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A modified knudsen manometer-an apparatus for the measurement of low gas pressures and the determination of thermal accommodation coefficients
A modified Knudsen manometer-an apparatus for the measurement of low gas pressures and the determination of thermal accommodation coefficients received
30 April
W Kreisel,
lnstitut
1976 fiir
Phys.
Chemie
der Universitat
Heidelberg’,
Heidelberg,
Germany
An instrument is described for measuring pressure in the range of 10 -= to 1 O-7 (torr) . This gauge is bakeable up to 350°C and is relatively insensitive to mechanical vibrations. The principle of the measuring method is based on an electromagnetic self-adjusting compensation of the radiometric force effected by a definite but non-uniform distribution of temperature in a rarefied gas. Moreover, an equation for the radiometric force is given. This equation is valid for %ny temperature ratios and arbitrary accommodation coefficients. Using this equation it is possible to calculate coefficients of the gases from the experimental results. Assuming the accommodation coefficients to be unity, the device can be used as an absolute manometer since all other quantities can be measured directly. Comparative measurements are carried out with a calibrated ionisation gauge, and, in addition, at higher pressures with a McLeod gauge. These measurements were taken for testing the Knudsen manometer. Helium, argon and krypton, as well as N2 and C02, serve as suitable measuring gases.
Introduction
Theory
The principle which allows one to measure very low gas pressures with the ‘absolute manometer’ first constructed by Knudsen’ is based on the radiometric forces. This manometer consists of two parallel plates at different temperatures, Tl and T2. According to the theory of rarefied gases, the heat-as well as the momentum transport-in the free molecular flow regime between the two plates is proportional to the pressure px. The formula derived from this theory was given by Knudsen *
Considerthe ‘absolutemanometer’constructed by Knudsen.’ Then the resultant radiometric force F per unit area acting on the light suspended plate AZ which isat temperatureT2 is given by F = P$
Vacuum/volume26fnumber8.
Pergemon
Press/Printed
in Great
(1)
where px is the gas pressureto be measuredand Tl is the temperatureof the heatedand stationary plate A,. The formula is seento be independentof the molecularweight of the gas.
where F is the resultant radiometric force per unit area. As was shown late? this equation is only valid assumingthe thermal accommodationcoefficients (hereafter abbreviated to a.~.) of the plates to be unit. (The a.~. is a measureof the average efficiency of the energy exchange per encounter of a gas moleculewith the solid at the gas-solidinterface.) Although, Knudsen’sapparatuswas modifiedad the technical shortcomingshave irrrecent years rarely justified its use.In particular, the Knudsen manometer gauge did not give a continuous record of pressurebut had to be ‘read’, and, furthermore it wasvery sensitiveto mechanicalvibrations. This paper describesa form of radiometer gaugewhich is free from the above mentioneddisadvantages. We also derive an equation for the radiometric force from Wu’s revisedtheory of thermal transpiration’ which is valid for all a.c.‘s and temperatureconditions on the boundary. Using this equation we calculate the Q.C.‘Semploying the resultsof specialmeasurements (to be specifiedlater) as input. * Present address: Institut fiir Umweltschutzder Universitlt Dortmund,46 Dortmund,Rosemeyerstr. 6, Germany.
KWG)“2 - 11
T,
d
A2 T,
A, Figure1. Knudsen’s absolutemanometer-planview. This assumption,however, is only valid provided the a.c.‘s are identical, viz. a, = u2 = 1 where aI and ~72are the u.c.‘s of the platesAl and A2, respectively.As wasshownby Knudsen the manometer will obviously cease to function correctly according to equation (1) unless the distance between the platesis smallcomparedwith their linear dimensionsand to the meanfree path of the molecules. Moreover, equation (1) is basedon the traditional original Knudsentheory for thermal transp&ation.” That is, the term plT’12
is a constantwithin a closedvacuum systemwith a non-uniform temperaturedistribution. This traditional theory, however, is a mereapproximation to reality aswasshownby Wu.’ According to Wu the term p/T’l’
Britain
339
W Kreisel: An apparatus
for the measurement
of low
gas pressures
and the determination
of thermal
accommodation
coefficients
is inversely proportional to a new quantity I(isotropy) which has been introduced as a measure for the isotropicity of the velocity distribution function in the non-equilibrium Knudsen-regime. With the inclusion of this new quantity I one fnds that the ratio
molecules reflected from the surface A and the vane, The isotropy in this flow field is given by
p//r’
Furthermore, using obtains the following
I2
is a constant p //T’12
for a closed
vacuum
system,
i.e.,
= p’ ]‘/T”12
(2)
Consider now the modified manometer as shown in the planview in Figure 2. The aluminium bodies A and A’ heated up to an arbitrary but constant temperature T, by a thermostatic controlled liquid are placed opposite to the surfaces B and B’ which are at the arbitrary but constant temperature T2 whereby T, > T2. The light suspended vane may have the temperature T, # T,, T2. With T,, , we denote the temperature characterizing the flow field existing between the vane and the surfaces A and A’, while I,. , characterizes its isotropy and pint its pressure.
respectively.
(5) the notations equations:
as
indicated
above
one
Tin 2 = CT,,,T,, ,)I” W-c, Tc-, ,I””
Iin 2 = With
T co“2 + Tj,,
regard
to the definition
of the o.c.~ it follows
that
(8) T ho
a3 =
I -
The
-
The
*=
a3 T3
T co I - r,o T3
-
Tco
whereby a, and a2 are the a.c.‘s of the surfaces A, A’ and B, B’, respectively, while a3 and a3* are the a.c.‘s of the collision surfaces of the vane (Figure 2). Simple calculations yield the solution for the gas temperature of the molecules reflected (or emitted) from the surfaces:
a,T, + a,(1 - alIT
(a) The =
a,
+ a3 -
a3T3 + a,(1 - a,)T,
(b) The I =
a,+a3-alu3
u~TJ +
(~1 Tc, =
u2
+
Figure
2. Modified
Knudsen
manometer
(plan
view)-flow
system.
The molecules reflected from the surfaces A and A’, respectively, possess the gas temperature The. Those molecules leaving the collision surfaces of the vane opposite to the surfaces A and A’ possess the gas temperature The ,. Analogous notations are used for the flow field existing between the vane and the surfaces B and B’. TX is the temperature of the vacuum system outside the flow fields, I, its isotropy and pX is its pressure which is to be determined. With regard to
T = Ti, 1 I = Iin 1 p =pin 1 T’ = Ti, 2 I’ = Ii, 2 P’ =Pio 2 one obtains
from
equation
Tc,
, =
Substituting one finds vane and
by
the
geometric
(3)
(4) mean
of
the
-~3*)T2
U2 + U3*
-
U2U3*
those expressions that the temperature the surfaces A and
In a similar
way
gas
temperatures
of
the
T,,. and Thol into equation in the flow field between is given by
for
A’
+ a,(1 - u~)T,]}“~
+a,-u,u,
one obtains
for
(4) the
(10)
Tin z :
Tin 2 = +
From
(2)
~2)T3
u2u3*
a,*(1
-
u2)T31
b73*7’3
u2 + a3*
Assuming the molecular flow between the aluminium body A and the vane to be stationary but anisotropic it follows from the revised theory of thermal transpiration (in the absence of intermolecular collisions) that the temperature T,. , of this flow field is given by
i.e.
-
+ u~(I
a,
Iin
equations
(5),
(9a)
-
and
+
a,(1
-
u3*)T21}“2
(1 1)
a2u3 *
(9b)
it follows
that
1 = 2Ua,T,
I = (T,,o Tim 11~‘~
-
i[a~Tt + ad1 - a,)T31 b3T3
{[“2T3
P.1” 1lin ~lTf, t = Pin 2 4” 2lTfn2
Tin
(9)
a3*(l u3*
~3*T3 (4
ala3
+
b,T,
+
and
from
zin
-
a,V’33
- a,)T,]“’
equations
(7),
(9c)
[u,T, + a,( 1 + [u,T, + a,(1 and
(9d)
-
u~)T,]}“~ u~)T,]“~
(12)
one obtains
2 =
2{[u,T2 [~2T2
a,(1
a,(1
+ u3*(1 +
a,*(1
-
-
u2)T3]
~2)T3]l’~
[~3*T3 + [~3*T3
+ a,(1
-
~3*)T2]}“~
+ a2 (1 -
~3*)T2]“~
(13)
W Kreisel: An apparatusfor the measurement of low gaspressures and the determinationof thermalaccommodationcoefficients
Using equations(3) and (IO)-(13) the pressurep,,, , in the flow held betweenlhc vane and the surfaces.4 and A’ is now given bY Pin
I =
Pin
a2 + aJ* - a2as* ‘I2 a, +a, -aza3 >
2
b,T, + aJ1 - a,)T,]“’ azTz + a,*( I - a,)T,]‘”
1 I, j
+ [NJTA + a,(] - a,)T,]“’ + [a3*T3 + a,(1 - a,*)T2]“z
(14)
i.c.,p,,, , = Pinz C*, whereC* = l I. Thus, for the resultant radiometric force F per unit area on the central vane we obtain F = (IJin I - Pin 2) = Pin 2tc*
- I)
(15)
However. the pressurepi,, z in this equationis not identical with the pressurepx which is to be measured.But, it is possibleto state the relation (16) analogousto equation (3):
M!r;sr
Figure
3. Vane.
Since the resultant radiometric force acting on this area A* is given by SinceI, = I (Maxwellian distribution) it followsfrom equations (I I), (13) and (16) that F,&
’
we obtain from equations(20) and (21):
2p
1 1J
f [a,T, + a,(1 - a,)T,]“’ + [ajT3 + a,(1 - a,)T,]“2 (a, + a, - a,a3)1’2 I[
[
[a?T, +rr,*(l
-az)T3]“’ + [a3*T3 +aJ I -a,*)Tz]“‘] (a2 + a3* - a2a,*)“’
\
(17)
i.e.,
(22) As described later this radiometric turning moment D,, is electromagneticallycompensatedby self-adjustment.Therefore, the systemwill be in equilibrium if the relationship De, = Dra
F = ‘A
*
2 ry’12
(174
x
where X is the term in brackets { }. Equation (17), as already given in I”, is the most general form for the resulting force per unit area on the central vane valid for all a.c.‘s and for arbitrary temperatureconditions. For a, = a2 = a3 = u3* = a one obtains F,& 2 G2
[(T, +T3 - uT,)“~ + (T, + T3 -aT,)‘12 -
is satisfied,where D,, is the electromagneticturning moment given by D,, = nA,,,Bi, whereII is the numberof windingsper unit length,A,., isthe surfaceareaof the coil and B the magnetic induction. Thus, it follows that .2tIA,,i B Px = 1 A*r
(T,“’
=
T,‘f2
-
T2112
(‘8)
In this casethe gaugecan be usedas an absolutemanometer sinceall other quantitiescan be measureddirectly.
- Tz”~)
(1%
Experimental
This equation is plausible when the temperature T, is equal to the temperature T, of the unheated surfacesB and B’, respectively, in which caseone obtains equation (1) as already given by Knudsen.’ With regard to the symmetry of the system(Figure 2) and the linear dimensions of the vane (Figure 3) the radiometric turning moment is given by r2 rdr=p,l(r,-r,)(r,+r,)=p,rA* s r1 whereA* is the total collision surfaceof the vane. D,, = 2p.J
(23)
- (T, + T3 - QTJ”~]
while for a, = a2 = a3 = a3* = 1 it follows that F,k 2 p ’
[Nm’l.
As long asall of the u.c.‘s are equal to 1, one can substitutefor X the expression X
(T2 + T3 - UT,)“’
~~‘I2
x
(20)
As with most of the componentsof this gaugethe manometer vesselis made of high grade steel. This vessel, 100 mm in diameter was constructed according to our specificationsby Vacuum GeneratorsLtd. The upper part of the vesselis vacuum sealedwith a special flange (FC 100) upon which are symmetrically mounted four heat-insulatedliquid feed-throughs(LFT 2) and, moreover, two electrical feed-throughs(EFT 5) eachhaving eight conductors. The LET 2’sare suitablefor thermostaticcontrolled liquids as well as for liquid nitrogen. A small tightly fitting ahuninium body is locatedon eachof the four insidetubesof the LET 2’s. The aluminium bodiesare reamedto fit the outer diameter of 341
W Kreisel:
An
apparatus for the measurement of low gas pressures and the determination of thermal accommodation coefficients MUFC
Colltslon
15 Vacuum pump
surfaces
Dlfferentlol pholoconductlve
cell
\/sew-port
L1gi;l
A
wll,ce
Torston-system mounting
Ertenslon LFT2-liquid throltghs Vessel
c: the feed
gouge Holf-sAvered Annular
I
Lens
mirror
spoccr
Mirror
LFT EFT5-elertrlccl feed th:oughs
Z-llqu:d throughs
feed
FC 100
Figure 4. Modified Knudsenmanometer.
the LFT 2’s insuring an optimal heat transmission. The electrical feed-throughsare usedfor the electrical power supply of the galvanometer coil as well as for the connections of resistancethermometersby meansof which the temperatureof the aluminium body and the temperature of the vacuum systeminsidethe gaugeare measured. Moreover, the vessel has a lateral view-port, 38 mm in diameterto record the slightestdeflectionof the vane by means of a light beam. The zero-point of this light suspended vane can be mechanically adjustedby accessthrough a bore-hole suitably placed in the middle of the FC 100 flange. Once adjusted it will never have to be touched again. This small bore-hole, 5 mm in diameterissealedwith a blind flangeusinga coppergasket. On the insideof the FC 100four blind bore-holesare located symmetrically around the central bore-hole. They serve as a mounting for the meter frame which iseasyto dismount. The galvanometersystemwas constructedaccording to our specificationsby SiemensLtd. Two plates of high grade steel joined with four vertical bars serve as the outer mounting. The thin filament is elastically suspendedbetweentwo torsion headswhich are ceramically insulated from the plates (Figure 5). The characteristicsof the filament are as follows: diameter. tensile strength ultimate tensile strength elastic hysteresis torsion modulus material 342
0.015 mm 274 k&/mm2 48.4 gf 0.05 % 7560 kgf/mm2 Nivaflex
The amount of the torsion stressis 10 gf. The filament carries the rectangular aluminium vane which is rigidly joined with a suitably constructedgalvanometercoil whosewinding consists of thin glasssilk insulatedplatinum thread. This galvanometer coil aswell as the ceramic insulationsendurean outgassingup to350T. In order to makethe surfacesof the vane and the surfacesof the aluminium bodiesasalike aspossiblethey are oxidized in a similar manner. Furthermore, in order to achieve high u.c.‘s we have usedrough surfaces. The deflection of the vane effected by the radiometric turning moment will be electromagnetically compensated by self-adjustment so that the vane remains in the stationary position, even under the influenceof the radiometric forces. This is accomplishedas follows: a light beam incident on the galvanometer mirror is reflected on to a differential photoconductive cell consisting of two light-sensitive halves with resistancesRI and R2 which form one part of an active bridge circuit of an amplifier. The other part of the bridge consistsof equal resistancesR3 and R+ When the vane is in its zero-position, i.e. exactly betweenthe heatedaluminium bodies and the unheatedbodies,respectively, the light beamreflected from the galvanometermirror will meet the photoconductive cell at its centre. In this position R, = R2. That is, the output voltage of the amplifier is zero. The resistanceR5 representsthe outer negative feedback path by meansof which the counterforceagainstthe mechanical deflections of the vane is produced. The capacitor C, is to Electra-optics.
W Kreisel:
An apparatus
for the measurement
of low
Torwon
Towon
gas pressures
head
and the determination
of thermal
accommodation
coefficients
The deflection of the vane from its zero-point effected by radiometric forces is approximately inversely proportional the amplification, just as the deviation from linearity of output voltage is inversely proportional to the radiometric forces. Consequently, a high amplification is desired. But, higher the amplification the higher is the risk for the vane start swinging. Therefore, the optimum conditions have to found empirically.
thread Vone
the to the the to be
Measuring Mirror
Permanent
Ceramic
mogne
l-l
I
e
e
Mounting permanent
bnsulatvon
of ihe mognet
agl
Torsion
Figure 5. Torsion
system
system.
prevent the self-excitation of the empirically optimized. Moreover, current by means of the adjustable zero-errors can be removed.
system, and CZ has to be one can add a boosting resistance P,. Thus, small
equipment-vacuum system. Figure 7 is a general diagram of the vacuum system as used for the comparison measurements: the portion within the dashed area at the left is an ultrahigh-vacuum system bakeable LLP to 350°C. and within the dashed area on the right is a gas-metering system. The gas pressure outside the ultrahigh-vacuum system is measured at higher pressures by the McLeod, and, at lower pressures, with the Bayard-Alpert gauge IM 2. The pressure inside the ultrahigh vacuum system on the one hand is measured with the modified Knudsen manometer, and, on the other hand with the Bayard-Alpert gauge IM I. The McLeod gauge approximately covers the range 10m2 - 5 x 10es torr, while the Bayard-Alpert gauges cover the range 10e9 - 10m4 torr. The McLeod gauge is designed, after Los and Morrison,” to be an absolute instrument. It is the standard against which IM I and IM 2 are calibrated. Besides, errors due to the pumping effect of the mercury diffusing from the reservoir to the trap were calculated as was proposed by Gaede.” The pressure inside the ultrahigh-vacuum system is raised by means of the metal leak valve VS. At higher pressures the valve V2 is opened and, therefore, an equilibrium state is reached in a few seconds. Thus, the pressure within the whole vacuum system could be measured with the McLeod gauge and, moreover, with the ionization gauges IM 1 and IM 2, respectively. Helium, argon, krypton as well as nitrogen and carbon dioxide served as suitable measuring gases which were spectroscopically pure.
Results Figures 8 and 9 represent the pressure as indicated by the ionization gauge IM I and, moreover, the current intensity used for compensating the radiometric turning moment. Both R52kL-L BPIA
Output
Figure 6. Amplifier. 343
W Kreisel: An apparatus
for the measurement
of low
gas pressures
and the determination
of thermal
HI, LdCOPII la1 y tube
-
KN MCLD
Knudsen-manometer Mcleod Boyord-Alpert gauges Valves Leak valve
1,3,4
IM1,2 V
VP K : ;zss Ii :;‘J3 P , z j 4 - Hg-Diffusion pumps Flosks(colibrot~ongases) Mi;ti,‘2- Exponston flask VE )
I
L--.---------
r
r?GK I MGK 2
Figure 7. Vacuumsystem.
Change
of the sensltlwty
accommodation
coefficients
-up;-H3 P4 “4 T Q
range
of the
I 35. 46. 61. 74. .-- 07. 100’
2.5x10+
“i
II3
=
4.5xlcT5.
’
6.5~16~.
&
6.5xl65
1
0
1.05x&
---Knudsen
1 i 0
2.5
5
7.5
IO
12.5
I5
17.5
20
22.5
25
27.5
30
32.5
set
Figure8. Pressure andcurrentintensityplottedduringexhaustionof the vacuumsystem. Figure 9. Scaleddown versionof the original plot of pressure andcurrentintensitycurves.
quantities were recorded by a 2-track recorder. One observes kinks in the pressureand current intensity plots (Figure 9) which synchronize with eachother, if one takesinto account a minimal delay time in the Knudsen manometerrelative to the ionization gauge. Moreover, due to the inherent physical properties of the Knudsen gauge it shows a ‘levelling-off’ period of a few seconds(Figure 9). But, in the current intensity plots of both Figures, it is worth noting their uniform courseseven for small current intensjties. 344
There is a close agreement in the pressurevalues measured with the ionization gaugepIMI, the McLeod gaugepMEL,,and the modified Knudsengauge pKn (Table 1). The data shown in Table 1 were taken using krypton gas. Column ‘A’ gives the percentageof deviation of the Knudsen manometerfrom the ionization gauge, ‘B’ that of the latter from the McLeod gaugeand ‘C’ that of the Knudsenapparatus from the McLeod. Provided that the ionization gauge measures correctly at pressures p < lo-’ torr it follows that in the rangep < 10m5
W Kreisel:
An apparatus
for the measurement
Table
of low
1. Comparison
pK. (torr) 4.1 4.6 1.3 9.7 1.6 4.1 7.7 I.3 1.4
: :, Y :\ -: .’ .’
lo-’ IO-” lo-S lo-S IO-’ lo-J IO-4 10-S lo-3
gas pressures
of pressure
intensity
accommodation
p1311 Uorr)
phleLd (torr)
A (%)
B(%)
CC%)
2.2 3.2 9.6 1.3 2.1 6.2 1.4
I.1 1.8 5.0 I.0 2.1 2.7
+ + + -
+ -k -I + -k +
-
‘.: 10-7 /I lo-” .: IO-” ‘< IO-J ‘%: 10-a x IO-+ b’ 10-a 2.6 :: lO-3 3.8 7 IO-3
curve
of thermal
of argon
coefficients
measurements
torr the values obtained are too high but by no means out of the admissible limit of errors. In the pressure range 10e5 < p c 5 >: IOe4 torr there are only very small deviations. A close agreement is remarkable with the McLeod gauge in the range 10e4 < p < 10m3 torr. But, the deviation of the Knudsen gauge from the McLeod becomes very high at pressures p > 10m3 torr. Since the accuracy of pressure measurements of the used McLeod in the range 10V2 > p > 10V3 torr is remarkable, the acceptable measuring range of the Knudsen gauge in measuring the noble gas Krypton must have been exceeded at pressures higher than approximately 10e3 torr. In general, the measuring range of this gauge has an upper bound which is determined by the ratio +i, where X is the mean free path for the particular gas and d the distance between the vane and the surfaces A, A’ and B, B’, respectively. Thus, at higher pressures the ratio X/d is so small that the linear relationship between the gas pressure and the current intensity is no longer valid (equation 23). In measuring lower pressures the measuring range is essentially limited by the characteristics of the amplifier described above. Figures 10 and I I represent on the one hand the as a function of the pressure PlMl and pMCLd, respectively, current intensity and, on the other hand the pressure PKn
Pressure-current
and the determination
:\ >: :: x x ?’
10-J IO-J lo-a 10-A 10-a lo-”
calculated equation
G”
86.3 43.7 35.4 25.4 23.8 33.9 45.0 50.0 63.2
18.2 16.6 24.0 40.0 23.7 40.7
1 II.8 - II.1 - 18.0 - 23.0 - 38.1 - 48.2
from the measured pkn = ick,, where l/2.
=
.
current intensity C&, is given by
7-72
_ ,) -I/z
by means
,
obtained from equation (23) for TX N T2 and a, = az = uj = u3 * = I. Figure IO shows the plots for krypton gas, and Figure I I represents those for helium. Referring to Figure 10 both curves run closely parallel up to a pressure of about 10m3 torr. Thereafter, the slope of the solid curve levels off and, at an approximate pressure of 2.5 x 10m3 torr runs roughly parallel to the p-axis. In this range pressure changes produce no change of current intensity. Comparing Figures IO and I1 we see that the slopes of the two curves in Figure I1 are steeper than their counterparts in Figure 10. Moreover, the equation pkn = iCk, is valid up to higher pressures compared with Figure 10. That is, the measuring range for helium is wider than that for krypton in accordance with the different mean free paths of these two gases. But, it is possible to increase the measuring range by decreasing the distanced, i.e. by increasing the value of the radio X:d. As is well known there are fundamental differences between
Pressure-current
intensity
curve of helium
l lonisotion gouge Leod gouge 0 Knudsen gouge ---
l
r-, 0.5
,---,-1---10 1.5
Figure 10. Pressure-current
2.0
2.5
3.0
p x IO“,
torr
intensity
curve
3.5
7
4.0
--_T
4.5
5.0
p x lo-3, of argon,
of the
Figure 11. Pressure-current
intensity
torr curve
of helium.
W
Kreisel: An apparatus
for the measurement
of low gas pressures
and the determination
the u.c.‘s
Accommodation
and 13 clearly
equal to I, equation (23)
of helium and other gases. Comparison of Figures I2 shows this fact. On the one hand, for COz, approximately 6.3 2: 10e5 [A] are needed for compensating a pressure of about 5 s 10e5 torr (Figure 12), while on the other hand, for helium, only about 2.4 x 10e5 [A] are necessary to compensate the radiometric turning moment effected by the same pressure. From this, one can conclude that the radiometric turning moment of helium is much smaller than that of other gases as explained by the smaller Q.c.‘s.
ps = i Now, n.c.‘s P~.~.,
of thermal
i.e.
2 II ncoi
accommodation
coefficients
coefficients. As long as each of the n.c.‘s are 0, =n2 =a3 =a3 * = I it follows from
B 7-v"2(T1112
_ 7-21/2)-
112
Wa)
A*/
denoting the pressure ps of those gases which have for those with a.c.‘s # I with = I by p,,.c.= ,, and , then we find using equation (23) and (23a) that
(24) 112
02g
+ a,*(1
-a,,:
(4
I
1[
a3*z
+
x
[
;
u3*
-
+ Qz(l - a3*g]“2j]
-(G>“‘l=
()
a2u3*y
L
J’
(24) shows that a given current intensity i will different pressures-so long as for one gas only the U.C. # 1 is satisfied. of p..=. = 1 and P,,.~.+ , we now introduce the ratio is known exactly: Using the ilp..,.= ,. The ihc. + , -value measured values for the quantities I’, II, A*, A,,, and B at the fixed temperature ratio T,/T, = 1,273 one obtains from equation (23a) with TX N T2 and px = pa.=.=,
Since there are a large number of paired values (P,,.~. + Ji) which were measured for each of the gases considered the problem consists in finding the slope S of the straight line
Equation compensate condition Instead
= i/p,... = 1 =
CK,* = l/C,, For obtains:
-[(y’
the
factor
C o.e. =h.+
1,322
P,,~.~ ,/P~.~.= ,
AC.
[A/torr] in equation
= I = Pa.=. + ,li
Using the methods of regression analysis one obtains with regard to the known value C&, the &.-values gathered in Table 2: Since the surfaces of the vane and the surfaces A, A’ and B, B’, respectively, are oxidized in the same manner it can be concluded with regard to T, N T, that a3 = a3* = a*. Moreover, one can readily state that T2 N TX as it was shown by the measurements. Using these presumptions one obtains from equation (24):
(24)
one
hence
Gn*
-2
C l2.c.
-K2> I
112
2
J Pressure-current of CO, in a lower
intensity pressure
curve range
1
-1
(25)
=o J Pressure-current Intensity of helium in o lower pressure
10.0
curve range
:
9.0
d
6.0
g5.0
” ^ 2.5 0 x - 2.0
." 4.0 3.0
-70 270310 torr
p x lo-‘,
Figure range.
346
12. Pressure-current
intensity
curve
of COz
px
in a lower
pressure
Figure 13. Pressure-current pressure
range.
;o’o
4:0-5~0~0~~~-& lo-‘,
intensity
torr
curve
of helium
in a lower
W Kreisel:
An apparatus
Table 2. Cocfficicnts
for the measurement
or equation
Gas
C;,.,.
Argon Carbon dioxide Helium Krypton Nitrogen
I .037, 1.0172 I .334* 1.021, 1.0489
of low
gas pressures
(25)
and the determination
of thermal
accommodation
coefficients
very little from unity. The n.c.‘s of those gases having a value much less than unity, however, must necessarily be determined in the light of the actual construction and types of materials used for the collision surfaces. These a.c.‘s then must be calculated in the same manner as described above. Moreover, the Knudsen manometer possesses essentially two notable advantages against the present known absolute pressure gauges: The Knudsen has a relatively large measuring range, i.e. from 10V3 to IO-’ torr. 2. It is easier to handle than, for instance, the McLeod. 1.
The problem which now arises consists in determining those a,- and a,-values for which the function has a minimum. These calculations were carried out on an IBM 370 computer using the fixed temperature ratio T,/T, mentioned above and the C’,,,.-values given in Table 2. One obtains the thermal accommodation coefficients given in Table 3 which are valid for oxidized aluminium surfaces.
Table 3. Accommodation Gas Argon Carbon dioxide Helium Krypton Nitrogen
coefficients
Acknowledgements I wish to thank Prof Dr Klaus Schafer, Physikalisch-Chemisches Institut der Universitat Heidelberg, and Prof Dr Dieter Schuller, Universitat Oldenburg, for their stimulating discussions and encouragement. This research was sponsored by Bergbauforschung, Essen.
U.C.
a1
a2
0.912 0.982 0.121 0.95 I 0.923
0.955 1 0.205 0.976 0.975
These results correspond very well with those obtained by Schafer and Gerstacker.r3 The small but measurable differences between the a.c.‘s for a, and a2 are generally explained by the negative temperature coefficients of the u.c.‘s Conclusion The Knudsen manometer as discussed proves to be an absolute pressure gauge for those gases which possess n.c.‘s differing
References ’ M Knudsen, Al~rr Physik, 32, 1910, 809. ’ M Knudsen, Am Physik, 34, 1911, 621. 3 M Smoluchowsky, Alrrr Physik, 35, 191 I, 1000. 4 J W M DuMond and W M Pickels, Reu Sci Instr, 6, 1935, 362. 5 A E Lockenvitz, Rev Sci Itlsfr, 9, 1938, 417. 6 A Klumb and H Schwarz, Z Phys, 122, 1944,418. ’ Y Wu, J Chm Phys 48, 1968, 889; Am Physik, 18, 1966, 321; Am Physik, 19, 1967, 144. * M Knudsen, Am Physik, 31, 1910, 633. 9 M Knudsen, Am Physik, 5,1930, 129. lo W Kreisel, Diplomarbeit 1971, Phys Chem Institut d Uni Heidelberg. ” J M Los and J A Morrison, Reo Sci Imfr, 22, 1952, 805. lz W Gaede, Am Physik, 46, 1915, 357. I3 K Schafer und H Gerstacker, Zfschr Elekfrochetnie, 60, S. 1960, 875.
347
30 April
W Kreisel,
lnstitut
1976 fiir
Phys.
Chemie
der Universitat
Heidelberg’,
Heidelberg,
Germany
An instrument is described for measuring pressure in the range of 10 -= to 1 O-7 (torr) . This gauge is bakeable up to 350°C and is relatively insensitive to mechanical vibrations. The principle of the measuring method is based on an electromagnetic self-adjusting compensation of the radiometric force effected by a definite but non-uniform distribution of temperature in a rarefied gas. Moreover, an equation for the radiometric force is given. This equation is valid for %ny temperature ratios and arbitrary accommodation coefficients. Using this equation it is possible to calculate coefficients of the gases from the experimental results. Assuming the accommodation coefficients to be unity, the device can be used as an absolute manometer since all other quantities can be measured directly. Comparative measurements are carried out with a calibrated ionisation gauge, and, in addition, at higher pressures with a McLeod gauge. These measurements were taken for testing the Knudsen manometer. Helium, argon and krypton, as well as N2 and C02, serve as suitable measuring gases.
Introduction
Theory
The principle which allows one to measure very low gas pressures with the ‘absolute manometer’ first constructed by Knudsen’ is based on the radiometric forces. This manometer consists of two parallel plates at different temperatures, Tl and T2. According to the theory of rarefied gases, the heat-as well as the momentum transport-in the free molecular flow regime between the two plates is proportional to the pressure px. The formula derived from this theory was given by Knudsen *
Considerthe ‘absolutemanometer’constructed by Knudsen.’ Then the resultant radiometric force F per unit area acting on the light suspended plate AZ which isat temperatureT2 is given by F = P$
Vacuum/volume26fnumber8.
Pergemon
Press/Printed
in Great
(1)
where px is the gas pressureto be measuredand Tl is the temperatureof the heatedand stationary plate A,. The formula is seento be independentof the molecularweight of the gas.
where F is the resultant radiometric force per unit area. As was shown late? this equation is only valid assumingthe thermal accommodationcoefficients (hereafter abbreviated to a.~.) of the plates to be unit. (The a.~. is a measureof the average efficiency of the energy exchange per encounter of a gas moleculewith the solid at the gas-solidinterface.) Although, Knudsen’sapparatuswas modifiedad the technical shortcomingshave irrrecent years rarely justified its use.In particular, the Knudsen manometer gauge did not give a continuous record of pressurebut had to be ‘read’, and, furthermore it wasvery sensitiveto mechanicalvibrations. This paper describesa form of radiometer gaugewhich is free from the above mentioneddisadvantages. We also derive an equation for the radiometric force from Wu’s revisedtheory of thermal transpiration’ which is valid for all a.c.‘s and temperatureconditions on the boundary. Using this equation we calculate the Q.C.‘Semploying the resultsof specialmeasurements (to be specifiedlater) as input. * Present address: Institut fiir Umweltschutzder Universitlt Dortmund,46 Dortmund,Rosemeyerstr. 6, Germany.
KWG)“2 - 11
T,
d
A2 T,
A, Figure1. Knudsen’s absolutemanometer-planview. This assumption,however, is only valid provided the a.c.‘s are identical, viz. a, = u2 = 1 where aI and ~72are the u.c.‘s of the platesAl and A2, respectively.As wasshownby Knudsen the manometer will obviously cease to function correctly according to equation (1) unless the distance between the platesis smallcomparedwith their linear dimensionsand to the meanfree path of the molecules. Moreover, equation (1) is basedon the traditional original Knudsentheory for thermal transp&ation.” That is, the term plT’12
is a constantwithin a closedvacuum systemwith a non-uniform temperaturedistribution. This traditional theory, however, is a mereapproximation to reality aswasshownby Wu.’ According to Wu the term p/T’l’
Britain
339
W Kreisel: An apparatus
for the measurement
of low
gas pressures
and the determination
of thermal
accommodation
coefficients
is inversely proportional to a new quantity I(isotropy) which has been introduced as a measure for the isotropicity of the velocity distribution function in the non-equilibrium Knudsen-regime. With the inclusion of this new quantity I one fnds that the ratio
molecules reflected from the surface A and the vane, The isotropy in this flow field is given by
p//r’
Furthermore, using obtains the following
I2
is a constant p //T’12
for a closed
vacuum
system,
i.e.,
= p’ ]‘/T”12
(2)
Consider now the modified manometer as shown in the planview in Figure 2. The aluminium bodies A and A’ heated up to an arbitrary but constant temperature T, by a thermostatic controlled liquid are placed opposite to the surfaces B and B’ which are at the arbitrary but constant temperature T2 whereby T, > T2. The light suspended vane may have the temperature T, # T,, T2. With T,, , we denote the temperature characterizing the flow field existing between the vane and the surfaces A and A’, while I,. , characterizes its isotropy and pint its pressure.
respectively.
(5) the notations equations:
as
indicated
above
one
Tin 2 = CT,,,T,, ,)I” W-c, Tc-, ,I””
Iin 2 = With
T co“2 + Tj,,
regard
to the definition
of the o.c.~ it follows
that
(8) T ho
a3 =
I -
The
-
The
*=
a3 T3
T co I - r,o T3
-
Tco
whereby a, and a2 are the a.c.‘s of the surfaces A, A’ and B, B’, respectively, while a3 and a3* are the a.c.‘s of the collision surfaces of the vane (Figure 2). Simple calculations yield the solution for the gas temperature of the molecules reflected (or emitted) from the surfaces:
a,T, + a,(1 - alIT
(a) The =
a,
+ a3 -
a3T3 + a,(1 - a,)T,
(b) The I =
a,+a3-alu3
u~TJ +
(~1 Tc, =
u2
+
Figure
2. Modified
Knudsen
manometer
(plan
view)-flow
system.
The molecules reflected from the surfaces A and A’, respectively, possess the gas temperature The. Those molecules leaving the collision surfaces of the vane opposite to the surfaces A and A’ possess the gas temperature The ,. Analogous notations are used for the flow field existing between the vane and the surfaces B and B’. TX is the temperature of the vacuum system outside the flow fields, I, its isotropy and pX is its pressure which is to be determined. With regard to
T = Ti, 1 I = Iin 1 p =pin 1 T’ = Ti, 2 I’ = Ii, 2 P’ =Pio 2 one obtains
from
equation
Tc,
, =
Substituting one finds vane and
by
the
geometric
(3)
(4) mean
of
the
-~3*)T2
U2 + U3*
-
U2U3*
those expressions that the temperature the surfaces A and
In a similar
way
gas
temperatures
of
the
T,,. and Thol into equation in the flow field between is given by
for
A’
+ a,(1 - u~)T,]}“~
+a,-u,u,
one obtains
for
(4) the
(10)
Tin z :
Tin 2 = +
From
(2)
~2)T3
u2u3*
a,*(1
-
u2)T31
b73*7’3
u2 + a3*
Assuming the molecular flow between the aluminium body A and the vane to be stationary but anisotropic it follows from the revised theory of thermal transpiration (in the absence of intermolecular collisions) that the temperature T,. , of this flow field is given by
i.e.
-
+ u~(I
a,
Iin
equations
(5),
(9a)
-
and
+
a,(1
-
u3*)T21}“2
(1 1)
a2u3 *
(9b)
it follows
that
1 = 2Ua,T,
I = (T,,o Tim 11~‘~
-
i[a~Tt + ad1 - a,)T31 b3T3
{[“2T3
P.1” 1lin ~lTf, t = Pin 2 4” 2lTfn2
Tin
(9)
a3*(l u3*
~3*T3 (4
ala3
+
b,T,
+
and
from
zin
-
a,V’33
- a,)T,]“’
equations
(7),
(9c)
[u,T, + a,( 1 + [u,T, + a,(1 and
(9d)
-
u~)T,]}“~ u~)T,]“~
(12)
one obtains
2 =
2{[u,T2 [~2T2
a,(1
a,(1
+ u3*(1 +
a,*(1
-
-
u2)T3]
~2)T3]l’~
[~3*T3 + [~3*T3
+ a,(1
-
~3*)T2]}“~
+ a2 (1 -
~3*)T2]“~
(13)
W Kreisel: An apparatusfor the measurement of low gaspressures and the determinationof thermalaccommodationcoefficients
Using equations(3) and (IO)-(13) the pressurep,,, , in the flow held betweenlhc vane and the surfaces.4 and A’ is now given bY Pin
I =
Pin
a2 + aJ* - a2as* ‘I2 a, +a, -aza3 >
2
b,T, + aJ1 - a,)T,]“’ azTz + a,*( I - a,)T,]‘”
1 I, j
+ [NJTA + a,(] - a,)T,]“’ + [a3*T3 + a,(1 - a,*)T2]“z
(14)
i.c.,p,,, , = Pinz C*, whereC* = l I. Thus, for the resultant radiometric force F per unit area on the central vane we obtain F = (IJin I - Pin 2) = Pin 2tc*
- I)
(15)
However. the pressurepi,, z in this equationis not identical with the pressurepx which is to be measured.But, it is possibleto state the relation (16) analogousto equation (3):
M!r;sr
Figure
3. Vane.
Since the resultant radiometric force acting on this area A* is given by SinceI, = I (Maxwellian distribution) it followsfrom equations (I I), (13) and (16) that F,&
’
we obtain from equations(20) and (21):
2p
1 1J
f [a,T, + a,(1 - a,)T,]“’ + [ajT3 + a,(1 - a,)T,]“2 (a, + a, - a,a3)1’2 I[
[
[a?T, +rr,*(l
-az)T3]“’ + [a3*T3 +aJ I -a,*)Tz]“‘] (a2 + a3* - a2a,*)“’
\
(17)
i.e.,
(22) As described later this radiometric turning moment D,, is electromagneticallycompensatedby self-adjustment.Therefore, the systemwill be in equilibrium if the relationship De, = Dra
F = ‘A
*
2 ry’12
(174
x
where X is the term in brackets { }. Equation (17), as already given in I”, is the most general form for the resulting force per unit area on the central vane valid for all a.c.‘s and for arbitrary temperatureconditions. For a, = a2 = a3 = u3* = a one obtains F,& 2 G2
[(T, +T3 - uT,)“~ + (T, + T3 -aT,)‘12 -
is satisfied,where D,, is the electromagneticturning moment given by D,, = nA,,,Bi, whereII is the numberof windingsper unit length,A,., isthe surfaceareaof the coil and B the magnetic induction. Thus, it follows that .2tIA,,i B Px = 1 A*r
(T,“’
=
T,‘f2
-
T2112
(‘8)
In this casethe gaugecan be usedas an absolutemanometer sinceall other quantitiescan be measureddirectly.
- Tz”~)
(1%
Experimental
This equation is plausible when the temperature T, is equal to the temperature T, of the unheated surfacesB and B’, respectively, in which caseone obtains equation (1) as already given by Knudsen.’ With regard to the symmetry of the system(Figure 2) and the linear dimensions of the vane (Figure 3) the radiometric turning moment is given by r2 rdr=p,l(r,-r,)(r,+r,)=p,rA* s r1 whereA* is the total collision surfaceof the vane. D,, = 2p.J
(23)
- (T, + T3 - QTJ”~]
while for a, = a2 = a3 = a3* = 1 it follows that F,k 2 p ’
[Nm’l.
As long asall of the u.c.‘s are equal to 1, one can substitutefor X the expression X
(T2 + T3 - UT,)“’
~~‘I2
x
(20)
As with most of the componentsof this gaugethe manometer vesselis made of high grade steel. This vessel, 100 mm in diameter was constructed according to our specificationsby Vacuum GeneratorsLtd. The upper part of the vesselis vacuum sealedwith a special flange (FC 100) upon which are symmetrically mounted four heat-insulatedliquid feed-throughs(LFT 2) and, moreover, two electrical feed-throughs(EFT 5) eachhaving eight conductors. The LET 2’sare suitablefor thermostaticcontrolled liquids as well as for liquid nitrogen. A small tightly fitting ahuninium body is locatedon eachof the four insidetubesof the LET 2’s. The aluminium bodiesare reamedto fit the outer diameter of 341
W Kreisel:
An
apparatus for the measurement of low gas pressures and the determination of thermal accommodation coefficients MUFC
Colltslon
15 Vacuum pump
surfaces
Dlfferentlol pholoconductlve
cell
\/sew-port
L1gi;l
A
wll,ce
Torston-system mounting
Ertenslon LFT2-liquid throltghs Vessel
c: the feed
gouge Holf-sAvered Annular
I
Lens
mirror
spoccr
Mirror
LFT EFT5-elertrlccl feed th:oughs
Z-llqu:d throughs
feed
FC 100
Figure 4. Modified Knudsenmanometer.
the LFT 2’s insuring an optimal heat transmission. The electrical feed-throughsare usedfor the electrical power supply of the galvanometer coil as well as for the connections of resistancethermometersby meansof which the temperatureof the aluminium body and the temperature of the vacuum systeminsidethe gaugeare measured. Moreover, the vessel has a lateral view-port, 38 mm in diameterto record the slightestdeflectionof the vane by means of a light beam. The zero-point of this light suspended vane can be mechanically adjustedby accessthrough a bore-hole suitably placed in the middle of the FC 100 flange. Once adjusted it will never have to be touched again. This small bore-hole, 5 mm in diameterissealedwith a blind flangeusinga coppergasket. On the insideof the FC 100four blind bore-holesare located symmetrically around the central bore-hole. They serve as a mounting for the meter frame which iseasyto dismount. The galvanometersystemwas constructedaccording to our specificationsby SiemensLtd. Two plates of high grade steel joined with four vertical bars serve as the outer mounting. The thin filament is elastically suspendedbetweentwo torsion headswhich are ceramically insulated from the plates (Figure 5). The characteristicsof the filament are as follows: diameter. tensile strength ultimate tensile strength elastic hysteresis torsion modulus material 342
0.015 mm 274 k&/mm2 48.4 gf 0.05 % 7560 kgf/mm2 Nivaflex
The amount of the torsion stressis 10 gf. The filament carries the rectangular aluminium vane which is rigidly joined with a suitably constructedgalvanometercoil whosewinding consists of thin glasssilk insulatedplatinum thread. This galvanometer coil aswell as the ceramic insulationsendurean outgassingup to350T. In order to makethe surfacesof the vane and the surfacesof the aluminium bodiesasalike aspossiblethey are oxidized in a similar manner. Furthermore, in order to achieve high u.c.‘s we have usedrough surfaces. The deflection of the vane effected by the radiometric turning moment will be electromagnetically compensated by self-adjustment so that the vane remains in the stationary position, even under the influenceof the radiometric forces. This is accomplishedas follows: a light beam incident on the galvanometer mirror is reflected on to a differential photoconductive cell consisting of two light-sensitive halves with resistancesRI and R2 which form one part of an active bridge circuit of an amplifier. The other part of the bridge consistsof equal resistancesR3 and R+ When the vane is in its zero-position, i.e. exactly betweenthe heatedaluminium bodies and the unheatedbodies,respectively, the light beamreflected from the galvanometermirror will meet the photoconductive cell at its centre. In this position R, = R2. That is, the output voltage of the amplifier is zero. The resistanceR5 representsthe outer negative feedback path by meansof which the counterforceagainstthe mechanical deflections of the vane is produced. The capacitor C, is to Electra-optics.
W Kreisel:
An apparatus
for the measurement
of low
Torwon
Towon
gas pressures
head
and the determination
of thermal
accommodation
coefficients
The deflection of the vane from its zero-point effected by radiometric forces is approximately inversely proportional the amplification, just as the deviation from linearity of output voltage is inversely proportional to the radiometric forces. Consequently, a high amplification is desired. But, higher the amplification the higher is the risk for the vane start swinging. Therefore, the optimum conditions have to found empirically.
thread Vone
the to the the to be
Measuring Mirror
Permanent
Ceramic
mogne
l-l
I
e
e
Mounting permanent
bnsulatvon
of ihe mognet
agl
Torsion
Figure 5. Torsion
system
system.
prevent the self-excitation of the empirically optimized. Moreover, current by means of the adjustable zero-errors can be removed.
system, and CZ has to be one can add a boosting resistance P,. Thus, small
equipment-vacuum system. Figure 7 is a general diagram of the vacuum system as used for the comparison measurements: the portion within the dashed area at the left is an ultrahigh-vacuum system bakeable LLP to 350°C. and within the dashed area on the right is a gas-metering system. The gas pressure outside the ultrahigh-vacuum system is measured at higher pressures by the McLeod, and, at lower pressures, with the Bayard-Alpert gauge IM 2. The pressure inside the ultrahigh vacuum system on the one hand is measured with the modified Knudsen manometer, and, on the other hand with the Bayard-Alpert gauge IM I. The McLeod gauge approximately covers the range 10m2 - 5 x 10es torr, while the Bayard-Alpert gauges cover the range 10e9 - 10m4 torr. The McLeod gauge is designed, after Los and Morrison,” to be an absolute instrument. It is the standard against which IM I and IM 2 are calibrated. Besides, errors due to the pumping effect of the mercury diffusing from the reservoir to the trap were calculated as was proposed by Gaede.” The pressure inside the ultrahigh-vacuum system is raised by means of the metal leak valve VS. At higher pressures the valve V2 is opened and, therefore, an equilibrium state is reached in a few seconds. Thus, the pressure within the whole vacuum system could be measured with the McLeod gauge and, moreover, with the ionization gauges IM 1 and IM 2, respectively. Helium, argon, krypton as well as nitrogen and carbon dioxide served as suitable measuring gases which were spectroscopically pure.
Results Figures 8 and 9 represent the pressure as indicated by the ionization gauge IM I and, moreover, the current intensity used for compensating the radiometric turning moment. Both R52kL-L BPIA
Output
Figure 6. Amplifier. 343
W Kreisel: An apparatus
for the measurement
of low
gas pressures
and the determination
of thermal
HI, LdCOPII la1 y tube
-
KN MCLD
Knudsen-manometer Mcleod Boyord-Alpert gauges Valves Leak valve
1,3,4
IM1,2 V
VP K : ;zss Ii :;‘J3 P , z j 4 - Hg-Diffusion pumps Flosks(colibrot~ongases) Mi;ti,‘2- Exponston flask VE )
I
L--.---------
r
r?GK I MGK 2
Figure 7. Vacuumsystem.
Change
of the sensltlwty
accommodation
coefficients
-up;-H3 P4 “4 T Q
range
of the
I 35. 46. 61. 74. .-- 07. 100’
2.5x10+
“i
II3
=
4.5xlcT5.
’
6.5~16~.
&
6.5xl65
1
0
1.05x&
---Knudsen
1 i 0
2.5
5
7.5
IO
12.5
I5
17.5
20
22.5
25
27.5
30
32.5
set
Figure8. Pressure andcurrentintensityplottedduringexhaustionof the vacuumsystem. Figure 9. Scaleddown versionof the original plot of pressure andcurrentintensitycurves.
quantities were recorded by a 2-track recorder. One observes kinks in the pressureand current intensity plots (Figure 9) which synchronize with eachother, if one takesinto account a minimal delay time in the Knudsen manometerrelative to the ionization gauge. Moreover, due to the inherent physical properties of the Knudsen gauge it shows a ‘levelling-off’ period of a few seconds(Figure 9). But, in the current intensity plots of both Figures, it is worth noting their uniform courseseven for small current intensjties. 344
There is a close agreement in the pressurevalues measured with the ionization gaugepIMI, the McLeod gaugepMEL,,and the modified Knudsengauge pKn (Table 1). The data shown in Table 1 were taken using krypton gas. Column ‘A’ gives the percentageof deviation of the Knudsen manometerfrom the ionization gauge, ‘B’ that of the latter from the McLeod gaugeand ‘C’ that of the Knudsenapparatus from the McLeod. Provided that the ionization gauge measures correctly at pressures p < lo-’ torr it follows that in the rangep < 10m5
W Kreisel:
An apparatus
for the measurement
Table
of low
1. Comparison
pK. (torr) 4.1 4.6 1.3 9.7 1.6 4.1 7.7 I.3 1.4
: :, Y :\ -: .’ .’
lo-’ IO-” lo-S lo-S IO-’ lo-J IO-4 10-S lo-3
gas pressures
of pressure
intensity
accommodation
p1311 Uorr)
phleLd (torr)
A (%)
B(%)
CC%)
2.2 3.2 9.6 1.3 2.1 6.2 1.4
I.1 1.8 5.0 I.0 2.1 2.7
+ + + -
+ -k -I + -k +
-
‘.: 10-7 /I lo-” .: IO-” ‘< IO-J ‘%: 10-a x IO-+ b’ 10-a 2.6 :: lO-3 3.8 7 IO-3
curve
of thermal
of argon
coefficients
measurements
torr the values obtained are too high but by no means out of the admissible limit of errors. In the pressure range 10e5 < p c 5 >: IOe4 torr there are only very small deviations. A close agreement is remarkable with the McLeod gauge in the range 10e4 < p < 10m3 torr. But, the deviation of the Knudsen gauge from the McLeod becomes very high at pressures p > 10m3 torr. Since the accuracy of pressure measurements of the used McLeod in the range 10V2 > p > 10V3 torr is remarkable, the acceptable measuring range of the Knudsen gauge in measuring the noble gas Krypton must have been exceeded at pressures higher than approximately 10e3 torr. In general, the measuring range of this gauge has an upper bound which is determined by the ratio +i, where X is the mean free path for the particular gas and d the distance between the vane and the surfaces A, A’ and B, B’, respectively. Thus, at higher pressures the ratio X/d is so small that the linear relationship between the gas pressure and the current intensity is no longer valid (equation 23). In measuring lower pressures the measuring range is essentially limited by the characteristics of the amplifier described above. Figures 10 and I I represent on the one hand the as a function of the pressure PlMl and pMCLd, respectively, current intensity and, on the other hand the pressure PKn
Pressure-current
and the determination
:\ >: :: x x ?’
10-J IO-J lo-a 10-A 10-a lo-”
calculated equation
G”
86.3 43.7 35.4 25.4 23.8 33.9 45.0 50.0 63.2
18.2 16.6 24.0 40.0 23.7 40.7
1 II.8 - II.1 - 18.0 - 23.0 - 38.1 - 48.2
from the measured pkn = ick,, where l/2.
=
.
current intensity C&, is given by
7-72
_ ,) -I/z
by means
,
obtained from equation (23) for TX N T2 and a, = az = uj = u3 * = I. Figure IO shows the plots for krypton gas, and Figure I I represents those for helium. Referring to Figure 10 both curves run closely parallel up to a pressure of about 10m3 torr. Thereafter, the slope of the solid curve levels off and, at an approximate pressure of 2.5 x 10m3 torr runs roughly parallel to the p-axis. In this range pressure changes produce no change of current intensity. Comparing Figures IO and I1 we see that the slopes of the two curves in Figure I1 are steeper than their counterparts in Figure 10. Moreover, the equation pkn = iCk, is valid up to higher pressures compared with Figure 10. That is, the measuring range for helium is wider than that for krypton in accordance with the different mean free paths of these two gases. But, it is possible to increase the measuring range by decreasing the distanced, i.e. by increasing the value of the radio X:d. As is well known there are fundamental differences between
Pressure-current
intensity
curve of helium
l lonisotion gouge Leod gouge 0 Knudsen gouge ---
l
r-, 0.5
,---,-1---10 1.5
Figure 10. Pressure-current
2.0
2.5
3.0
p x IO“,
torr
intensity
curve
3.5
7
4.0
--_T
4.5
5.0
p x lo-3, of argon,
of the
Figure 11. Pressure-current
intensity
torr curve
of helium.
W
Kreisel: An apparatus
for the measurement
of low gas pressures
and the determination
the u.c.‘s
Accommodation
and 13 clearly
equal to I, equation (23)
of helium and other gases. Comparison of Figures I2 shows this fact. On the one hand, for COz, approximately 6.3 2: 10e5 [A] are needed for compensating a pressure of about 5 s 10e5 torr (Figure 12), while on the other hand, for helium, only about 2.4 x 10e5 [A] are necessary to compensate the radiometric turning moment effected by the same pressure. From this, one can conclude that the radiometric turning moment of helium is much smaller than that of other gases as explained by the smaller Q.c.‘s.
ps = i Now, n.c.‘s P~.~.,
of thermal
i.e.
2 II ncoi
accommodation
coefficients
coefficients. As long as each of the n.c.‘s are 0, =n2 =a3 =a3 * = I it follows from
B 7-v"2(T1112
_ 7-21/2)-
112
Wa)
A*/
denoting the pressure ps of those gases which have for those with a.c.‘s # I with = I by p,,.c.= ,, and , then we find using equation (23) and (23a) that
(24) 112
02g
+ a,*(1
-a,,:
(4
I
1[
a3*z
+
x
[
;
u3*
-
+ Qz(l - a3*g]“2j]
-(G>“‘l=
()
a2u3*y
L
J’
(24) shows that a given current intensity i will different pressures-so long as for one gas only the U.C. # 1 is satisfied. of p..=. = 1 and P,,.~.+ , we now introduce the ratio is known exactly: Using the ilp..,.= ,. The ihc. + , -value measured values for the quantities I’, II, A*, A,,, and B at the fixed temperature ratio T,/T, = 1,273 one obtains from equation (23a) with TX N T2 and px = pa.=.=,
Since there are a large number of paired values (P,,.~. + Ji) which were measured for each of the gases considered the problem consists in finding the slope S of the straight line
Equation compensate condition Instead
= i/p,... = 1 =
CK,* = l/C,, For obtains:
-[(y’
the
factor
C o.e. =h.+
1,322
P,,~.~ ,/P~.~.= ,
AC.
[A/torr] in equation
= I = Pa.=. + ,li
Using the methods of regression analysis one obtains with regard to the known value C&, the &.-values gathered in Table 2: Since the surfaces of the vane and the surfaces A, A’ and B, B’, respectively, are oxidized in the same manner it can be concluded with regard to T, N T, that a3 = a3* = a*. Moreover, one can readily state that T2 N TX as it was shown by the measurements. Using these presumptions one obtains from equation (24):
(24)
one
hence
Gn*
-2
C l2.c.
-K2> I
112
2
J Pressure-current of CO, in a lower
intensity pressure
curve range
1
-1
(25)
=o J Pressure-current Intensity of helium in o lower pressure
10.0
curve range
:
9.0
d
6.0
g5.0
” ^ 2.5 0 x - 2.0
." 4.0 3.0
-70 270310 torr
p x lo-‘,
Figure range.
346
12. Pressure-current
intensity
curve
of COz
px
in a lower
pressure
Figure 13. Pressure-current pressure
range.
;o’o
4:0-5~0~0~~~-& lo-‘,
intensity
torr
curve
of helium
in a lower
W Kreisel:
An apparatus
Table 2. Cocfficicnts
for the measurement
or equation
Gas
C;,.,.
Argon Carbon dioxide Helium Krypton Nitrogen
I .037, 1.0172 I .334* 1.021, 1.0489
of low
gas pressures
(25)
and the determination
of thermal
accommodation
coefficients
very little from unity. The n.c.‘s of those gases having a value much less than unity, however, must necessarily be determined in the light of the actual construction and types of materials used for the collision surfaces. These a.c.‘s then must be calculated in the same manner as described above. Moreover, the Knudsen manometer possesses essentially two notable advantages against the present known absolute pressure gauges: The Knudsen has a relatively large measuring range, i.e. from 10V3 to IO-’ torr. 2. It is easier to handle than, for instance, the McLeod. 1.
The problem which now arises consists in determining those a,- and a,-values for which the function has a minimum. These calculations were carried out on an IBM 370 computer using the fixed temperature ratio T,/T, mentioned above and the C’,,,.-values given in Table 2. One obtains the thermal accommodation coefficients given in Table 3 which are valid for oxidized aluminium surfaces.
Table 3. Accommodation Gas Argon Carbon dioxide Helium Krypton Nitrogen
coefficients
Acknowledgements I wish to thank Prof Dr Klaus Schafer, Physikalisch-Chemisches Institut der Universitat Heidelberg, and Prof Dr Dieter Schuller, Universitat Oldenburg, for their stimulating discussions and encouragement. This research was sponsored by Bergbauforschung, Essen.
U.C.
a1
a2
0.912 0.982 0.121 0.95 I 0.923
0.955 1 0.205 0.976 0.975
These results correspond very well with those obtained by Schafer and Gerstacker.r3 The small but measurable differences between the a.c.‘s for a, and a2 are generally explained by the negative temperature coefficients of the u.c.‘s Conclusion The Knudsen manometer as discussed proves to be an absolute pressure gauge for those gases which possess n.c.‘s differing
References ’ M Knudsen, Al~rr Physik, 32, 1910, 809. ’ M Knudsen, Am Physik, 34, 1911, 621. 3 M Smoluchowsky, Alrrr Physik, 35, 191 I, 1000. 4 J W M DuMond and W M Pickels, Reu Sci Instr, 6, 1935, 362. 5 A E Lockenvitz, Rev Sci Itlsfr, 9, 1938, 417. 6 A Klumb and H Schwarz, Z Phys, 122, 1944,418. ’ Y Wu, J Chm Phys 48, 1968, 889; Am Physik, 18, 1966, 321; Am Physik, 19, 1967, 144. * M Knudsen, Am Physik, 31, 1910, 633. 9 M Knudsen, Am Physik, 5,1930, 129. lo W Kreisel, Diplomarbeit 1971, Phys Chem Institut d Uni Heidelberg. ” J M Los and J A Morrison, Reo Sci Imfr, 22, 1952, 805. lz W Gaede, Am Physik, 46, 1915, 357. I3 K Schafer und H Gerstacker, Zfschr Elekfrochetnie, 60, S. 1960, 875.
347