# Knudsen effusion mass spectrometry: Current and future approaches

Nathan Jacobson, Jean-yves Colle, Valentina Stolyarova, Torsten Markus, Ioana Nuta

► To cite this version:

Nathan Jacobson, Jean-yves Colle, Valentina Stolyarova, Torsten Markus, Ioana Nuta. Knudsen effusion mass spectrometry: Current and future approaches. Rapid Communications in Mass Spectrometry, 2024, 38 (14), pp.e9744. 10.1002/rcm.9744. hal-04602114

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<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
# Knudsen effusion mass spectrometry: Current and future approaches

Nathan Jacobson, Jean-yves Colle, Valentina Stolyarova, Torsten Markus, Ioana Nuta

► To cite this version:

Nathan Jacobson, Jean-yves Colle, Valentina Stolyarova, Torsten Markus, Ioana Nuta. Knudsen effusion mass spectrometry: Current and future approaches. Rapid Communications in Mass Spectrometry, 2024, 38 (14), pp.e9744. 10.1002/rcm.9744. hal-04602114

HAL Id: hal-04602114

https://hal.science/hal-04602114v1

Submitted on 5 Jun 2024

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
REVIEW ARTICLE

<figure><img src="image_2.png" /></figure>

Rapid
Communications in
Mass Spectrometry

WILEY

# Knudsen effusion mass spectrometry: Current and future approaches

Nathan S. Jacobson<fnref n="1" /> | Jean-Yves Colle<fnref n="2" />

Jean-Yves Colle<fnref n="2" /> | Valentina Stolyarova<fnref n="3" />,<fnref n="4" />

Torsten Markus<fnref n="5" /> | Ioana Nuta<fnref n="6" />

**Correspondence**

N. S. Jacobson, Material and Structures Division, NASA Glenn Research Center/HX5, LLC, Cleveland, OH 44135, USA.
Email: njacobson248@gmail.com

Jean-Yves Colle, European Commission, Joint Research Centre (JRC), Karlsruhe, Germany.
Email: jean-yves.colle@ec.europa.eu

**Funding information**

NASA Transformative Tools and Technologies

**Rationale:** Knudsen effusion mass spectrometry (KEMS) has been a powerful tool in physical chemistry since 1954. There are many excellent reviews of the basic principles of KEMS in the literature. In this review, we focus on the current status and potential growth areas for this instrumental technique.

**Methods:** We discuss (1) instrumentation, (2) measurement techniques, and (3) selected novel applications of the technique. Improved heating methods and temperature measurement allow for better control of the Knudsen cell effusive source. Accurate computer models of the effusive beam and its introduction to the ionizer allow optimization of such parameters as sensitivity and removal of background signals. Computer models of the ionizer allow for optimized sensitivity and resolution. Additionally, data acquisition systems specifically tailored to a KEMS system permit improved quantity and quality of data.

**Results:** KEMS is traditionally utilized for thermodynamic measurements of pure compounds and solutions. These measurements can now be strengthened using first principles and model-based computational thermochemistry. First principles can be used to calculate accurate Gibbs energy functions (gefs) for improving third law calculations. Calculated enthalpies of formation and dissociation energies from ab initio methods can be compared to those measured using KEMS. For model-based thermochemistry, solution parameters can be derived from measured thermochemical data on metallic and nonmetallic solutions. Beyond thermodynamic measurements, KEMS has been used for many specific applications. We select examples for discussion: measurements of phase changes, measurement/control of low-oxygen potential systems, thermochemistry of ultrahigh-temperature ceramics, geological applications, nuclear applications, applications to organic and organometallic compounds, and thermochemistry of functional room temperature materials, such as lithium ion batteries.

**Conclusions:** We present an overview of the current status of KEMS and discuss ideas for improving KEMS instrumentation and measurements. We discuss selected KEMS studies to illustrate future directions of KEMS.

This is an open access article under the terms of the [Creative Commons Attribution-NonCommercial-NoDerivs](https://creativecommons.org/licenses/by-nc-nd/4.0/) License, which permits use and distribution in any medium, provided the original work is properly cited, the use is non-commercial and no modifications or adaptations are made.

© 2024 The Authors. Rapid Communications in Mass Spectrometry published by John Wiley & Sons Ltd. This article has been contributed to by U.S. Government employees and their work is in the public domain in the USA.

<footnote n="1">¹Material and Structures Division, NASA Glenn Research Center/HX5, LLC, Cleveland, Ohio, USA</footnote>

<footnote n="2">²European Commission, Joint Research Centre (JRC), Karlsruhe, Germany</footnote>

<footnote n="3">³Saint Petersburg State University, Saint Petersburg, Russian Federation</footnote>

<footnote n="4">⁴Institute of Silicate Chemistry of the Russian Academy of Sciences, Saint Petersburg, Russian Federation</footnote>

<footnote n="5">⁵Institute for Materials Science and Engineering, Mannheim University of Applied Sciences, Mannheim, Germany</footnote>

<footnote n="6">⁶University Grenoble Alpes, CNRS, Grenoble INP-SIMaP, Grenoble, France</footnote>
REVIEW ARTICLE

<figure><img src="image_2.png" /></figure>

WILEY

# Knudsen effusion mass spectrometry: Current and future approaches

Nathan S. Jacobson<fnref n="1" /> | Jean-Yves Colle<fnref n="2" />

Valentina Stolyarova<fnref n="3" />,<fnref n="4" />

Torsten Markus<fnref n="5" /> | Ioana Nuta<fnref n="6" />

**Correspondence**

N. S. Jacobson, Material and Structures Division, NASA Glenn Research Center/HX5, LLC, Cleveland, OH 44135, USA.
Email: njacobson248@gmail.com

Jean-Yves Colle, European Commission, Joint Research Centre (JRC), Karlsruhe, Germany.
Email: jean-yves.colle@ec.europa.eu

**Funding information**

NASA Transformative Tools and Technologies

**Rationale:** Knudsen effusion mass spectrometry (KEMS) has been a powerful tool in physical chemistry since 1954. There are many excellent reviews of the basic principles of KEMS in the literature. In this review, we focus on the current status and potential growth areas for this instrumental technique.

**Methods:** We discuss (1) instrumentation, (2) measurement techniques, and (3) selected novel applications of the technique. Improved heating methods and temperature measurement allow for better control of the Knudsen cell effusive source. Accurate computer models of the effusive beam and its introduction to the ionizer allow optimization of such parameters as sensitivity and removal of background signals. Computer models of the ionizer allow for optimized sensitivity and resolution. Additionally, data acquisition systems specifically tailored to a KEMS system permit improved quantity and quality of data.

**Results:** KEMS is traditionally utilized for thermodynamic measurements of pure compounds and solutions. These measurements can now be strengthened using first principles and model-based computational thermochemistry. First principles can be used to calculate accurate Gibbs energy functions (gefs) for improving third law calculations. Calculated enthalpies of formation and dissociation energies from ab initio methods can be compared to those measured using KEMS. For model-based thermochemistry, solution parameters can be derived from measured thermochemical data on metallic and nonmetallic solutions. Beyond thermodynamic measurements, KEMS has been used for many specific applications. We select examples for discussion: measurements of phase changes, measurement/control of low-oxygen potential systems, thermochemistry of ultrahigh-temperature ceramics, geological applications, nuclear applications, applications to organic and organometallic compounds, and thermochemistry of functional room temperature materials, such as lithium ion batteries.

**Conclusions:** We present an overview of the current status of KEMS and discuss ideas for improving KEMS instrumentation and measurements. We discuss selected KEMS studies to illustrate future directions of KEMS.

This is an open access article under the terms of the [Creative Commons Attribution-NonCommercial-NoDerivs](https://creativecommons.org/licenses/by-nc-nd/4.0/) License, which permits use and distribution in any medium, provided the original work is properly cited, the use is non-commercial and no modifications or adaptations are made.

© 2024 The Authors. Rapid Communications in Mass Spectrometry published by John Wiley & Sons Ltd. This article has been contributed to by U.S. Government employees and their work is in the public domain in the USA.

<footnote n="1">¹Material and Structures Division, NASA Glenn Research Center/HX5, LLC, Cleveland, Ohio, USA</footnote>

<footnote n="2">²European Commission, Joint Research Centre (JRC), Karlsruhe, Germany</footnote>

<footnote n="3">³Saint Petersburg State University, Saint Petersburg, Russian Federation</footnote>

<footnote n="4">⁴Institute of Silicate Chemistry of the Russian Academy of Sciences, Saint Petersburg, Russian Federation</footnote>

<footnote n="5">⁵Institute for Materials Science and Engineering, Mannheim University of Applied Sciences, Mannheim, Germany</footnote>

<footnote n="6">⁶University Grenoble Alpes, CNRS, Grenoble INP-SIMaP, Grenoble, France</footnote>
<figure><img src="image_3.png" /></figure>

# 1 | INTRODUCTION

Knudsen effusion mass spectrometry (KEMS) has long been an important experimental tool in physical chemistry. The Knudsen cell is a small enclosure in which a near-condensed phase/vapor equilibria is established. A well-defined orifice directs a molecular beam (nearly collision free, directed flow), representative of the vapor at low pressures, to a mass spectrometer operating at pressures near the ultrahigh vacuum (UHV) range. This technique has found numerous applications in thermochemistry, geology, metallurgy, ceramics, nuclear materials, and a range of other fields.¹⁻¹¹ Although the use of this technique peaked in the 1960s-1970s, the need for KEMS data continues today. There are currently about 10 active groups using this method. At the time of writing, the authors are aware of several groups in Europe, several in Russia, one in the United States, and one in India.

There are many excellent reviews on the KEMS technique¹⁻¹¹ that provide the relevant background and explain the development of the technique. We shall not repeat such information here but rather focus on new and fundamental developments in the field and the importance of the technique to current and future research in the physical sciences.

We first discuss instrumentation in a typical KEMS setup. We pay particular attention to the advances in sample introduction, which include accurate positioning of the Knudsen cell, precise temperature measurements, attaining equilibrium in the cell, and molecular beam collimation. Advances in the actual instrumentation include computer simulation of the molecular beam into the ionizer and the actual ionizer, which allow optimization of the instrument geometry and tuning of the ionizer for improved sensitivity and resolution. Overall system performance is also improved with advanced data acquisition systems, which allow better instrument control and signal averaging/signal collection.

Next, we discuss actual measurements obtained using KEMS, highlighting several areas for potential growth of this technique. Traditionally, KEMS has been used for precision thermochemical measurements of both pure compounds and solutions. We review these measurements as well as the methods used to make them. An exciting growth area for KEMS is the integration of these methods with computational chemistry. Computational chemistry can be divided into two categories: first principles-based calculations (which are most often applied to gaseous species) and model-based solution calculations (which are most often applied to complex condensed-phase systems with many components). Bond energy measurements using KEMS can verify or refute ab initio calculations. Gibbs energy functions (gefs) derived from first principles allow more accurate KEMS determinations of the enthalpy of formation via the third law. In regard to model-based computations of solutions, the KEMS measurements can directly provide the constants for the fitted polynomials.

As previously noted, KEMS has been used in such diverse fields as thermochemistry, geology, ceramics, metallurgy, nuclear materials, and more recently organic chemistry. We discuss some selected areas

that are currently of interest. These include phase change measurements, control/measurement in processes with low P(O₂), thermochemistry of ultrahigh-temperature materials, applications to geochemistry, cosmochemistry, nuclear applications, thermodynamics of organic and organometallic compounds, and thermodynamics of lithium ion batteries.

# 2 | INSTRUMENTATION

Figure 1 shows the basic layout of a KEMS instrument. We shall discuss each component as it relates to the unique case of effusate vapor sampling from a Knudsen cell.

1. Vapor source (Knudsen cell) and furnace.

2. Molecular beam sampling and shutter design.

3. Ionizer and subsequent ion focusing plate assembly. Due to the high temperatures, electron impact is nearly universally used in KEMS. The classic design has remained unchanged for many years. However, as will be discussed, there is now room for much improvement in the ionizer and focusing plate assembly.

4. Ion sorter/mass filter—could be magnetic, electrostatic, or time of flight. Generally, a magnetic sector is preferred due to its high resolution and lack of mass discrimination. However, modern quadrupole mass filters exhibit much better resolution and limited mass discrimination compared to their early counterparts. Most of the active KEMS groups use magnetic sector instruments; however, there are several using quadrupole instruments.

5. Detector—involves an electron multiplier with discrete dynodes or a continuous dynode channeltron. Both types are typically used with a Faraday cup for comparison to the raw signal or in the case of high signals when using pulse counting.

6. Vacuum system—great strides have been made in vacuum technology in recent years. It is possible to build an entirely oil-free system at a reasonable price now. A clean vacuum system leads to less background peaks, lower intensities for the background peaks that remain, and clarity in identifying signals from the Knudsen cell.

The heart of any KEMS machine is the Knudsen cell and furnace, which generates the vapor to examine. We shall discuss this first.

## 2.1 | Knudsen cell and furnace flange

### 2.1.1 | Knudsen cells

As described in other publications,³,¹² the Knudsen cell is a small enclosure in which, ideally, a condensed phase equilibrates with its vapor. A small, well-defined orifice allows a representative small portion of the vapor to escape under vacuum as a molecular beam that is directed into the mass spectrometer. The required condition is that the escaping vapor does not perturb the vapor phase/condensed
<figure><img src="image_3.png" /></figure>

# 1 | INTRODUCTION

Knudsen effusion mass spectrometry (KEMS) has long been an important experimental tool in physical chemistry. The Knudsen cell is a small enclosure in which a near-condensed phase/vapor equilibria is established. A well-defined orifice directs a molecular beam (nearly collision free, directed flow), representative of the vapor at low pressures, to a mass spectrometer operating at pressures near the ultrahigh vacuum (UHV) range. This technique has found numerous applications in thermochemistry, geology, metallurgy, ceramics, nuclear materials, and a range of other fields.¹⁻¹¹ Although the use of this technique peaked in the 1960s-1970s, the need for KEMS data continues today. There are currently about 10 active groups using this method. At the time of writing, the authors are aware of several groups in Europe, several in Russia, one in the United States, and one in India.

There are many excellent reviews on the KEMS technique¹⁻¹¹ that provide the relevant background and explain the development of the technique. We shall not repeat such information here but rather focus on new and fundamental developments in the field and the importance of the technique to current and future research in the physical sciences.

We first discuss instrumentation in a typical KEMS setup. We pay particular attention to the advances in sample introduction, which include accurate positioning of the Knudsen cell, precise temperature measurements, attaining equilibrium in the cell, and molecular beam collimation. Advances in the actual instrumentation include computer simulation of the molecular beam into the ionizer and the actual ionizer, which allow optimization of the instrument geometry and tuning of the ionizer for improved sensitivity and resolution. Overall system performance is also improved with advanced data acquisition systems, which allow better instrument control and signal averaging/signal collection.

Next, we discuss actual measurements obtained using KEMS, highlighting several areas for potential growth of this technique. Traditionally, KEMS has been used for precision thermochemical measurements of both pure compounds and solutions. We review these measurements as well as the methods used to make them. An exciting growth area for KEMS is the integration of these methods with computational chemistry. Computational chemistry can be divided into two categories: first principles-based calculations (which are most often applied to gaseous species) and model-based solution calculations (which are most often applied to complex condensed-phase systems with many components). Bond energy measurements using KEMS can verify or refute ab initio calculations. Gibbs energy functions (gefs) derived from first principles allow more accurate KEMS determinations of the enthalpy of formation via the third law. In regard to model-based computations of solutions, the KEMS measurements can directly provide the constants for the fitted polynomials.

As previously noted, KEMS has been used in such diverse fields as thermochemistry, geology, ceramics, metallurgy, nuclear materials, and more recently organic chemistry. We discuss some selected areas

that are currently of interest. These include phase change measurements, control/measurement in processes with low P(O₂), thermochemistry of ultrahigh-temperature materials, applications to geochemistry, cosmochemistry, nuclear applications, thermodynamics of organic and organometallic compounds, and thermodynamics of lithium ion batteries.

# 2 | INSTRUMENTATION

Figure 1 shows the basic layout of a KEMS instrument. We shall discuss each component as it relates to the unique case of effusate vapor sampling from a Knudsen cell.

1. Vapor source (Knudsen cell) and furnace.

2. Molecular beam sampling and shutter design.

3. Ionizer and subsequent ion focusing plate assembly. Due to the high temperatures, electron impact is nearly universally used in KEMS. The classic design has remained unchanged for many years. However, as will be discussed, there is now room for much improvement in the ionizer and focusing plate assembly.

4. Ion sorter/mass filter—could be magnetic, electrostatic, or time of flight. Generally, a magnetic sector is preferred due to its high resolution and lack of mass discrimination. However, modern quadrupole mass filters exhibit much better resolution and limited mass discrimination compared to their early counterparts. Most of the active KEMS groups use magnetic sector instruments; however, there are several using quadrupole instruments.

5. Detector—involves an electron multiplier with discrete dynodes or a continuous dynode channeltron. Both types are typically used with a Faraday cup for comparison to the raw signal or in the case of high signals when using pulse counting.

6. Vacuum system—great strides have been made in vacuum technology in recent years. It is possible to build an entirely oil-free system at a reasonable price now. A clean vacuum system leads to less background peaks, lower intensities for the background peaks that remain, and clarity in identifying signals from the Knudsen cell.

The heart of any KEMS machine is the Knudsen cell and furnace, which generates the vapor to examine. We shall discuss this first.

## 2.1 | Knudsen cell and furnace flange

### 2.1.1 | Knudsen cells

As described in other publications,³,¹² the Knudsen cell is a small enclosure in which, ideally, a condensed phase equilibrates with its vapor. A small, well-defined orifice allows a representative small portion of the vapor to escape under vacuum as a molecular beam that is directed into the mass spectrometer. The required condition is that the escaping vapor does not perturb the vapor phase/condensed
<figure><img src="image_4.png" /><figcaption>FIGURE 1 NASA Glenn magnetic sector KEMS (Knudsen effusion mass spectrometry) instrument. [Color figure can be viewed at wileyonlinelibrary.com]</figcaption></figure>

phase equilibrium in the cell. This is generally indicated by the relationship:

$$BW_B/S \le 10^{-2} \qquad (1)$$

Here B is the surface area of the cell orifice; W<sub>B</sub> is the Clausing or transmission factor for the cell orifice,<sup>13</sup> which will be discussed later; and S is the vaporizing surface area of the condensed phase sample that may be often reduced to the cell section.

Different types of cells are shown in Figure 2. The cell should ideally be chemically inert to the contained condensed phase and the vapor; however, in practice there is likely to be some reaction, particularly at high temperatures. For some sample/cell systems, material interactions above a certain temperature impose a use limit. A wide range of cell materials have been used, including refractory metals, refractory oxides, graphite, and various ceramics. Improved processing and machining methods allow the possibility of more exotic cell materials and cell liners. Sample/cell material chemical reactions are the most obvious concern, but structural issues may also be a problem. For example, liquid Au in Al<sub>2</sub>O<sub>3</sub> appears to penetrate only grain boundaries, and the system performs well for several runs until the cell walls are weakened. Another approach is to use the cell as a reactant. For example, a Mo cell has been used to study Mo (powder) mixed with rare-earth silicates.<sup>14</sup> Additionally, a Hf liner has been used to study the reaction of Hf and HfO<sub>2</sub>.<sup>15</sup> Silicon in dense graphite forms a compact SiC layer inside the cell, which limits further reaction.<sup>16</sup>

There are many variations in the simple cell in Figure 2A,B. These include a gas-leak cell (Figure 2C),<sup>17</sup> a multiple cell (Figure 2D),<sup>18,19</sup> an electrochemical cell (Figure 2E),<sup>20</sup> and a tandem cell with a cracker (Figure 2F).<sup>21</sup> The gas-leak cell allows another vapor or gas to be introduced to the cell and reacted with the contained sample, and the resulting gas species can be studied. The multiple cell consists of two or more cells in an isothermal block that can be translated in and out of the sampling region to compare a measured ion intensity in a sample under study to a reference ion intensity in a short time period; it is an instantaneous calibration. This will be discussed in detail later in this paper. An electrochemical cell allows the chemical potential of a species (e.g., O<sub>2</sub> or S<sub>2</sub>) to be fixed above the sample.<sup>20</sup> The tandem cell with a cracker is a specially designed cell for studying reactions of air-sensitive organometallics. This will be discussed later in this paper.

## 2.1.2 | Furnace and temperature measurements

A variety of methods have been used to heat Knudsen cells. Figure 3A,B shows resistance heating: Figure 3A shows the system at NASA Glenn, USA<sup>18</sup>; Figure 3B shows the system at SIMaP, France<sup>19</sup>; Figure 3C shows electron bombardment at Saint Petersburg State University, Russia; and Figure 3D shows laser heating at JRC Karlsruhe, Germany. It is essential that the cell is heated uniformly-thermodynamic measurements require a constant temperature throughout the cell. The challenge is to design a compact furnace with a tight heat shield pack, in which the cell orifice is as close as possible to the ionizer.
<figure><img src="image_4.png" /><figcaption>FIGURE 1 NASA Glenn magnetic sector KEMS (Knudsen effusion mass spectrometry) instrument. [Color figure can be viewed at wileyonlinelibrary.com]</figcaption></figure>

phase equilibrium in the cell. This is generally indicated by the relationship:

$$BW_B/S \le 10^{-2} \qquad (1)$$

Here B is the surface area of the cell orifice; W<sub>B</sub> is the Clausing or transmission factor for the cell orifice,<sup>13</sup> which will be discussed later; and S is the vaporizing surface area of the condensed phase sample that may be often reduced to the cell section.

Different types of cells are shown in Figure 2. The cell should ideally be chemically inert to the contained condensed phase and the vapor; however, in practice there is likely to be some reaction, particularly at high temperatures. For some sample/cell systems, material interactions above a certain temperature impose a use limit. A wide range of cell materials have been used, including refractory metals, refractory oxides, graphite, and various ceramics. Improved processing and machining methods allow the possibility of more exotic cell materials and cell liners. Sample/cell material chemical reactions are the most obvious concern, but structural issues may also be a problem. For example, liquid Au in Al<sub>2</sub>O<sub>3</sub> appears to penetrate only grain boundaries, and the system performs well for several runs until the cell walls are weakened. Another approach is to use the cell as a reactant. For example, a Mo cell has been used to study Mo (powder) mixed with rare-earth silicates.<sup>14</sup> Additionally, a Hf liner has been used to study the reaction of Hf and HfO<sub>2</sub>.<sup>15</sup> Silicon in dense graphite forms a compact SiC layer inside the cell, which limits further reaction.<sup>16</sup>

There are many variations in the simple cell in Figure 2A,B. These include a gas-leak cell (Figure 2C),<sup>17</sup> a multiple cell (Figure 2D),<sup>18,19</sup> an electrochemical cell (Figure 2E),<sup>20</sup> and a tandem cell with a cracker (Figure 2F).<sup>21</sup> The gas-leak cell allows another vapor or gas to be introduced to the cell and reacted with the contained sample, and the resulting gas species can be studied. The multiple cell consists of two or more cells in an isothermal block that can be translated in and out of the sampling region to compare a measured ion intensity in a sample under study to a reference ion intensity in a short time period; it is an instantaneous calibration. This will be discussed in detail later in this paper. An electrochemical cell allows the chemical potential of a species (e.g., O<sub>2</sub> or S<sub>2</sub>) to be fixed above the sample.<sup>20</sup> The tandem cell with a cracker is a specially designed cell for studying reactions of air-sensitive organometallics. This will be discussed later in this paper.

## 2.1.2 | Furnace and temperature measurements

A variety of methods have been used to heat Knudsen cells. Figure 3A,B shows resistance heating: Figure 3A shows the system at NASA Glenn, USA<sup>18</sup>; Figure 3B shows the system at SIMaP, France<sup>19</sup>; Figure 3C shows electron bombardment at Saint Petersburg State University, Russia; and Figure 3D shows laser heating at JRC Karlsruhe, Germany. It is essential that the cell is heated uniformly-thermodynamic measurements require a constant temperature throughout the cell. The challenge is to design a compact furnace with a tight heat shield pack, in which the cell orifice is as close as possible to the ionizer.
<figure><img src="image_5.png" /></figure>

<figure><img src="image_6.png" /><figcaption>**FIGURE 2** Types of Knudsen cells. (A) Basic Knudsen cell. (B) Knudsen cell with isothermal metal jacket. (C) Gas-leak Knudsen cell where the external gases admitted to the cell may be from either a secondary cell or an external connection to a permanent gas. Source: Adapted from Meschi et al.¹⁷ with permission from AIP Publishing. (D) Multiple cell with two or more cells in an isothermal block. (E) Electrochemical cell where an external chemical potential (e.g., O₂, S₂) is imposed on the sample under study. Source: Reprinted from Hilpert⁷ with permission from John Wiley and Sons. (F) Tandem cell with a cracker. Source: Reprinted with permission from Violet et al.²¹ with permission from John Wiley and Sons. [Color figure can be viewed at wileyonlinelibrary.com]</figcaption></figure>

Resistance heating requires large power feedthroughs to handle the high currents needed to obtain the high temperatures; electron bombardment requires less current as only the cell is heated, but arcing from the high voltage may be a problem. Newer designs utilize laser heating of the Knudsen cell.²² Figure 3B and 3D show that the furnace is surrounded by a tight heat shield pack (Ta, Mo, or W). Contamination of the effusing molecular beam from the trapped vapor in the heat shield cavity is an important issue, and corrections must be applied.²³ The restricted collimation scheme, which will be described, helps minimize this effect.

*The importance of accurate temperature measurements to the KEMS method cannot be emphasized enough.* Because vapor pressure is an

exponential function of temperature, small errors in temperature can lead to large errors in vapor pressure and resultant thermodynamic parameters. Cater²⁴ describes the significant errors that can result from small temperature variations. Thermocouple and pyrometer measurements are the two most common temperature measurement methods used with KEMS. Both methods offer a high degree of accuracy when used properly and have advanced in recent years.

A thermocouple must be firmly attached to a Knudsen cell with clear contact with the cell's surface. This can be accomplished using a metal strap either placed to the side of the cell or clamped in a well at the bottom of the cell. It is easily connected to a data collection system, which automatically adds corrections for the “cold” junction.
<figure><img src="image_5.png" /></figure>

<figure><img src="image_6.png" /><figcaption>**FIGURE 2** Types of Knudsen cells. (A) Basic Knudsen cell. (B) Knudsen cell with isothermal metal jacket. (C) Gas-leak Knudsen cell where the external gases admitted to the cell may be from either a secondary cell or an external connection to a permanent gas. Source: Adapted from Meschi et al.¹⁷ with permission from AIP Publishing. (D) Multiple cell with two or more cells in an isothermal block. (E) Electrochemical cell where an external chemical potential (e.g., O₂, S₂) is imposed on the sample under study. Source: Reprinted from Hilpert⁷ with permission from John Wiley and Sons. (F) Tandem cell with a cracker. Source: Reprinted with permission from Violet et al.²¹ with permission from John Wiley and Sons. [Color figure can be viewed at wileyonlinelibrary.com]</figcaption></figure>

Resistance heating requires large power feedthroughs to handle the high currents needed to obtain the high temperatures; electron bombardment requires less current as only the cell is heated, but arcing from the high voltage may be a problem. Newer designs utilize laser heating of the Knudsen cell.²² Figure 3B and 3D show that the furnace is surrounded by a tight heat shield pack (Ta, Mo, or W). Contamination of the effusing molecular beam from the trapped vapor in the heat shield cavity is an important issue, and corrections must be applied.²³ The restricted collimation scheme, which will be described, helps minimize this effect.

*The importance of accurate temperature measurements to the KEMS method cannot be emphasized enough.* Because vapor pressure is an

exponential function of temperature, small errors in temperature can lead to large errors in vapor pressure and resultant thermodynamic parameters. Cater²⁴ describes the significant errors that can result from small temperature variations. Thermocouple and pyrometer measurements are the two most common temperature measurement methods used with KEMS. Both methods offer a high degree of accuracy when used properly and have advanced in recent years.

A thermocouple must be firmly attached to a Knudsen cell with clear contact with the cell's surface. This can be accomplished using a metal strap either placed to the side of the cell or clamped in a well at the bottom of the cell. It is easily connected to a data collection system, which automatically adds corrections for the “cold” junction.
<figure><img src="image_7.png" /></figure>

<figure><img src="image_8.png" /></figure>

<figure><img src="image_9.png" /></figure>

<figure><img src="image_10.png" /></figure>

<figure><img src="image_11.png" /><figcaption>**FIGURE 3** Knudsen cell heating configurations: (A) resistance heating with helix element, (B) resistance heating with sheet element, (C) electron bombardment heating, and (D) laser heating at JRC Karlsruhe. [Color figure can be viewed at wileyonlinelibrary.com]</figcaption></figure>

The drawbacks of a thermocouple are that it draws heat away from the cell, so its location adjacent to the sample and in the homogeneous temperature portion of the sample is critical. Further, thermocouples can easily be degraded by the vacuum and/or vapor environment after repetitive experiments.

Pyrometry is a noncontact temperature measurement. Sighting is generally on a dead-end black body hole drilled into the body (side or bottom) of the Knudsen cell, as shown in Figure 2A. This should have a length-to-diameter ratio ≥10. It may be necessary to put a shield around the hole to prevent reflected light from the heating element from interfering with the measurement. Note that due to the imperfections of optics and the presence of windows (vacuum interface), a phenomenon known as the size-of-source effect may appear.²⁵ This phenomenon has the effect that light slightly outside of the measurement field may partially be measured. To avoid this, it is preferable to have a measurement spot clearly smaller than the image of the black body hole. At NASA Glenn, we have used a tube around the hole to limit the effect of spurious light. This is particularly true in the case of Knudsen cell temperature measurement where the surface of the cell may appear brighter (reflection from heating element) than the black body hole.

Ideally, one could see into the Knudsen cell orifice, which is itself a black body. However, in practice, the vapor emerging from the cell

will interfere with the measurement and quickly deposit on the pyrometer window. Also, when temperature is measured on a liquid surface through the effusion orifice, the pyrometer records the brightness of the observed orifice. This results from the addition of two emissivities—that of the liquid in the effusion cell and that of the reflected image of the effusion orifice. If the surface of the liquid is concave, a darker image of the orifice is observed insider a brighter corona; this is the true temperature of the black body. However, if the surface of the liquid is convex, the darker image is larger than the effusion orifice, and the temperature measured will necessarily be lower. Errors of 30 K were noted in calibration runs at CEA-Saclay (S. Chatain and E. Lizon, private communication). Ackerman et al.,²⁶ Bhatt et al.,²⁷ and Balakrishnan et al.,²⁸ used the appearance of the dark spot to determine the liquidus temperature in metal/metal-carbide systems. Thus, it is best not to use the effusion orifice as a black body hole for temperature measurements.

Older pyrometers measured radiance/brightness using a “disappearing filament” where the operator would match the color of a filament in the pyrometer to that of the black body hole. Modern pyrometers measure the bright spot with a photon counting device. With a rigid mount, such an automatic device is suitable for repeated measurements and computer data acquisition. A critical issue with pyrometry is keeping the optical path clear and window clean.
<figure><img src="image_7.png" /></figure>

<figure><img src="image_8.png" /></figure>

<figure><img src="image_9.png" /></figure>

<figure><img src="image_10.png" /></figure>

<figure><img src="image_11.png" /><figcaption>**FIGURE 3** Knudsen cell heating configurations: (A) resistance heating with helix element, (B) resistance heating with sheet element, (C) electron bombardment heating, and (D) laser heating at JRC Karlsruhe. [Color figure can be viewed at wileyonlinelibrary.com]</figcaption></figure>

The drawbacks of a thermocouple are that it draws heat away from the cell, so its location adjacent to the sample and in the homogeneous temperature portion of the sample is critical. Further, thermocouples can easily be degraded by the vacuum and/or vapor environment after repetitive experiments.

Pyrometry is a noncontact temperature measurement. Sighting is generally on a dead-end black body hole drilled into the body (side or bottom) of the Knudsen cell, as shown in Figure 2A. This should have a length-to-diameter ratio ≥10. It may be necessary to put a shield around the hole to prevent reflected light from the heating element from interfering with the measurement. Note that due to the imperfections of optics and the presence of windows (vacuum interface), a phenomenon known as the size-of-source effect may appear.²⁵ This phenomenon has the effect that light slightly outside of the measurement field may partially be measured. To avoid this, it is preferable to have a measurement spot clearly smaller than the image of the black body hole. At NASA Glenn, we have used a tube around the hole to limit the effect of spurious light. This is particularly true in the case of Knudsen cell temperature measurement where the surface of the cell may appear brighter (reflection from heating element) than the black body hole.

Ideally, one could see into the Knudsen cell orifice, which is itself a black body. However, in practice, the vapor emerging from the cell

will interfere with the measurement and quickly deposit on the pyrometer window. Also, when temperature is measured on a liquid surface through the effusion orifice, the pyrometer records the brightness of the observed orifice. This results from the addition of two emissivities—that of the liquid in the effusion cell and that of the reflected image of the effusion orifice. If the surface of the liquid is concave, a darker image of the orifice is observed insider a brighter corona; this is the true temperature of the black body. However, if the surface of the liquid is convex, the darker image is larger than the effusion orifice, and the temperature measured will necessarily be lower. Errors of 30 K were noted in calibration runs at CEA-Saclay (S. Chatain and E. Lizon, private communication). Ackerman et al.,²⁶ Bhatt et al.,²⁷ and Balakrishnan et al.,²⁸ used the appearance of the dark spot to determine the liquidus temperature in metal/metal-carbide systems. Thus, it is best not to use the effusion orifice as a black body hole for temperature measurements.

Older pyrometers measured radiance/brightness using a “disappearing filament” where the operator would match the color of a filament in the pyrometer to that of the black body hole. Modern pyrometers measure the bright spot with a photon counting device. With a rigid mount, such an automatic device is suitable for repeated measurements and computer data acquisition. A critical issue with pyrometry is keeping the optical path clear and window clean.
Corrections for the window may be made using a separate calibration stand or an in situ calibration. It has been found that single-wavelength pyrometers work best for the small hole machined in a Knudsen cell; multiwavelength pyrometers require a larger sample emitting more energy. However, this requirement may be changing given the rapid development of commercial pyrometers and increasing cell size and sighting hole in improved mass spectrometers.

With both methods, frequent calibrations are a must. Generally, Ag, Au, or Pt melting point is used in appropriate cell materials where no eutectic forms. The first two are ITS-90 standards,²⁹ but container materials and compatibility are often not discussed. A typical melting point calibration for Au is shown in Figure 4, with a ramp rate of 1 K/min. These data were obtained from an automatic brightness pyrometer and during repeated scanning of the Au peak with signal averaging. The region with constant intensity versus temperature is the three-phase solid-liquid-gas invariant. The melting temperature can be derived from a mean value, when using heating and freezing ramps to evaluate the temperature lag between the thermocouple (or pyrometric cavity) and the sample. This time lag is rate dependent. With a slow ramp rate, the calibration point is generally taken as the onset of melting. It is best to use a consistent ramp rate for every calibration. It should be noted that there are no ITS melting point standards above that of Au, and various high-temperature eutectics have been proposed. The use of high-temperature eutectics as fixed points for calibration to high temperature is a growing area in metrology³⁰ with applications to KEMS.

Other methods of temperature calibration include the complete vaporization method, in which a small amount of a known vapor pressure standard (Au, Ag, and CaF₂) is vaporized and the net vapor flux is compared to a known value for different temperatures.⁵

### 2.1.3 | Attaining equilibrium in a Knudsen cell

It is critical that chemical equilibrium be attained in the cell for reliable thermodynamic measurements.³,³¹⁻³³ The immediate issue is the

presence of the orifice, which means that the system is not truly a closed system because molecules rebounding from cell lid collisions do not equal the vapor flux from the sample. Whitman³² and Motzfeld³³ have developed an expression to extrapolate to zero orifice size, based on balancing the upward and downward fluxes in a Knudsen cell; a good derivation is provided in the chapter by Paule and Margrave.³⁴ Extrapolation to zero orifice size is generally used in torsion effusion³⁵ and weight loss measurements³⁴ but has more recently been applied to KEMS.³⁶ The central equation is

$$P_m \left[ 1 + f \left( \frac{1}{\alpha_c} + \frac{1}{W_A} - 2 \right) \right] = \frac{\alpha_L}{\alpha_c} P_{eq} f = \frac{W_B B}{A} \quad (2)$$

In this equation, the subscripts A and B indicate the cell body and orifice cross sections, respectively, $P_m$ is the measured vapor pressure, $W_A$ is the transmission factor (Clausing coefficient between 0 and 1) for the cell body, $W_B$ is the transmission factor (Clausing coefficient between 0 and 1) for the orifice, $P_{eq}$ is the equilibrium pressure, $\alpha_c$ is the condensation coefficient (fraction of molecules that stick to the sample surface on striking), and $\alpha_L$ is the vaporization coefficient (fraction of molecules leaving the vaporizing surface toward the vapor compared to equilibrium flow). Transmission or Clausing factors represent the fraction of molecules that go through a given geometric opening; they may be calculated analytically¹³,³⁷,³⁸ or using a Monte Carlo method.³⁹ This will be discussed in detail later. Generally, the condensation coefficient is taken as equal to the vaporization coefficient; this is correct only for molecules that obey “micro-reversibility” (evaporation and condensation paths are the same) but can be taken as a reasonable assumption. Now apply Equation (2) to a cylindrical cell with the orifice on top and an evaporation coefficient of unity (no kinetic barrier to vaporization) and dimensions such that $f = 1/100$. Equation (2) then indicates that the measured equilibrium pressure is 0.99 of the equilibrium pressure, an error less than other uncertainties in KEMS.

A plot of $P_m$ versus $P_m \left[ f \left( \frac{1}{\alpha_c} + \frac{1}{W_A} - 2 \right) \right]$ should yield a line with intercept of $P_{eq}$. Furthermore, the vaporization coefficient, which

<figure><img src="image_12.png" /><figcaption>**FIGURE 4** Melting of gold temperature and pressure calibration. Ramp rate of 1 K/min (Copland and Jacobson¹⁸). [Color figure can be viewed at wileyonlinelibrary.com]</figcaption></figure>
Corrections for the window may be made using a separate calibration stand or an in situ calibration. It has been found that single-wavelength pyrometers work best for the small hole machined in a Knudsen cell; multiwavelength pyrometers require a larger sample emitting more energy. However, this requirement may be changing given the rapid development of commercial pyrometers and increasing cell size and sighting hole in improved mass spectrometers.

With both methods, frequent calibrations are a must. Generally, Ag, Au, or Pt melting point is used in appropriate cell materials where no eutectic forms. The first two are ITS-90 standards,²⁹ but container materials and compatibility are often not discussed. A typical melting point calibration for Au is shown in Figure 4, with a ramp rate of 1 K/min. These data were obtained from an automatic brightness pyrometer and during repeated scanning of the Au peak with signal averaging. The region with constant intensity versus temperature is the three-phase solid-liquid-gas invariant. The melting temperature can be derived from a mean value, when using heating and freezing ramps to evaluate the temperature lag between the thermocouple (or pyrometric cavity) and the sample. This time lag is rate dependent. With a slow ramp rate, the calibration point is generally taken as the onset of melting. It is best to use a consistent ramp rate for every calibration. It should be noted that there are no ITS melting point standards above that of Au, and various high-temperature eutectics have been proposed. The use of high-temperature eutectics as fixed points for calibration to high temperature is a growing area in metrology³⁰ with applications to KEMS.

Other methods of temperature calibration include the complete vaporization method, in which a small amount of a known vapor pressure standard (Au, Ag, and CaF₂) is vaporized and the net vapor flux is compared to a known value for different temperatures.⁵

### 2.1.3 | Attaining equilibrium in a Knudsen cell

It is critical that chemical equilibrium be attained in the cell for reliable thermodynamic measurements.³,³¹⁻³³ The immediate issue is the

presence of the orifice, which means that the system is not truly a closed system because molecules rebounding from cell lid collisions do not equal the vapor flux from the sample. Whitman³² and Motzfeld³³ have developed an expression to extrapolate to zero orifice size, based on balancing the upward and downward fluxes in a Knudsen cell; a good derivation is provided in the chapter by Paule and Margrave.³⁴ Extrapolation to zero orifice size is generally used in torsion effusion³⁵ and weight loss measurements³⁴ but has more recently been applied to KEMS.³⁶ The central equation is

$$P_m \left[ 1 + f \left( \frac{1}{\alpha_c} + \frac{1}{W_A} - 2 \right) \right] = \frac{\alpha_L}{\alpha_c} P_{eq} f = \frac{W_B B}{A} \quad (2)$$

In this equation, the subscripts A and B indicate the cell body and orifice cross sections, respectively, $P_m$ is the measured vapor pressure, $W_A$ is the transmission factor (Clausing coefficient between 0 and 1) for the cell body, $W_B$ is the transmission factor (Clausing coefficient between 0 and 1) for the orifice, $P_{eq}$ is the equilibrium pressure, $\alpha_c$ is the condensation coefficient (fraction of molecules that stick to the sample surface on striking), and $\alpha_L$ is the vaporization coefficient (fraction of molecules leaving the vaporizing surface toward the vapor compared to equilibrium flow). Transmission or Clausing factors represent the fraction of molecules that go through a given geometric opening; they may be calculated analytically¹³,³⁷,³⁸ or using a Monte Carlo method.³⁹ This will be discussed in detail later. Generally, the condensation coefficient is taken as equal to the vaporization coefficient; this is correct only for molecules that obey “micro-reversibility” (evaporation and condensation paths are the same) but can be taken as a reasonable assumption. Now apply Equation (2) to a cylindrical cell with the orifice on top and an evaporation coefficient of unity (no kinetic barrier to vaporization) and dimensions such that $f = 1/100$. Equation (2) then indicates that the measured equilibrium pressure is 0.99 of the equilibrium pressure, an error less than other uncertainties in KEMS.

A plot of $P_m$ versus $P_m \left[ f \left( \frac{1}{\alpha_c} + \frac{1}{W_A} - 2 \right) \right]$ should yield a line with intercept of $P_{eq}$. Furthermore, the vaporization coefficient, which

<figure><img src="image_12.png" /><figcaption>**FIGURE 4** Melting of gold temperature and pressure calibration. Ramp rate of 1 K/min (Copland and Jacobson¹⁸). [Color figure can be viewed at wileyonlinelibrary.com]</figcaption></figure>
gives a measure of the kinetic barrier to equilibrium, can be extracted from the slope of such a line. Note that the use of this equation to determine the evaporation coefficient requires that the same retarded mechanism occur in the range of chosen orifice sizes and is related to the saturation degree of the surface with vapor, as stated by Rosenblatt.⁴⁰ Thus, vaporization coefficients measured this way depend on the selected orifice sizes and degree of saturation of the vapor.⁴⁰,⁴¹ In general, metals have vaporization coefficients close to unity. Compounds, such as oxides, carbides, and nitrides, may have vaporization coefficients much less than unity.⁴² A Knudsen cell study of the reactions of SiC and SiO₂ indicated measured vapor pressures had a strong dependence on orifice size and hence a small vaporization coefficient and high kinetic barrier to equilibration.⁴³ Note also that Equation (2) can be used to design a cell geometry that is best for studying a system with a low vaporization coefficient. In summary, Equation (2) makes corrections to the measured vapor pressure and provides information on the vaporization coefficient.⁴⁴ More details on vaporization coefficients can be found in Stolyarova and Semenov⁵ and Rosenblatt.⁴⁰,⁴⁵

Hildenbrand³¹ has discussed the problem of equilibria in Knudsen cells in some detail. He further references the study of Golden et al.,⁴⁶ which discusses the importance of collision number, that is, the number of collisions a molecule undergoes in the cell. To obtain sufficient internal energy for a unimolecular reaction, about 50 collisions are needed. Golden et al. estimates that 10 000 or more collisions are needed for full thermodynamic equilibrium, with these numbers from the ratio of the orifice area to sample surface area. A Monte Carlo simulation of Knudsen or molecular flow can provide better values for collision numbers.⁴⁷ Hildenbrand discusses the special case of a gas-leak cell, where the secondary vapor is leaked into a reaction cell. The secondary vapor can be from either a vessel outside the vacuum system or a second Knudsen cell at a different temperature (see Figure 2C,F).¹⁷ Hildenbrand proposes two tests to check for equilibrium in such a system: either the reactant flow or pressure is changed while maintaining the reaction cell at the same temperature, or equilibrium is approached from the reverse direction. In either case, the change should not alter the measured equilibrium constant. In their studies with a gas-leak Knudsen cell, Gorokhov et al.⁴⁸ have used the method based on changing H vapor pressure to study the reactions:

$$ \mathrm{CrOH(g) + O(g) = CrO_2(g) + H(g)} \quad (3a) $$

$$ \mathrm{CrOH(g) + Cr(g) + O(g) = 2CrO(g) + H(g)} \quad (3b) $$

Varying P(H) resulted in very little change in equilibrium constant for these reactions, verifying that equilibrium was reached.

In summary, the best test for equilibrium in the cell is to measure an equilibrium constant and check that it is truly constant with changes in reactant flow or pressure. Note that for some compounds with very low vaporization coefficients, like silica, it may be very hard to attain equilibrium in the cell. This will be discussed in more detail later in this paper.

Related to a check for equilibrium is the necessity to completely characterize the system. Important tools are X-ray diffraction (XRD) and electron optical/microanalysis methods. Careful characterization of the solid reactants before and after a KEMS vaporization study is critical. It should be noted that the reactants cannot be truly quenched in a KEMS experiment, and it might be best to supplement KEMS studies with hot-stage XRD.

### 2.1.4 | Positioning of the Knudsen cell

The Knudsen cell and furnace are typically mounted on a vacuum flange. The cell must be rigidly mounted on the furnace yet be able to be translated so that the emerging molecular beam can be directed into the mass spectrometer to maintain a constant ionization volume. This is important in a single-cell instrument (Figure 5A) but essential in a multiple-cell instrument (Figure 5B,C). Chatillon et al.⁵⁰ have developed a system that holds five cells in an isothermal block. This isothermal block is held by an elastic rod, and each cell is pulled into the sampling region by jacks controlled from outside the vacuum. This system has been more recently refined by Heyrman et al.,⁴⁹,⁵¹ as shown in Figure 5C. The flange is fitted with a large bellow assembly and positioned by x and y jacks, located outside the vacuum system, which tilt the cell into position. This system has a reported positioning accuracy of better than 0.05 mm. The system at JRC Karlsruhe uses a similar system to that of Chatillon et al. but with a smaller bellow and mounting on a vertical door. Johnston and Burley⁵² have used a rotatable block holding four cells. Other investigators have used an x-y table, either inside or outside of the vacuum chamber.⁵³ Figure 5B shows this configuration as used on the NASA Glenn KEMS.¹⁸

### 2.1.5 | Molecular beam

The essence of the KEMS method is the well-defined molecular beam that emerges from the Knudsen cell orifice. The diameter of the orifice must be such that Knudsen or molecular flow through the orifice is assured. This is typically taken as the mean free path of the gas escaping from the cell ≥10 times the orifice diameter. Generally, the smallest orifice has a diameter of 0.1 mm, so the maximum useable pressure for this technique is ~1 × 10⁻⁴ bar. Some investigators have extended the method beyond the molecular flow region into the continuum region and developed correction formulae.⁵⁴ The lower limit for the KEMS method typically depends on the sensitivity of the instrument. Some highly sensitive instruments can measure into the 10⁻¹² bar region. This very large dynamic range is one of the many advantages of KEMS.

We have thus far implicitly assumed molecular flow through the Knudsen cell orifice. This means the vapor species strikes the orifice wall and is reemitted in a completely random direction. Molecular flow can be modeled analytically⁵⁵,⁵⁶ or simulated using a Monte Carlo technique.³⁹,⁵⁷⁻⁶¹ Today, Monte Carlo is probably the most useful technique. A valid simulation can be run on a desktop
gives a measure of the kinetic barrier to equilibrium, can be extracted from the slope of such a line. Note that the use of this equation to determine the evaporation coefficient requires that the same retarded mechanism occur in the range of chosen orifice sizes and is related to the saturation degree of the surface with vapor, as stated by Rosenblatt.⁴⁰ Thus, vaporization coefficients measured this way depend on the selected orifice sizes and degree of saturation of the vapor.⁴⁰,⁴¹ In general, metals have vaporization coefficients close to unity. Compounds, such as oxides, carbides, and nitrides, may have vaporization coefficients much less than unity.⁴² A Knudsen cell study of the reactions of SiC and SiO₂ indicated measured vapor pressures had a strong dependence on orifice size and hence a small vaporization coefficient and high kinetic barrier to equilibration.⁴³ Note also that Equation (2) can be used to design a cell geometry that is best for studying a system with a low vaporization coefficient. In summary, Equation (2) makes corrections to the measured vapor pressure and provides information on the vaporization coefficient.⁴⁴ More details on vaporization coefficients can be found in Stolyarova and Semenov⁵ and Rosenblatt.⁴⁰,⁴⁵

Hildenbrand³¹ has discussed the problem of equilibria in Knudsen cells in some detail. He further references the study of Golden et al.,⁴⁶ which discusses the importance of collision number, that is, the number of collisions a molecule undergoes in the cell. To obtain sufficient internal energy for a unimolecular reaction, about 50 collisions are needed. Golden et al. estimates that 10 000 or more collisions are needed for full thermodynamic equilibrium, with these numbers from the ratio of the orifice area to sample surface area. A Monte Carlo simulation of Knudsen or molecular flow can provide better values for collision numbers.⁴⁷ Hildenbrand discusses the special case of a gas-leak cell, where the secondary vapor is leaked into a reaction cell. The secondary vapor can be from either a vessel outside the vacuum system or a second Knudsen cell at a different temperature (see Figure 2C,F).¹⁷ Hildenbrand proposes two tests to check for equilibrium in such a system: either the reactant flow or pressure is changed while maintaining the reaction cell at the same temperature, or equilibrium is approached from the reverse direction. In either case, the change should not alter the measured equilibrium constant. In their studies with a gas-leak Knudsen cell, Gorokhov et al.⁴⁸ have used the method based on changing H vapor pressure to study the reactions:

$$ \mathrm{CrOH(g) + O(g) = CrO_2(g) + H(g)} \quad (3a) $$

$$ \mathrm{CrOH(g) + Cr(g) + O(g) = 2CrO(g) + H(g)} \quad (3b) $$

Varying P(H) resulted in very little change in equilibrium constant for these reactions, verifying that equilibrium was reached.

In summary, the best test for equilibrium in the cell is to measure an equilibrium constant and check that it is truly constant with changes in reactant flow or pressure. Note that for some compounds with very low vaporization coefficients, like silica, it may be very hard to attain equilibrium in the cell. This will be discussed in more detail later in this paper.

Related to a check for equilibrium is the necessity to completely characterize the system. Important tools are X-ray diffraction (XRD) and electron optical/microanalysis methods. Careful characterization of the solid reactants before and after a KEMS vaporization study is critical. It should be noted that the reactants cannot be truly quenched in a KEMS experiment, and it might be best to supplement KEMS studies with hot-stage XRD.

### 2.1.4 | Positioning of the Knudsen cell

The Knudsen cell and furnace are typically mounted on a vacuum flange. The cell must be rigidly mounted on the furnace yet be able to be translated so that the emerging molecular beam can be directed into the mass spectrometer to maintain a constant ionization volume. This is important in a single-cell instrument (Figure 5A) but essential in a multiple-cell instrument (Figure 5B,C). Chatillon et al.⁵⁰ have developed a system that holds five cells in an isothermal block. This isothermal block is held by an elastic rod, and each cell is pulled into the sampling region by jacks controlled from outside the vacuum. This system has been more recently refined by Heyrman et al.,⁴⁹,⁵¹ as shown in Figure 5C. The flange is fitted with a large bellow assembly and positioned by x and y jacks, located outside the vacuum system, which tilt the cell into position. This system has a reported positioning accuracy of better than 0.05 mm. The system at JRC Karlsruhe uses a similar system to that of Chatillon et al. but with a smaller bellow and mounting on a vertical door. Johnston and Burley⁵² have used a rotatable block holding four cells. Other investigators have used an x-y table, either inside or outside of the vacuum chamber.⁵³ Figure 5B shows this configuration as used on the NASA Glenn KEMS.¹⁸

### 2.1.5 | Molecular beam

The essence of the KEMS method is the well-defined molecular beam that emerges from the Knudsen cell orifice. The diameter of the orifice must be such that Knudsen or molecular flow through the orifice is assured. This is typically taken as the mean free path of the gas escaping from the cell ≥10 times the orifice diameter. Generally, the smallest orifice has a diameter of 0.1 mm, so the maximum useable pressure for this technique is ~1 × 10⁻⁴ bar. Some investigators have extended the method beyond the molecular flow region into the continuum region and developed correction formulae.⁵⁴ The lower limit for the KEMS method typically depends on the sensitivity of the instrument. Some highly sensitive instruments can measure into the 10⁻¹² bar region. This very large dynamic range is one of the many advantages of KEMS.

We have thus far implicitly assumed molecular flow through the Knudsen cell orifice. This means the vapor species strikes the orifice wall and is reemitted in a completely random direction. Molecular flow can be modeled analytically⁵⁵,⁵⁶ or simulated using a Monte Carlo technique.³⁹,⁵⁷⁻⁶¹ Today, Monte Carlo is probably the most useful technique. A valid simulation can be run on a desktop
<figure><img src="image_13.png" /></figure>

computer, and a number of factors can be easily explored. Both methods allow calculation of the transmission or Clausing factor through the orifice. This factor is defined as the number of molecules that escape from the orifice divided by the total number of molecules that enter the orifice. More details on Clausing factors can be found in Margrave.⁶² Tabulations and equations for Clausing factors for cylinders with different length-to-radius ratios are given in Santeler.¹³

Of particular interest in KEMS is the distribution of the effusate or molecules emerging from the orifice. This is shown in Figure 6, calculated by sorting the number of molecules emerging at a specific angle relative to the surface of the cell. The solid black-lined sphere is for a very thin-channel or a knife-edged orifice. This is the classical cosine distribution, which means the fraction of molecules is proportional to the cosine of the angle that a molecule's trajectory makes with the surface. Thus, the largest fraction of molecules emerges from the orifice at 0° from the normal. Note that as the channel becomes more elongated (length-to-radius ratio increases), the distribution becomes an ovoid. Thus, the effusate is more directed. The elongation may not increase the axial flux, but it would (1) limit contamination of the furnace/heat shield assembly and (2) decrease the escaping flux from the cell and force equilibrium in the cell.

The Monte Carlo simulations allow counting collisions with the wall and design Knudsen cell orifices accordingly. Some investigators prefer a thin channel or knife edge that gives a near-unity transmission factor and a few molecule wall collisions. Other investigators prefer a channel that gives a lower transmission factor and several wall collisions for each emerging molecule, but a more directed beam toward the ionizing chamber. Monte Carlo simulations additionally allow the investigator to trace the history of each vaporizing species. It can be shown, as discussed by Cater,³ that for a cell with the orifice on top and a near-unity transmission coefficient for that orifice, most of the molecules that escape are vaporized from the sample surface and travel directly out of the orifice. This is particularly true for low pressures ($<10^{-7}$ bar) and is a desirable situation for directly probing a vaporization process.

This discussion assumes no interaction with the cell surface and no movement along the cell surface (surface diffusion of adsorbed molecules or atoms). In application, there are likely deviations from this behavior because the interior surface of the cell is likely covered by adsorbed molecules that play a role in the condensed phase/vapor equilibria. As noted in Section 2.1.1, no two materials are truly “inert” to each other. Cater³ has discussed the effects of surface diffusion on Knudsen effusion in some detail. We shall briefly summarize his important conclusions here. In some cases, the molecule may strike the wall and move along this wall before desorbing near the orifice edge. This surface diffusion would necessarily change the net flux, add a contribution to material loss, and also alter the vapor composition emerging from the cell orifice. It is difficult to correct for surface diffusion, as the needed parameters of diffusion path lengths and surface diffusion coefficient are very system specific and not generally available. Monte Carlo simulations offer some insight into

<figure><img src="image_14.png" /><figcaption>**FIGURE 5** (A) Typical single-cell flange with resistance heating furnace; (B) multi-cell furnace/Knudsen cell positioner at NASA Glenn. The computer-controlled x and y stepper motors position the cell and furnace assembly. Note the foil heating element that can be W or Ta. (C) Multi-cell positioner at SiMaP. Source: Reprinted from Heyrman et al.⁴⁹ with permission from John Wiley and Sons.</figcaption></figure>
<figure><img src="image_13.png" /></figure>

computer, and a number of factors can be easily explored. Both methods allow calculation of the transmission or Clausing factor through the orifice. This factor is defined as the number of molecules that escape from the orifice divided by the total number of molecules that enter the orifice. More details on Clausing factors can be found in Margrave.⁶² Tabulations and equations for Clausing factors for cylinders with different length-to-radius ratios are given in Santeler.¹³

Of particular interest in KEMS is the distribution of the effusate or molecules emerging from the orifice. This is shown in Figure 6, calculated by sorting the number of molecules emerging at a specific angle relative to the surface of the cell. The solid black-lined sphere is for a very thin-channel or a knife-edged orifice. This is the classical cosine distribution, which means the fraction of molecules is proportional to the cosine of the angle that a molecule's trajectory makes with the surface. Thus, the largest fraction of molecules emerges from the orifice at 0° from the normal. Note that as the channel becomes more elongated (length-to-radius ratio increases), the distribution becomes an ovoid. Thus, the effusate is more directed. The elongation may not increase the axial flux, but it would (1) limit contamination of the furnace/heat shield assembly and (2) decrease the escaping flux from the cell and force equilibrium in the cell.

The Monte Carlo simulations allow counting collisions with the wall and design Knudsen cell orifices accordingly. Some investigators prefer a thin channel or knife edge that gives a near-unity transmission factor and a few molecule wall collisions. Other investigators prefer a channel that gives a lower transmission factor and several wall collisions for each emerging molecule, but a more directed beam toward the ionizing chamber. Monte Carlo simulations additionally allow the investigator to trace the history of each vaporizing species. It can be shown, as discussed by Cater,³ that for a cell with the orifice on top and a near-unity transmission coefficient for that orifice, most of the molecules that escape are vaporized from the sample surface and travel directly out of the orifice. This is particularly true for low pressures ($<$10⁻⁷ bar) and is a desirable situation for directly probing a vaporization process.

This discussion assumes no interaction with the cell surface and no movement along the cell surface (surface diffusion of adsorbed molecules or atoms). In application, there are likely deviations from this behavior because the interior surface of the cell is likely covered by adsorbed molecules that play a role in the condensed phase/vapor equilibria. As noted in Section 2.1.1, no two materials are truly “inert” to each other. Cater³ has discussed the effects of surface diffusion on Knudsen effusion in some detail. We shall briefly summarize his important conclusions here. In some cases, the molecule may strike the wall and move along this wall before desorbing near the orifice edge. This surface diffusion would necessarily change the net flux, add a contribution to material loss, and also alter the vapor composition emerging from the cell orifice. It is difficult to correct for surface diffusion, as the needed parameters of diffusion path lengths and surface diffusion coefficient are very system specific and not generally available. Monte Carlo simulations offer some insight into

<figure><img src="image_14.png" /><figcaption>**FIGURE 5** (A) Typical single-cell flange with resistance heating furnace; (B) multi-cell furnace/Knudsen cell positioner at NASA Glenn. The computer-controlled x and y stepper motors position the cell and furnace assembly. Note the foil heating element that can be W or Ta. (C) Multi-cell positioner at SiMaP. Source: Reprinted from Heyrman et al.⁴⁹ with permission from John Wiley and Sons.</figcaption></figure>
<figure><img src="image_15.png" /><figcaption>FIGURE 6 Monte Carlo-generated distributions of molecules emerging from orifices of different geometry. Note that L is the length of the cylindrical orifice and r is the radius. [Color figure can be viewed at wileyonlinelibrary.com]</figcaption></figure>

this phenomena, and Ward et al.⁵⁹ have derived a general expression to correct transmission factors for surface diffusion. However, it is best to avoid such material combinations that lead to surface diffusion if possible. According to Ward et al.⁵⁹ surface diffusion can be minimized using larger cylindrical orifices instead of thin-edged orifices.

An extreme case of surface diffusion is the phenomenon of liquid creeping up the edges of a cell. This is particularly troublesome with glasses and some salts, and the volume of material that escapes renders a KEMS experiment impossible. Cater³ recommends the use of cells with sharp-edged orifices or an inner crucible with a sharp edge that cuts the creeping film, the use of single-crystal cell materials, and the avoidance systems where the sample wets the cell material. This is a big problem in studying glasses; the selection of cell material becomes very important.

Coupling of the Knudsen cell to the mass spectrometer is a critical aspect in a KEMS instrument. This coupling has a major effect on the sensitivity of the instrument, the degree of background interference, and the general stability. Recent instruments show considerable advances in this area. Figure 1 shows that a copper plate separates the Knudsen cell vacuum chamber and the ionizer. The copper plate is typically cooled by water or even liquid nitrogen. The basic function of the plate is to shield the ionizer from the high temperatures of the Knudsen cell furnace. Nearly all instruments contain a moveable shutter to help distinguish the beam emerging from the Knudsen cell from the background.¹ This is simply a moveable plate to interrupt the beam emerging from the cell. It may be a part of the copper plate or a separate smaller moveable plate. The shape and location of the shutter is important, particularly in the case of permanent gases (e.g., N₂, O₂, and CO) that exist in the vacuum, as analyzed by Froment et al.⁶³ It should not act as a valve between the Knudsen cell vacuum chamber and the ionizer chamber, which would interfere with pumping patterns

and would lead to erroneous measurements for permanent gases. A small free-standing plate is best, to interrupt only the useful molecular beam.

The copper plate is also used to collimate the full molecular beam emerging from the cell and limit effusing gases from reaching and contaminating ionizer walls. Chatillon et al.¹⁹,⁶⁴,⁶⁵ have developed the “restricted collimation method” to direct the effusing vapor into the ionization region. Other groups have applied this method as well.¹⁸,⁶⁶,⁶⁷ An aperture in the copper plate with a diameter smaller than the diameter of the Knudsen cell orifice coupled to a second ionization chamber aperture effectively allows the ionizer to only “see” inside the Knudsen cell, allowing only the vapors from the inner-cell gas phase to reach the ionization region. This is shown in Figure 7. According to the nomenclature of Chatillon, the aperture in the copper plate is the “field aperture,” and the aperture adjacent to the ionizing region is the “source aperture.” The geometry of the NASA Glenn system is shown in Figure 8A, and modeling parameters (aperture spacing and orifice diameters) are shown in Figure 8B.⁶¹ Chatillon et al. have derived equations for the flux through these different apertures based on the analogy of light transmission between two disks, which decreases inversely with the square of the distance between the two disks.⁶⁸,⁶⁹ The fraction of molecules that leave a radiating disc and arrive at a receiving disk is given by

$$F(r_1, r_2, a) = \frac{(r_1^2 + r_2^2 + a^2) - \sqrt{(r_1^2 + r_2^2 + a^2) - 4r_1^2 r_2^2}}{2r_1^2} \quad (4)$$

Here a is the distance between the two disks, r₁ is the radius of the radiating disk, and r₂ is the radius of the receiving disk. This fraction is the unit of solid angle that reaches the second aperture. The flux reaching the second aperture is given as
<figure><img src="image_15.png" /><figcaption>FIGURE 6 Monte Carlo-generated distributions of molecules emerging from orifices of different geometry. Note that L is the length of the cylindrical orifice and r is the radius. [Color figure can be viewed at wileyonlinelibrary.com]</figcaption></figure>

this phenomena, and Ward et al.⁵⁹ have derived a general expression to correct transmission factors for surface diffusion. However, it is best to avoid such material combinations that lead to surface diffusion if possible. According to Ward et al.⁵⁹ surface diffusion can be minimized using larger cylindrical orifices instead of thin-edged orifices.

An extreme case of surface diffusion is the phenomenon of liquid creeping up the edges of a cell. This is particularly troublesome with glasses and some salts, and the volume of material that escapes renders a KEMS experiment impossible. Cater³ recommends the use of cells with sharp-edged orifices or an inner crucible with a sharp edge that cuts the creeping film, the use of single-crystal cell materials, and the avoidance systems where the sample wets the cell material. This is a big problem in studying glasses; the selection of cell material becomes very important.

Coupling of the Knudsen cell to the mass spectrometer is a critical aspect in a KEMS instrument. This coupling has a major effect on the sensitivity of the instrument, the degree of background interference, and the general stability. Recent instruments show considerable advances in this area. Figure 1 shows that a copper plate separates the Knudsen cell vacuum chamber and the ionizer. The copper plate is typically cooled by water or even liquid nitrogen. The basic function of the plate is to shield the ionizer from the high temperatures of the Knudsen cell furnace. Nearly all instruments contain a moveable shutter to help distinguish the beam emerging from the Knudsen cell from the background.¹ This is simply a moveable plate to interrupt the beam emerging from the cell. It may be a part of the copper plate or a separate smaller moveable plate. The shape and location of the shutter is important, particularly in the case of permanent gases (e.g., N₂, O₂, and CO) that exist in the vacuum, as analyzed by Froment et al.⁶³ It should not act as a valve between the Knudsen cell vacuum chamber and the ionizer chamber, which would interfere with pumping patterns

and would lead to erroneous measurements for permanent gases. A small free-standing plate is best, to interrupt only the useful molecular beam.

The copper plate is also used to collimate the full molecular beam emerging from the cell and limit effusing gases from reaching and contaminating ionizer walls. Chatillon et al.¹⁹,⁶⁴,⁶⁵ have developed the “restricted collimation method” to direct the effusing vapor into the ionization region. Other groups have applied this method as well.¹⁸,⁶⁶,⁶⁷ An aperture in the copper plate with a diameter smaller than the diameter of the Knudsen cell orifice coupled to a second ionization chamber aperture effectively allows the ionizer to only “see” inside the Knudsen cell, allowing only the vapors from the inner-cell gas phase to reach the ionization region. This is shown in Figure 7. According to the nomenclature of Chatillon, the aperture in the copper plate is the “field aperture,” and the aperture adjacent to the ionizing region is the “source aperture.” The geometry of the NASA Glenn system is shown in Figure 8A, and modeling parameters (aperture spacing and orifice diameters) are shown in Figure 8B.⁶¹ Chatillon et al. have derived equations for the flux through these different apertures based on the analogy of light transmission between two disks, which decreases inversely with the square of the distance between the two disks.⁶⁸,⁶⁹ The fraction of molecules that leave a radiating disc and arrive at a receiving disk is given by

$$F(r_1, r_2, a) = \frac{(r_1^2 + r_2^2 + a^2) - \sqrt{(r_1^2 + r_2^2 + a^2) - 4r_1^2 r_2^2}}{2r_1^2} \quad (4)$$

Here a is the distance between the two disks, r₁ is the radius of the radiating disk, and r₂ is the radius of the receiving disk. This fraction is the unit of solid angle that reaches the second aperture. The flux reaching the second aperture is given as
<figure><img src="image_16.png" /><figcaption>FIGURE 7 “Ray diagram” illustrating the advantages of restricted collimation. [Color figure can be viewed at wileyonlinelibrary.com]</figcaption></figure>

$$J_A = \left( \frac{p_A}{\sqrt{2\pi M_A R T}} \right) F(r_1, r_2, a) \quad (5)$$

Factor F is the proportion of molecules exchanged between the surfaces of the two apertures. The cross section of the orifice, which is larger than the source (Figure 7), is replaced by the surface of the field aperture. The F(r₁,r₂,a) term contains the emitted surface (field aperture) emitting into the vacuum. Nuta and Chatillon⁶⁵,⁷⁰ have performed calculations using different distances between the sample and field apertures and showed the flow, Jₐ, from the surface of the sample (the surface that is viewed by the detector in the restricted molecular beam device) is constant, even when tilting the sample. This means the resulting flow arriving at the ionizer remains the same. Thus, the restricted collimation device is (1) really the sampling of the useful molecular beam in place of the effusion orifice and (2) independent of the cell (tilting, distance, orifice, etc.) and sample.

Chatillon and Nuta⁶⁵,⁷¹ use their derived equations to optimize the distances between apertures and diameters of apertures. This problem can also be handled utilizing the Monte Carlo simulations discussed earlier.⁶¹ A direct comparison of the Monte Carlo simulation of flux transmission to the analytical simulation results of Chatillon et al.⁶⁵,⁷⁰ gives qualitatively similar results from both approaches, as shown in Figure 8C.

Another relatively recent advancement at the Knudsen cell-ionizer interface is the introduction of a valve between the Knudsen cell furnace chamber and the ionizer chamber.⁷,⁷² The inclusion of this valve is technically challenging, as it must be able to withstand very high temperatures and also be a “low profile,” in that it does not add unnecessary distance between the Knudsen cell and the ionizer. However, such a valve has many advantages. It allows the ionizer and detector to remain powered at all times. This results in improved filament and detector longevity and, most importantly, constant sensitivity and calibration constants for successive experiments. Additionally, it allows rapid changing of specimens.

## 2.2 | Ionizer

### 2.2.1 | Ionizer/ion extraction and acceleration

The next major section of a KEMS instrument is the ionizer. Novel ionization methods allow removal of electrons from a large molecule without fragmentation. However, many of these novel methods are not applicable to high-temperature vapors; as such electron impact continues to be the predominant method of ionization in most instruments, because ionization yields (~1 in 10⁴) are quite constant
<figure><img src="image_16.png" /><figcaption>FIGURE 7 “Ray diagram” illustrating the advantages of restricted collimation. [Color figure can be viewed at wileyonlinelibrary.com]</figcaption></figure>

$$J_A = \left( \frac{p_A}{\sqrt{2\pi M_A R T}} \right) F(r_1, r_2, a) \quad (5)$$

Factor F is the proportion of molecules exchanged between the surfaces of the two apertures. The cross section of the orifice, which is larger than the source (Figure 7), is replaced by the surface of the field aperture. The F(r₁,r₂,a) term contains the emitted surface (field aperture) emitting into the vacuum. Nuta and Chatillon⁶⁵,⁷⁰ have performed calculations using different distances between the sample and field apertures and showed the flow, Jₐ, from the surface of the sample (the surface that is viewed by the detector in the restricted molecular beam device) is constant, even when tilting the sample. This means the resulting flow arriving at the ionizer remains the same. Thus, the restricted collimation device is (1) really the sampling of the useful molecular beam in place of the effusion orifice and (2) independent of the cell (tilting, distance, orifice, etc.) and sample.

Chatillon and Nuta⁶⁵,⁷¹ use their derived equations to optimize the distances between apertures and diameters of apertures. This problem can also be handled utilizing the Monte Carlo simulations discussed earlier.⁶¹ A direct comparison of the Monte Carlo simulation of flux transmission to the analytical simulation results of Chatillon et al.⁶⁵,⁷⁰ gives qualitatively similar results from both approaches, as shown in Figure 8C.

Another relatively recent advancement at the Knudsen cell-ionizer interface is the introduction of a valve between the Knudsen cell furnace chamber and the ionizer chamber.⁷,⁷² The inclusion of this valve is technically challenging, as it must be able to withstand very high temperatures and also be a “low profile,” in that it does not add unnecessary distance between the Knudsen cell and the ionizer. However, such a valve has many advantages. It allows the ionizer and detector to remain powered at all times. This results in improved filament and detector longevity and, most importantly, constant sensitivity and calibration constants for successive experiments. Additionally, it allows rapid changing of specimens.

## 2.2 | Ionizer

### 2.2.1 | Ionizer/ion extraction and acceleration

The next major section of a KEMS instrument is the ionizer. Novel ionization methods allow removal of electrons from a large molecule without fragmentation. However, many of these novel methods are not applicable to high-temperature vapors; as such electron impact continues to be the predominant method of ionization in most instruments, because ionization yields (~1 in 10⁴) are quite constant
<figure><img src="image_17.png" /><figcaption>FIGURE 8 (A) Knudsen cell-generated molecular beam introduction to ionizer in NASA Glenn KEMS (Knudsen effusion mass spectrometry) instrument. (B) Geometry for computer model of above. (C) Results of Monte Carlo simulation and analytical formulation for optimizing dimensions. Source: Reprinted with permission from Radke et al.,⁶¹ John Wiley and Sons. [Color figure can be viewed at wileyonlinelibrary.com]</figcaption></figure>

for different species (i.e., with cross sections in the same range). Figure 9A,B shows the classic design of the ionizer filament and the high-voltage focusing plates used in the NASA Glenn magnetic sector instrument (modified Nuclide/MAAS/PATCO 12-90-HT). The apex of the filament emits the electrons toward the ionization box under a fixed voltage. These electrons are collimated as a quasi-monoenergetic beam to a well-defined ionization (but unknown) volume (Figure 8A). Note that the entire ionization assembly is at a high voltage, typically 5–10 kV. Typically, the filament is heated with a few volts and several amperes up to 2000–3000 K for emission to start.⁷³ The relationship between the electron current density, $i_e$, and temperature, $T$, and the properties of the filament are given by the Richardson equation⁷³:

$$i_e \propto T^2 \exp\left(-\frac{\beta\phi}{T}\right) \qquad (6)$$

Here $\beta$ is a proportionality constant, and $\phi$ is the work function. The high-kinetic-energy electrons collide with the incoming molecular beam to form ions:

$$A + e^{-} = A^{+} + 2e^{-} \qquad (7)$$

The trap collects a small portion (usually ~5%) of the electrons that do not interact with the molecular beam. The filament current is

controlled by maintaining either a constant trap current or, more commonly, a constant emission. The energy of ionizing electrons (or electron-accelerating potential) is important and, in a well-designed ionizing chamber, controlled to ±0.1 eV. At low electron energies and constant emission, high filament current is necessary. Typically, ionizer circuits have an adjustable current limit control to protect the filament from burning out.

It should be noted that thermal ionization may be important at very high temperatures. Kudin and colleagues⁷⁴,⁷⁵ have studied thermal ionization of a number of metal halides by turning off the electron impact ionizer and carefully examining the thermal ions produced in the Knudsen cell.

The filament box, including the filament, filament aperture, trap, and repeller (Figure 9A), and the ion entrance aperture and focusing lenses (Figure 9B) are all at different potentials. Klemperer and Barnett⁷⁶ present a useful discussion of the fields formed when two plane conductors are placed on either side of an aperture and the penetration of the fields through that aperture. Figure 10A,B show this situation with equipotential lines. Figure 10A is for the ionization box where the two conductors are the filament and the filament box. Because the filament is negative relative to the box, the conductors are at opposite potentials and the equipotential lines show a saddle-point-shaped pattern. This imparts a bent trajectory to the electrons. Further additional energy is added to the electrons. Figure 10B is for
<figure><img src="image_17.png" /><figcaption>FIGURE 8 (A) Knudsen cell-generated molecular beam introduction to ionizer in NASA Glenn KEMS (Knudsen effusion mass spectrometry) instrument. (B) Geometry for computer model of above. (C) Results of Monte Carlo simulation and analytical formulation for optimizing dimensions. Source: Reprinted with permission from Radke et al.,⁶¹ John Wiley and Sons. [Color figure can be viewed at wileyonlinelibrary.com]</figcaption></figure>

for different species (i.e., with cross sections in the same range). Figure 9A,B shows the classic design of the ionizer filament and the high-voltage focusing plates used in the NASA Glenn magnetic sector instrument (modified Nuclide/MAAS/PATCO 12-90-HT). The apex of the filament emits the electrons toward the ionization box under a fixed voltage. These electrons are collimated as a quasi-monoenergetic beam to a well-defined ionization (but unknown) volume (Figure 8A). Note that the entire ionization assembly is at a high voltage, typically 5–10 kV. Typically, the filament is heated with a few volts and several amperes up to 2000–3000 K for emission to start.⁷³ The relationship between the electron current density, $i_e$, and temperature, $T$, and the properties of the filament are given by the Richardson equation⁷³:

$$i_e \propto T^2 \exp\left(-\frac{\beta\phi}{T}\right) \quad (6)$$

Here $\beta$ is a proportionality constant, and $\phi$ is the work function. The high-kinetic-energy electrons collide with the incoming molecular beam to form ions:

$$A + e^- = A^+ + 2e^- \quad (7)$$

The trap collects a small portion (usually ~5%) of the electrons that do not interact with the molecular beam. The filament current is

controlled by maintaining either a constant trap current or, more commonly, a constant emission. The energy of ionizing electrons (or electron-accelerating potential) is important and, in a well-designed ionizing chamber, controlled to ±0.1 eV. At low electron energies and constant emission, high filament current is necessary. Typically, ionizer circuits have an adjustable current limit control to protect the filament from burning out.

It should be noted that thermal ionization may be important at very high temperatures. Kudin and colleagues⁷⁴,⁷⁵ have studied thermal ionization of a number of metal halides by turning off the electron impact ionizer and carefully examining the thermal ions produced in the Knudsen cell.

The filament box, including the filament, filament aperture, trap, and repeller (Figure 9A), and the ion entrance aperture and focusing lenses (Figure 9B) are all at different potentials. Klemperer and Barnett⁷⁶ present a useful discussion of the fields formed when two plane conductors are placed on either side of an aperture and the penetration of the fields through that aperture. Figure 10A,B show this situation with equipotential lines. Figure 10A is for the ionization box where the two conductors are the filament and the filament box. Because the filament is negative relative to the box, the conductors are at opposite potentials and the equipotential lines show a saddle-point-shaped pattern. This imparts a bent trajectory to the electrons. Further additional energy is added to the electrons. Figure 10B is for
<figure><img src="image_18.png" /></figure>

<figure><img src="image_19.png" /></figure>

<figure><img src="image_20.png" /><figcaption>**FIGURE 9** Ionizer and ion beam focusing plates for NASA Glenn KEMS instrument: (A) electron impact ionizer and (B) ion beam focusing plates. [Color figure can be viewed at wileyonlinelibrary.com]</figcaption></figure>

the formed ions, and two conductors are now the repeller plate and the extraction plate. These fields are both in the same direction, and the equipotential lines are now all in the same direction. This directs the positive ions through the ion entrance aperture. The important points are as follows: (1) the unavoidable nonuniform fields in the filament box necessarily change the energy of the ionizing electrons and must be corrected for, and (2) those fields set the formed ions on different trajectories and lead to crossover in the ion beam. To correct

for the changes in the energy of ionizing electrons, background species ($N_2$ or CO, $H_2O$, Ar) can be used to calibrate the electron energy scale.⁷⁷

Figure 10C is a plot of positive ion intensity versus repeller potential. Note the repeller is typically 10 000 ± 10 V. Ideally at 0 V (10 000 V) the ion intensity reaches a sharp maximum. The maximum is also obtained with a positive “push” of the ions into the extractor. The repeller together with the extractor is termed the “immersion lens.” Improvements to the ionizer and immersion lens can lead to improved ionization efficiency and sensitivity.⁷⁸ Some investigators have added magnets to their ionizers, which increases sensitivity but may introduce some mass dependence on sensitivity.

Current modeling tools allow design of the ionizer with regard to the physical arrangement of the focusing plates and their potentials.⁷⁸,⁷⁹ The design and optimization of ionizers have been revolutionized by computer simulation software, most notably SIMION.⁷⁹ This software has been in use for over 30 years and allows the physical and electrical characteristics of an ionizer to be optimized with respect to sensitivity, ion yield, and other desired parameters. SIMION models of the ionizer in the NASA Glenn instrument and resultant electron and ion trajectories are shown in Figures 11A-C. Figure 11B shows the electrons in the box set for maximum intersection with the ion beam. Note that the repeller is often at a nonzero potential for maximum signal, and this may impart additional energy to the electrons. The electron energy thus needs to be calibrated with known gases.⁸⁰ Figure 11C shows the optimum ion beam through the focusing plates. Note the ideal, thin ion beam and the crossover points.

### 2.2.2 | Ionization efficiency curves

The ionization of a neutral species by an energetic electron is rarely as simple as that depicted in Equation (7). Often, ionization is accompanied by bond breaking (fragmentation), multiple ionization processes, and kinetic energy gain of the resultant ions or neutrals. These processes can, at least partially, be separated according to their ionization efficiency curve, shown in Figure 12. Ionization efficiency curves are described in detail in the reviews of Drowart and Goldfinger² and Grimley.⁴ The appearance potential can be extracted from this curve by using one of several methods.¹⁸ The linear extrapolation method has been used for many years and is shown in Figure 12. The linear portion of the curve is simply extrapolated to the x-axis. This x-axis intercept is the appearance potential, which may be calibrated by comparison to the well-established appearance potentials of known vapors.⁸¹,⁸² The curvature of the ionization efficiency curve (Figure 12) may be due to several reasons: (1) non-monoenergetic electrons, ~0.1 eV, spread from the temperature distribution along the emitting filament, which increases at low electron energies due to a loss of emission regulation, as discussed earlier; (2) different ionization locations in the immersion lens region;
<figure><img src="image_18.png" /></figure>

<figure><img src="image_19.png" /></figure>

<figure><img src="image_20.png" /><figcaption>**FIGURE 9** Ionizer and ion beam focusing plates for NASA Glenn KEMS instrument: (A) electron impact ionizer and (B) ion beam focusing plates. [Color figure can be viewed at wileyonlinelibrary.com]</figcaption></figure>

the formed ions, and two conductors are now the repeller plate and the extraction plate. These fields are both in the same direction, and the equipotential lines are now all in the same direction. This directs the positive ions through the ion entrance aperture. The important points are as follows: (1) the unavoidable nonuniform fields in the filament box necessarily change the energy of the ionizing electrons and must be corrected for, and (2) those fields set the formed ions on different trajectories and lead to crossover in the ion beam. To correct

for the changes in the energy of ionizing electrons, background species ($N_2$ or CO, $H_2O$, Ar) can be used to calibrate the electron energy scale.⁷⁷

Figure 10C is a plot of positive ion intensity versus repeller potential. Note the repeller is typically 10 000 ± 10 V. Ideally at 0 V (10 000 V) the ion intensity reaches a sharp maximum. The maximum is also obtained with a positive “push” of the ions into the extractor. The repeller together with the extractor is termed the “immersion lens.” Improvements to the ionizer and immersion lens can lead to improved ionization efficiency and sensitivity.⁷⁸ Some investigators have added magnets to their ionizers, which increases sensitivity but may introduce some mass dependence on sensitivity.

Current modeling tools allow design of the ionizer with regard to the physical arrangement of the focusing plates and their potentials.⁷⁸,⁷⁹ The design and optimization of ionizers have been revolutionized by computer simulation software, most notably SIMION.⁷⁹ This software has been in use for over 30 years and allows the physical and electrical characteristics of an ionizer to be optimized with respect to sensitivity, ion yield, and other desired parameters. SIMION models of the ionizer in the NASA Glenn instrument and resultant electron and ion trajectories are shown in Figures 11A-C. Figure 11B shows the electrons in the box set for maximum intersection with the ion beam. Note that the repeller is often at a nonzero potential for maximum signal, and this may impart additional energy to the electrons. The electron energy thus needs to be calibrated with known gases.⁸⁰ Figure 11C shows the optimum ion beam through the focusing plates. Note the ideal, thin ion beam and the crossover points.

### 2.2.2 | Ionization efficiency curves

The ionization of a neutral species by an energetic electron is rarely as simple as that depicted in Equation (7). Often, ionization is accompanied by bond breaking (fragmentation), multiple ionization processes, and kinetic energy gain of the resultant ions or neutrals. These processes can, at least partially, be separated according to their ionization efficiency curve, shown in Figure 12. Ionization efficiency curves are described in detail in the reviews of Drowart and Goldfinger² and Grimley.⁴ The appearance potential can be extracted from this curve by using one of several methods.¹⁸ The linear extrapolation method has been used for many years and is shown in Figure 12. The linear portion of the curve is simply extrapolated to the x-axis. This x-axis intercept is the appearance potential, which may be calibrated by comparison to the well-established appearance potentials of known vapors.⁸¹,⁸² The curvature of the ionization efficiency curve (Figure 12) may be due to several reasons: (1) non-monoenergetic electrons, ~0.1 eV, spread from the temperature distribution along the emitting filament, which increases at low electron energies due to a loss of emission regulation, as discussed earlier; (2) different ionization locations in the immersion lens region;
<figure><img src="image_21.png" /></figure>

<figure><img src="image_22.png" /><figcaption>**FIGURE 10** Diagram of filament box with lines showing relative magnitude of electric field and trajectories of (A) ionizing electrons and (B) formed ions. Source: Adapted from Klemperer and Barnett.⁷⁶ (C) A plot of ion current versus repeller voltage. [Color figure can be viewed at wileyonlinelibrary.com]</figcaption></figure>

and (3) molecules/species with closely spaced energy levels. Note that in Figure 12, the measured appearance potential for Al is 6.83 eV, compared to tabulated ionization potentials of 5.77–5.98 eV. Al does not have closely spaced electronic levels, but other issues that form the curvature may explain the difference between the tabulated ionization potential of 5.77–5.98 and the measured appearance potential of 6.83 eV.

Drowart and Goldfinger² and Grimley⁴ have shown the various ionization efficiency curves that result from fragmentation. These

ionization efficiency curves are a valuable tool in understanding the ionization process. In some cases, when formation energies are known and no kinetic energy exchange occurs, Born-Haber cycles can be used to calculate dissociative ionization thresholds.⁸³

Often, fragmentation of a simple molecule can be minimized by working at an electron energy between the appearance potential and the summation of the appearance potential and dissociation energy of that molecule. For example, the appearance potential of SO₂⁺ is 12.32 eV⁸² and the bond energy of the S=O bond is 5.51 eV.⁸⁴ Thus
<figure><img src="image_21.png" /></figure>

<figure><img src="image_22.png" /><figcaption>**FIGURE 10** Diagram of filament box with lines showing relative magnitude of electric field and trajectories of (A) ionizing electrons and (B) formed ions. Source: Adapted from Klemperer and Barnett.⁷⁶ (C) A plot of ion current versus repeller voltage. [Color figure can be viewed at wileyonlinelibrary.com]</figcaption></figure>

and (3) molecules/species with closely spaced energy levels. Note that in Figure 12, the measured appearance potential for Al is 6.83 eV, compared to tabulated ionization potentials of 5.77–5.98 eV. Al does not have closely spaced electronic levels, but other issues that form the curvature may explain the difference between the tabulated ionization potential of 5.77–5.98 and the measured appearance potential of 6.83 eV.

Drowart and Goldfinger² and Grimley⁴ have shown the various ionization efficiency curves that result from fragmentation. These

ionization efficiency curves are a valuable tool in understanding the ionization process. In some cases, when formation energies are known and no kinetic energy exchange occurs, Born-Haber cycles can be used to calculate dissociative ionization thresholds.⁸³

Often, fragmentation of a simple molecule can be minimized by working at an electron energy between the appearance potential and the summation of the appearance potential and dissociation energy of that molecule. For example, the appearance potential of SO₂⁺ is 12.32 eV⁸² and the bond energy of the S=O bond is 5.51 eV.⁸⁴ Thus
<figure><img src="image_23.png" /><figcaption>FIGURE 11 SIMION model of ionizer and ion beam focusing plates: (A) three-dimensional view; (B) electron beam intersection with molecular beam (green circle); (C) cross section showing optimized beam shape, with repeller at 10 000 V, J2 9560 V, J3 9390 V, and J4 6500 V (Copland and Jacobson¹⁸). [Color figure can be viewed at wileyonlinelibrary.com]</figcaption></figure>

<figure><img src="image_24.png" /><figcaption>FIGURE 12 Ionization efficiency plot for Al.¹⁹</figcaption></figure>

it would be best to work with an electron energy of 12.32–17.83 to avoid fragmentation of SO₂.⁸⁵,⁸⁶ A similar method may be used to distinguish O⁺ from O(g) and not from O₂(g) dissociation.

Conversely, some investigators choose to use a high electron energy to maximize signal strength. In such cases, fragmentation occurs, and therefore the fragmentation pattern must be known. An example is studies on Am, U, and Pu oxide vapors.⁷⁷,⁸⁰

In summary, electron impact ionization of molecules is the sum of several different processes:

$$AB + e^{-} \rightarrow AB^{+} + e^{-} \rightarrow AB^{+} + 2e^{-} \qquad \sigma_{AB}^{+}/AB \qquad \text{IP — ionization potential} \qquad (8a)$$

$$\rightarrow A^{+} + B(g) + 2e^{-} \qquad \sigma_{A}^{+}/AB \qquad \text{AP — appearance potential} \qquad (8b)$$
<figure><img src="image_23.png" /><figcaption>FIGURE 11 SIMION model of ionizer and ion beam focusing plates: (A) three-dimensional view; (B) electron beam intersection with molecular beam (green circle); (C) cross section showing optimized beam shape, with repeller at 10 000 V, J2 9560 V, J3 9390 V, and J4 6500 V (Copland and Jacobson¹⁸). [Color figure can be viewed at wileyonlinelibrary.com]</figcaption></figure>

<figure><img src="image_24.png" /><figcaption>FIGURE 12 Ionization efficiency plot for Al.¹⁹</figcaption></figure>

it would be best to work with an electron energy of 12.32–17.83 to avoid fragmentation of SO₂.⁸⁵,⁸⁶ A similar method may be used to distinguish O⁺ from O(g) and not from O₂(g) dissociation.

Conversely, some investigators choose to use a high electron energy to maximize signal strength. In such cases, fragmentation occurs, and therefore the fragmentation pattern must be known. An example is studies on Am, U, and Pu oxide vapors.⁷⁷,⁸⁰

In summary, electron impact ionization of molecules is the sum of several different processes:

$$AB + e^{-} \rightarrow AB^{+} + e^{-} \rightarrow AB^{+} + 2e^{-} \quad \sigma_{AB}^{+}/AB \quad \text{IP — ionization potential} \quad (8a)$$

$$\rightarrow A^{+} + B(g) + 2e^{-} \quad \sigma_{A}^{+}/AB \quad \text{AP — appearance potential} \quad (8b)$$
$$ \rightarrow B^{+} + A(g) + 2e^{-} \qquad \sigma_{B^{+}/AB} \text{ AP} \text{— appearance potential (8c)} $$

Here $\sigma$ is the ionization cross section, which will be discussed in subsequent sections of this paper and are covered in detailed in Drowart et al.⁸⁷

### 2.2.3 | Relationship between ion intensity and partial pressure of vapor species

The relationship between partial pressure in the cell and measured ion intensity is central to KEMS. This relationship has been derived in Drowart and Goldfinger,² Grimley,⁴ and Chatillon et al.,¹⁹,⁸⁸ and will be briefly summarized¹⁸ here due to its importance. The number of ions in an infinitesimal unit of volume is given by the following expression:

$$ N_{\text{A}}^{+}(E) d\nu = I_{\text{e}}(E) I_{\text{e}} \rho_{\text{A}} \sigma_{\text{A}}(E) d\nu = I_{\text{e}}(E) I_{\text{e}} \sigma_{\text{A}}(E) \frac{J_{\text{A}}}{c_{\text{A}}} d\nu \quad (9) $$

Here $I_{\text{e}}(E)$ is the intensity of the ionizing electrons, which is a function of their energy, $E$; $I_{\text{e}}$ is the electrons' path length; $\rho_{\text{A}}$ is the density of neutral atoms A in the ionization volume $d\nu$; and $\sigma_{\text{A}}(E)$ is the ionization cross section, which is a function of the ionizing electron energy, $E$. The instantaneous density of the neutral atoms is given by the flux of atoms A from the Knudsen cell orifice, $j_{\text{A}}$, divided by the average molecular speed, $c_{\text{A}}$. This flux reaching the ionizer is scaled by the inverse of distance from the orifice to the ionizer, as discussed previously, and given by the following integral over the surface of the molecular beam entering the ionizer:

$$ j_{\text{A}} = \int_{S} \left( \frac{W_{c}p_{\text{A}}}{\sqrt{2\pi M_{\text{A}}RT}} \right) \frac{1}{a^2} ds \quad (10) $$

Here $a$ is the distance from the cell orifice to the ionization region. Equation (10) can be substituted into Equation (9) and integrated over the ionization volume to give

$$ N_{\text{A}}^{+}(E) = \int_{V} \int_{S} \left( \frac{W_{c}p_{\text{A}}}{\sqrt{2\pi M_{\text{A}}RT}} \right) \frac{1}{a^2} \sigma_{\text{A}}(E) \sqrt{\frac{\pi M_{\text{A}}}{8RT}} ds d\nu \quad (11) $$

In practice, the volume and surface integrals are absorbed into a constant, and the density of ionized $A^+$ species is written as

$$ N_{\text{A}}^{+}(E) = \frac{k p_{\text{A}} \sigma_{\text{A}}(E)}{T} \quad (12) $$

The measured ion intensity can be expressed in terms of the ionizer and detector parameters:

$$ I_{\text{A}}(E) = N_{\text{A}}^{+}(E) \varepsilon(x, y, z) \tau_{\text{ion}} \gamma_{\text{A}} f_{\text{A}} \quad (13) $$

Here $\varepsilon(x,y,z)$ is the ion-extraction efficiency, $\tau_{\text{ion}}$ is the transmission probability of the mass analyzer (to be discussed in the

next section), $\gamma_{\text{A}}$ is the detector efficiency (to be discussed in subsequent sections), and $f_{\text{A}}$ is the isotopic abundance. Combining Equations (12) and (13) gives

$$ I_{\text{A}}(E) = \frac{k p_{\text{A}} \sigma_{\text{A}}(E)}{T} \varepsilon(x, y, z) \tau_{\text{ion}} \gamma_{\text{A}} f_{\text{A}} \quad (14) $$

This can be rearranged to

$$ p_{\text{A}} = \frac{k' I_{\text{A}}(E) T}{\gamma_{\text{A}} f_{\text{A}} \sigma_{\text{A}}(E)} \quad (15) $$

The factors $\varepsilon(x,y,z)$ and $\tau_{\text{ion}}$ are absorbed into $k'$. In this expression, $k'$ is commonly referred to as the machine constant and is generally determined from a well-known standard material, such as Au or Ag. Equation (15) is the central equation in KEMS, which relates partial pressure in the Knudsen cell to measured ion intensity.

### 2.2.4 | Ionization cross sections of vapor species

An important quantity in Equation (15) is the $\sigma_{\text{A}}(E)$, the ionization cross section. Since the beginnings of MS, the accuracy of cross sections has been the limiting factor in determining absolute pressure. They are not known with accuracy, and their dependency on parameters such as temperature and ionizing electron energy is not clearly established. Drowart et al.⁸⁷ discuss both atomic and molecular cross sections in detail in their 2005 review of KEMS. Here we shall briefly summarize calculations and measurements of atomic and molecular cross sections and current quantum methods of calculation. Atomic cross sections are better known than molecular cross sections. There are several datasets of atomic cross sections in the literature.⁸⁹⁻⁹² The early calculations of atomic cross sections of Otvos and Stevenson⁹³ and Mann⁹⁰,⁹¹—mainly at their maximum near 70 eV—are based on Bethe's results of cross-sectional proportionality of mean square electron shell radius.⁹³ They use hydrogen-like wave functions to calculate that radius. Then these calculations were scaled using measured ionization cross sections. Bell et al.⁹⁴ have developed an expression to describe the dependence of atomic cross section on ionizing electron energy:

$$ \sigma_{\text{ij}}(E) = \frac{1}{E E_{i+j}} \left[ A_{\text{ij}} \ln \left( \frac{E}{E_{ij}} \right) + \sum_{i=1}^{5} B_{\text{jk}} \left( 1 - \frac{E_{ij}}{E} \right)^{k} \right] \quad (16) $$

Here, $E$ is the electron energy, $E_{ij}$ is the binding energy for orbital $j$, and $A_{ij}$ and $B_{\text{jk}}$ are constants. Bonnell and Hastie have developed a computer program based on Mann's atomic cross sections to calculate the cross section as a function of ionizing electron energy.⁸⁷ This program is currently available on the KEMS web page.⁹⁵ An example of the calculated cross section of Au as a function of electron energy is shown in Figure 13. Many measured cross sections are given in the literature,⁸⁷,⁹⁶ which can be compared to the calculations.
$$ \rightarrow B^{+} + A(g) + 2e^{-} \qquad \sigma_{B^{+}/AB} \text{ AP} \text{— appearance potential (8c)} $$

Here $\sigma$ is the ionization cross section, which will be discussed in subsequent sections of this paper and are covered in detailed in Drowart et al.⁸⁷

### 2.2.3 | Relationship between ion intensity and partial pressure of vapor species

The relationship between partial pressure in the cell and measured ion intensity is central to KEMS. This relationship has been derived in Drowart and Goldfinger,² Grimley,⁴ and Chatillon et al.,¹⁹,⁸⁸ and will be briefly summarized¹⁸ here due to its importance. The number of ions in an infinitesimal unit of volume is given by the following expression:

$$ N_{\text{A}}^{+}(E) d\nu = I_{\text{e}}(E) I_{\text{e}} \rho_{\text{A}} \sigma_{\text{A}}(E) d\nu = I_{\text{e}}(E) I_{\text{e}} \sigma_{\text{A}}(E) \frac{J_{\text{A}}}{c_{\text{A}}} d\nu \quad (9) $$

Here $I_e(E)$ is the intensity of the ionizing electrons, which is a function of their energy, $E$; $I_e$ is the electrons' path length; $\rho_A$ is the density of neutral atoms A in the ionization volume $d\nu$; and $\sigma_A(E)$ is the ionization cross section, which is a function of the ionizing electron energy, $E$. The instantaneous density of the neutral atoms is given by the flux of atoms A from the Knudsen cell orifice, $j_A$, divided by the average molecular speed, $c_A$. This flux reaching the ionizer is scaled by the inverse of distance from the orifice to the ionizer, as discussed previously, and given by the following integral over the surface of the molecular beam entering the ionizer:

$$ j_A = \int_S \left( \frac{W_c p_A}{\sqrt{2\pi M_A R T}} \right) \frac{1}{a^2} ds \quad (10) $$

Here $a$ is the distance from the cell orifice to the ionization region. Equation (10) can be substituted into Equation (9) and integrated over the ionization volume to give

$$ N_{\text{A}}^{+}(E) = \int_{V} \int_{S} \left( \frac{W_c p_A}{\sqrt{2\pi M_A R T}} \right) \frac{1}{a^2} \sigma_A(E) \sqrt{\frac{\pi M_A}{8RT}} ds d\nu \quad (11) $$

In practice, the volume and surface integrals are absorbed into a constant, and the density of ionized $A^+$ species is written as

$$ N_{\text{A}}^{+}(E) = \frac{k p_{\text{A}} \sigma_{\text{A}}(E)}{T} \quad (12) $$

The measured ion intensity can be expressed in terms of the ionizer and detector parameters:

$$ I_{\text{A}}(E) = N_{\text{A}}^{+}(E) \varepsilon(x, y, z) \tau_{\text{ion}} \gamma_{\text{A}} f_{\text{A}} \quad (13) $$

Here $\varepsilon(x,y,z)$ is the ion-extraction efficiency, $\tau_{\text{ion}}$ is the transmission probability of the mass analyzer (to be discussed in the

next section), $\gamma_A$ is the detector efficiency (to be discussed in subsequent sections), and $f_A$ is the isotopic abundance. Combining Equations (12) and (13) gives

$$ I_{\text{A}}(E) = \frac{k p_{\text{A}} \sigma_{\text{A}}(E)}{T} \varepsilon(x, y, z) \tau_{\text{ion}} \gamma_{\text{A}} f_{\text{A}} \quad (14) $$

This can be rearranged to

$$ p_{\text{A}} = \frac{k' I_{\text{A}}(E) T}{\gamma_{\text{A}} f_{\text{A}} \sigma_{\text{A}}(E)} \quad (15) $$

The factors $\varepsilon(x,y,z)$ and $\tau_{\text{ion}}$ are absorbed into $k'$. In this expression, $k'$ is commonly referred to as the machine constant and is generally determined from a well-known standard material, such as Au or Ag. Equation (15) is the central equation in KEMS, which relates partial pressure in the Knudsen cell to measured ion intensity.

### 2.2.4 | Ionization cross sections of vapor species

An important quantity in Equation (15) is the $\sigma_A(E)$, the ionization cross section. Since the beginnings of MS, the accuracy of cross sections has been the limiting factor in determining absolute pressure. They are not known with accuracy, and their dependency on parameters such as temperature and ionizing electron energy is not clearly established. Drowart et al.⁸⁷ discuss both atomic and molecular cross sections in detail in their 2005 review of KEMS. Here we shall briefly summarize calculations and measurements of atomic and molecular cross sections and current quantum methods of calculation. Atomic cross sections are better known than molecular cross sections. There are several datasets of atomic cross sections in the literature.⁸⁹⁻⁹² The early calculations of atomic cross sections of Otvos and Stevenson⁸⁹ and Mann⁹⁰,⁹¹—mainly at their maximum near 70 eV—are based on Bethe's results of cross-sectional proportionality of mean square electron shell radius.⁹³ They use hydrogen-like wave functions to calculate that radius. Then these calculations were scaled using measured ionization cross sections. Bell et al.⁹⁴ have developed an expression to describe the dependence of atomic cross section on ionizing electron energy:

$$ \sigma_j(E) = \frac{1}{E E_{i+j}} \left[ A_j \ln\left(\frac{E}{E_{i,j}}\right) + \sum_{i=1}^{5} B_{jk} \left(1 - \frac{E_{i,j}}{E}\right)^k \right] \quad (16) $$

Here, $E$ is the electron energy, $E_{i,j}$ is the binding energy for orbital $j$, and $A_j$ and $B_{jk}$ are constants. Bonnell and Hastie have developed a computer program based on Mann's atomic cross sections to calculate the cross section as a function of ionizing electron energy.⁸⁷ This program is currently available on the KEMS web page.⁹⁵ An example of the calculated cross section of Au as a function of electron energy is shown in Figure 13. Many measured cross sections are given in the literature,⁸⁷,⁹⁶ which can be compared to the calculations.
<figure><img src="image_25.png" /></figure>

<figure><img src="image_26.png" /><figcaption>**FIGURE 13** Calculated cross section as a function of electron energy. [Color figure can be viewed at wileyonlinelibrary.com]</figcaption></figure>

The ionization cross sections of molecules are much less well understood. Otvos and Stevenson⁸⁹ first proposed the “additivity rule” where ionization cross sections are summed, and then some investigators have applied a “correction factor” for a particular molecule. The “correction factor” is based on measurements of dimer to monomer ratios. Most commonly, a factor of 0.75 is used.⁵ However other factors have been used.⁸⁷ Drowart et al.⁸⁷ have discussed the additivity in regard to known molecules and particularly saturated hydrocarbons and found that it is likely correct and may be related to fragmentation patterns. Younés et al.⁹⁷ and Gotcu-Freis et al.⁷⁷,⁹⁸ have successfully utilized the additivity rule with a full knowledge of the fragmentation pattern in their studies on UO₂ and Pu-Am-O vapors, respectively.

The eventual solution to these uncertainties in both atomic and molecular cross sections may lie in refining the description of the electron-neutral species collision and the application of modern quantum methods to calculations of total and partial cross sections. Kim and Rudd⁹⁹ have developed the binary encounter theory to more accurately describe the interaction between the atom/molecule and an electron. This was further developed by Deutsch and Märk¹⁰⁰⁻¹⁰² in a series of papers. Their approach improves on the “correction” factor in molecular cross sections by using Mulliken orbital population distributions.¹⁰³ These concepts continue to evolve and show promise for resolving this “weak link” in many types of MS.

## 2.3 | Ion filtering/mass analysis

Once a beam of ionized species has been produced and accelerated, the species are sorted based on their mass-to-charge ratio. Three types of ion sorters have been used to analyze high-temperature inorganic vapors: magnetic sector, quadrupole mass filters, and time-of-flight mass filters. Details of these different types of systems can be found in the literature.⁷³,¹⁰⁴ In the past, magnetic sector instruments have been preferred due to their mass-independent

transmission and high resolution. Time-of-flight mass spectrometers do not have mass-dependent sensitivity; however, their overall sensitivity is lower than that of magnetic sector instruments. Ionization efficiency curves can be obtained with accuracy on these instruments due to null electric fields. In regard to quadrupole mass spectrometers, recent improvements have led to instruments with much less mass-dependent sensitivity and higher resolution. Many excellent KEMS studies have been reported using quadrupole mass spectrometers.¹⁰⁵⁻¹⁰⁸

## 2.4 | Ion detectors and signal measurement

### 2.4.1 | Ion detectors

Generally, ions are detected by either a Faraday cup or an electron multiplier. Electron multipliers may be in discrete dynode form or continuous dynode form. The output of either may be measured as ion current or ion counts.

First, consider a Faraday cup.⁷³,¹⁰⁹,¹¹⁰ This is simply a cup or cage (typically several millimeters) that collects ions from the ion sorter, generally with a front electrode to prevent the generated electrons on impact from escaping. The ions hitting the metal produce a small current as they neutralize. This current is the ion intensity and is measured as a potential drop across a precision resistor. A simple Faraday cup provides the raw signal with no amplification.

Most instruments use some type of electron multiplier.⁷³,¹⁰⁹ A conventional electron multiplier contains a conversion dynode, which converts the stream of ions to electrons. This is followed by a series of dynodes (usually 16–20) set up for a “cascading” effect where each dynode ejects more electrons than strike it. Typically, the dynodes are arranged as coupled opened cylinders or as coupled slats in a “venetian blind” arrangement. Cu-Be alloys were the most commonly used dynode materials in the past. The gain, $\gamma$, from the multiplier can be calculated as
<figure><img src="image_25.png" /><figcaption>**FIGURE 13** Calculated cross section as a function of electron energy. [Color figure can be viewed at wileyonlinelibrary.com]</figcaption></figure>

The ionization cross sections of molecules are much less well understood. Otvos and Stevenson⁸⁹ first proposed the “additivity rule” where ionization cross sections are summed, and then some investigators have applied a “correction factor” for a particular molecule. The “correction factor” is based on measurements of dimer to monomer ratios. Most commonly, a factor of 0.75 is used.⁵ However other factors have been used.⁸⁷ Drowart et al.⁸⁷ have discussed the additivity in regard to known molecules and particularly saturated hydrocarbons and found that it is likely correct and may be related to fragmentation patterns. Younés et al.⁹⁷ and Gotcu-Freis et al.⁷⁷,⁹⁸ have successfully utilized the additivity rule with a full knowledge of the fragmentation pattern in their studies on UO₂ and Pu-Am-O vapors, respectively.

The eventual solution to these uncertainties in both atomic and molecular cross sections may lie in refining the description of the electron-neutral species collision and the application of modern quantum methods to calculations of total and partial cross sections. Kim and Rudd⁹⁹ have developed the binary encounter theory to more accurately describe the interaction between the atom/molecule and an electron. This was further developed by Deutsch and Märk¹⁰⁰⁻¹⁰² in a series of papers. Their approach improves on the “correction” factor in molecular cross sections by using Mulliken orbital population distributions.¹⁰³ These concepts continue to evolve and show promise for resolving this “weak link” in many types of MS.

## 2.3 | Ion filtering/mass analysis

Once a beam of ionized species has been produced and accelerated, the species are sorted based on their mass-to-charge ratio. Three types of ion sorters have been used to analyze high-temperature inorganic vapors: magnetic sector, quadrupole mass filters, and time-of-flight mass filters. Details of these different types of systems can be found in the literature.⁷³,¹⁰⁴ In the past, magnetic sector instruments have been preferred due to their mass-independent

transmission and high resolution. Time-of-flight mass spectrometers do not have mass-dependent sensitivity; however, their overall sensitivity is lower than that of magnetic sector instruments. Ionization efficiency curves can be obtained with accuracy on these instruments due to null electric fields. In regard to quadrupole mass spectrometers, recent improvements have led to instruments with much less mass-dependent sensitivity and higher resolution. Many excellent KEMS studies have been reported using quadrupole mass spectrometers.¹⁰⁵⁻¹⁰⁸

## 2.4 | Ion detectors and signal measurement

### 2.4.1 | Ion detectors

Generally, ions are detected by either a Faraday cup or an electron multiplier. Electron multipliers may be in discrete dynode form or continuous dynode form. The output of either may be measured as ion current or ion counts.

First, consider a Faraday cup.⁷³,¹⁰⁹,¹¹⁰ This is simply a cup or cage (typically several millimeters) that collects ions from the ion sorter, generally with a front electrode to prevent the generated electrons on impact from escaping. The ions hitting the metal produce a small current as they neutralize. This current is the ion intensity and is measured as a potential drop across a precision resistor. A simple Faraday cup provides the raw signal with no amplification.

Most instruments use some type of electron multiplier.⁷³,¹⁰⁹ A conventional electron multiplier contains a conversion dynode, which converts the stream of ions to electrons. This is followed by a series of dynodes (usually 16–20) set up for a “cascading” effect where each dynode ejects more electrons than strike it. Typically, the dynodes are arranged as coupled opened cylinders or as coupled slats in a “venetian blind” arrangement. Cu-Be alloys were the most commonly used dynode materials in the past. The gain, $\gamma$, from the multiplier can be calculated as
$$ \gamma = CD^n \qquad (17) $$

Here C is the conversion efficiency of the first dynode, D is the conversion efficiency of subsequent dynodes, and n is the number of dynodes. If C is taken as 1 and D as 2, for a 20-dynode multiplier the resultant gain is on the order of $10^6$. The output of the multiplier is measured as a voltage drop across a precision resistor measured using an electrometer. The usual maximum multiplier gain is ~$10^6$ and is reached by increasing the applied high voltage until the appearance of a “dark” or background current at the most sensitive setting when no ions impinge on the multiplier. The experimental gain is measured for a high-incident ionic current by comparing the Faraday cup signal with the multiplier signal. Degradation of the multiplier, which is typically due to oxidation of the Cu-Be dynodes, is indicated by a loss in gain, that is, the appearance of the “dark” current at lower applied voltages. Ag-Mg dynodes have replaced Cu-Be dynodes on many multipliers due their superior oxidation resistance.¹¹¹

Continuous dynode multipliers are used in many quadrupole instruments.¹⁰⁴ They are constructed of a semiconducting material that is generally more oxidation resistant than the Cu-Be dynodes of a discrete dynode multiplier. They are also more compact and less expensive than discrete dynode multipliers.

Sensitivity independence of mass in multipliers can be achieved using ion counting. Each ion impact produces an electron pulse that is recorded by a counting chain (preamplifier and counter). The height of the pulses is maximized using a high-voltage limit, which sets a gain of ~$10^8$. The counting chain can be set to minimize pulses from the dark current, which are often referred to as “electronic ripple.” The “dark” current, which consists of low-amplitude pulses, is eliminated by an amplitude threshold that must be set. Most Ag-Mg multipliers have a dark current of <1 cps. The counted pulses are the number of ions that impact the first dynode of the multiplier. For optimum performance the multiplier should be protected with shields of Cu, Armco iron, and magnetic µ-metal. As mentioned, the multiplier should be in a portion of the spectrometer that is under UHV.

### 2.4.2 | Ion current and ion counting measurements in the mass spectrum of a vapor

As noted, the multiplier output can be measured as the voltage drop across a precision resistor using an electrometer. The major disadvantage of this method is that the multiplier gain, $γ_A$, is dependent on mass number, $M_A$. This is expressed for mass numbers greater than ~10 as¹¹²

$$ \gamma_A = K\sqrt{M_A} \qquad (18) $$

Here K is a constant. This dependence on mass number needs to be considered when converting ion intensity to partial pressure (Equation 15).

Many modern KEMS instruments use ion counting¹¹³,¹¹⁴ as described in the previous section. This is primarily due to high sensitivity of the ion counting and lack of mass dependence on counting sensitivity. Ion counters typically operate at high speed (≥$10^8$ Hz), and impedances of the ion counting chain-multiplier connections, vacuum flange feedthroughs, preamplifier, and counter have to be matched and suited to high-speed operation. As noted, when plotting gain versus voltage for a multiplier, there is typically a plateau. This is the “counting plateau.” At this voltage counts can be measured. Ion counting has a large dynamic range. In theory, very low count numbers (1 cps) can be accurately measured by extending the counting interval to long times. At the other extreme, high count rates must be corrected for dead time. This is related to the amount of time needed for the counter to process an event or pile-up limit. It is defined as

$$ R_T = \frac{R_m}{(1 - \tau R_m)} \qquad (19) $$

Here $R_T$ and $R_m$ are the true count rate and measured count rate, respectively. $τ$ is the dead time. In practice the dead time is determined from different isotopes of the same material¹⁸,¹¹⁰ and using the known isotopic ratio to determine the dead time correction. Note that the highest count rates should be measured using the Faraday cup.

## 2.5 | Assignment of ion peaks in the mass spectra of a vapor to parent molecules

The assignment of an ion signal to its parent molecule is a critical issue in every application of MS. KEMS has many tools for assigning an ion signal to a particular parent:

1. The ion signal's mass-to-charge ratio. This is the basic mass spectrometric analysis.

2. The isotope distribution of a collected ion. Commercially available computer codes can calculate the isotopic distribution for a species containing atoms with several stable isotopes. This is particularly useful for species with several abundant isotopes, such as Cl or Mo.

3. The “shutterability” of a particular signal. “Shutterability” indicates how much of a signal is background and how much is from the cell. Permanent gases, such as $O_2$, $CO_2$, and $N_2$, tend to be less “shutterable.” Nonetheless, in a well-designed system, a permanent gas whose intensity decreases on closing the shutter emerges from the cell. As noted earlier, this shutter must be a “free plate” that does not valve off the Knudsen cell chamber from the ionization chamber in any way.⁶³

4. The appearance potential. These are tabulated and can be compared to the unknown.
$$ \gamma = CD^n \qquad (17) $$

Here C is the conversion efficiency of the first dynode, D is the conversion efficiency of subsequent dynodes, and n is the number of dynodes. If C is taken as 1 and D as 2, for a 20-dynode multiplier the resultant gain is on the order of $10^6$. The output of the multiplier is measured as a voltage drop across a precision resistor measured using an electrometer. The usual maximum multiplier gain is ~$10^6$ and is reached by increasing the applied high voltage until the appearance of a “dark” or background current at the most sensitive setting when no ions impinge on the multiplier. The experimental gain is measured for a high-incident ionic current by comparing the Faraday cup signal with the multiplier signal. Degradation of the multiplier, which is typically due to oxidation of the Cu-Be dynodes, is indicated by a loss in gain, that is, the appearance of the “dark” current at lower applied voltages. Ag-Mg dynodes have replaced Cu-Be dynodes on many multipliers due their superior oxidation resistance.¹¹¹

Continuous dynode multipliers are used in many quadrupole instruments.¹⁰⁴ They are constructed of a semiconducting material that is generally more oxidation resistant than the Cu-Be dynodes of a discrete dynode multiplier. They are also more compact and less expensive than discrete dynode multipliers.

Sensitivity independence of mass in multipliers can be achieved using ion counting. Each ion impact produces an electron pulse that is recorded by a counting chain (preamplifier and counter). The height of the pulses is maximized using a high-voltage limit, which sets a gain of ~$10^8$. The counting chain can be set to minimize pulses from the dark current, which are often referred to as “electronic ripple.” The “dark” current, which consists of low-amplitude pulses, is eliminated by an amplitude threshold that must be set. Most Ag-Mg multipliers have a dark current of <1 cps. The counted pulses are the number of ions that impact the first dynode of the multiplier. For optimum performance the multiplier should be protected with shields of Cu, Armco iron, and magnetic µ-metal. As mentioned, the multiplier should be in a portion of the spectrometer that is under UHV.

### 2.4.2 | Ion current and ion counting measurements in the mass spectrum of a vapor

As noted, the multiplier output can be measured as the voltage drop across a precision resistor using an electrometer. The major disadvantage of this method is that the multiplier gain, $γ_A$, is dependent on mass number, $M_A$. This is expressed for mass numbers greater than ~10 as¹¹²

$$ \gamma_A = K\sqrt{M_A} \qquad (18) $$

Here K is a constant. This dependence on mass number needs to be considered when converting ion intensity to partial pressure (Equation 15).

Many modern KEMS instruments use ion counting¹¹³,¹¹⁴ as described in the previous section. This is primarily due to high sensitivity of the ion counting and lack of mass dependence on counting sensitivity. Ion counters typically operate at high speed (≥$10^8$ Hz), and impedances of the ion counting chain-multiplier connections, vacuum flange feedthroughs, preamplifier, and counter have to be matched and suited to high-speed operation. As noted, when plotting gain versus voltage for a multiplier, there is typically a plateau. This is the “counting plateau.” At this voltage counts can be measured. Ion counting has a large dynamic range. In theory, very low count numbers (1 cps) can be accurately measured by extending the counting interval to long times. At the other extreme, high count rates must be corrected for dead time. This is related to the amount of time needed for the counter to process an event or pile-up limit. It is defined as

$$ R_T = \frac{R_m}{(1 - \tau R_m)} \qquad (19) $$

Here $R_T$ and $R_m$ are the true count rate and measured count rate, respectively. $τ$ is the dead time. In practice the dead time is determined from different isotopes of the same material¹⁸,¹¹⁰ and using the known isotopic ratio to determine the dead time correction. Note that the highest count rates should be measured using the Faraday cup.

## 2.5 | Assignment of ion peaks in the mass spectra of a vapor to parent molecules

The assignment of an ion signal to its parent molecule is a critical issue in every application of MS. KEMS has many tools for assigning an ion signal to a particular parent:

1. The ion signal's mass-to-charge ratio. This is the basic mass spectrometric analysis.

2. The isotope distribution of a collected ion. Commercially available computer codes can calculate the isotopic distribution for a species containing atoms with several stable isotopes. This is particularly useful for species with several abundant isotopes, such as Cl or Mo.

3. The “shutterability” of a particular signal. “Shutterability” indicates how much of a signal is background and how much is from the cell. Permanent gases, such as $O_2$, $CO_2$, and $N_2$, tend to be less “shutterable.” Nonetheless, in a well-designed system, a permanent gas whose intensity decreases on closing the shutter emerges from the cell. As noted earlier, this shutter must be a “free plate” that does not valve off the Knudsen cell chamber from the ionization chamber in any way.⁶³

4. The appearance potential. These are tabulated and can be compared to the unknown.
5. The fragmentation pattern, which is known for some types of species. If the fragmentation pattern is not known, it must be studied by monitoring the ionization efficiency curves for all the ions in the system. Changes in these curves must be known with changes in the system composition.

6. Comparison to calculated/estimated vapor pressures for the condensed phase in the Knudsen cell. For a complex system with multiple vapor species, a Gibbs free energy minimization code can predict many of the expected species. FactSage¹¹⁵ and ThermoCalc¹¹⁶ include extensive solution databases and therefore give good approximations for complex alloys and oxide systems. It should be noted that these calculated compositions are for a perfect equilibrium, which does exist in a Knudsen cell. Nonetheless, these calculations provide a good guideline for which species might be observed. As an example, take a complex lunar basalt¹¹⁷ and calculate the expected vapor species using the SLAG database in FactSage¹¹⁵ at 2500 K. The results are presented in Table 1. The experimental study of De Maria et al.¹¹⁸ reports observations of many of the predicted vapor species.

## 2.6 | Vacuum system

KEMS experiments necessarily must be conducted under high vacuum.¹¹⁹ KEMS instruments have at least two differentially pumped stages—one for the Knudsen cell and furnace and one for the ionizer. The Knudsen cell chamber should have a higher-speed pump and, as noted, is separated by a Cu plate and/or cold trap. The speed of this pump is determined by the necessity to pump away effusing permanent gases from the Knudsen cell. Condensates from the Knudsen cell should collect in this chamber and not the ionizer chamber, which limits any “memory” effects. The ionizer chamber is typically much cleaner.

The significant advances made in vacuum technology over recent years have enhanced modern KEMS instruments.¹²⁰ Vacuum systems—including the high-vacuum and fore-vacuum pumps—are now available as completely oil free, which will minimize background in a sensitive mass spectrometer. The pumping system shown in Figure 1 is illustrative of the number of pumps needed for a magnetic sector instrument: there is a large turbomolecular pump (500 L/s) in

<table><caption>TABLE 1 Predicted vapor species of a lunar basalt at 2500 K (from FactSage¹¹⁵ calculations) compared to experimentally observed species reported by De Maria et al.¹¹⁸</caption><thead><tr><th colspan="2">Chemical analysis (Snyder et al.<sup>117</sup>)<sup>a</sup></th><th>Species</th><th>Predicted and observed<br>species from lunar<br>basalt sample 12022</th><th></th></tr></thead><tbody><tr><td colspan="2">SiO<sub>2</sub></td><td>43.2</td><td>PO<sub>2</sub>(g)</td><td>0.004569408</td></tr><tr><td colspan="2">TiO<sub>2</sub></td><td>5.16</td><td>SiO(g)</td><td>0.004541139</td><td>✓</td></tr><tr><td colspan="2">Al<sub>2</sub>O<sub>3</sub></td><td>9.04</td><td>Fe(g)</td><td>0.003317841</td><td>✓</td></tr><tr><td colspan="2">FeO</td><td>21.44</td><td>Na(g)</td><td>0.002800994</td><td>✓</td></tr><tr><td colspan="2">MnO</td><td>0.25</td><td>PO(g)</td><td>0.001925179</td><td></td></tr><tr><td colspan="2">MgO</td><td>10.43</td><td>K(g)</td><td>0.000794608</td><td>✓</td></tr><tr><td colspan="2">CaO</td><td>9.56</td><td>FeO(g)</td><td>0.00031189</td><td></td></tr><tr><td colspan="2">Na<sub>2</sub>O</td><td>0.47</td><td>O(g)</td><td>0.000156368</td><td>✓</td></tr><tr><td colspan="2">K<sub>2</sub>O</td><td>0.07</td><td>O<sub>2</sub>(g)</td><td>0.000116427</td><td>✓</td></tr><tr><td colspan="2">P<sub>2</sub>O<sub>5</sub></td><td>0.13</td><td>Mg(g)</td><td>0.000107833</td><td>✓</td></tr><tr><td colspan="2"></td><td></td><td>SiO<sub>2</sub>(g)</td><td>6.14219E-05</td><td>✓</td></tr><tr><td colspan="2"></td><td></td><td>TiO<sub>2</sub>(g)</td><td>7.34285E-06</td><td>✓</td></tr><tr><td colspan="2"></td><td></td><td>P(g)</td><td>3.85922E-06</td><td></td></tr><tr><td colspan="2"></td><td></td><td>KO(g)</td><td>1.08059E-06</td><td></td></tr><tr><td colspan="2"></td><td></td><td>TiO(g)</td><td>2.00284E-07</td><td>✓</td></tr><tr><td colspan="2"></td><td></td><td>Ca(g)</td><td>1.0955E-07</td><td>✓</td></tr><tr><td colspan="2"></td><td></td><td>CaO(g)</td><td>1.8925E-08</td><td></td></tr><tr><td colspan="2"></td><td></td><td>Al(g)</td><td>3.66595E-09</td><td>✓</td></tr><tr><td colspan="2"></td><td></td><td>Si(g)</td><td>3.17871E-09</td><td></td></tr><tr><td colspan="2"></td><td></td><td>K<sub>2</sub>(g)</td><td>8.01489E-10</td><td></td></tr><tr><td colspan="2"></td><td></td><td>Ti(g)</td><td>1.85167E-11</td><td></td></tr><tr><td colspan="2"></td><td></td><td></td><td>Cr</td><td></td></tr><tr><td colspan="2"></td><td></td><td></td><td>AlO</td><td></td></tr><tr><td colspan="2"></td><td></td><td></td><td>Al<sub>2</sub>O</td><td></td></tr></tbody></table>

<footnote>a3300 ppm by weight Cr (form not clear and not included in calculations).</footnote>
5. The fragmentation pattern, which is known for some types of species. If the fragmentation pattern is not known, it must be studied by monitoring the ionization efficiency curves for all the ions in the system. Changes in these curves must be known with changes in the system composition.

6. Comparison to calculated/estimated vapor pressures for the condensed phase in the Knudsen cell. For a complex system with multiple vapor species, a Gibbs free energy minimization code can predict many of the expected species. FactSage¹¹⁵ and ThermoCalc¹¹⁶ include extensive solution databases and therefore give good approximations for complex alloys and oxide systems. It should be noted that these calculated compositions are for a perfect equilibrium, which does exist in a Knudsen cell. Nonetheless, these calculations provide a good guideline for which species might be observed. As an example, take a complex lunar basalt¹¹⁷ and calculate the expected vapor species using the SLAG database in FactSage¹¹⁵ at 2500 K. The results are presented in Table 1. The experimental study of De Maria et al.¹¹⁸ reports observations of many of the predicted vapor species.

## 2.6 Vacuum system

KEMS experiments necessarily must be conducted under high vacuum.¹¹⁹ KEMS instruments have at least two differentially pumped stages—one for the Knudsen cell and furnace and one for the ionizer. The Knudsen cell chamber should have a higher-speed pump and, as noted, is separated by a Cu plate and/or cold trap. The speed of this pump is determined by the necessity to pump away effusing permanent gases from the Knudsen cell. Condensates from the Knudsen cell should collect in this chamber and not the ionizer chamber, which limits any “memory” effects. The ionizer chamber is typically much cleaner.

The significant advances made in vacuum technology over recent years have enhanced modern KEMS instruments.¹²⁰ Vacuum systems—including the high-vacuum and fore-vacuum pumps—are now available as completely oil free, which will minimize background in a sensitive mass spectrometer. The pumping system shown in Figure 1 is illustrative of the number of pumps needed for a magnetic sector instrument: there is a large turbomolecular pump (500 L/s) in

<table><caption>TABLE 1 Predicted vapor species of a lunar basalt at 2500 K (from FactSage¹¹⁵ calculations) compared to experimentally observed species reported by De Maria et al.¹¹⁸</caption><thead><tr><th colspan="2">Chemical analysis (Snyder et al.<sup>117</sup>)<sup>a</sup></th><th>Species</th><th>Predicted and observed<br>species from lunar<br>basalt sample 12022</th><th></th></tr></thead><tbody><tr><td colspan="2">SiO<sub>2</sub></td><td>43.2</td><td>PO<sub>2</sub>(g)</td><td>0.004569408</td></tr><tr><td colspan="2">TiO<sub>2</sub></td><td>5.16</td><td>SiO(g)</td><td>0.004541139</td><td>✓</td></tr><tr><td colspan="2">Al<sub>2</sub>O<sub>3</sub></td><td>9.04</td><td>Fe(g)</td><td>0.003317841</td><td>✓</td></tr><tr><td colspan="2">FeO</td><td>21.44</td><td>Na(g)</td><td>0.002800994</td><td>✓</td></tr><tr><td colspan="2">MnO</td><td>0.25</td><td>PO(g)</td><td>0.001925179</td><td></td></tr><tr><td colspan="2">MgO</td><td>10.43</td><td>K(g)</td><td>0.000794608</td><td>✓</td></tr><tr><td colspan="2">CaO</td><td>9.56</td><td>FeO(g)</td><td>0.00031189</td><td></td></tr><tr><td colspan="2">Na<sub>2</sub>O</td><td>0.47</td><td>O(g)</td><td>0.000156368</td><td>✓</td></tr><tr><td colspan="2">K<sub>2</sub>O</td><td>0.07</td><td>O<sub>2</sub>(g)</td><td>0.000116427</td><td>✓</td></tr><tr><td colspan="2">P<sub>2</sub>O<sub>5</sub></td><td>0.13</td><td>Mg(g)</td><td>0.000107833</td><td>✓</td></tr><tr><td colspan="2"></td><td></td><td>SiO<sub>2</sub>(g)</td><td>6.14219E-05</td><td>✓</td></tr><tr><td colspan="2"></td><td></td><td>TiO<sub>2</sub>(g)</td><td>7.34285E-06</td><td>✓</td></tr><tr><td colspan="2"></td><td></td><td>P(g)</td><td>3.85922E-06</td><td></td></tr><tr><td colspan="2"></td><td></td><td>KO(g)</td><td>1.08059E-06</td><td></td></tr><tr><td colspan="2"></td><td></td><td>TiO(g)</td><td>2.00284E-07</td><td>✓</td></tr><tr><td colspan="2"></td><td></td><td>Ca(g)</td><td>1.0955E-07</td><td>✓</td></tr><tr><td colspan="2"></td><td></td><td>CaO(g)</td><td>1.8925E-08</td><td></td></tr><tr><td colspan="2"></td><td></td><td>Al(g)</td><td>3.66595E-09</td><td>✓</td></tr><tr><td colspan="2"></td><td></td><td>Si(g)</td><td>3.17871E-09</td><td></td></tr><tr><td colspan="2"></td><td></td><td>K<sub>2</sub>(g)</td><td>8.01489E-10</td><td></td></tr><tr><td colspan="2"></td><td></td><td>Ti(g)</td><td>1.85167E-11</td><td></td></tr><tr><td colspan="2"></td><td></td><td></td><td>Cr</td><td></td></tr><tr><td colspan="2"></td><td></td><td></td><td>AlO</td><td></td></tr><tr><td colspan="2"></td><td></td><td></td><td>Al<sub>2</sub>O<sub>3</sub></td><td></td></tr></tbody></table>

<footnote>a3300 ppm by weight Cr (form not clear and not included in calculations).</footnote>
<figure><img src="image_27.png" /></figure>

the Knudsen cell/furnace chamber, another turbomolecular pump (240 L/s) in the ionization chamber, an ion pump (50 L/s) in the ion sorting region, and an ion pump (50 L/s) in the detector.

Most newer systems use turbomolecular pumps. However, modern diffusion pumps and oils are still less expensive and offer the advantage of uniform pumping speeds for all gases. With proper liquid nitrogen traps and a liquid nitrogen cold finger, the background is minimal due to only normal outgassing of the housing materials. As noted, pumping speed in the Knudsen cell chamber is dictated by the requirement to pump permanent gases from the effusion cells. The SIMaP installation uses 800-L/s diffusion pumps each in the Knudsen cell and the ionization chamber. With a multiple cell effusing $4 \times 10^{-4}$ mbar (Knudsen limit), the pressure in the Knudsen cell chamber reaches $4 \times 10^{-5}$ mbar, and the ionization chamber remains undisturbed at $2 \times 10^{-8}$ mbar. Note that the restricted collimation strongly limits gas exchange between the Knudsen cell chamber and the ionization chamber.

Many systems also have liquid nitrogen cold fingers in the ionization chamber that provide additional local pumping. High-speed, ultraclean cryo-pumps have been used in some KEMS systems with good success (Hildenbrand, private communication). Cryo-pumps may help with improving background gas pressures to obtain a larger range in the measurements of permanent gases such as oxygen and nitrogen effusing from within the Knudsen cell. A special case for trapping background gases in use is at SIMaP. Here a liquid nitrogen cold finger is in contact with the µ-metal ionization box. This setup, in association with restricted collimation, allows (1) the trapping of any high volatile vapors from the molecular beam, for example, $P_4(g)$ and $P_2(g)$ molecules in the vapor of $InP$,¹²¹,¹²² and (2) the pumping of residual vapors that enter the ionization zone; for example, it was found that the addition of liquid nitrogen to this device decreases the peak at mass 28 by a factor of 80, and no peaks were detectable above mass 80.

Additionally, chambers of the instrument can be valved off and remain under clean vacuum when the rest of the system is opened to air; the system at NASA Glenn allows the ion sorting and detector regions to be valved off. As discussed earlier, other systems allow the region from the ionizer to the detector to be fully valved off and kept under vacuum at all times. This not only keeps the system cleaner and lowers the background spectrum but also allows the electronics to remain on and gives the instrument enhanced stability.

## 2.7 | KEMS systems in glove boxes

As previously noted, KEMS is particularly useful in nuclear applications, which necessarily involve the handling of radioactive materials. Several KEMS instruments in glove boxes exist worldwide.⁷²,¹²³⁻¹²⁵ A glove box can be used to either protect an air-sensitive sample from exposure before and during loading in the mass spectrometer or protect the user and environment from a hazardous sample. Glove boxes for the former are widely commercially available. Generally the glove box operates at a slight overpressure so that any

gas flows outside and thus avoids glove box atmosphere contamination in case of leak or accidental opening of the confinement. Usually, an inert gas like nitrogen or argon circulates in a loop in the glove box and through passive or active filters cleaning the circulating gas from contaminants such as oxygen, water vapor, and solvent.

Next consider nuclear applications in which the glove box is necessary to protect the user from inhalation or ingestion of hazards. Nuclear-safe glove boxes are manufactured by specialized companies. These glove boxes are typically kept at a pressure that is lower than the laboratory environment. The under-pressure results in flow to inside the box in case of leak or accidental opening of the box and avoids the contamination of the outside atmosphere. As previously described for sample-protecting glove boxes, a cover gas circulates through the glove box with passive or active filters. It has been demonstrated that such systems can keep atmosphere to ppm levels for $O_2$ and $H_2O$.¹²⁶ Those glove boxes protect both users and samples.

Figure 14 shows the coupling of a glove box with a mass spectrometer. There are several things to consider in such a configuration:

1. The stability of the gas circulation system may have an impact on the stability of auxiliary instruments such as balances.

2. The use of a glove box induces severe limitations in terms of physical manipulation. The use of gloves limits operator dexterity in loading/unloading delicate samples and servicing the precision components, such as the ionizer.

3. The introduction of tools and instrumentation into the glove box may render them no longer useable for other tasks.

4. Some spectrometers may not be compatible with the glove box atmosphere. A very dry atmosphere can lead to electrostatic loading issues that would not occur at room humidity levels. This may affect electronics. Atmospheres such as argon may lead to insufficient cooling of the spectrometer due to the low thermal conductivity of argon. Very low oxygen potentials can even have an impact on spectrometer construction materials.

Points 1 and 4 can be addressed by monitoring and improving the stability of the circulating system as well as auxiliary cooling of the mass spectrometer. Alarms are especially important for contaminants in the circulating gas and for overtemperature.

## 2.8 | Data acquisition systems in KEMS

Data acquisition systems have developed rapidly in recent years. Many control aspects of a KEMS system can be automated, and data analysis can be run using a computer. Given the unique aspects of each system, KEMS systems tend to have custom-written software.¹⁸,¹²⁷,¹²⁸

Computer control of vacuum systems is well established. Many vacuum systems have automated pump-down procedures and alarms/
<figure><img src="image_26.png" /></figure>

the Knudsen cell/furnace chamber, another turbomolecular pump (240 L/s) in the ionization chamber, an ion pump (50 L/s) in the ion sorting region, and an ion pump (50 L/s) in the detector.

Most newer systems use turbomolecular pumps. However, modern diffusion pumps and oils are still less expensive and offer the advantage of uniform pumping speeds for all gases. With proper liquid nitrogen traps and a liquid nitrogen cold finger, the background is minimal due to only normal outgassing of the housing materials. As noted, pumping speed in the Knudsen cell chamber is dictated by the requirement to pump permanent gases from the effusion cells. The SIMaP installation uses 800-L/s diffusion pumps each in the Knudsen cell and the ionization chamber. With a multiple cell effusing $4 \times 10^{-4}$ mbar (Knudsen limit), the pressure in the Knudsen cell chamber reaches $4 \times 10^{-5}$ mbar, and the ionization chamber remains undisturbed at $2 \times 10^{-8}$ mbar. Note that the restricted collimation strongly limits gas exchange between the Knudsen cell chamber and the ionization chamber.

Many systems also have liquid nitrogen cold fingers in the ionization chamber that provide additional local pumping. High-speed, ultraclean cryo-pumps have been used in some KEMS systems with good success (Hildenbrand, private communication). Cryo-pumps may help with improving background gas pressures to obtain a larger range in the measurements of permanent gases such as oxygen and nitrogen effusing from within the Knudsen cell. A special case for trapping background gases in use is at SIMaP. Here a liquid nitrogen cold finger is in contact with the μ-metal ionization box. This setup, in association with restricted collimation, allows (1) the trapping of any high volatile vapors from the molecular beam, for example, $P_4(g)$ and $P_2(g)$ molecules in the vapor of $InP$,¹²¹,¹²² and (2) the pumping of residual vapors that enter the ionization zone; for example, it was found that the addition of liquid nitrogen to this device decreases the peak at mass 28 by a factor of 80, and no peaks were detectable above mass 80.

Additionally, chambers of the instrument can be valved off and remain under clean vacuum when the rest of the system is opened to air; the system at NASA Glenn allows the ion sorting and detector regions to be valved off. As discussed earlier, other systems allow the region from the ionizer to the detector to be fully valved off and kept under vacuum at all times. This not only keeps the system cleaner and lowers the background spectrum but also allows the electronics to remain on and gives the instrument enhanced stability.

## 2.7 | KEMS systems in glove boxes

As previously noted, KEMS is particularly useful in nuclear applications, which necessarily involve the handling of radioactive materials. Several KEMS instruments in glove boxes exist worldwide.⁷²,¹²³⁻¹²⁵ A glove box can be used to either protect an air-sensitive sample from exposure before and during loading in the mass spectrometer or protect the user and environment from a hazardous sample. Glove boxes for the former are widely commercially available. Generally the glove box operates at a slight overpressure so that any

gas flows outside and thus avoids glove box atmosphere contamination in case of leak or accidental opening of the confinement. Usually, an inert gas like nitrogen or argon circulates in a loop in the glove box and through passive or active filters cleaning the circulating gas from contaminants such as oxygen, water vapor, and solvent.

Next consider nuclear applications in which the glove box is necessary to protect the user from inhalation or ingestion of hazards. Nuclear-safe glove boxes are manufactured by specialized companies. These glove boxes are typically kept at a pressure that is lower than the laboratory environment. The under-pressure results in flow to inside the box in case of leak or accidental opening of the box and avoids the contamination of the outside atmosphere. As previously described for sample-protecting glove boxes, a cover gas circulates through the glove box with passive or active filters. It has been demonstrated that such systems can keep atmosphere to ppm levels for $O_2$ and $H_2O$.¹²⁶ Those glove boxes protect both users and samples.

Figure 14 shows the coupling of a glove box with a mass spectrometer. There are several things to consider in such a configuration:

1. The stability of the gas circulation system may have an impact on the stability of auxiliary instruments such as balances.

2. The use of a glove box induces severe limitations in terms of physical manipulation. The use of gloves limits operator dexterity in loading/unloading delicate samples and servicing the precision components, such as the ionizer.

3. The introduction of tools and instrumentation into the glove box may render them no longer useable for other tasks.

4. Some spectrometers may not be compatible with the glove box atmosphere. A very dry atmosphere can lead to electrostatic loading issues that would not occur at room humidity levels. This may affect electronics. Atmospheres such as argon may lead to insufficient cooling of the spectrometer due to the low thermal conductivity of argon. Very low oxygen potentials can even have an impact on spectrometer construction materials.

Points 1 and 4 can be addressed by monitoring and improving the stability of the circulating system as well as auxiliary cooling of the mass spectrometer. Alarms are especially important for contaminants in the circulating gas and for overtemperature.

## 2.8 | Data acquisition systems in KEMS

Data acquisition systems have developed rapidly in recent years. Many control aspects of a KEMS system can be automated, and data analysis can be run using a computer. Given the unique aspects of each system, KEMS systems tend to have custom-written software.¹⁸,¹²⁷,¹²⁸

Computer control of vacuum systems is well established. Many vacuum systems have automated pump-down procedures and alarms/
<figure><img src="image_28.png" /><figcaption>FIGURE 14 KEMS system in a nuclearized glove box with lead shields. [Color figure can be viewed at wileyonlinelibrary.com]</figcaption></figure>

set points controlled by a computer. Commercial quadrupoles today nearly always include computer control of ion optics, the mass filter, and the detector. This can be combined with KEMS-specific software that controls the furnace, reads temperature from a thermocouple or pyrometer, controls cell position, and operates the shutter.¹²⁹ Most data systems transfer the data to a text file for analysis using commercially available spreadsheet and/or graphing programs.

Larger systems may use a single custom-written code. The data system for the NASA Glenn system has been described in Copland and Jacobson.¹⁸ The current code is written in LabView (National Instruments, Austin, TX, USA), which offers much flexibility in the control of the instrumentation. In the NASA Glenn system, the following devices are controlled/monitored via computer:

1. Furnace. The control voltage and current to a resistively heated furnace can be controlled. Ramping of temperature is also possible.

2. Temperature measurement via pyrometer or thermocouple.

3. Stepper motors—for opening/closing the shutter and positioning cells.

4. Electron energy selector.

5. Magnet for selecting mass. The mass-to-charge ratio is

$$ \frac{m}{q} = \frac{r^2 B^2}{2V} \qquad (20) $$

Here m is the mass, q is the charge, r is the radius of the drift tube, B is the magnetic field strength, and V is the accelerating voltage. Note that this equation is for an ideal situation; in practice the expression is more complex due to magnetic leaks. Generally the mass-to-charge scale is calibrated using standards of known mass

numbers (e.g., inert gases) and fitted to a polynomial. Some instruments first select an approximate mass-to-charge ratio (m/q) with the magnetic field and then scan over a particular mass range with voltage. Care must be taken to scan over only a narrow voltage range to avoid sensitivity changes.

6. Electrometer for ion current signal measurement.

7. Ion counting for signal measurement. This includes control of the count interval (an accurate clock is part of the counting chain) and number of repetitive counts.

A KEMS/LabView application has also been developed at JRC Karlsruhe.¹²⁷ It is available in open source on GitHub (https://github.com). It includes the following:

* A user interface for live overview, experiment management, equipment configuration and control, metadata management, and text file saving.

* Instrument acquisition/control loops—simple acquisition, synchronized acquisition, complex control acquisition (e.g., furnace temperature control with furnace movement), and a mass spectrometer management loop.

* Full documentation of software.

The newly designed KEMS device which is currently developed at Mannheim University of Applied Sciences will use a software based on Siemens SPS System. The goal besides the aforementioned is to meet some standard procedures which can be reused for other groups who want to build up own slightly modified devices based on our experiences. The software allows maintenance from a distance as well as live demonstrations of the experiments which are advantageous also for teaching and academia.
<figure><img src="image_27.png" /><figcaption>FIGURE 14 KEMS system in a nuclearized glove box with lead shields. [Color figure can be viewed at wileyonlinelibrary.com]</figcaption></figure>

set points controlled by a computer. Commercial quadrupoles today nearly always include computer control of ion optics, the mass filter, and the detector. This can be combined with KEMS-specific software that controls the furnace, reads temperature from a thermocouple or pyrometer, controls cell position, and operates the shutter.¹²⁹ Most data systems transfer the data to a text file for analysis using commercially available spreadsheet and/or graphing programs.

Larger systems may use a single custom-written code. The data system for the NASA Glenn system has been described in Copland and Jacobson.¹⁸ The current code is written in LabView (National Instruments, Austin, TX, USA), which offers much flexibility in the control of the instrumentation. In the NASA Glenn system, the following devices are controlled/monitored via computer:

1. Furnace. The control voltage and current to a resistively heated furnace can be controlled. Ramping of temperature is also possible.

2. Temperature measurement via pyrometer or thermocouple.

3. Stepper motors—for opening/closing the shutter and positioning cells.

4. Electron energy selector.

5. Magnet for selecting mass. The mass-to-charge ratio is

$$ \frac{m}{q} = \frac{r^2 B^2}{2V} \qquad (20) $$

Here m is the mass, q is the charge, r is the radius of the drift tube, B is the magnetic field strength, and V is the accelerating voltage. Note that this equation is for an ideal situation; in practice the expression is more complex due to magnetic leaks. Generally the mass-to-charge scale is calibrated using standards of known mass

numbers (e.g., inert gases) and fitted to a polynomial. Some instruments first select an approximate mass-to-charge ratio (m/q) with the magnetic field and then scan over a particular mass range with voltage. Care must be taken to scan over only a narrow voltage range to avoid sensitivity changes.

6. Electrometer for ion current signal measurement.

7. Ion counting for signal measurement. This includes control of the count interval (an accurate clock is part of the counting chain) and number of repetitive counts.

A KEMS/LabView application has also been developed at JRC Karlsruhe.¹²⁷ It is available in open source on GitHub (https://github.com). It includes the following:

* A user interface for live overview, experiment management, equipment configuration and control, metadata management, and text file saving.

* Instrument acquisition/control loops—simple acquisition, synchronized acquisition, complex control acquisition (e.g., furnace temperature control with furnace movement), and a mass spectrometer management loop.

* Full documentation of software.

The newly designed KEMS device which is currently developed at Mannheim University of Applied Sciences will use a software based on Siemens SPS System. The goal besides the aforementioned is to meet some standard procedures which can be reused for other groups who want to build up own slightly modified devices based on our experiences. The software allows maintenance from a distance as well as live demonstrations of the experiments which are advantageous also for teaching and academia.
In principle, many KEMS measurements can be fully automated. The instrument at KFA-Juelich has this capability for second law enthalpies. However, KEMS remains a research technique with many measurements that are quite system specific. Every thermochemical experiment must be run by the experimenter as a function of successive observations. Automation would be based on elementary actions specifically selected at each run. Full automation for every experiment remains elusive because these actions are specific to each experiment.

## 3 | MEASUREMENTS OF PARTIAL PRESSURES AND THERMODYNAMIC PROPERTIES USING KEMS

In this section we discuss actual KEMS measurements. The primary application of KEMS has been in physical chemistry, specifically the chemical thermodynamics of inorganic and some organic compounds. This is summarized in Figure 15. Chemical thermodynamics has changed in recent years due to the dramatic advances in computational quantum chemistry. We will show how modern quantum chemical calculations can be used in conjunction with KEMS to generate better data. In addition, KEMS can be a powerful tool in solution thermodynamics and used to generate model parameters.

### 3.1 | Thermochemistry of pure compounds

The principal reaction under study in KEMS is the transformation of a condensed phase into a vapor phase. Consider, for example, the

vaporization of a solid pure element or compound A into its gas, with partial pressure, pA. We assume congruent vaporization, in which the vapor has the same composition as the solid:

$$A(s) = A(g) \qquad (21)$$

$$\Delta_v G^\circ_T = -RT \ln K_p = -RT \ln p_A = \Delta_v H^\circ_T - T \Delta_v S^\circ_T \qquad (22)$$

Here ΔvG°T is the free energy of vaporization, R is the gas constant, T is the absolute temperature, Kp is the equilibrium constant for reaction (21), ΔvH°T is the enthalpy change associated with vaporization, and ΔvS°T is the entropy change associated with vaporization. Thus a variety of thermodynamic quantities can be extracted from measurements of partial pressures of each species in the vapor. There are two primary methods to do this, the “second law” and the “third law” method. We shall discuss each of these.

#### 3.1.1 | Second law thermodynamic method

The second law thermodynamic method to obtain an enthalpy of vaporization is probably the most common method used in KEMS. Taking the differential of Equation (22) gives the van't Hoff equation:

$$\frac{d\ln p_A}{d(1/T)} = -\frac{\Delta_v H^\circ_T}{R} \qquad (23a)$$

Equation (15) can be combined with Equation (23a) leading to

<figure><img src="image_29.png" /><figcaption>FIGURE 15 Thermodynamic properties commonly measured using KEMS (Knudsen effusion mass spectrometry). [Color figure can be viewed at wileyonlinelibrary.com]</figcaption></figure>
In principle, many KEMS measurements can be fully automated. The instrument at KFA-Juelich has this capability for second law enthalpies. However, KEMS remains a research technique with many measurements that are quite system specific. Every thermochemical experiment must be run by the experimenter as a function of successive observations. Automation would be based on elementary actions specifically selected at each run. Full automation for every experiment remains elusive because these actions are specific to each experiment.

## 3 | MEASUREMENTS OF PARTIAL PRESSURES AND THERMODYNAMIC PROPERTIES USING KEMS

In this section we discuss actual KEMS measurements. The primary application of KEMS has been in physical chemistry, specifically the chemical thermodynamics of inorganic and some organic compounds. This is summarized in Figure 15. Chemical thermodynamics has changed in recent years due to the dramatic advances in computational quantum chemistry. We will show how modern quantum chemical calculations can be used in conjunction with KEMS to generate better data. In addition, KEMS can be a powerful tool in solution thermodynamics and used to generate model parameters.

### 3.1 | Thermochemistry of pure compounds

The principal reaction under study in KEMS is the transformation of a condensed phase into a vapor phase. Consider, for example, the

vaporization of a solid pure element or compound A into its gas, with partial pressure, $p_A$. We assume congruent vaporization, in which the vapor has the same composition as the solid:

$$A(s) = A(g) \qquad (21)$$

$$\Delta_v G^\circ_T = -RT \ln K_p = -RT \ln p_A = \Delta_v H^\circ_T - T \Delta_v S^\circ_T \qquad (22)$$

Here $\Delta_v G^\circ_T$ is the free energy of vaporization, R is the gas constant, T is the absolute temperature, $K_p$ is the equilibrium constant for reaction (21), $\Delta_v H^\circ_T$ is the enthalpy change associated with vaporization, and $\Delta_v S^\circ_T$ is the entropy change associated with vaporization. Thus a variety of thermodynamic quantities can be extracted from measurements of partial pressures of each species in the vapor. There are two primary methods to do this, the “second law” and the “third law” method. We shall discuss each of these.

#### 3.1.1 | Second law thermodynamic method

The second law thermodynamic method to obtain an enthalpy of vaporization is probably the most common method used in KEMS. Taking the differential of Equation (22) gives the van't Hoff equation:

$$\frac{d \ln p_A}{d(1/T)} = - \frac{\Delta_v H^\circ_T}{R} \qquad (23a)$$

Equation (15) can be combined with Equation (23a) leading to

<figure><img src="image_28.png" /><figcaption>FIGURE 15 Thermodynamic properties commonly measured using KEMS (Knudsen effusion mass spectrometry). [Color figure can be viewed at wileyonlinelibrary.com]</figcaption></figure>
<figure><img src="image_30.png" /></figure>

$$ \frac{d \ln I_A T}{d(1/T)} = - \frac{\Delta_v H_T^o}{R} \qquad (23b) $$

Thus, the machine constant and ionization cross section become unnecessary to obtain an enthalpy of vaporization. Note the reliability of the slope is related to the stability of the slope and that depends on the constancy of MS sensitivity. If the machine constant and ionization cross section are known to sufficient accuracy that a vapor pressure can be accurately determined, then the y-intercept will give $\Delta_v S_T^o/R$.

Note that the enthalpy of vaporization is determined at the average temperature of the particular dataset. For comparison to other measurements and use in databases, we need the enthalpy of vaporization at 298.15 K, $\Delta_v H_{298}^o$. This is the heat capacity correction:

$$ \Delta_v H_{298}^o = \Delta_v H_T^o - \int_{T}^{298.15} \Delta_v C_p dT \qquad (24) $$

Here, $C_p$ is the heat capacity, which must be known for the gas and condensed phases.

Cater²⁴ describes the sigma method,¹³⁰ and other investigators have used a modified sigma method¹³¹,¹³² as ways to extract $\Delta_v H_{298}^o$ from a dataset at higher temperatures. In the sigma method, a heat capacity change must be known as a polynomial. Consider a linear polynomial:

$$ \Delta C_p = \Delta a + \Delta bT \qquad (25) $$

Here $\Delta a$ and $\Delta b$ are symbols that define the heat capacity polynomial. A sigma function is then defined as

$$ \Sigma = -R \ln K_p + \Delta a \ln T + \Delta b T/2 \qquad (26) $$

A plot of $\Sigma$ versus $1/T$ leads to a line with slope $\Delta H_I$ and intercept $I$. Then enthalpy and entropy at $T$ can be calculated from

$$ \Delta_v H_T^o = \Delta H_I + T \Delta a + \Delta b T^2/2 \qquad \Delta_v S_T^o = -I + \Delta a (1 - \ln T) + \Delta b T \qquad (27) $$

Cubiciotti introduced a “modified sigma method,”¹³¹ which is based on gefs.

$$ gef_{298} = \left( \frac{G_T^o - H_{298}^T}{T} \right) \qquad (28) $$

Thus, the equation can be written

$$ \Delta[-(gef_{298})] - R \ln(p_A) = P + \frac{Q}{T} \qquad (29) $$

Thus, a plot of the left-hand side of Equation (29) versus $1/T$ gives a slope $Q$ equal to $\Delta_v H_{298}^o$ and intercept $P$ equal to $\Delta_v S_{298}^o$.

These methods require heat capacities and/or gefs. In the earlier days of KEMS, many investigators obtained very good results by estimating heat capacities and gefs from estimated spectroscopic constants.¹³³,¹³⁴ However, today these values are almost universally calculated using quantum chemistry.¹³⁵ In many cases the geometry, vibrational frequencies, and electronic states of the gas phase molecule in question¹³⁵ can be calculated using a relatively low level of theory. These are used to calculate the partition functions, which in turn are used to calculate heat capacities and gefs.¹³⁰ It was found that the gefs for metal hydroxide vapors were adequately calculated with density functional theory (DFT) using the B3LYP functional and augmented correlation consistent triple-$\zeta$ (aug-cc-pVTZ) basis sets.¹³⁶⁻¹³⁸ More computationally intensive methods are required to directly calculate enthalpies of reaction and enthalpies of formation.

The best example of the second law thermodynamic method is the simple vaporization of a pure metal, as shown in Figure 16. The vaporization of Au and second law enthalpy is used as a frequent check for instrument operation—stability, temperature correction, and linearity of instrument between vapor pressure and the product of intensity and absolute temperature.

## 3.1.2 | Third law thermodynamic method

gefs are also necessary for the third law method. In essence, the third law thermodynamic method involves calculating the Gibbs free energy change at each temperature and subtracting the entropy to get an enthalpy. This can be written as

$$ T\{\Delta[-(gef_{298})] - R\ln(p_A)\} = \Delta_v H_{298}^o \qquad (30) $$

The major advantage of this method is that for every vapor pressure data point measured a $\Delta_v H_{298}^o$ is generated and that enthalpy is at 298 K. Typically, there are numerous measurements of $\Delta_v H_{298}^o$, which establish error limits. The quality of a third law measurement is, of course, dependent on the accuracy of the vapor pressure and temperature measured. It is also dependent on the quality of the gef. As previously noted, early studies would estimate gefs for molecules where spectroscopic data were not available. However, today these can be derived from quantum chemistry calculations with relative ease. A recent transpiration study on $CrO_2(OH)_2(g)$ shows that consistent second law, third law, and ab initio enthalpies were obtained using only gefs rigorously derived from current ab initio methods.¹³⁹

It is best to do a second law thermodynamic analysis and a third law thermodynamic analysis. Standard deviations for both methods are generally published as proposed by Drowart et al.⁸⁷ However, these deviations do not reflect the total uncertainty that has to be analyzed and added to this standard deviation for the final results.

The results of a recent study are presented in Table 2.¹⁵ In this study the following reaction was studied:

$$ 0.5 \text{Hf(s)} + 0.5 \text{HfO}_2 \text{(s)} = \text{HfO(g)} \qquad (31) $$
<figure><img src="image_29.png" /></figure>

$$ \frac{d \ln I_A T}{d(1/T)} = - \frac{\Delta_v H_T^o}{R} \qquad (23b) $$

Thus, the machine constant and ionization cross section become unnecessary to obtain an enthalpy of vaporization. Note the reliability of the slope is related to the stability of the slope and that depends on the constancy of MS sensitivity. If the machine constant and ionization cross section are known to sufficient accuracy that a vapor pressure can be accurately determined, then the y-intercept will give $\Delta_v S_T^o/R$.

Note that the enthalpy of vaporization is determined at the average temperature of the particular dataset. For comparison to other measurements and use in databases, we need the enthalpy of vaporization at 298.15 K, $\Delta_v H_{298}^o$. This is the heat capacity correction:

$$ \Delta_v H_{298}^o = \Delta_v H_T^o - \int_{T}^{298.15} \Delta_v C_p dT \qquad (24) $$

Here, $C_p$ is the heat capacity, which must be known for the gas and condensed phases.

Cater²⁴ describes the sigma method,¹³⁰ and other investigators have used a modified sigma method¹³¹,¹³² as ways to extract $\Delta_v H_{298}^o$ from a dataset at higher temperatures. In the sigma method, a heat capacity change must be known as a polynomial. Consider a linear polynomial:

$$ \Delta C_p = \Delta a + \Delta bT \qquad (25) $$

Here $\Delta a$ and $\Delta b$ are symbols that define the heat capacity polynomial. A sigma function is then defined as

$$ \Sigma = -R \ln K_p + \Delta a \ln T + \Delta b T/2 \qquad (26) $$

A plot of $\Sigma$ versus $1/T$ leads to a line with slope $\Delta H_I$ and intercept $I$. Then enthalpy and entropy at $T$ can be calculated from

$$ \Delta_v H_T^o = \Delta H_I + T \Delta a + \Delta b T^2/2 \qquad \Delta_v S_T^o = -I + \Delta a (1 - \ln T) + \Delta b T \qquad (27) $$

Cubiciotti introduced a “modified sigma method,”¹³¹ which is based on gefs.

$$ gef_{298} = \left( \frac{G_T^o - H_{298}^T}{T} \right) \qquad (28) $$

Thus, the equation can be written

$$ \Delta[-(gef_{298})] - R \ln(p_A) = P + \frac{Q}{T} \qquad (29) $$

Thus, a plot of the left-hand side of Equation (29) versus $1/T$ gives a slope $Q$ equal to $\Delta_v H_{298}^o$ and intercept $P$ equal to $\Delta_v S_{298}^o$.

These methods require heat capacities and/or gefs. In the earlier days of KEMS, many investigators obtained very good results by estimating heat capacities and gefs from estimated spectroscopic constants.¹³³,¹³⁴ However, today these values are almost universally calculated using quantum chemistry.¹³⁵ In many cases the geometry, vibrational frequencies, and electronic states of the gas phase molecule in question¹³⁵ can be calculated using a relatively low level of theory. These are used to calculate the partition functions, which in turn are used to calculate heat capacities and gefs.¹³⁰ It was found that the gefs for metal hydroxide vapors were adequately calculated with density functional theory (DFT) using the B3LYP functional and augmented correlation consistent triple-$\zeta$ (aug-cc-pVTZ) basis sets.¹³⁶⁻¹³⁸ More computationally intensive methods are required to directly calculate enthalpies of reaction and enthalpies of formation.

The best example of the second law thermodynamic method is the simple vaporization of a pure metal, as shown in Figure 16. The vaporization of Au and second law enthalpy is used as a frequent check for instrument operation—stability, temperature correction, and linearity of instrument between vapor pressure and the product of intensity and absolute temperature.

## 3.1.2 | Third law thermodynamic method

gefs are also necessary for the third law method. In essence, the third law thermodynamic method involves calculating the Gibbs free energy change at each temperature and subtracting the entropy to get an enthalpy. This can be written as

$$ T\{\Delta[-(gef_{298})] - R\ln(p_A)\} = \Delta_v H_{298}^o \qquad (30) $$

The major advantage of this method is that for every vapor pressure data point measured a $\Delta_v H_{298}^o$ is generated and that enthalpy is at 298 K. Typically, there are numerous measurements of $\Delta_v H_{298}^o$, which establish error limits. The quality of a third law measurement is, of course, dependent on the accuracy of the vapor pressure and temperature measured. It is also dependent on the quality of the gef. As previously noted, early studies would estimate gefs for molecules where spectroscopic data were not available. However, today these can be derived from quantum chemistry calculations with relative ease. A recent transpiration study on $CrO_2(OH)_2(g)$ shows that consistent second law, third law, and ab initio enthalpies were obtained using only gefs rigorously derived from current ab initio methods.¹³⁹

It is best to do a second law thermodynamic analysis and a third law thermodynamic analysis. Standard deviations for both methods are generally published as proposed by Drowart et al.⁸⁷ However, these deviations do not reflect the total uncertainty that has to be analyzed and added to this standard deviation for the final results.

The results of a recent study are presented in Table 2.¹⁵ In this study the following reaction was studied:

$$ 0.5 \text{Hf(s)} + 0.5 \text{HfO}_2(\text{s}) = \text{HfO(g)} \qquad (31) $$
<figure><img src="image_31.png" /><figcaption>**TABLE 2** Third law thermodynamic calculations for the reaction ½Hf(s) + ½HfO₂(s) = HfO(g), using σ = 8.2 × 10⁻¹⁰ m² and k = 1.68 × 10⁻²³ bar-m²/cps·K.</figcaption></figure>

<table><thead><tr><th>T (K)</th><th>I (cps)</th><th>Standard deviation</th><th>IT (cps K)</th><th>kIT/σ (bar)</th><th>-RT ln Kp (J/mol)</th><th>Δgef (J/mol)</th><th>Δ<sub>v</sub>H°(298) = -RT ln Kp - T Δgef (J/mol)</th></tr></thead><tbody><tr><td>1934</td><td>170.4</td><td>8.3</td><td>329,553.6</td><td>6.75E-09</td><td>302,506.7</td><td>-162.57</td><td>616,914.8</td></tr><tr><td>1986</td><td>457.6</td><td>14.5</td><td>908,753.9</td><td>1.86E-08</td><td>293,891.9</td><td>-162.22</td><td>616,051.4</td></tr><tr><td>1957</td><td>296.1</td><td>18.7</td><td>579,448.1</td><td>1.19E-08</td><td>296,922.2</td><td>-162.41</td><td>614,763.3</td></tr><tr><td>1945</td><td>228.0</td><td>11.6</td><td>443,460.0</td><td>9.09E-09</td><td>299,426.7</td><td>-162.49</td><td>615,477.5</td></tr><tr><td>1994</td><td>519.9</td><td>25.7</td><td>1,036,640.7</td><td>2.12E-08</td><td>292,893.1</td><td>-162.16</td><td>616,241.7</td></tr><tr><td>2034</td><td>896.0</td><td>21.4</td><td>1,822,464.0</td><td>3.73E-08</td><td>289,227.5</td><td>-161.89</td><td>618,509.3</td></tr><tr><td>1952</td><td>264.5</td><td>14.4</td><td>516,245.4</td><td>1.06E-08</td><td>298,037.9</td><td>-162.45</td><td>615,133.3</td></tr><tr><td>2004</td><td>622.2</td><td>18.1</td><td>1,246,848.7</td><td>2.55E-08</td><td>291,285.7</td><td>-162.09</td><td>616,119.7</td></tr></tbody></table>

Ion intensities of HfO⁺ were recorded from 1934 to 2034 K. A second law plot of ln(IT) versus 1/T yielded an enthalpy at 1976 K of -548.5 ± 22 kJ/mol. Using Equation (24) and the calculated heat capacity for HfO¹⁵ and the tabulated heat capacity for Hf¹¹⁵,¹⁴⁰ and HfO₂,¹¹⁵,¹⁴¹ an enthalpy of reaction at 298.15 K was calculated to be -633.0 ± 22 kJ/mol. The average of the third law enthalpies of reaction (taken at each temperature) was -616.2 ± 1.2 kJ/mol. Agreement is with 2.7%; however, the third law thermodynamic measurement is preferred as there are not sufficient data points or temperature range for a reliable second law enthalpy of reaction.

The enthalpy of reaction (31) can be combined with a reliable enthalpy of formation of HfO₂(s)¹⁴¹ to calculate an enthalpy of formation of HfO(g). This can be compared to other measurements and theoretical calculations. The computed enthalpy of formation of HfO(g) using CCSD(T) (coupled cluster, single, doubles, with approximate triples) methods is 63.19 ± 10 kJ/mol compared to 58.4 ± 12.3 kJ/mol using third law methods with KEMS.¹⁵

### 3.1.3 | Dissociation energy of diatomic vapor species

One of the most useful thermochemical parameters derived from KEMS is the bond dissociation energy or simply the dissociation energy.¹⁴²⁻¹⁴⁴ Consider the simple case of an element forming both a monomer (M) and a dimer (D) in the gaseous phase:

$$M(s) = M(g)\Delta_v H_0^°(M) \quad (32a)$$

$$2M(s) = M2(g)\Delta_v H_0^°(D) \quad (32b)$$

The enthalpies can be determined using either the second or third thermodynamic law methods described earlier. The dissociation energy, D₀°, is thus given by
<figure><img src="image_30.png" /></figure>

<figure><img src="image_31.png" /><figcaption>**TABLE 2** Third law thermodynamic calculations for the reaction ½Hf(s) + ½HfO₂(s) = HfO(g), using σ = 8.2 × 10⁻¹⁰ m² and k = 1.68 × 10⁻²³ bar-m²/cps·K.</figcaption></figure>

<table><thead><tr><th>T (K)</th><th>I (cps)</th><th>Standard deviation</th><th>IT (cps K)</th><th>kIT/σ (bar)</th><th>-RT ln Kp (J/mol)</th><th>Δgef (J/mol)</th><th>Δ<sub>v</sub>H°(298) = -RT ln Kp - T Δgef (J/mol)</th></tr></thead><tbody><tr><td>1934</td><td>170.4</td><td>8.3</td><td>329,553.6</td><td>6.75E-09</td><td>302,506.7</td><td>-162.57</td><td>616,914.8</td></tr><tr><td>1986</td><td>457.6</td><td>14.5</td><td>908,753.9</td><td>1.86E-08</td><td>293,891.9</td><td>-162.22</td><td>616,051.4</td></tr><tr><td>1957</td><td>296.1</td><td>18.7</td><td>579,448.1</td><td>1.19E-08</td><td>296,922.2</td><td>-162.41</td><td>614,763.3</td></tr><tr><td>1945</td><td>228.0</td><td>11.6</td><td>443,460.0</td><td>9.09E-09</td><td>299,426.7</td><td>-162.49</td><td>615,477.5</td></tr><tr><td>1994</td><td>519.9</td><td>25.7</td><td>1,036,640.7</td><td>2.12E-08</td><td>292,893.1</td><td>-162.16</td><td>616,241.7</td></tr><tr><td>2034</td><td>896.0</td><td>21.4</td><td>1,822,464.0</td><td>3.73E-08</td><td>289,227.5</td><td>-161.89</td><td>618,509.3</td></tr><tr><td>1952</td><td>264.5</td><td>14.4</td><td>516,245.4</td><td>1.06E-08</td><td>298,037.9</td><td>-162.45</td><td>615,133.3</td></tr><tr><td>2004</td><td>622.2</td><td>18.1</td><td>1,246,848.7</td><td>2.55E-08</td><td>291,285.7</td><td>-162.09</td><td>616,119.7</td></tr></tbody></table>

Ion intensities of HfO⁺ were recorded from 1934 to 2034 K. A second law plot of ln(IT) versus 1/T yielded an enthalpy at 1976 K of -548.5 ± 22 kJ/mol. Using Equation (24) and the calculated heat capacity for HfO¹⁵ and the tabulated heat capacity for Hf¹¹⁵,¹⁴⁰ and HfO₂,¹¹⁵,¹⁴¹ an enthalpy of reaction at 298.15 K was calculated to be -633.0 ± 22 kJ/mol. The average of the third law enthalpies of reaction (taken at each temperature) was -616.2 ± 1.2 kJ/mol. Agreement is with 2.7%; however, the third law thermodynamic measurement is preferred as there are not sufficient data points or temperature range for a reliable second law enthalpy of reaction.

The enthalpy of reaction (31) can be combined with a reliable enthalpy of formation of HfO₂(s)¹⁴¹ to calculate an enthalpy of formation of HfO(g). This can be compared to other measurements and theoretical calculations. The computed enthalpy of formation of HfO(g) using CCSD(T) (coupled cluster, single, doubles, with approximate triples) methods is 63.19 ± 10 kJ/mol compared to 58.4 ± 12.3 kJ/mol using third law methods with KEMS.¹⁵

### 3.1.3 | Dissociation energy of diatomic vapor species

One of the most useful thermochemical parameters derived from KEMS is the bond dissociation energy or simply the dissociation energy.¹⁴²⁻¹⁴⁴ Consider the simple case of an element forming both a monomer (M) and a dimer (D) in the gaseous phase:

$$M(s) = M(g)\Delta_v H_0^°(M) \quad (32a)$$

$$2M(s) = M2(g)\Delta_v H_0^°(D) \quad (32b)$$

The enthalpies can be determined using either the second or third thermodynamic law methods described earlier. The dissociation energy, D₀°, is thus given by
<figure><img src="image_32.png" /></figure>

$$M2(g) = 2M(g) \qquad D_0^o = 2\Delta_v H_0^o(M) - \Delta_v H_0^o(D) \quad (33)$$

Similar equations can be written for gaseous molecules, AB(g). For example, consider the dissociation energy of SiO(g):

$$\text{SiO}_2(\text{s}) = \text{SiO(g)} + \frac{1}{2}\text{O}_2(\text{g}) \quad \text{KEMS study} \quad (34)$$

$$\text{Si(s)} + \text{O}_2(\text{g}) = \text{SiO}_2(\text{s}) \quad \text{tabulated calorimetric data} \quad (35)$$

$$\text{Si(g)} = \text{Si(s)} \quad \text{tabulated or KEMS study} \quad (36)$$

$$\text{Si(g)} + \frac{1}{2}\text{O}_2(\text{g}) = \text{SiO(g)} - D_0^o(\text{SiO}) \quad (37)$$

Drowart and colleagues have made use of isomolecular-exchange reactions to determine dissociation energies,¹⁴⁵ such as

$$\text{YO(g)} + \text{Sc(g)} = \text{Y(g)} + \text{ScO(g)} \quad (38)$$

This reaction was set up with mixtures of Sc₂O₃ + Y in the Knudsen cell. The dissociation of YO(g) is known, and thus the dissociation energy of ScO(g) could be measured.

There are many KEMS studies on dissociation energies in the literature that report dissociation energies.¹⁴³,¹⁴⁶,¹⁴⁷ These can be directly compared to the values obtained using various computational methods.¹⁴⁸,¹⁴⁹ In the case of Csl(g) and its dimer, Roki et al.¹⁵⁰ compiled experimental enthalpy values, assessed these values, and compared to different ab initio calculations. The evaluated experimental uncertainty is within the range of values for the calculations.

## 3.2 | Thermochemistry of solutions

Vapor pressure measurements lead to fundamental solution properties. We shall first discuss these measurements for metallic alloys. There are several reviews specifically on this topic.⁶,¹⁸,¹⁵¹,¹⁵²

The primary solution parameter to be measured is the thermodynamic activity of components. Activity of component A(a_A) is defined as the change in chemical potential of component A(μ_A) as component A moves from solution phase β to its pure phase, designated by superscript o:

$$\Delta_{\text{mix}} \bar{G}_{\text{A}}^{\beta} = \Delta_{\text{mix}} \bar{H}_{\text{A}}^{\beta} - T \Delta_{\text{mix}} \bar{S}_{\text{A}}^{\beta} = \mu_{\text{A}}^{\beta} - \mu_{\text{A}}^{o} = RT \ln \frac{f_{\text{A}}^{\beta}}{f_{\text{A}}^{o}} = RT \ln \frac{p_{\text{A}}^{\beta}}{p_{\text{A}}^{o}} = RT \ln a_{\text{A}}^{\beta} \quad (39)$$

The reference state is arbitrary but must be clearly defined. It is often in the form of the pure element at the temperature of interest, which may be a particular crystalline structure (e.g., face-centered cubic Ni) or liquid (e.g., Al(l)). For the component A in a mixture (A,B), let us assume the reference state as the A(δ phase). We can write the appropriate chemical reactions as follows. For the pure component,

$$A(\delta) = A(g) \qquad \Delta_{\text{vap}} G^o(\delta) = -RT \ln \frac{p_A^o}{a_A(\delta)} = -RT \ln p_A^o \quad (40)$$

For the mixture at the same temperature, we have

$$A(\text{in A, B, γ}) = A(g) \qquad \Delta_{\text{vap}} \bar{G}^o(A(\gamma)) = -RT \ln K_p(T) = -RT \ln \frac{p_A}{a_A} \quad (41)$$

Combining Equations (39)–(41) gives

$$a_A(\text{in A, B, γ, wrt γ}) = \frac{p_A}{p_A^o} \quad (42)$$

Thus, any measurement of the A(g) species partial pressure over a mixture must be reported with a convenient reference to determine the activity of A. Applying thermodynamic relations, the variation in the logarithm of a_A versus 1/T leads to the partial mixing enthalpy of A, ΔmixH̅_A:̇:

$$\frac{d \ln a_i}{d(1/\gamma)} = - \frac{\Delta_{\text{mix}} \bar{H}_A^\gamma}{R} \quad (43)$$

This relation is similar to the second thermodynamic law equation for enthalpy of reaction of a pure substance, as discussed earlier. The calculated partial enthalpy of mixing, together with its uncertainty, may be compared to other types of measurements (e.g., calorimetry). Note that in optimizing procedures for phase diagram and associated data calculation, the activities of components and their derivatives cannot be used together as they are not independent.

If the partial molar enthalpy of mixing is known for all n components, the integral molar enthalpy of mixing may be derived:

$$\Delta_{\text{mix}} H^{\beta} = \sum_{k=1}^{n} x_k \Delta_{\text{mix}} \bar{H}_i^{\beta} \quad (44)$$

Note that the excess partial enthalpy is given by¹⁵³

$$\Delta_{\text{exc}} \bar{G}_{\text{A}}^{\beta} = RT \ln \gamma_{\text{A}}^{\beta} = \Delta_{\text{exc}} \bar{H}_{\text{A}}^{\beta} - T \Delta_{\text{exc}} \bar{S}_{\text{A}}^{\beta} \quad (45)$$

Here γ_A^β is the activity coefficient of A in phase β, defined as γ_A^β = a_A^β / x_A^β. The partial molar excess enthalpy of component A can be determined from

$$\frac{d \ln \gamma_A}{d(1/\gamma)} = - \frac{\Delta_{\text{exc}} \bar{H}_A^{\beta}}{R} \quad (46)$$

The integral molar excess enthalpy is thus given by

$$\Delta_{\text{exc}} H^{\beta} = \sum_{k=1}^{n} x_k \Delta_{\text{exc}} \bar{H}_i^{\beta} \quad (47)$$

Equations (42)–(47) show the range of key solution data that can be obtained via KEMS. Further, Equations (44) and (47) show that
<figure><img src="image_32.png" /></figure>

$$M2(g) = 2M(g) \qquad D_0^o = 2\Delta_v H_0^o(M) - \Delta_v H_0^o(D) \quad (33)$$

Similar equations can be written for gaseous molecules, AB(g). For example, consider the dissociation energy of SiO(g):

$$\text{SiO}_2(\text{s}) = \text{SiO(g)} + \frac{1}{2}\text{O}_2(\text{g}) \text{ KEMS study} \quad (34)$$

$$\text{Si(s)} + \text{O}_2(\text{g}) = \text{SiO}_2(\text{s}) \text{ tabulated calorimetric data} \quad (35)$$

$$\text{Si(g)} = \text{Si(s)} \text{ tabulated or KEMS study} \quad (36)$$

$$\text{Si(g)} + \frac{1}{2}\text{O}_2(\text{g}) = \text{SiO(g)} - D_0^o(\text{SiO}) \quad (37)$$

Drowart and colleagues have made use of isomolecular-exchange reactions to determine dissociation energies,¹⁴⁵ such as

$$\text{YO(g)} + \text{Sc(g)} = \text{Y(g)} + \text{ScO(g)} \quad (38)$$

This reaction was set up with mixtures of Sc₂O₃ + Y in the Knudsen cell. The dissociation of YO(g) is known, and thus the dissociation energy of ScO(g) could be measured.

There are many KEMS studies on dissociation energies in the literature that report dissociation energies.¹⁴³,¹⁴⁶,¹⁴⁷ These can be directly compared to the values obtained using various computational methods.¹⁴⁸,¹⁴⁹ In the case of Csl(g) and its dimer, Roki et al.¹⁵⁰ compiled experimental enthalpy values, assessed these values, and compared to different ab initio calculations. The evaluated experimental uncertainty is within the range of values for the calculations.

## 3.2 | Thermochemistry of solutions

Vapor pressure measurements lead to fundamental solution properties. We shall first discuss these measurements for metallic alloys. There are several reviews specifically on this topic.⁶,¹⁸,¹⁵¹,¹⁵²

The primary solution parameter to be measured is the thermodynamic activity of components. Activity of component A(a_A) is defined as the change in chemical potential of component A(μ_A) as component A moves from solution phase β to its pure phase, designated by superscript o:

$$\Delta_{\text{mix}} \bar{G}_{\text{A}}^{\beta} = \Delta_{\text{mix}} \bar{H}_{\text{A}}^{\beta} - T \Delta_{\text{mix}} \bar{S}_{\text{A}}^{\beta} = \mu_{\text{A}}^{\beta} - \mu_{\text{A}}^{\text{o}} = RT \ln \frac{f_{\text{A}}^{\beta}}{f_{\text{A}}^{\text{o}}} = RT \ln \frac{p_{\text{A}}^{\beta}}{p_{\text{A}}^{\text{o}}} = RT \ln a_{\text{A}}^{\beta} \quad (39)$$

The reference state is arbitrary but must be clearly defined. It is often in the form of the pure element at the temperature of interest, which may be a particular crystalline structure (e.g., face-centered cubic Ni) or liquid (e.g., Al(l)). For the component A in a mixture (A,B), let us assume the reference state as the A(δ phase). We can write the appropriate chemical reactions as follows. For the pure component,

$$A(\delta) = A(g) \qquad \Delta_{\text{vap}} G^o(\delta) = -RT \ln \frac{p_A^o}{a_A(\delta)} = -RT \ln p_A^o \quad (40)$$

For the mixture at the same temperature, we have

$$A(\text{in A, B, γ}) = A(g) \qquad \Delta_{\text{vap}} \bar{G}^o(A(\gamma)) = -RT \ln K_p(T) = -RT \ln \frac{p_A}{a_A} \quad (41)$$

Combining Equations (39)–(41) gives

$$a_A(\text{in A, B, γ, wrt γ}) = \frac{p_A}{p_A^o} \quad (42)$$

Thus, any measurement of the A(g) species partial pressure over a mixture must be reported with a convenient reference to determine the activity of A. Applying thermodynamic relations, the variation in the logarithm of a_A versus 1/T leads to the partial mixing enthalpy of A, ΔmixH̄_A:

$$\frac{d \ln a_i}{d(1/\tau)} = - \frac{\Delta_{\text{mix}} \bar{H}_A^\gamma}{R} \quad (43)$$

This relation is similar to the second thermodynamic law equation for enthalpy of reaction of a pure substance, as discussed earlier. The calculated partial enthalpy of mixing, together with its uncertainty, may be compared to other types of measurements (e.g., calorimetry). Note that in optimizing procedures for phase diagram and associated data calculation, the activities of components and their derivatives cannot be used together as they are not independent.

If the partial molar enthalpy of mixing is known for all n components, the integral molar enthalpy of mixing may be derived:

$$\Delta_{\text{mix}} H^{\beta} = \sum_{k=1}^{n} x_k \Delta_{\text{mix}} \bar{H}_i^{\beta} \quad (44)$$

Note that the excess partial enthalpy is given by¹⁵³

$$\Delta_{\text{exc}} \bar{G}_{\text{A}}^{\beta} = RT \ln \gamma_{\text{A}}^{\beta} = \Delta_{\text{exc}} \bar{H}_{\text{A}}^{\beta} - T \Delta_{\text{exc}} \bar{S}_{\text{A}}^{\beta} \quad (45)$$

Here γ_A^β is the activity coefficient of A in phase β, defined as γ_A^β = a_A^β/x_A^β. The partial molar excess enthalpy of component A can be determined from

$$\frac{d \ln \gamma_A}{d(1/\tau)} = - \frac{\Delta_{\text{exc}} \bar{H}_A^{\beta}}{R} \quad (46)$$

The integral molar excess enthalpy is thus given by

$$\Delta_{\text{exc}} H^{\beta} = \sum_{k=1}^{n} x_k \Delta_{\text{exc}} \bar{H}_i^{\beta} \quad (47)$$

Equations (42)–(47) show the range of key solution data that can be obtained via KEMS. Further, Equations (44) and (47) show that