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The experimental gain G of a dynode detector is given by the average ratio of the output current at the detector end to the input current at the detector entrance (G0 = Iout/Iin). In pulse counting mode, this reduces to the number of generated secondary electrons in a

3.1. Basic CEM operation parameters

pulse height amplitude [mV]

counts

(b)

average pulse height U0

(a)

pulse height amplitude [mV]

counts

low pulse height

contribution shifted, average_{height U}

0

linear amplification space charge saturation

Figure 3.2:(a) Illustrated pulse height distribution in the CEM corresponding to the calculated compound Poisson distribution Pm(n) as shown in fig. 2.2. Derived pulse height distribution prior to the last stages of amplification up in the CEM from, e.g., gm−5. Due to the effect of space charge saturation, at the last stages only lower pulse heights are still further amplified. (b) Resulting final, compressed pulse height amplitude distribution after gm. By space charge saturation, the initial compound Poisson distributionPm(n)evolves into a narrow quasi-Gaussian distribution of pulse amplitudes.

single output pulse per incident primary particle at the detector entrance. In the particular case of CEMs, the experimental gain G0 is a function of the applied CEM gain voltage UCEM over the detector (fig. 2.1), and the length-to-diameter ratio l/d of the CEM tube [47]. Experimentally, these two parameters will significantly affect the secondary emission properties in the CEM and therewith associated secondary electron yieldδ1..m in the dynode detector (see subsection 2.4.1). However, the length to diameter ratiol/dis intrinsically given by the spatial dimensions and the design of the particular CEM model. Therefore, for a given detector especially the applied CEM gain voltage UCEM is an important parameter which influences the gain of a CEM detector.

In fig. 3.1(a), a commonly measured, experimental CEM gainG0 versus gain voltageUCEM characteristic is illustrated (e.g. [85]). For comparably low values ofUCEM, the CEM detector stays in analogue mode acting as a perfect linear amplifier for incident primary particles as stated by eq. 3.1. The observed pulse height distribution at the CEM detector end strictly corresponds to the calculated compound Poisson distributionPm(n) as illustrated in fig. 2.2 (see subsection 3.1.1). However, for considerable high values of UCEM, the CEM gain G0 evolves into highly non-linear amplification mode due to the effect of space charge saturation at the CEM channel end [47, 53, 59, 136–138]. With the onset of space charge saturation, the CEM detector still amplifies incident primary particles but the pulse height distribution at the CEM output is significantly changed in shape with only the low pulse height contribution of the overall pulse height distribution being further amplified (fig. 3.2). Note that this non- linear amplification behaviour is particularly exploited for the pulse counting mode of CEM detectors.

Space charge saturation

To illustrate the effect of space charge saturation on the observed pulse height amplitude distribution2, in fig. 3.2 the pulse height distribution prior to the last stages of amplification in the CEM from, e.g., gm−5 is shown (fig. 2.1). In space charge saturation, the initially 2_{Note that according to subsection 3.1.1, the pulse height distribution at the CEM anode will directly cor-}

gain G

counts

average gain G0

DGfwhm

Figure 3.3:Illustrated distribution of gain valuesGof a CEM detector operating in space charge saturation, showing the modal gain G0 of a CEM. In the experiment, the gain curve Gsimply corresponds to the respective pulse height distribution as depicted in fig. 3.2 as stated by eq. 3.4.

uniform secondary emission yield value δ= 2 decreases toδ≤1for the last few stages gm of amplification due to positive wall charging of the channel tube [47, 137] (eq. 2.3 and eq. 2.4; subsection 2.1.2). The accumulated secondary electron avalanche in the CEM detector will thus experience no further amplification above a certain maximum value at the end of the CEM for macroscopic charge values. More specific, at the last multiplication stages gm only lower pulse heights (fig. 3.2(a); shaded area) are still amplified whereas already higher pulse heights are only maintained in charge [53, 59, 136, 137]. This results in CEM output pulses which all exhibit nearly the same amplitude. It further leads to a shifted and compressed pulse height amplitude distribution compared to the initial compound Poisson distribution Pm(n) (fig. 3.2(b)).

Therefore, by the effect of space charge saturation the initial compound Poisson distribution Pm(n) evolves into a narrow quasi-Gaussian distribution of pulse amplitudes centered at the average pulse height U0 (fig. 3.2(b)). Moreover, by the particular Gaussian shape, a rather defined pulse height distribution of signal pulses is obtained which is easily discriminated against any low spurious background signals, feedback pulses, or any background noise in the signal cable. This results as for the Gaussian pulse height distribution, the discriminator level for any successive current or pulse processing electronics can be adjusted to a much higher threshold level without losing any significant distribution of pulses which will otherwise remain undetected (see subsection 2.2.1; fig. 2.3).

Experimental CEM gain

Interpreted in experimentally measureable quantities, the modal gain G0 of a CEM detector is defined as the average number of electrons per pulse at the anode of the CEM. In pulse counting mode, it therefore represents the integrated charge in an output pulse with respect to an incident primary particle. As the CEM gain Gis proportional to the electric charge in the output pulse (see subsection 3.1.1), the modal gainG0 is thus given by the corresponding average pulse height value U0 in the observed pulse height distribution (fig. 3.2).

By definition, the amplification factor for an incident primary particle in a CEM is denoted as the CEM gain G, and reads

G=Qpulse/e0= 1 e0R ˆ U(t) dt, (3.2) where Qpulse = e0 ´

n(t) dt is the integrated number of secondary electrons within the generated output pulse, U(t) is the pulse height amplitude at the time t, and R denotes

3.1. Basic CEM operation parameters

the ohmic output resistance at the CEM anode (50 Ω; fig. 4.8(b), see section 4.2). Due to the Poissonian nature of the secondary emission processes in the CEM and the Maxwellian velocity distribution of the emitted secondary electrons at any stage gm within the cascade [47, 60, 61], the temporal amplitude shapeU(t) of the output pulse in eq. 3.2 arriving at the CEM anode is expected to be Gaussian [139], corresponding to

U(t) =U0·exp − 1 2 t−t0 σ 2! , (3.3)

where U0 is the amplitude of the Gaussian pulse at the time t0 (fig. 3.3), and σ = tfwhm/(2

√

2 ln 2) represents a single standard deviation of the Gaussian. Assuming t0 = 0, the gainGfor a particular single primary particle incidence is then given by

G= U0 e0R ˆ ∞ −∞ exp −1 2 t σ 2! dt= √ 2π e0R U0σ= √ π 2√ln 2 U0tfwhm e0R . (3.4)

As the CEM gain G is thus linearly proportional to the pulse height amplitude U0, the modal gain G0 is given by the corresponding average pulse height value U0 in the observed pulse height distribution for several particle incidences (e.g., fig. 3.2). The modal gainG0 of a CEM detector therefore yields the most probable gain value (fig. 3.3), and is correspondingly defined as G0 = √ π 2√ln 2 U0tfwhm e0R , (3.5)

whereU0 represents the average pulse height of the observed pulse height distribution, and tfwhm denotes the average pulse width at half maximum, respectively. In contrast to eq. 3.5, in the literature often a simplifying rectangular approximation of the Gaussian pulse shape to determine the experimental CEM gainG0 is used, with

G0,linear=

U0tfwhm e0R

. (3.6)

However, the linear approximation slightly underestimates the CEM gain compared to the observed Gaussian shaped pulses of eq. 3.5 by a factor ofG0 ≈1.06447G0,linear.

CEM detector gain degradation and detector age

Intrinsically, CEM detectors display a stable operation only over a certain period of time. In particular, gradual degradation of the active surface of the secondary emitting layer in the CEM worsens the detector performance as the detector deteriorates by extensive use [48, 90, 139]. Therefore, if one considers the lifetime of stable operation for a CEM detector there are some potential sources and unfavourable operation conditions which will promote an enhanced detector degradation [139, 140]. Of these, high vacuum pressuresp >10−6mbar

[89, 90, 141], multiple vacuum bakeout [142–146], extensive atmospheric/alkaline exposure, contamination with hydrocarbons presumably from vacuum pump oils [112, 140, 147], ultra- high vacuum fatigue [148–150], intensive particle bombardment [144], and accumulated counts greater than 1010 [90, 151] are the most prominent representatives. As a result, an aged or heavily used detector needs a higher gain voltage UCEM to sustain the magnitude of the

counted particles

CEM gain voltage [kV] linear gain plateau region ion feedback

Figure 3.4:Illustrated, characteristic response of a CEM detector to the CEM gain voltageUCEM assuming a constant flux of incident particles (’knee-curve’, [85]). For optimum pulse counting in pulse counting mode, UCEM is set to a value at the plateau region where the detector is operated in space charge saturation.

initially measured gain values G0 for operation [152]. Moreover, also the knee curve (see subsection 3.1.3) of the corresponding detector is shifted to considerably higher gain voltages UCEM(e.g., [85]). Consequently, as the CEM degrades and ages, the gainG=G(UCEM)of the CEM has frequently to be monitored to ensure stable event counting. However, readjusting the CEM to the plateau region (fig. 3.4) will maintain a sufficient high pulse height distribution which can clearly be discriminated against the signal background.