential orbits

        -   It requires an experienced person, and it takes \~2 days for
            the entire Tevatron.

    -   Fitting differential orbits in the model allows one to determine
        linear optics and dispersions with X-Y coupling taken into
        account.

        -   The accuracy is about 10-15% for both the beta-functions and
            the dispersions.

    -   Presently, we acquired data at

        -   Injection energy -- analysis is done for many different
            measurements

        -   Flat top -- single measurement, analyzed

        -   Low beta -- single measurement, analysis is not finished

![](media/image1.wmf){width="6.488194444444445in"
height="1.5180555555555555in"}

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height="1.5208333333333333in"}

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height="1.551388888888889in"}

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![](media/image5.wmf){width="6.495138888888889in"
height="1.6194444444444445in"}

F0 A0 B0 C0

Differential orbits for the first Tevatron half after Tevatron optics
correction

![](media/image6.wmf){width="6.488194444444445in"
height="1.5180555555555555in"}

![](media/image7.wmf){width="6.496527777777778in"
height="1.5208333333333333in"}

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height="1.551388888888889in"}

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![](media/image10.wmf){width="6.495138888888889in"
height="1.6194444444444445in"}

C0 D0 E0 F0

Differential orbits for the second Tevatron half after Tevatron optics
correction

##### [Findings]{.underline}

-   BPMs

    -   HA34 had wrong polarity and was fixed

    -   In average the measured beam displacement is about 8% below the
        actual one

    -   There is a number of BPMs which differential responses are up to
        10-20% different from the average

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-   Correctors

    -   Correctors used for measurements are tilted up to a few degrees

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-   Dipoles

    -   Edge focusing correction

        -   0.057 units (0.039 deg edge focusing) at injection energy

        -   -0.033 units (0.023 deg edge focusing) at top energy

    -   Coherent skew-quadrupole term in all dipoles

        -   A~1~ ≈ 1.4 unit at injection energy

        -   A~1~ ≈ 2 unit at top energy

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-   Quadrupoles

    -   Focusing correction for quads on the main bus

        -   0.177% at injection energy

        -   0.261% at top energy

    -   To fit the measurements \~30 focusing corrections need to be
        applied

        -   Single correction can be as large as 2.5%

        -   Quadrupole and skew-quadrupole corrections are of the same
            order

        -   For most of the errors the origin is not understood

            -   Measurements taken at different times (separated by a
                few weeks) require quite different corrections to be
                applied

            -   Orbit effect ???

                -   2% focus. error corresponds to \~1 cm offset in a
                    chrom. sext.

                -   For many dipoles non-linearity is worse than for
                    sextupoles

        -   Errors in focusing at S6 family feeddown sextupoles were
            well visible before its current was reduced from \~20 A to
            \~7 A

            -   Optics correction for 1 mm displacement in S6 sextupole

                -   1% before current reduction

                -   0.35% after current reduction

### Point optics corrections for August 22 & 29 

  ----------------- -------------- ------------------- -------------------
  Corrections for                                      
  the regular                                          
  quads^\*^                                            

  Name^\*\*^        Value, kG      B42                 1

  A11s              3.5            B42s                -2

  A13               1              B44                 5

  A15               4              C13                 1

  A17               -3             C13s                5

  A28               -2             C17                 -2

  A32               1              C17s                -1

  A34               2              C45                 2

  A42s              -4             C46s                1

  A42               -2             D16s                -5

  A44               1              D45                 4

  A46               1              Corrections for the 
                                   final focus quads   

  A46s              -3                                 

  B15               -1             Name                value

  B15s              -2             F_B0Q3              0.6%

  B17s              -2             Roll_B0Q3D          -0.05 deg

  B23               2              Roll_B0Q3F          -0.02 deg

  B28               -1             F_D0Q3              0.7%

  B38s              1                                  
  ----------------- -------------- ------------------- -------------------

^\*^ Integral strength of regular quad at 150 is 191.2 kG

^\*\*^ Character "s" at the name end denotes skew-quad

###### There is no optics corrections in E and F sectors !!!

## 

### [Problems with differential orbit measurements and plans for the future]{.underline}

-   There is no sufficient data redundancy in the present differential
    orbit measurements and therefore the manual data analysis is
    preferable

    -   Measured points 5\*236 = 1180 (236 BPMs, 5 measurements)

    -   Unknowns 3\*236+5 =713 (skew & normal quad errors, BPM dif.
        resp.)

-   In collaboration with ANL we began building more advanced software
    for differential orbit data taking

    -   More differential orbits will be acquired

        -   20 to 50 instead of 5

        -   Measurement time will grow up from \~5 min to 15-30 min

    -   Software will compute

        -   Quad focusing errors and rotations

        -   Differential responses for BPMs

        -   Rotation and strength of the correctors used for excitation

    -   First tests are expected to start at the beginning 2004

## 

## [2. Formalism for X-Y coupled betatron motion]{.underline}

Eigen-vectors for uncoupled betatron motion

> $${\hat{v}}_{1}(s) = \begin{bmatrix}
> \sqrt{\beta_{x}} \\
>  - \frac{i + \alpha_{x}}{\sqrt{\beta_{x}}} \\
> 0 \\
> 0
> \end{bmatrix}\ ,\ {\hat{v}}_{2}(s) = \begin{bmatrix}
> 0 \\
> 0 \\
> \sqrt{\beta_{y}} \\
>  - \frac{i + \alpha_{y}}{\sqrt{\beta_{y}}}
> \end{bmatrix}\ ,$$

Development of Mais-Ripken representation

> $${\hat{v}}_{1} = \begin{bmatrix}
> \sqrt{\beta_{1x}} \\
>  - \frac{i(1 - u) + \alpha_{1x}}{\sqrt{\beta_{1x}}} \\
> \sqrt{\beta_{1y}(s)}e^{{i\nu}_{1}} \\
>  - \frac{\text{iu} + \alpha_{1y}}{\sqrt{\beta_{1y}(s)}}e^{{i\nu}_{1}}
> \end{bmatrix}\ ,\ {\hat{v}}_{2} = \begin{bmatrix}
> \sqrt{\beta_{2x}}e^{{i\nu}_{2}} \\
>  - \frac{\text{iu} + \alpha_{2x}}{\sqrt{\beta_{2x}}}e^{{i\nu}_{2}} \\
> \sqrt{\beta_{2y}} \\
>  - \frac{i(1 - u) + \alpha_{2y}}{\sqrt{\beta_{2y}}}
> \end{bmatrix}\ ,$$

-   11 optics functions but only 8 of them are independent because of
    symplecticity.

-   3 symplecticity conditions were already used to bind up terms
    proportional to *i*.

![](media/image11.wmf){width="8.998611111111112in"
height="2.872916666666667in"}

F0 A0 B0 C0

![](media/image12.wmf){width="8.998611111111112in"
height="2.6881944444444446in"}

C0 D0 E0 F0

β-functions computed from differential orbit measurements of Aug. 22 and
29, 2003; central orbit, corrected optics

## ![](media/image13.wmf){width="8.748611111111112in" height="2.8756944444444446in"}

F0 A0 B0 C0

![](media/image14.wmf){width="8.629861111111111in"
height="2.7736111111111112in"}

C0 D0 E0 F0

Dispersions restored from differential orbit measurements; central
orbit, corrected optics

## [3. On-line optics measurements and correction]{.underline} 

-   Differential orbit measurements results

    -   There are no single strong source of optics discrepancy

    -   There is no point corrections in vicinity of inj. point (E and F
        sectors)

    -   Overall betatron mismatch is reasonably small

        -   Before correction in E and F sectors

            -   Vertically - ∆β~y~/β ~y~ \~±15%

            -   Horizontally - ∆β ~x~/β ~x~ \~±5%

    -   Horizontal mismatch is significantly smaller

-   The aim of optics correction

    -   To minimize emittance growth at transfers

-   Optics correction method

    -   The tunes are close to high integer and therefore there is a
        resonant amplification of beta-function mismatch

    -   Correcting betatron wave in E and F sectors reduce optics
        mismatch through entire machine

        -   It also slightly improves helical beam separation

    -   2 orthogonal quads in each plane anywhere in the ring will do
        the job

    -   4 quads around D0 interaction point were chosen

        -   D0 is the place of strongest optics distortions

-   On-line optics measurements in E and F sectors

    -   Courant --Snyder invariants built from differential orbit
        measurements

        -   2 correctors orthogonal in betatron phase space for each
            plane

    -   Tune shifts due to focusing change in a single quadrupole

        -   2 quads orthogonal in betatron phase space for each plane

##  

**O**ptics measurements and optics corrections performed at August 22

+--------------+-----------+-------------+------------+---------------+
|              |           | Initial     | 1^st^      | 2^nd^ iter    |
|              |           |             | i          |               |
|              |           |             | ter(delta) |               |
+--------------+-----------+-------------+------------+---------------+
| Optics       | C:CQ9     | 63.92       | 45.9(-18)  | 63.92(0)      |
| corrections  | \[A\]     |             |            |               |
+--------------+-----------+-------------+------------+---------------+
|              | C:CQ7     | 98.07       | 125.1(27)  | 98.07(0)      |
|              | \[A\]     |             |            |               |
+--------------+-----------+-------------+------------+---------------+
|              | C:DQ7     | 99.65       | 117.6(18)  | 108.65(9)     |
|              | \[A\]     |             |            |               |
+--------------+-----------+-------------+------------+---------------+
|              | C:DQ9     | 89.65       | 64.6(-25)  | 76.79(-12.86) |
|              | \[A\]     |             |            |               |
+--------------+-----------+-------------+------------+---------------+
|              | T:SQ      | -2.827      | -3.023     | -2.852(-.025) |
|              | \[A\]     |             |            |               |
+--------------+-----------+-------------+------------+---------------+
|              | T:SQA0    | 4.185       | 6.680      | 4.361(0.176)  |
|              | \[A\]     |             |            |               |
+--------------+-----------+-------------+------------+---------------+
|              | QY(set)   | 20.568      | 20.551     | 20.55514      |
+--------------+-----------+-------------+------------+---------------+
|              | QX(set)   | 20.605      | 20.58315   | 20.6096       |
+--------------+-----------+-------------+------------+---------------+
| Measured     | QY(meas)  | .575        | .5752      | .5756         |
| tunes        |           |             |            |               |
+--------------+-----------+-------------+------------+---------------+
|              | QX(meas)  | .583        | .5831      | .5831         |
+--------------+-----------+-------------+------------+---------------+
| Optics meas. | QE17      | .           | .0         | .0208/-.0071  |
| by 10 A quad |           | 0213/-.0075 | 249/-.0061 |               |
| current      |           |             |            |               |
| change,      |           |             |            |               |
|              |           |             |            |               |
| ∆Qx/∆Qy      |           |             |            |               |
+--------------+-----------+-------------+------------+---------------+
|              | QE19      | .           | .0         | .0208/-.0067  |
|              |           | 0207/-.0076 | 187/-.0080 |               |
+--------------+-----------+-------------+------------+---------------+
|              | QE47      | .           | .0         | .0061/-.0287  |
|              |           | 0065/-.0255 | 068/-.0310 |               |
+--------------+-----------+-------------+------------+---------------+
|              | QF33      | .           | .0         | .0059/-.0275  |
|              |           | 0059/-.0310 | 072/-.0239 |               |
+--------------+-----------+-------------+------------+---------------+
| ∆Qx/∆Qy for  | QE17      |             |            | .0225/-.0077  |
| ∆I=10 A      |           |             |            |               |
| predi-cted   |           |             |            |               |
| by model^\*^ |           |             |            |               |
+--------------+-----------+-------------+------------+---------------+
|              | QE19      |             |            | .0210/-.0079  |
+--------------+-----------+-------------+------------+---------------+
|              | QE47      |             |            | .0070/-.0287  |
+--------------+-----------+-------------+------------+---------------+
|              | QF33      |             |            | .0069/-.0290  |
+--------------+-----------+-------------+------------+---------------+

^\*^ The model uses periodic beta-functions in E and F sectors with
coupling taken into account

  --------------- ------------ -------------- ------------- ----------------
                               Initial        1^st^         2^nd^ iter
                                              iter(delta)   

  Optics meas.    Ax1          0.75           0.72          0.76
  with C.-S.                                                
  invariants                                                

                  Ax2          0.77 (+0.02)   0.79          0.77 (+0.01)

                  Ay1          0.73           0.80          0.74

                  Ay2          0.79 (+0.06)   0.74          0.77 (+0.03)
  --------------- ------------ -------------- ------------- ----------------

![](media/image15.wmf){width="6.5in" height="1.7986111111111112in"}

![](media/image16.wmf){width="6.500694444444444in"
height="1.7986111111111112in"}

![](media/image17.wmf){width="6.5in" height="1.7986111111111112in"}

## [4. Emittance growth due to coupling and optics mismatches]{.underline}

a\) Emittance growth due to betatron and dispersion mismatch from a
lattice with *β*~1~, *α*~1~, *D*~1~ and $D_{1}^{'}$ to a lattice with
*β*~2~, *α*~2~, *D*~2~ and $D_{2}^{'}$ is

#### $$\begin{array}{r}
\varepsilon^{'} = \frac{\varepsilon}{2}\left( \frac{\beta_{1}}{\beta_{2}}\left\lbrack 1 + \alpha_{2^{2}} \right\rbrack + \frac{\beta_{2}}{\beta_{1}}\left\lbrack 1 + \alpha_{1^{2}} \right\rbrack - 2\alpha_{1}\alpha_{2} \right) + \\
\ \frac{\sigma_{p^{2}}}{2}\left( \beta_{2}\left( D_{0}^{'} - D_{1}^{'} \right)^{2} + 2\alpha_{2}\left( D_{0}^{'} - D_{1}^{'} \right)\left( D_{0} - D_{1} \right) + \frac{\left( D_{0} - D_{1} \right)^{2}}{\beta_{2}}\left( 1 + \alpha_{2^{2}} \right) \right)
\end{array}$$

> ⇒
> $\frac{\text{δε}}{\varepsilon} \approx \ \frac{1}{2}\left( \frac{\Delta\beta}{\beta}|_{\text{max}} \right)^{2} + \ \frac{\left( \sigma_{p}{\delta D}_{\text{max}} \right)^{2}}{2\text{εβ}_{\text{max}}}$

b\) Emittance growth for beam transfer from an uncoupled lattice with
*β*~x~, *α*~x~, *β*~y~ and *α*~y~, to a coupled lattice described by
*β*~1x~, *α*~1x~, *β*~1y~, *α*~1y~, *β*~2x~, *α*~2x~, *β*~2y~ and
*α*~2y~ with the eigen-vectors

> $$\begin{array}{r}
> \varepsilon_{1^{\prime}} = \varepsilon_{1}A_{\text{11}} + \varepsilon_{2}A_{\text{12}} \\
> \varepsilon_{2^{\prime}} = \varepsilon_{1}A_{\text{21}} + \varepsilon_{2}A_{\text{22}}
> \end{array}$$

#### 

#### 

-   Measurements of P1 and A1 lines performed at the end of 2002 could
    not explain the reason of large emittance growth in the round trip
    emittance measurements

-   Tevatron optics distortions were most probable reason

    -   but there were not clear understanding how

    -   and what particular phenomenon is responsible for this

![](media/image22.wmf)

Measured round trip emittances before optics correction

![](media/image23.wmf)

Measured round trip emittances after optics correction

**Measured and computed^\*^** round trip emittance growth

+--------------+-----------+-------------+-------------+---------------+
|              | **Un      |             | **Corrected |               |
|              | corrected |             | (old)       |               |
|              | (old)     |             | Tevatron    |               |
|              | Tevatron  |             | optics**    |               |
|              | optics**  |             |             |               |
+--------------+-----------+-------------+-------------+---------------+
|              | **M       | *           | *           | **Computed**  |
|              | easured** | *Computed** | *Measured** |               |
+--------------+-----------+-------------+-------------+---------------+
| δε~x~ \[mm,  | **4.      | **2.54**    | **          | **3.08**      |
| mrad\]       | 22±0.21** |             | 4.07±0.12** |               |
+--------------+-----------+-------------+-------------+---------------+
| δε~y~ \[mm,  | **1.      | **3.40**    | **          | **2.98**      |
| mrad\]       | 58±1.47** |             | 2.83±0.61** |               |
+--------------+-----------+-------------+-------------+---------------+
| (δε~x~ +     | **2.9**   | **2.97**    | **3.45**    | **3.03**      |
| δε~y~ )/2,   |           |             |             |               |
|              |           |             |             |               |
| \[mm, mrad\] |           |             |             |               |
+--------------+-----------+-------------+-------------+---------------+

^\*^ Computations were carried out for equal horiz. and vert. emittances
of 11 mm mrad.

-   Careful Tevatron tuning reduced emittance growth in comparison with
    the end of 2002 measurements

    -   ∆ε~x~ ≈ 7 mm mrad ⇒ ∆ε~x~ = 4 mm mrad

    -   ∆ε~y~ ≈ 4 mm mrad ⇒ ∆ε~y~ = 3 mm mrad

-   Tevatron optics correction made barely visible improvement for
    transfers

-   Strong coupling in Tevatron is found to be the major offender

    -   ∼14% emittance growth in each plane per transfer

-   Model prediction of machine optics on helices is not sufficiently
    accurate

    -   It yields a discrepancy between measurement and predictions

    -   There is no reliable information about machine non-linearities

        -   The model uses uniform distribution of dipole
            non-linearities

### 

### [Tevatron Emittance monitors]{.underline}

### Tevatron emittance measurements were carried out after optics correction

### The machine beta-functions should be close to the design beta-functions

### Horizontal emittance monitor is consistent with MI emittance monitor

### Vertical emittance monitor reports 16 mm mrad emittance versus 12 mm mrad following from MI measurements (1.33 times higher)

### There was scraping on the level of 2-3% when helix was open

### It could reduce the emittance of the beam coming back to MI by 1-1.5 mm mrad. That takes out less than 1 mm mrad out of 4 mm mad discrepancy!!!

### The question is open 

### [Correction of A1 for 106 Tevatron dipoles]{.underline} 

![](media/image26.wmf){width="8.988888888888889in"
height="1.8083333333333333in"}

E0 F0 A0

![](media/image27.wmf){width="8.988888888888889in"
height="1.804861111111111in"}

B0 C0 D0

4D beta-functions and positions of skew-quads

-   Presently, there are many dipoles, which do not have nearby
    skew-quads

    -   That prevents suppression of coupling to sufficiently small
        level

        -   Small tune split does not automatically mean small coupling

-   Compensation of skew-quadrupole term in 106 of 774 Tevatron dipoles
    will allow significantly decrease coupling

    -   Dipoles are located around B0 and D0 IPs (A44-B19 and C44-D19)

![](media/image28.wmf)![](media/image29.wmf)

Projections of betatron modes to X-Y plan at proton injection point for
present Tevatron (left) and after correction of skew-quadrupole
component (right).

![](media/image30.wmf){width="8.875in" height="2.8027777777777776in"}

![](media/image31.wmf){width="8.877777777777778in"
height="2.5368055555555555in"}

Projections of betatron functions to orthogonal plan for present
Tevatron (top) and after correction of skew-quadrupole component
(bottom).

## [Conclusions]{.underline}

1.  Measured and predicted round trip emittance growths are in good
    agreement

    -   Coupling makes major contribution: ∆ε ≈ 1.5 mm mrad per transfer
        for ε = 11 mm mrad

    -   Simulations predict that there is no significant difference for
        transfers to the central orbit and p-bar helix

2.  Correction of skew-quadrupole term in 106 Tevatron dipoles reduces
    the emittance growth related to coupling from 13% to 3% per
    transfer. That will yield:

    -   \~ 1 mm mrad improvement for pbars

    -   \~ 2 mm mrad improvement for protons

        -   \~ 10% luminosity increase due to smaller beam emit, and
            \~5% due to loss

3.  There is a difference of 4 mm mrad between vertical MI and Tevatron
    flying wires

    -   Instrumental error needs to be understood and fixed

4.  Preliminary results for the differential orbit measurements at Flat
    top

    -   Skew-quad field grows from \~1.5 units to \~2 units

    -   Optics discrepancies are of the same order as at the injection

5.  There is an evidence that B11V high voltage separator deflects the
    beam by 8% more than its computed strength

6.  Shots after optics correction exhibited higher gain in luminosity
    than it predicted

    -   Possible reason can be that there are other discrepancies not
        taken into account in the model (MI, transfer lines, etc.). Then
        we have been far away from the optimum where sensitivity to
        errors is mach higher.