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A dynamic Knudsen vacuum gauge
Volume 103A, number 1,2
PHYSICS LETTERS
18 June 1984
A DYNAMIC KNUDSEN VACUUM GAUGE H. BIALAS Abteilung Festk6rperphysik, Universitiit Ulm, D-7900 Ulm, Fed. Rep. Germany Received 4 January 1984 Revised manuscript received 9 April 1984
A vacuum gauge is described which utilizes the radiometer effect but avoids the disadvantages of a conventional Knudsen manometer by modulating the temperature of the heater.
According to the kinetic theory of rarefied gases the m o m e n t u m transfer normal to surfaces is proportional to the gas pressure. If this momentum transfer can be measured in terms of mechanical forces (radiometer forces) this can be used as a means of measuring low pressures, as demonstrated by Knudsen [1]. The basic element of the Knudsen radiometer gauge consists of two parallel plates (fig. 1) one o f which is heated, separated by a distance d '~ A (A: mean free path of the gas molecules). The unheated plate is supported on a sensitive suspension so that a small force acting upon it can be measured by its deflection. It can be shown [2,3], that the mechanical force F acting on the movable plate is given by F = ~ CGAp [(T l / T 0 ) 1/ 2 _ 1 ]
(1)
(A : area o f the plate, T1, T O: temperature of the two plates, p: pressure). G is a factor depending on the geometry of the system [4,5]. It describes edge effects for arbitrary values of d and a. (d: distance, a: linear dimension of the plates). For d ~ a G has a value of unity. The constant C contains the accommodation
Fig. 1. The basic elements of a Knudsen manometer. 0.375-9601/84/$ 03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
coefficients a i for plate 0 and 1 C = 1 - (or1 - ot0)/(ot0 + ot1 - Otla0) .
(2)
If the plates were made o f the same material and the temperature difference AT = T 1 - T O is small compared with TO, a 1 and a 0 are equal in good approximation, i.e. C = 1. In this limit the force F does not depend on the type of gas. In principle such a gauge can be calibrated absolutely without using a standard. Therefore, Knudsen called it an "absolute manometer". In spite of these advantages the Knudsen manometer did not succeed in vacuum laboratories, because of its delicate suspension which is comparable to a mirror galvanometer. The gauge can only be handled with extreme care, and sudden pressure rises may even destroy the instrument. Moreover, zero-set drifts and electrostatic loads may falsify the display. The main problem in such conventional Knudsen gauges arises from the static method of measurement (the deflection o f a small mirror or the compensating dc voltage). Proposals for a dynamic method of measurement (resulting in an ac voltage) were given by E~,rad et al. [6], Robinson [7] and Schwarz [8]. In ref. [6] a small graphite disc is used which is frictionfree suspended by diamagnetic levitation in an inhomogeneous magnetic field (Bma x = 1.2 T). It rotates in a cooled turbine stator. By means of the radiometer effect the disc is accelerated or decelerated. Thus the frequency change per time interval is proportional to the gas pressure. In ref. [7] a quartz disc intermits 35
Volume 103A, number 1,2
PHYSICS LETTERS
©
periodically the path of molecules between the heater and a small aluminium ribbon which serves as a capacity probe. Since the quartz disc is transparent for the electromagnetic radiation of the heater, the displacement of the ribbon by the radiation pressure (dc signal) can be separated from that due to the radiometer pressure (ac signal). In ref. [8] it is proposed to operate the heater with periodically varying temperature. Thus the radiometer force acting on an elastic membrane should vary at the same rate. No experimental results are reported. In the present work also a varying temperature of the heater is used. The heater is driven by an ac current with an off-set /(t) =J0(1 + cos cot).
18 June 1984
Urn(t)
r/////////////////A
1
d;
--3
Fig. 2. Schematic sketch of the dynamic K n u d s e n gauge; 1: evaporated heater, 2: thermal isolator (150/~m glass), 3: bulk copper substrate, 4: m e m b r a n e , 5: electrode; d = 0.5 mm.
(3) bulk copper substrate. The time dependent temperature of the heater (resulting from an ac current) can be calculated by solving the appropriate differential equation [10]. A frequency of 2 Hz was chosen for tile heater current. This value represents a compromise between the thermal time constant r of the heater/ substrate system (demanding a low frequency) and the frequency response of the microphone (demanding a high frequency). The time constant ~ was measured by means of an IR-microscope with built-in bolometer, and yielded ~-= 0.3 s. The maximum attainable temperature modulation was AT = 40 K.
The periodic radiometer force is detected by a condensor microphone [9]. If the temperature modulation AT = Tma x Tmin is small compared to TO, eq. (1) can be developed, and one gets for the time dependent radiometer pressure (4)
2xp(t) = F ( t ) / A = [ A T ( t ) / 4 T o ] p .
Thus the microphone signal Um(t ) = S2xp(t) (S: sensitivity of the microphone) is proportional to the gas pressure p. Fig. 2 shows the detector head schematically. The evaporated heater is thermally isolated by a thin glass (150/am thick) cemented on a water cooled
100
jo
Um[P) -UmiP e ) [mV]
/o oJ °
/
0,1
o~° i
10 -~"
p [mbar] i
10-3
i
10-2
i
10q
i
10o
i
101
Fig. 3. Microphone output minus background versus pressure.
36
102
Volume 103A, number 1,2
PHYSICS LETTERS
In fig. 3 the amplitude of the microphone voltage minus microphone voltage at the lowest system pressure P0 = 6 × 10 - 6 mbar is plotted versus pressure p. There is a linear relation from p = 10 - 4 mbar to about 0.5 mbar. At p > 0.5 mbar the condition A >> d is no longer fulfilled. The pressure-independent signal Urn(p0 ) ~ 20 mV is caused by two effects: primary by cross-talk of the heater voltage to the microphone electrode, and to a much lower amount by radiation pressure [4,11]. It can be completely compensated by adding a small fraction of inverted heating voltage to the output signal. By applying more sophisticated electronics it seems possible to extend the lower limit to at least 10 - 5 mbar. I am indebted to Professor O. Weis who draw my attention to the radiometer effect and to Dr. M. Pietralla for the measurements with the IR-microscope.
18 June 1984
References [1] M. Knudsen, Ann. Phys. (Leipzig) 32 (1910) 809. [2] A. Roth, Vacuum technology (North-Holland, Amsterdam, 1982). [3] C. Edelmann, ed., Vakuumphysik und-technik (Akad. Verlagsges., Leipzig, 1978). [4] W. Steckelmacher, Vacuum 1 (1951) 266. [5] G. Spiwak, Phys. Z. Sowjetunion 2 (1932) 101. [6] R. Evrad and G.A. Boutry, J. Vac. Sci. Technol. 6 (1969) 279. [7] N.W. Robinson, Le Vide 17 (1962) 570. [8] H. Schwarz, Rev. Sci. Instrum. 31 (1960) 433. [9] W. Kuhl, G.R. Schr6der and F.G. SchriSder, Acustica 4 (1954) 519. [10] H. Bialas, to be published. [ 11] S. Schalkowsky and T. Marshall, Rev. Sci. Instrum. 35 (1964) 908.
37
PHYSICS LETTERS
18 June 1984
A DYNAMIC KNUDSEN VACUUM GAUGE H. BIALAS Abteilung Festk6rperphysik, Universitiit Ulm, D-7900 Ulm, Fed. Rep. Germany Received 4 January 1984 Revised manuscript received 9 April 1984
A vacuum gauge is described which utilizes the radiometer effect but avoids the disadvantages of a conventional Knudsen manometer by modulating the temperature of the heater.
According to the kinetic theory of rarefied gases the m o m e n t u m transfer normal to surfaces is proportional to the gas pressure. If this momentum transfer can be measured in terms of mechanical forces (radiometer forces) this can be used as a means of measuring low pressures, as demonstrated by Knudsen [1]. The basic element of the Knudsen radiometer gauge consists of two parallel plates (fig. 1) one o f which is heated, separated by a distance d '~ A (A: mean free path of the gas molecules). The unheated plate is supported on a sensitive suspension so that a small force acting upon it can be measured by its deflection. It can be shown [2,3], that the mechanical force F acting on the movable plate is given by F = ~ CGAp [(T l / T 0 ) 1/ 2 _ 1 ]
(1)
(A : area o f the plate, T1, T O: temperature of the two plates, p: pressure). G is a factor depending on the geometry of the system [4,5]. It describes edge effects for arbitrary values of d and a. (d: distance, a: linear dimension of the plates). For d ~ a G has a value of unity. The constant C contains the accommodation
Fig. 1. The basic elements of a Knudsen manometer. 0.375-9601/84/$ 03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
coefficients a i for plate 0 and 1 C = 1 - (or1 - ot0)/(ot0 + ot1 - Otla0) .
(2)
If the plates were made o f the same material and the temperature difference AT = T 1 - T O is small compared with TO, a 1 and a 0 are equal in good approximation, i.e. C = 1. In this limit the force F does not depend on the type of gas. In principle such a gauge can be calibrated absolutely without using a standard. Therefore, Knudsen called it an "absolute manometer". In spite of these advantages the Knudsen manometer did not succeed in vacuum laboratories, because of its delicate suspension which is comparable to a mirror galvanometer. The gauge can only be handled with extreme care, and sudden pressure rises may even destroy the instrument. Moreover, zero-set drifts and electrostatic loads may falsify the display. The main problem in such conventional Knudsen gauges arises from the static method of measurement (the deflection o f a small mirror or the compensating dc voltage). Proposals for a dynamic method of measurement (resulting in an ac voltage) were given by E~,rad et al. [6], Robinson [7] and Schwarz [8]. In ref. [6] a small graphite disc is used which is frictionfree suspended by diamagnetic levitation in an inhomogeneous magnetic field (Bma x = 1.2 T). It rotates in a cooled turbine stator. By means of the radiometer effect the disc is accelerated or decelerated. Thus the frequency change per time interval is proportional to the gas pressure. In ref. [7] a quartz disc intermits 35
Volume 103A, number 1,2
PHYSICS LETTERS
©
periodically the path of molecules between the heater and a small aluminium ribbon which serves as a capacity probe. Since the quartz disc is transparent for the electromagnetic radiation of the heater, the displacement of the ribbon by the radiation pressure (dc signal) can be separated from that due to the radiometer pressure (ac signal). In ref. [8] it is proposed to operate the heater with periodically varying temperature. Thus the radiometer force acting on an elastic membrane should vary at the same rate. No experimental results are reported. In the present work also a varying temperature of the heater is used. The heater is driven by an ac current with an off-set /(t) =J0(1 + cos cot).
18 June 1984
Urn(t)
r/////////////////A
1
d;
--3
Fig. 2. Schematic sketch of the dynamic K n u d s e n gauge; 1: evaporated heater, 2: thermal isolator (150/~m glass), 3: bulk copper substrate, 4: m e m b r a n e , 5: electrode; d = 0.5 mm.
(3) bulk copper substrate. The time dependent temperature of the heater (resulting from an ac current) can be calculated by solving the appropriate differential equation [10]. A frequency of 2 Hz was chosen for tile heater current. This value represents a compromise between the thermal time constant r of the heater/ substrate system (demanding a low frequency) and the frequency response of the microphone (demanding a high frequency). The time constant ~ was measured by means of an IR-microscope with built-in bolometer, and yielded ~-= 0.3 s. The maximum attainable temperature modulation was AT = 40 K.
The periodic radiometer force is detected by a condensor microphone [9]. If the temperature modulation AT = Tma x Tmin is small compared to TO, eq. (1) can be developed, and one gets for the time dependent radiometer pressure (4)
2xp(t) = F ( t ) / A = [ A T ( t ) / 4 T o ] p .
Thus the microphone signal Um(t ) = S2xp(t) (S: sensitivity of the microphone) is proportional to the gas pressure p. Fig. 2 shows the detector head schematically. The evaporated heater is thermally isolated by a thin glass (150/am thick) cemented on a water cooled
100
jo
Um[P) -UmiP e ) [mV]
/o oJ °
/
0,1
o~° i
10 -~"
p [mbar] i
10-3
i
10-2
i
10q
i
10o
i
101
Fig. 3. Microphone output minus background versus pressure.
36
102
Volume 103A, number 1,2
PHYSICS LETTERS
In fig. 3 the amplitude of the microphone voltage minus microphone voltage at the lowest system pressure P0 = 6 × 10 - 6 mbar is plotted versus pressure p. There is a linear relation from p = 10 - 4 mbar to about 0.5 mbar. At p > 0.5 mbar the condition A >> d is no longer fulfilled. The pressure-independent signal Urn(p0 ) ~ 20 mV is caused by two effects: primary by cross-talk of the heater voltage to the microphone electrode, and to a much lower amount by radiation pressure [4,11]. It can be completely compensated by adding a small fraction of inverted heating voltage to the output signal. By applying more sophisticated electronics it seems possible to extend the lower limit to at least 10 - 5 mbar. I am indebted to Professor O. Weis who draw my attention to the radiometer effect and to Dr. M. Pietralla for the measurements with the IR-microscope.
18 June 1984
References [1] M. Knudsen, Ann. Phys. (Leipzig) 32 (1910) 809. [2] A. Roth, Vacuum technology (North-Holland, Amsterdam, 1982). [3] C. Edelmann, ed., Vakuumphysik und-technik (Akad. Verlagsges., Leipzig, 1978). [4] W. Steckelmacher, Vacuum 1 (1951) 266. [5] G. Spiwak, Phys. Z. Sowjetunion 2 (1932) 101. [6] R. Evrad and G.A. Boutry, J. Vac. Sci. Technol. 6 (1969) 279. [7] N.W. Robinson, Le Vide 17 (1962) 570. [8] H. Schwarz, Rev. Sci. Instrum. 31 (1960) 433. [9] W. Kuhl, G.R. Schr6der and F.G. SchriSder, Acustica 4 (1954) 519. [10] H. Bialas, to be published. [ 11] S. Schalkowsky and T. Marshall, Rev. Sci. Instrum. 35 (1964) 908.
37