C.M. Ginsburg

September 15, 2006

**Statistical analysis of cell best gradient 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 different
gradient than others, when separated into individual production numbers.

**1. Data Selection and Analysis Technique**

The data selection and analysis technique for finding the cell(s)
gradient from the mode measurements are the same as described in the
breakdown-cell note of Sept.6. The cell gradients are calculated by the
software during the test, assuming field flatness, and may or may not be
corrected afterward.

In the present note, I have included all 105 Q vs. E~acc~ cavity tests,
independent of test limitation: 26 tests for Production 1, 27 tests for
Production 2, 33 tests for Production 3, and 19 tests for Production 4.

**2. Gradient Distributions**

The individual best-test maximum cell gradients, separated by cell(s)
and by production number, are shown in Figures 1-5. The average and
standard deviation for each distribution are written on the plots, and
are summarized in Table 1 and Figure 6. In all cases, the bin size is 2
MV/m. The averages and standard deviations are calculated from the
unbinned data.

The gradient dependence on production number is far more significant
than the dependence on cell number(s).

**2. Gradient Distributions, plotted differently**

The individual best-test maximum gradients for the cells, separated by
cell(s) and production number, are shown in Figure 7. These are the same
data shown in Figures 1-5. The bin size is 2 MV/m. For completeness, all
four production batches are lumped together in Figure 8.

**3. Results**

No matter how the data are plotted, the gradient dependence on
production number is far more significant than the dependence on cell
number(s). To quantify this effect, I calculated the Kolmogorov-Smirnov
(K-S) probability that pairs of (unbinned) gradient distributions are
compatible with being random samplings of the same unknown parent
distribution. The K-S probability ranges from zero to one, with one
(zero) meaning most (least) probably deriving from the same parent
distribution. Table 2 shows the K-S probability that the gradient
distributions from the different productions are compatible with
representing the same parent distribution; the probabilities are
negligible, except, interestingly, for that between production 3 and 4
(and the check that the diagonal elements are one). By construction, the
matrix is symmetric; this is another check. Table 3 shows the K-S
probability that the gradient distributions from the different cell(s)
are compatible with representing the same parent distribution; most
elements indicate a very high probability of coming from the same parent
distribution.

**4. Summary**

I analyzed DESY/TTF vertical CW test data to study the cell dependence
of cavity gradient. A clear improvement in gradient occurred for all
cell(s) with each subsequent production batch. No pair of cells (or cell
5) reaches significantly higher (or lower) gradient than the others. The
gradient data support the previous hypothesis of random breakdown cell
location.

![](media/image1.wmf){width="2.9694444444444446in"
height="1.7826388888888889in"}![](media/image2.wmf){width="2.9805555555555556in"
height="1.7909722222222222in"}![](media/image3.wmf){width="2.997916666666667in"
height="1.8in"}![](media/image4.wmf){width="2.997916666666667in"
height="1.8in"}

Figure 1: The distributions of maximum gradient in cell 1 or 9,
separated by production number. The average and standard deviation for
each distribution are shown.

![](media/image5.wmf){width="2.9694444444444446in"
height="1.7826388888888889in"}![](media/image6.wmf){width="2.9625in"
height="1.7763888888888888in"}

![](media/image7.wmf){width="2.997916666666667in"
height="1.8in"}![](media/image8.wmf){width="2.997916666666667in"
height="1.8in"}

Figure 2: The distributions of maximum gradient in cell 2 or 8,
separated by production number. The average and standard deviation for
each distribution are shown.

![](media/image9.wmf){width="2.98125in"
height="1.7895833333333333in"}![](media/image10.wmf){width="2.9625in"
height="1.7763888888888888in"}

![](media/image11.wmf){width="2.997916666666667in"
height="1.8in"}![](media/image12.wmf){width="2.9916666666666667in"
height="1.8in"}

Figure 3: The distributions of maximum gradient in cell 3 or 7,
separated by production number. The average and standard deviation for
each distribution are shown.

![](media/image13.wmf){width="2.9694444444444446in"
height="1.7826388888888889in"}![](media/image14.wmf){width="2.9625in"
height="1.7763888888888888in"}![](media/image15.wmf){width="2.997916666666667in"
height="1.8in"}![](media/image16.wmf){width="2.997916666666667in"
height="1.8in"}

Figure 4: The distributions of maximum gradient in cell 4 or 6,
separated by production number. The average and standard deviation for
each distribution are shown.

![](media/image17.wmf){width="2.9694444444444446in"
height="1.7826388888888889in"}![](media/image18.wmf){width="2.9805555555555556in"
height="1.7909722222222222in"}
![](media/image19.wmf){width="2.997916666666667in"
height="1.8in"}![](media/image20.wmf){width="2.9916666666666667in"
height="1.8in"}

Figure 5: The distributions of maximum gradient in cell 5, separated by
production number. The average and standard deviation for each
distribution are shown.

![](media/image21.wmf){width="5.75in" height="3.625in"}

Figure 6: The average best gradient, separated by cell(s) and by
production number. The error bar is the standard deviation of the best
gradient distribution.

![](media/image22.wmf){width="2.9965277777777777in"
height="1.8090277777777777in"}![](media/image23.wmf){width="3.002083333333333in"
height="1.8104166666666666in"}
![](media/image24.wmf){width="3.002083333333333in"
height="1.8104166666666666in"}![](media/image25.wmf){width="3.0083333333333333in"
height="1.8173611111111112in"}

Figure 7: Distributions of maximum gradients for all cells, separated by
production number.

![](media/image26.wmf){width="5.997916666666667in"
height="3.627083333333333in"}

Figure 8: Distribution of maximum gradients for all cells, all
production numbers.

Table 1: Average and standard deviation for each of the best-test
maximum gradient distributions.

Table 2: Kolmogorov-Smirnov probability for gradient distributions with
all cells combined, separated by production number.

  -------------------- ----------- ------------- ------------ -----------
  **K-S probability**  **Prod 1**  **Prod 2**    **Prod 3**   **Prod 4**

  **Production 1**     1.000000    0.000483      0.000127     0.001883

  **Production 2**     0.000483    1.000000      0.000406     0.026475

  **Production 3**     0.000127    0.000406      1.000000     0.422991

  **Production 4**     0.001883    0.026475      0.422991     1.000000
  -------------------- ----------- ------------- ------------ -----------

Table 3: Kolmogorov-Smirnov probability for gradient distributions with
all production numbers combined, separated by cell(s).

  -------------------- -------------------- -------------------- -------------------- -------------------- ------------------
  **K-S probability**  **[cells             **[cells             **[cells             **[cells             **[cell
                       1&9]{.underline}**   2&8]{.underline}**   3&7]{.underline}**   4&6]{.underline}**   5]{.underline}**

  **[cells             1.000000             0.968798             0.909878             0.818339             0.373931
  1&9]{.underline}**                                                                                       

  **[cells             0.968798             1.000000             0.994219             0.373931             0.114970
  2&8]{.underline}**                                                                                       

  **[cells             0.909878             0.994219             1.000000             0.588237             0.216172
  3&7]{.underline}**                                                                                       

  **[cells             0.818339             0.373931             0.588237             1.000000             0.818339
  4&6]{.underline}**                                                                                       

  **[cell              0.373931             0.114970             0.216172             0.818339             1.000000
  5]{.underline}**                                                                                         
  -------------------- -------------------- -------------------- -------------------- -------------------- ------------------