Vacuum Gauges

1. Theory

• Definition: Vacuum gauges measure total pressure in process and analysis chambers, using force on a diaphragm at high pressure, gas heat conduction or rotor drag at intermediate pressure, and ionization at low pressure (Jousten lecture; Edwards note).
• Direct and indirect: Only force-per-area gauges measure true total pressure independent of gas; thermal, drag and ionization gauges depend on gas species (Jousten lecture; Edwards note).
• Gas dependence: Indirect readings are corrected by relative sensitivity to nitrogen (Edwards note).
• Ranges: About fifteen decades are covered by overlapping gauges, from $1\times 10^5$ $\mathrm{Pa}$ down to $1\times 10^{-10}$ $\mathrm{Pa}$ (Jousten lecture).

2. Working Principle

• Capacitance diaphragm: Thin INVAR or ceramic membrane deflecting by fractions of a nanometre, read capacitively (Jousten lecture).
• Pirani: Heated wire or MEMS sheet losing heat to the gas (Jousten lecture).
• Spinning rotor: Magnetically levitated ball slowed by molecular drag (Jousten lecture; background knowledge, Fremerey).
• Ionization: Hot-cathode Bayard-Alpert or cold-cathode Penning and magnetron gauges (Edwards note; Jousten lecture).

3. Applications

• Process pressure control: Heated capacitance manometers on etch and CVD chambers (background knowledge).
• Pump-down and load locks: Pirani and combination gauges (Edwards note).
• High and ultra-high vacuum: Ionization gauges in sputter, MBE and analysis tools (Jousten lecture).
• Calibration: Spinning rotor gauges as transfer standards (Jousten lecture).

Direct Gauges and Ranges

Name	Formula	Related Formulae	Variables/Constants
Capacitance diaphragm gauge	$C=\frac{{\epsilon }_0{A}_e}{g},\frac{\Delta C}{C}\approx \frac{w}{g}$	$w_0=\frac{p{a}^4}{64D},D=\frac{E{t}^3}{12\left(1-{\nu }^2\right)}$	$C:electrodecapacitance$ $A_e:electrodearea$ $g:gap$ $w:diaphragmdeflection$ $w_0:centredeflectionofaclampedplate$ $a:radius$ $t:thickness$ $\begin{array}{r}D:flexuralrigidity;Joustenlecture:CDG\\ range1\times 10^{-4}to1\times 10^5\mathrm{Pa},normaluncertainty\\ 0.1to0.003,lowest0.006to0.001,\\ deflectionsensitivity0.4\mathrm{nm},INVARor\\ ceramicmembranesasthinas25\mu \mathrm{m},two\\ capacitorsandthermostatedhousing;plate\\ formula(backgroundknowledge,Timoshenko)\end{array}$ $\begin{array}{r}(assumed:E=141\mathrm{GPa},\nu =0.26,t=25\mu \mathrm{m},\\ a=10mm;computed:0.79\mu m/Pa,so0.4\mathrm{nm}\\ is5\times 10^{-4}\mathrm{Pa})\end{array}$
Thermal transpiration	$\frac{p_1}{p_2}=\sqrt{\frac{T_1}{T_2}}$	$\text{applies when}\lambda \gg d_{tube}$	$\begin{array}{r}{p}_1,p_2,T_1,T_2:pressuresandtemperatures\\ oftwovolumesconnectedinthemolecular\\ regime;Joustenlecture:relationand\\ exampleratio1.036for296and318K,\\ realtotalpressurefromforce-per-area\\ gaugesonlyabove100\mathrm{Pa},transpiration\\ neededbelow;condition(background\\ knowledge,O^{\prime }Hanlon)\end{array}$ $(computed:\sqrt{318/296}=1.0365)$
Range and uncertainty	$p_{true}=p_{ind}\left(1\pm U_{rel}\right)$	$\text{direct gauges}\gtrsim 1e-4\mathrm{mbar}$	$\begin{array}{r}{U}_{rel}:relativeexpandeduncertainty(k=2);\\ Joustenlecture:Pirani1\times 10^{-1}to1\times 10^4\mathrm{Pa}\\ (1to0.1),SRG1\times 10^{-5}to10\mathrm{Pa}(0.1to\\ 0.007),Penning1\times 10^{-7}to1\mathrm{Pa}(0.5to\\ 0.2),magnetron1\times 10^{-8}to1\mathrm{Pa},\\ emission-cathodeionization1\times 10^{-10}to1\times 10^{-2}\\ \mathrm{Pa}(1to0.05),errorsabove100percent\\ possible;Edwardsnote:practicallower\\ limitofdirectgauges1\times 10^{-4}\mathrm{mbar}\end{array}$ $\begin{array}{r}(computed:1\times 10^{-4}\mathrm{Pa}is1\times 10^{-6}\mathrm{mbar},two\\ decadesbelowtheEdwardslimit)\end{array}$

Thermal and Drag Gauges

Name	Formula	Related Formulae	Variables/Constants
Pirani heat balance	$P_{el}=P_{rad}+P_{end}+P_{gas}\left(p\right)$	$P_{gas}\propto p\left(Kn\gg 1\right),P_{gas}\approx \text{const}\left(Kn\ll 1\right)$	$P_{el}:electricalpower$ $P_{rad}:radiation$ $P_{end}:conductiontosupports$ $P_{gas}:gasconduction$ $\begin{array}{r}Kn:Knudsennumber;Joustenlecture:\\ radiation,conductionandgastransport\\ withconvectionathighpressure,\\ constant-temperatureorconstant-voltage,\\ currentorpowermodes,MEMSsheetat60\\ {}^{\circ }CwithhigherKnandnoconvection;\\ balance(backgroundknowledge,O^{\prime }Hanlon)\end{array}$ $\begin{array}{r}(myreading:theusefulrangelieswhere\\ gasconductioniscomparabletoradiation\\ andendlossesandstillgrowswith\\ pressure)\end{array}$
Knudsen number of the sensor	$Kn=\frac{\lambda }{d_s},\lambda =\frac{k_{B}T}{\sqrt{2}\pi d_m^{2}p}$	$\lambda \approx \frac{6.5\mathrm{mm}\mathrm{Pa}}{p}{\text{(N}}_2\text{)}$	$d_s:sensorgaporwirediameter$ $\begin{array}{r}{d}_m:moleculardiameter;relations\\ (backgroundknowledge,O^{\prime }Hanlon);Jousten\\ lecture:Piranistabilityas\\ repeat-calibrationstandarddeviations,\\ forexample0.03to3.30percentbetween\\ 0.05and30\mathrm{mbar}acrossfourmodels\end{array}$ $\begin{array}{r}(assumed:N2,0.375\mathrm{nm},296K,25\mu \mathrm{m}\\ dimension;computed:\lambda =6.5mmat1\mathrm{Pa},\\ Kn=262at1\mathrm{Pa}and0.026at1\times 10^4\mathrm{Pa})\end{array}$
Spinning rotor gauge	$p=\frac{\pi \rho d\bar{c}}{20\sigma }\left(-\frac{\dot{\omega }}{\omega }\right)-p_{RD}$	$\bar{c}=\sqrt{\frac{8k_{B}T}{\pi m}}$	$\rho ,d:rotordensityanddiameter$ $\bar{c}:meanmolecularspeed$ $\sigma :accommodationcoefficient$ $\begin{array}{r}{p}_{RD}:residual-dragequivalent;Fremerey\\ form(backgroundknowledge,Fremerey);\\ Joustenlecture:viscosity-basedrelation\\ (garbled),residualdragDCRabout\\ 1.85\times 10^{-7}to2.15\times 10^{-7}at404to416Hz,no\\ gasconsumptionordissociation,high\\ stability,effectivemassformixtures(\sum \\ a_i{m}_i^{1/2}{)}^2\end{array}$ $\begin{array}{r}(assumed:steel7700kgpercubicm,4.5\\ mm,\sigma =1,N2at296K;computed:\bar{c}=473\\ m/s,deceleration3.9\times 10^{-7}persat1\times 10^{-3}\\ \mathrm{Pa},soa2\times 10^{-7}residualdragequalsabout\\ 5\times 10^{-4}\mathrm{Pa})\end{array}$

Ionization Gauges and Gas Correction

Name	Formula	Related Formulae	Variables/Constants
Hot-cathode ionization gauge	$I^{+}=S{I}^{-}p$	$S=\frac{I^{+}/I^{-}}{p}\approx 10\text{to}25\mathrm{mba}{\mathrm{r}}^{-1}$	$I^{+}:collectorioncurrent$ $I^{-}:emissioncurrent$ $\begin{array}{r}S:gaugesensitivity;Edwardsnote:GS=\\ (I+/I-)/P,typicalBayard-Alpertvalues10\\ to25\mathrm{mba}{\mathrm{r}}^{-1},constantemission;Jousten\\ lecture:emission-cathoderange1\times 10^{-10}to\\ 1\times 10^{-2}\mathrm{Pa}\end{array}$ $\begin{array}{r}(assumed:S=10\mathrm{mba}{\mathrm{r}}^{-1},4\mathrm{mA};computed:\\ 4\times 10^{-10}Aat1\times 10^{-8}\mathrm{mbar})\end{array}$
Gas correction	$p_{true}=\frac{p_{ind,N_2}}{S_{gas}}$	$p_{true}=p_{ind}\frac{\sum r_i}{\sum s_i{r}_i},C{F}_{eff}=\sum a_{i}C{F}_i$	$S_{gas}:relativesensitivity(N2=1)$ $r_i,a_i:fractions$ $s_i:sensitivities$ $\begin{array}{r}CF:correctionfactors;Edwardsnote:air\\ 0.97to1.00,CO21.3to1.45,He0.14to\\ 0.25,H20.36to0.60,CH41.40to1.62,\\ Kr1.72to1.94,H2O0.86to1.25,divide\\ byS;Joustenlecture:mixtureruleand\\ Piraniheliumfactorsfromabout0to1.4\\ over1\times 10^{-3}to1\times 10^2\mathrm{mbar};RGApage:Hiden\\ He0.14andH20.44\end{array}$ $\begin{array}{r}(assumed:S_{He}=0.18;computed:a50/50\\ He-N2mixreadslowby1.69times)\end{array}$
Cold-cathode and X-ray limit	$I^{+}=K{p}^m,m\approx 1\text{to}1.2$	$I_X\approx S{I}^{-}{p}_X$	$K,m:Penningormagnetronconstants$ $\begin{array}{r}{I}_X:photoelectroncurrentfromsoftX-rays\\ atthecollector\end{array}$ $\begin{array}{r}{p}_X:X-raylimit;relations(background\\ knowledge,O^{\prime }Hanlon);Joustenlecture:\\ Penning1\times 10^{-7}to1\mathrm{Pa},magnetron1\times 10^{-8}to1\\ \mathrm{Pa};Bayard-Alpertthincollectorreducing\\ p_X(backgroundknowledge)\end{array}$ $\begin{array}{r}(myreading:below{p}_{X}theioncurrentno\\ longerfallswithpressure,sothegauge\\ readsafloorratherthanthetruevalue)\end{array}$

Important Questions

Ques: Why are several gauge types needed?

Because no single principle covers the whole range. The Jousten lecture lists capacitance diaphragm gauges from $1\times 10^{-4}$ to $1\times 10^5$ $\mathrm{Pa}$, Pirani from $1\times 10^{-1}$ to $1\times 10^4$ $\mathrm{Pa}$, spinning rotor gauges from $1\times 10^{-5}$ to 10 $\mathrm{Pa}$, cold-cathode gauges down to $1\times 10^{-7}$ or $1\times 10^{-8}$ $\mathrm{Pa}$ and emission-cathode ionization gauges from $1\times 10^{-10}$ to $1\times 10^{-2}$ $\mathrm{Pa}$. Only force-per-area gauges such as the diaphragm measure true total pressure independent of gas, and Jousten notes that above 100 $\mathrm{Pa}$ only these do so without transpiration corrections. The others infer pressure from heat conduction, drag or ionization, which depend on the gas. Normal uncertainties range from 0.3 per cent for good diaphragm gauges to 100 per cent for ionization gauges, and errors above 100 per cent are possible. Reference: K. Jousten (PTB), ‘Vacuum gauges for the fine and high vacuum’, CERN Accelerator School, Platja d’Aro (2006) (fetch-tool summary; some equations garbled).

Ques: Can a capacitance manometer read $1\times 10^{-4}$ Pa, or only $1\times 10^{-4}$ mbar?

The sources differ by two decades. Jousten lists a capacitance diaphragm range down to $1\times 10^{-4}$ $\mathrm{Pa}$, which is $1\times 10^{-6}$ $\mathrm{mbar}$ (computed), while the Edwards note gives a practical lower limit of $1\times 10^{-4}$ $\mathrm{mbar}$ for direct gauges. Jousten’s 0.4 $\mathrm{nm}$ deflection sensitivity supports the lower figure: an assumed 10 mm radius, 25 $\mu \mathrm{m}$ INVAR membrane deflects 0.79 $\mu m/Pa$, so 0.4 $\mathrm{nm}$ is about $5\times 10^{-4}$ $\mathrm{Pa}$ (computed). However, his normal uncertainty at the low end is 10 per cent, and zero drift dominates there. The Edwards value is therefore more likely the practical limit for routine process gauges, while $1\times 10^{-4}$ $\mathrm{Pa}$ is the resolution of the most sensitive heads with thermostated housings (my reading). Reference: K. Jousten (PTB), ‘Vacuum gauges for the fine and high vacuum’, CERN Accelerator School, Platja d’Aro (2006) (fetch-tool summary; some equations garbled); Edwards Vacuum, ‘Calibration Factors for Vacuum Pressure Gauges’, application note 3601 2064 01 (2024) (fetch-tool summary); J. F. O’Hanlon, A User’s Guide to Vacuum Technology, 3rd ed., Wiley 2003; K. Jousten (ed.), Handbook of Vacuum Technology, Wiley-VCH 2016; J. K. Fremerey, J. Vac. Sci. Technol. A 3, 1715 (1985); S. Timoshenko and S. Woinowsky-Krieger, Theory of Plates and Shells, McGraw-Hill 1959 (background knowledge; not opened).

Ques: Why do ion gauges misread helium and hydrogen?

Because ionization probability depends on the gas. An ion gauge measures $I^{+}=S{I}^{-}p$ with a sensitivity calibrated for nitrogen, typically 10 to 25 $\mathrm{mba}{\mathrm{r}}^{-1}$ for Bayard-Alpert gauges (Edwards note). Helium ionizes far less readily: Edwards gives a relative sensitivity of 0.14 to 0.25 and hydrogen 0.36 to 0.60, against Hiden’s single values of 0.14 and 0.44 for an RGA ion source (RGA page). The true pressure is the indicated reading divided by the factor, so pure helium read on a nitrogen-calibrated gauge can be four to seven times higher than shown (computed). For mixtures, Edwards divides by the fraction-weighted sensitivity: an equal helium-nitrogen mix with an assumed factor of 0.18 reads low by 1.69 times (computed). Reference: Edwards Vacuum, ‘Calibration Factors for Vacuum Pressure Gauges’, application note 3601 2064 01 (2024) (fetch-tool summary); Residual Gas Analyzer (RGA) page of this vault.

Ques: How does a heated capacitance manometer cause transpiration error?

In the molecular flow regime, connected volumes at different temperatures reach different pressures, $p_1/p_2=\sqrt{T_1/T_2}$ (Jousten lecture). Process manometers are often heated to avoid condensation, so a gauge at 318 K reading a 296 K chamber is 1.036 times high, a 3.6 per cent error (computed), matching Jousten’s example. The correction applies when the mean free path is long compared with the connecting tube; at 1 $\mathrm{Pa}$ the nitrogen mean free path is 6.5 mm (assumed; computed), comparable to a tube, so the error is partial and must be calibrated (my reading). Jousten notes that below 100 $\mathrm{Pa}$ the transpiration effect must be considered even for force-per-area gauges. Reference: K. Jousten (PTB), ‘Vacuum gauges for the fine and high vacuum’, CERN Accelerator School, Platja d’Aro (2006) (fetch-tool summary; some equations garbled); J. F. O’Hanlon, A User’s Guide to Vacuum Technology, 3rd ed., Wiley 2003; K. Jousten (ed.), Handbook of Vacuum Technology, Wiley-VCH 2016; J. K. Fremerey, J. Vac. Sci. Technol. A 3, 1715 (1985); S. Timoshenko and S. Woinowsky-Krieger, Theory of Plates and Shells, McGraw-Hill 1959 (background knowledge; not opened).

Ques: Why is the spinning rotor gauge a transfer standard?

Because its reading depends on known kinetic theory rather than gas ionization. Molecules striking a levitated steel ball slow its rotation, and pressure is proportional to the relative deceleration through $p=(\pi \rho d\bar{c}/20\sigma )(-\dot{\omega }/\omega )$ (background knowledge, Fremerey). For an assumed 4.5 mm steel ball in nitrogen at 296 K, $\bar{c}$ is 473 m/s and the deceleration rate at $1\times 10^{-3}$ $\mathrm{Pa}$ is $3.9\times 10^{-7}$ per second (computed). Jousten’s residual-drag data of about $2\times 10^{-7}$ therefore correspond to about $5\times 10^{-4}$ $\mathrm{Pa}$, which must be subtracted (computed). He cites no gas consumption, no dissociation, low outgassing and high long-term stability, with best uncertainties of 0.4 to 0.6 per cent near $1\times 10^{-3}$ to $1\times 10^{-1}$ $\mathrm{Pa}$. Reference: J. F. O’Hanlon, A User’s Guide to Vacuum Technology, 3rd ed., Wiley 2003; K. Jousten (ed.), Handbook of Vacuum Technology, Wiley-VCH 2016; J. K. Fremerey, J. Vac. Sci. Technol. A 3, 1715 (1985); S. Timoshenko and S. Woinowsky-Krieger, Theory of Plates and Shells, McGraw-Hill 1959 (background knowledge; not opened); K. Jousten (PTB), ‘Vacuum gauges for the fine and high vacuum’, CERN Accelerator School, Platja d’Aro (2006) (fetch-tool summary; some equations garbled).

Ques: What limits a Pirani gauge?

Heat conduction stops depending on pressure at both ends of its range. The wire or MEMS sheet loses power to radiation, its supports and the gas, and only the gas part carries pressure information (Jousten lecture). At low pressure gas conduction is proportional to pressure but small against fixed losses; at high pressure the mean free path becomes short compared with the sensor dimension and conduction saturates. For an assumed 25 $\mu \mathrm{m}$ dimension, the Knudsen number is 262 at 1 $\mathrm{Pa}$ and 0.026 at $1\times 10^4$ $\mathrm{Pa}$ (computed). Jousten notes that MEMS designs at 60 ${}^{\circ }\mathrm{C}$ have higher Knudsen number and avoid convection. Stability varies: his repeat calibrations range from 0.03 to 3.30 per cent between models. Reference: K. Jousten (PTB), ‘Vacuum gauges for the fine and high vacuum’, CERN Accelerator School, Platja d’Aro (2006) (fetch-tool summary; some equations garbled); J. F. O’Hanlon, A User’s Guide to Vacuum Technology, 3rd ed., Wiley 2003; K. Jousten (ed.), Handbook of Vacuum Technology, Wiley-VCH 2016; J. K. Fremerey, J. Vac. Sci. Technol. A 3, 1715 (1985); S. Timoshenko and S. Woinowsky-Krieger, Theory of Plates and Shells, McGraw-Hill 1959 (background knowledge; not opened).

References

1. K. Jousten, “Vacuum gauges for the fine and high vacuum,” CERN Accelerator School on Vacuum, Platja d’Aro, 2006. Available: https://cas.web.cern.ch/sites/default/files/lectures/platjadaro-2006/jousten-1.pdf
2. Edwards Vacuum, “Calibration Factors for Vacuum Pressure Gauges,” application note 3601 2064 01, 2024. Available: https://www.edwardsvacuum.com/content/dam/brands/edwards-vacuum/general-vacuum/gated-downloads/application-notes/3601-2064-01_Calibration-factors-for-vacuum-pressure-gauges.pdf
3. J. F. O’Hanlon, A User’s Guide to Vacuum Technology, 3rd ed., Wiley, 2003; K. Jousten (ed.), Handbook of Vacuum Technology, Wiley-VCH, 2016; J. K. Fremerey, “Spinning rotor vacuum gauges,” J. Vac. Sci. Technol. A 3, 1715 (1985); S. Timoshenko and S. Woinowsky-Krieger, Theory of Plates and Shells, McGraw-Hill, 1959 (cited from background knowledge).
4. Residual Gas Analyzer (RGA) page of this vault.

Related pages in this vault: Residual Gas Analyzer (RGA), Langmuir Probe, Quartz Crystal Microbalance (QCM), Optical Emission Spectroscopy Endpoint (OES), Electrostatics and Capacitance, Thermal and Electro-Thermal Simulation, Reflection High-Energy Electron Diffraction (RHEED), Calibration and Traceability, Thermal Evaporator.