wherec1 and c2 are determined experimentally and found to be roughly constant over a range of voltages and pressures for any given gas. Lisovskiy and co-workers [77] measured potential minima in the Paschen curves of argon at a Pd∗ value of about
1.1 Torr cm, which is far below the values expected to be achieved in this work. One may make a rough estimate of the expected breakdown voltage according to [76] by taking c1 = 84.78 VPa−1m−1, c2 = 7.93 Pa−1m−1 and γ = 0.068. For an electrode gap of d∗ = 0.4 cm, the breakdown voltage amounts to 354 V at 40 mbar argon, which
represents another limitation of the E/n parameter achievable in the drift region, see also AppendixC.
3.3.3 Choosing a proper extraction nozzle
The background pressures in the detection part of the spectrometer are defined by the buffer gas pressure inside the cell as well as by the nozzle throat diameter. Generally, higher cell pressures and larger orifice diameters are preferred, which result in higher extraction efficiencies of the sample ions. In such cases, however, an efficient pumping of the detection part has to be ensured in order to keep the transit timett as small as
possible with respect to the drift time of the ions. Otherwise, lower buffer gas pressures and smaller orifices have to be used.
The gas jet at the nozzle
Assuming a steady, isentropic and one-dimensional ideal gas flow, the ambient pressure in the nozzle throat p1 remains equal to the background pressure in the extraction chamber until p1 =p∗, the critical pressure, which is given by [78]
p∗ =p 0 µ 2 κ+ 1 ¶ κ κ−1 , (3.4)
with the isentropic exponent κ and the stagnation pressure at the converging nozzle
p0. Thereafter, further reduction of the background pressure has no influence on the static pressure in the nozzle throat; the latter remains constant at the value p1 = p∗. Taking p0 equal to the buffer gas pressure inside the cell Pcell = 40 mbar, the resulting
critical pressure amounts to p∗ = 19.5 mbar, with the isentropic exponent for argon
κ= 1.658 at 298 K. Thus, the ions of interest are extracted from the nozzle in an argon gas jet of supersonic velocities. Remember that the sonic velocity for argon at room temperature amounts to ([78], p. 156) v∗ = q2κRT
κ+1 = 276 m/s with the gas constant for argon R= 208.15 Jkg−1K−1.
For an estimation of the mass flow rate Q (in mbar l/s) through the nozzle, one may use the simple expression given by [79]
Q= 14.2 · Pcell·d
2
N oz √
MT , (3.5)
with the nozzle throat diameter dN oz in mm, the atomic mass M of the gas in u, the
gas pressure Pcell in mbar and the gas temperatureT in K. Of course, viscosity effects
in real gases are not considered here. For two different nozzles of dN oz = 1 mm and
dN oz = 0.5 mm, one gets gas flow rates for T = 293 K of about Q = 5.2 mbar l/s and
Q= 1.3 mbar l/s at an argon pressure of 40 mbar, respectively. The mean free path in the extraction chamber
Usually, the background pressure drastically decreases within the differential pumping section of the spectrometer, and the pressure inside the extraction chamber is the most critical one when seeking shortest transit time tt of the ions. Therefore, the mean free
path of the ions inside that chamber has to be as large as possible. This can be achieved by keeping the background pressure in the extraction chamber below 10−1 mbar Ar. According to [80] the mean free path of an ion λion in a specific gas is given by
λion =
1
nπ(ξgas+ξion)2
, (3.6)
with the gas number density n and the collision radii ξgas and ξion of the colliding
spherical particles1. This equation holds as long as the ions move relatively fast, and
1ξ
gasandξioncan be determined by means of viscosity, mobility or X-ray diffraction measurements.
They are in the range between 100 pm and 200 pm for most elements of the periodic table [80, 81, 82, 83, 84, 85].
Chapter 3: Preliminary considerations about the ion mobility spectrometer 25
the gas can be considered at rest. Let us assume a Maxwellian velocity distribution of the particles and that the relative movement of the ions and the gas atoms is small in the emerging gas jet inside the extraction chamber; in this case the mean free path decreases to
λion = √ 1
2nπ(ξgas+ξion)2
. (3.7)
In order to get an idea of λion, the coarse assumption of comparable collision radii
of the particles is made. Hence, the product of both the pressure and the mean free path is taken to be a gas specific parameter [86,87] at room temperatures, which leads to ([80], p. 611) e.g. λion = 6.4·10−3mbar cm/P1section = 0.13 cm for a background
pressure of 0.05 mbar inside the extraction chamber. Consequently, the length of the extraction chamber has to be as short as possible.