C.M. Ginsburg

September 6, 2006

**Statistical analysis of breakdown cells in DESY/TTF data**

**Abstract**

I analyzed DESY/TTF/Vertical (CW) passband mode data to determine
whether any cell pair (or cell 5) showed a statistically higher
probability to cause cavity breakdown than others.

**1. Data Selection**

To get the data, I went to the webpage:

<http://tesla-new.desy.de/content/cavitydatabank/index_eng.html>

I selected "RF Tests" in the left index, and then clicked on "Statistics
of Best & Last CW-Tests." I included all 117 cavities from Production
Batches 1, 2, 3, and 4. Then I clicked "Accept Choice" and selected the
"Best & Last" page. Clicking "PRINT" brought up a page called "Define
REPORT Parameters" where I selected "TEXT File" as the output format and
"Selected Records" as the records for output. The resulting text file
was imported into Excel for the analysis. I used only the "best test"
results. The data were extracted from the data base on July 24, 2006.

Of the 117 cavities, 105 cavity records included Q vs. E~acc~ tests.
Sixty-one of the "best" cavity tests list the limiting factor of the
π-mode test as breakdown (BD), and were included in the following
analysis.

The test record lists a gradient limit for all cells (from π-mode
measurement), and the maximum gradient seen by cells 1&9, 2&8, 3&7, 4&6,
and 5 (from other fundamental passband mode measurements), and each
measurement's limiting factor, e.g., breakdown, field emission or power
limit. The limiting factor is inserted into the database by hand at the
end of the test, and may show a bias on the part of the experimenter.

**2. Analysis Technique**

The cell(s) responsible for cavity breakdown may be isolated using the
measurement of gradient limits for the non-π-mode passbands. The
relative gradient for a given cell and mode is

![](media/image1.png){width="2.96875in" height="0.8854166666666666in"}

Equation 1: *E~cell~* definition

where mode \[1:9\], cell\[1:9\], and the number of cells is 9; the
π-mode would be mode=9 in this nomenclature. The resulting *E~cell~*
factors are given in Table 1.

  --------------- ------- ---------- ---------- ---------- ---------- ---------- ---------- ---------- ---------
  **cell/mode**   **π**   **8π/9**   **7π/9**   **6π/9**   **5π/9**   **4π/9**   **3π/9**   **2π/9**   **π/9**

  **1 (or 9)**    1       1          1          1          1          1          1          1          1

  **2 (or 8)**    1       0.88       0.53       0          0.65       1.35       2          2.53       2.88

  **3 (or 7)**    1       0.65       0.18       1          1.23       0.53       1          2.88       4.41

  **4 (or 6)**    1       0.35       0.82       1          0.23       1.53       1          1.88       5.41

  **5**           1       0.00       1.06       0          1.31       0          2          0          5.76
  --------------- ------- ---------- ---------- ---------- ---------- ---------- ---------- ---------- ---------

Table 1: *E~cell~* factors from Equation 1.

For each mode, the gradient measured by the pick-up probe is that seen
by the end-cell. The gradient seen by pairs of cells (or cell 5) may be
determined by scaling the measured gradient in the end-cell by the
relevant *E~cell~* factor above. The maximum gradient seen by pairs of
cells (or cell 5), determined in this manner, in any mode measurement,
is recorded in the database. Note that the completeness of this analysis
depends on the assumption of field flatness in all cells.

I assume that the lowest maximum gradient in a pair of cells (or cell 5)
indicates that the cause of the limitation is physically located in that
pair of cells (or cell 5). In many cases, the lowest maximum gradient
was evident in more than one pair of cells (or cell 5).

**3. Results**

First, I included all potentially limiting cells in the analysis; there
are 139 such cell(s) breakdowns out of 61 tests, so that on average
139/61=2.3 cells breakdown for each measurement. For random breakdown
cell location, the probability of breakdown in any given cell is
(139/61)\*(1/9). The number of breakdowns for each cell pair (or cell
5), the probability of breakdown of that cell pair (or cell 5) in any
given test and the corresponding binomial error, and the probability
normalized by number of cells, are shown in Table 2. The probabilities
normalized by cell are plotted in Figure 1, where the correlation among
the cells is evident in the small point-to-point variation with respect
to the error bars. From Figure 1, there is no evidence for a particular
cell pair (or cell 5) to most frequently break down, and the data agree
very well with the hypothesis of random BD cell location.

Table 2: Statistics of the breakdowns by cell(s), for 61 total tests,
including all potentially limiting cells.

  ------------------ -------------------- --------------- -----------------
  **Cell(s) causing  **\# tests w/BD in   **cell(s) BD    **BD prob/cell**
  BD**               cell(s)**            prob**          

  **1 or 9**         32                   0.52460.0639   0.26230.0320

  **2 or 8**         30                   0.49180.0640   0.24590.0320

  **3 or 7**         31                   0.50820.0640   0.25410.0320

  **4 or 6**         29                   0.47540.0639   0.23770.0320

  **5**              17                   0.27870.0574   0.27870.0574
  ------------------ -------------------- --------------- -----------------

![](media/image2.wmf){width="5.995833333333334in"
height="3.814583333333333in"}

Figure 1: Probability/cell of causing breakdown, by cell(s), including
all potentially limiting cells. The solid line indicates the probability
of random cell location, probability/cell= (139/61)\*(1/9).

Second, I restricted the analysis to include only those tests for which
a unique pair of cells (or cell 5) demonstrated the limiting gradient;
this sample contained 32 tests. The number of breakdowns for each cell
pair (or cell 5), the probability of breakdown of that cell pair (or
cell 5) in any given test and the corresponding binomial error, and the
probability normalized by number of cells, are shown in Table 3. The
probabilities normalized by cell are plotted in Figure 2. No cell pair
(or cell 5) can be considered significantly more likely to cause
breakdown than the others. Also in this case, the data agree very well
with the hypothesis of random BD cell location.

For completeness, I include in Figure 3 a plot showing the unique
breakdown cell(s) by production batch, though the statistics are too low
to draw any conclusions.

Table 3: Statistics of the breakdowns by cell(s), for 32 total tests,
including only uniquely limiting cell(s).

  ------------------- --------------------- --------------- ---------------
  **Cell(s) causing   **\# tests w/BD in    **cell(s) BD    **BD
  BD**                cell(s)**             prob**          prob/cell**

  **1 or 9**          10                    0.31250.0819   0.15630.0410

  **2 or 8**          9                     0.28130.0795   0.14060.0397

  **3 or 7**          6                     0.18750.0690   0.09380.0345

  **4 or 6**          4                     0.12500.0585   0.06250.0292

  **5**               3                     0.09380.0515   0.09380.0515
  ------------------- --------------------- --------------- ---------------

![](media/image3.wmf){width="5.998611111111111in"
height="3.8201388888888888in"}

Figure 2: Probability/cell of causing breakdown, by cell(s), including
only uniquely limiting cells. The solid line indicates the probability
of random cell location, probability/cell= (1/9).

![](media/image4.wmf){width="5.998611111111111in"
height="3.841666666666667in"}

Figure 3: Normalized number of breakdowns, showing production batch,
including only uniquely limiting cells.

**4. Summary**

I analyzed DESY/TTF vertical CW test data to study the cell dependence
of cavity breakdown. No pair of cells (or cell 5) is significantly more
likely to cause breakdown than the others, and the data agree very well
with the hypothesis of random breakdown cell location.