![](media/image1.png){width="2.6354166666666665in"
height="0.4791666666666667in"}

**November 30, 2005**

**[Technical Specification for]{.underline}**

the Radio Frequency Quadrupole (RFQ)

Vane Modulation for the Proton Driver

**[Fermilab Specification: 5500-ES-371026]{.underline}**

![](media/image2.jpeg){width="6.5in" height="0.23958333333333334in"}

+-------+-------+--------+----------------------+----------+----------+
| **    | ## E  | #      | **Description**      | **Or     | **       |
| Revis | R/ECO | # Date |                      | iginated | Approved |
| ion** |       |        |                      | By**     | By**     |
+-------+-------+--------+----------------------+----------+----------+
| None  | ER \# | 21-    | Original Issue       | *T.      | *G.      |
|       | 8220  | Dec-05 |                      | Page*    | Apo      |
|       |       |        |                      |          | llinari* |
+-------+-------+--------+----------------------+----------+----------+
|       |       |        |                      |          |          |
+-------+-------+--------+----------------------+----------+----------+
|       |       |        |                      |          |          |
+-------+-------+--------+----------------------+----------+----------+
|       |       |        |                      |          |          |
+-------+-------+--------+----------------------+----------+----------+
|       |       |        |                      |          |          |
+-------+-------+--------+----------------------+----------+----------+

**1.0 [GENERAL]{.underline}**

This document describes the vane modulation for the Radio Frequency
Quadrupole (RFQ) whose specifications are given in Fermilab
Specification 5500-ES-371025. The technical drawing that compliments
this document is 5520-ME-440197.

# [GOVERNING EQUATIONS]{.underline}

> **2.1 Input Radial Matcher**
>
> The distance *r* from the RFQ axis to the electrode in the input
> radial matcher is given by the expression:
>
> $$r(z) = \frac{R_{0}}{\text{sin}\left( \frac{\pi}{2N + 1}\ \left( \frac{z}{L_{c}} + 0\text{.}5 \right) \right)}$$
>
> where:
>
> $R_{0}$ = distance from the axis to the electrode at cross section
> with exact quadrupole symmetry = 0.34 cm
>
> $L_{c}$ = cell length $= \frac{\beta_{\text{in}}\lambda}{2}$
>
> $\beta_{\text{in}}$ = relative velocity of injected beam
>
> $\lambda$ = RF field wavelength
>
> $N$ = number of cells = 6.
>
> Figure 2.1.1 shows the input matcher electrode profile. Figure 2.1.2
> shows the EM Studio model on the input matcher.
>
> ![](media/image3.jpeg){width="3.359027777777778in"
> height="2.7958333333333334in"}

Figure 2.1.1. Input matcher electrode profile.

> ![](media/image4.jpeg){width="3.2402777777777776in"
> height="2.7534722222222223in"}

Figure 2.1.2. EM Studio model of input matcher.

> **2.2 Main RFQ**
>
> The modulation of the electrodes in the main part of the RFQ is
> sinusoidal with a constant vane tip radius, $R_{e}$= 0.255 cm. The
> distance from the axis to the electrode surface along the *i*-th cell
> is given by the following expressions:
>
> $r_{y}(z) = R_{0}\left\lbrack 1 - ( - 1)^{i} \cdot \frac{m_{i} - 1}{m_{i} + 1} \cdot \text{sin}\left( \frac{\pi}{L_{\text{ci}}} \cdot \left( z - Z_{i} \right) \right) \right\rbrack$
> for vertical plane,
>
> $r_{x}(z) = R_{0}\left\lbrack 1 + ( - 1)^{i} \cdot \frac{m_{i} - 1}{m_{i} + 1} \cdot \text{sin}\left( \frac{\pi}{L_{\text{ci}}} \cdot \left( z - Z_{i} \right) \right) \right\rbrack$
> for horizontal plane,
>
> where:
>
> $z$ = distance along the vanes
>
> $Z_{i}$ = the coordinate of the beginning of the *i*-th cell; the
> beginning of the cell is the point with exact quadrupole symmetry
>
> $R_{0}$ = distance from the axis to the electrode at cross section
> with exact quadrupole symmetry
>
> $L_{\text{ci}}$ = *i*-th cell length; for constant synchronous phase
> $L_{c} = \frac{\beta\ \lambda}{2}$
>
> $m$ = modulation factor = $\frac{2R_{0}}{a} - 1$
>
> $a$ = minimum aperture (radius)
>
> $\beta$ = relative velocity of particle
>
> $\lambda$ = RF field wavelength.
>
> The modulation factor, $m$, and cell length, $L_{c}$, for each cell
> has been calculated with DESRFQ code and are shown in Section 3, Table
> 3.1 of this document.
>
> Figure 2.2.1 shows the EM Studio model of the vane modulation while
> Figure 2.2.2 shows the cross section of the vane tips.
>
> ![](media/image5.jpeg){width="3.345138888888889in"
> height="2.2736111111111112in"}

Figure 2.2.1. EM Studio model of the vane modulation in main RFQ.

> ![](media/image6.jpeg){width="1.6716535433070867in"
> height="1.5708661417322836in"}

Figure 2.2.2. Vane tip cross section.

> **2.3 Output radial matcher**
>
> The distance *r* from the RFQ axis to the electrode in the output
> radial matcher is given by the expressions:
>
> ![](media/image7.wmf)
>
> where:
>
> $L_{1}$ = 2.2168 cm

*D* = 12.5664 cm

> $L_{2}$ = 13.7779 cm; this is the length of the output matcher.
>
> Figure 2.3.1 shows the output matcher electrode profile. Figure 2.3.2
> shows the EM Studio model on the output matcher.
>
> ![](media/image8.wmf){width="3.6909722222222223in"
> height="2.9944444444444445in"}

Figure 2.3.1. Output matcher electrode profile.

![](media/image9.jpeg){width="4.998611111111111in" height="2.375in"}

Figure 2.3.2. EM Studio model of input matcher.

# [DESIGN PARAMETERS]{.underline}

Table 3.1 lists the modulation factor, $m$, and cell length, $L_{c}$,
for each cell by cell number.

Table 3.1 Values of modulation factor and cell length by cell number.

  ----------------- --------------- -------------------- -----------------
                    Cell number     Modulation, $m_{i}$  Cell length,
                                                         $L_{c}$ \[cm\]

  Input radial      1                                    2.8554
  matcher                                                

  Main RFQ          2               1.025                0.4762

                    3               1.025                0.4762

                    4               1.025                0.4763

                    5               1.025                0.4763

                    6               1.025                0.4763

                    7               1.025                0.4763

                    8               1.025                0.4763

                    9               1.025                0.4763

                    10              1.025                0.4763

                    11              1.025                0.4763

                    12              1.025                0.4763

                    13              1.025                0.4764

                    14              1.025                0.4764

                    15              1.025                0.4764

                    16              1.025                0.4764

                    17              1.025                0.4764

                    18              1.028                0.4765

                    19              1.031                0.4765

                    20              1.034                0.4766

                    21              1.037                0.4766

                    22              1.04                 0.4767

                    23              1.043                0.4768

                    24              1.046                0.4768

                    25              1.049                0.4769

                    26              1.052                0.4769

                    27              1.055                0.4770

                    28              1.058                0.4771

                    29              1.061                0.4772

                    30              1.064                0.4772

                    31              1.067                0.4773

                    32              1.07                 0.4774

                    33              1.071                0.4774

                    34              1.072                0.4775

                    35              1.073                0.4775

                    36              1.074                0.4776

                    37              1.075                0.4776

                    38              1.076                0.4777

                    39              1.077                0.4777

                    40              1.078                0.4778

                    41              1.079                0.4782

                    42              1.08                 0.4783

                    43              1.081                0.4784

                    44              1.082                0.4785

                    45              1.083                0.4786

                    46              1.084                0.4787

                    47              1.085                0.4789

                    48              1.086                0.4791

                    49              1.087                0.4793

                    50              1.088                0.4796

                    51              1.089                0.4798

                    52              1.09                 0.4801

                    53              1.091                0.4804

                    54              1.092                0.4807

                    55              1.093                0.4811

                    56              1.094                0.4815

                    57              1.095                0.4819

                    58              1.095                0.4823

                    59              1.096                0.4828

                    60              1.096                0.4832

                    61              1.097                0.4837

                    62              1.097                0.4843

                    63              1.098                0.4848

                    64              1.098                0.4854

                    65              1.098                0.4860

                    66              1.098                0.4866

                    67              1.098                0.4873

                    68              1.098                0.4879

                    69              1.098                0.4886

                    70              1.099                0.4893

                    71              1.099                0.4901

                    72              1.099                0.4909

                    73              1.099                0.4917

                    74              1.099                0.4925

                    75              1.099                0.4933

                    76              1.0993               0.4942

                    77              1.0996               0.4951

                    78              1.0999               0.4960

                    79              1.1002               0.4970

                    80              1.1005               0.4980

                    81              1.1008               0.4990

                    82              1.1011               0.5000

                    83              1.1014               0.5011

                    84              1.1017               0.5022

                    85              1.102                0.5035

                    86              1.1023               0.5046

                    87              1.1026               0.5058

                    88              1.1029               0.5071

                    89              1.1032               0.5084

                    90              1.1035               0.5097

                    91              1.1038               0.5111

                    92              1.1041               0.5125

                    93              1.1044               0.5139

                    94              1.1047               0.5154

                    95              1.105                0.5169

                    96              1.1053               0.5185

                    97              1.1056               0.5201

                    98              1.1059               0.5218

                    99              1.1062               0.5235

                    100             1.1065               0.5252

                    101             1.1068               0.5270

                    102             1.1071               0.5289

                    103             1.1074               0.5307

                    104             1.1077               0.5327

                    105             1.108                0.5346

                    106             1.108217             0.5366

                    107             1.108434             0.5387

                    108             1.108651             0.5408

                    109             1.108868             0.5430

                    110             1.109085             0.5451

                    111             1.109302             0.5474

                    112             1.109519             0.5497

                    113             1.109736             0.5520

                    114             1.109953             0.5544

                    115             1.11017              0.5568

                    116             1.110387             0.5593

                    117             1.110604             0.5618

                    118             1.110821             0.5644

                    119             1.111038             0.5671

                    120             1.111255             0.5697

                    121             1.111472             0.5725

                    122             1.111689             0.5752

                    123             1.111906             0.5781

                    124             1.112123             0.5810

                    125             1.11234              0.5839

                    126             1.112557             0.5869

                    127             1.112774             0.5899

                    128             1.112991             0.5923

                    129             1.114233             0.5961

                    130             1.115466             0.5993

                    131             1.116699             0.6026

                    132             1.117932             0.6060

                    133             1.119165             0.6094

                    134             1.120398             0.6130

                    135             1.121631             0.6166

                    136             1.122864             0.6203

                    137             1.124097             0.6241

                    138             1.12533              0.6280

                    139             1.126563             0.6320

                    140             1.127796             0.6360

                    141             1.129029             0.6402

                    142             1.130262             0.6444

                    143             1.131495             0.6488

                    144             1.132728             0.6532

                    145             1.133961             0.6577

                    146             1.135194             0.6623

                    147             1.136427             0.6671

                    148             1.13766              0.6719

                    149             1.138893             0.6768

                    150             1.140126             0.6818

                    151             1.141359             0.6869

                    152             1.142592             0.6927

                    153             1.143825             0.6980

                    154             1.145058             0.7035

                    155             1.146291             0.7092

                    156             1.147524             0.7149

                    157             1.148757             0.7208

                    158             1.14999              0.7269

                    159             1.15299              0.7331

                    160             1.15599              0.7396

                    161             1.15899              0.7463

                    162             1.16199              0.7532

                    163             1.16499              0.7603

                    164             1.16799              0.7676

                    165             1.17099              0.7752

                    166             1.17399              0.7829

                    167             1.17699              0.7909

                    168             1.17999              0.7992

                    169             1.18529              0.8077

                    170             1.19059              0.8170

                    171             1.19589              0.8262

                    172             1.20119              0.8358

                    173             1.20649              0.8458

                    174             1.21179              0.8561

                    175             1.21709              0.8668

                    176             1.22239              0.8778

                    177             1.22769              0.8893

                    178             1.23299              0.9011

                    179             1.23829              0.9132

                    180             1.24359              0.9257

                    181             1.24889              0.9386

                    182             1.25419              0.9519

                    183             1.25949              0.9655

                    184             1.26479              0.9794

                    185             1.27009              0.9943

                    186             1.27539              1.0090

                    187             1.28069              1.0241

                    188             1.28599              1.0395

                    189             1.29599              1.0555

                    190             1.30599              1.0721

                    191             1.31599              1.0892

                    192             1.32599              1.1073

                    193             1.33599              1.1256

                    194             1.34599              1.1444

                    195             1.35599              1.1639

                    196             1.37099              1.1840

                    197             1.38599              1.2049

                    198             1.40099              1.2265

                    199             1.41599              1.2495

                    200             1.43099              1.2726

                    201             1.44599              1.2964

                    202             1.46799              1.3211

                    203             1.48999              1.3467

                    204             1.51199              1.3740

                    205             1.53399              1.4015

                    206             1.55599              1.4299

                    207             1.57799              1.4591

                    208             1.59999              1.4891

                    209             1.62999              1.5201

                    210             1.65999              1.5520

                    211             1.68999              1.5849

                    212             1.71999              1.6186

                    213             1.74999              1.6532

                    214             1.77999              1.6885

                    215             1.80999              1.7246

                    216             1.83999              1.7613

                    217             1.86999              1.7986

                    218             1.89999              1.8364

                    219             1.92999              1.8748

                    220             1.95999              1.9136

                    221             1.98999              1.9528

                    222             1.98                 1.9919

                    223             1.98                 2.0260

                    224             1.98                 2.0595

                    225             1.98                 2.0969

                    226             1.98                 2.1337

                    227             1.98                 2.1700

                    228             1.98                 2.2057

                    229             1.98                 2.2410

                    230             1.98                 2.2758

                    231             1.98                 2.3101

                    232             1.98                 2.3440

                    233             1.98                 2.3774

                    234             1.98                 2.4105

                    235             1.98                 2.4431

                    236             1.98                 2.4753

                    237             1.98                 2.5072

                    238             1.98                 2.5387

                    239             1.98                 2.5699

                    240             1.98                 2.6007

                    241             1.98                 2.6312

                    242             1.98                 2.6613

                    243             1.98                 2.6912

                    244             1.98                 2.7208

                    245             1.98                 2.7500

                    246             1.98                 2.7790

                    247             1.98                 2.8077

                    248             1.98                 2.8362

                    249             1.98                 2.8643

                    250             1.98                 2.8923

                    251             1.98                 2.9199

                    252             1.98                 2.9474

                    253             1.98                 2.9746

                    254             1.98                 3.0015

                    255             1.98                 3.0283

                    256             1.98                 3.0548

                    257             1.98                 3.0811

                    258             1.98                 3.1072

                    259             1.98                 3.1331

                    260             1.98                 3.1588

                    261             1.98                 3.1843

                    262             1.98                 3.2096

                    263             1.98                 3.2348

                    264             1.98                 3.2597

                    265             1.98                 3.2845

                    266             1.98                 3.3091

                    267             1.98                 3.3335

  Transition cell   268             1                    4.5867

  Output matcher    269             1                    13.7779

                                    Total electrode      302.7199
                                    length               
  ----------------- --------------- -------------------- -----------------

# 

# [VANE MODULATION DATA]{.underline}

Table 4.1 is a sample of the data points that define the vane modulation
geometry. This data will be provided to in a separate spreadsheet.

Table 4.1 Vane modulation data points.

  -------- ------------------- --------------- --------------------------
  Cell \#  Longitudinal        Distance from   Distance from axis to
           coordinate \[cm\]   axis to         horizontal electrode
                               vertical        \[cm\]
                               electrode       
                               \[cm\]          

  End of   228.24026           0.34            0.34
  cell 249                                     

  Cell 250 228.38487           0.35749         0.32251

           228.52948           0.37455         0.30545

           228.6741            0.39076         0.28924

           228.81871           0.40572         0.27428

           228.96332           0.41906         0.26094

           229.10794           0.43046         0.24954

           229.25255           0.43963         0.24037

           229.39716           0.44634         0.23366

           229.54178           0.45044         0.22956

           229.68639           0.45181         0.22819

           229.83101           0.45044         0.22956

           229.97562           0.44634         0.23366

           230.12023           0.43963         0.24037

           230.26485           0.43046         0.24954

           230.40946           0.41906         0.26094

           230.55407           0.40572         0.27428

           230.69869           0.39076         0.28924

           230.8433            0.37455         0.30545

           230.98791           0.35749         0.32251

           231.13253           0.34            0.34

  Cell 251 231.27852           0.32251         0.35749

           231.42452           0.30545         0.37455

           231.57052           0.28924         0.39076

           231.71652           0.27428         0.40572

           231.86251           0.26094         0.41906

           232.00851           0.24954         0.43046

           232.15451           0.24037         0.43963

           232.3005            0.23366         0.44634

           232.4465            0.22956         0.45044

           232.5925            0.22819         0.45181

           232.7385            0.22956         0.45044

           232.88449           0.23366         0.44634

           233.03049           0.24037         0.43963

           233.17649           0.24954         0.43046

           233.32249           0.26094         0.41906

           233.46848           0.27428         0.40572

           233.61448           0.28924         0.39076

           233.76048           0.30545         0.37455

           233.90647           0.32251         0.35749

           234.05247           0.34            0.34

  Cell 252 234.19984           0.35749         0.32251

           234.34721           0.37455         0.30545

           234.49458           0.39076         0.28924

           234.64195           0.40572         0.27428

           234.78932           0.41906         0.26094

           234.93669           0.43046         0.24954

           235.08405           0.43963         0.24037

           235.23142           0.44634         0.23366

           235.37879           0.45044         0.22956

           235.52616           0.45181         0.22819

           235.67353           0.45044         0.22956

           235.8209            0.44634         0.23366

           235.96827           0.43963         0.24037

           236.11564           0.43046         0.24954

           236.26301           0.41906         0.26094

           236.41038           0.40572         0.27428

           236.55774           0.39076         0.28924

           236.70511           0.37455         0.30545

           236.85248           0.35749         0.32251

           236.99985           0.34            0.34
  -------- ------------------- --------------- --------------------------

# [Inspection Parameters]{.underline}

+--------------------------+-----------------+-------------------------+
| # Azimut                 | #               | #                       |
| hal Field Non-uniformity |  +/- 1.0% (max) |  *Bead pull at factory* |
+--------------------------+-----------------+-------------------------+
| # Longitudi              | #               | #                       |
| nal Field Non-uniformity |  +/- 1.0% (max) |                         |
+--------------------------+-----------------+-------------------------+
| # Quality Factor         | # \>80%         | # *Factory Measurement* |
|                          | of calculated Q |                         |
+--------------------------+-----------------+-------------------------+
| # Mech                   | #               | # *C                    |
| anical Accuracy of Vanes | +/- 50 μm (max) | MM data from machining* |
+--------------------------+-----------------+-------------------------+
| # Vane                   | #               | # *Ven                  |
|  Accuracy after assembly | +/- 30 μm (RMS) | dor Q/A procedures TBD* |
+--------------------------+-----------------+-------------------------+
| # Min                    | # 4 MHz         | # *Measured at factory* |
| imum Separation Between  |                 |                         |
|                          |                 |                         |
| # Q                      |                 |                         |
| uadrupole & Dipole Modes |                 |                         |
+--------------------------+-----------------+-------------------------+