rus
DOI: https://doi.org/10.22184/1993-8578.2025.18.6.346.354
In modern metrology, the size of an atom is determined statistically as half the interatomic distance in a crystal lattice, rather than through direct measurement of an isolated atom. Scanning tunneling microscopy (STM) and atomic force microscopy (AFM) are key tools for directly observing and measuring the geometry of individual atoms weakly bound to a substrate. The image obtained in STM is the spatial distribution of the electron density of the outer shell of the atom. Although STM is traditionally considered a more detailed method, AFM with modified probes can in some cases provide greater resolution and the ability to chemically identify atoms. In this article, we have collected data on the observation of atoms and their size measurements using scanning tunneling and atomic force microscopy.
In modern metrology, the size of an atom is determined statistically as half the interatomic distance in a crystal lattice, rather than through direct measurement of an isolated atom. Scanning tunneling microscopy (STM) and atomic force microscopy (AFM) are key tools for directly observing and measuring the geometry of individual atoms weakly bound to a substrate. The image obtained in STM is the spatial distribution of the electron density of the outer shell of the atom. Although STM is traditionally considered a more detailed method, AFM with modified probes can in some cases provide greater resolution and the ability to chemically identify atoms. In this article, we have collected data on the observation of atoms and their size measurements using scanning tunneling and atomic force microscopy.
Теги: atomic force microscopy atoms nanometer standard scanning tunneling microscopy атомно-силовая микроскопия атомы сканирующая туннельная микроскопия эталон нанометра
Received: 5.09.2025 | Accepted: 10.09.2025 | DOI: https://doi.org/10.22184/1993-8578.2025.18.6.346.354
Original paper
ABOUT DIMENSION OF THE ATOM
D.I.Yaminsky1, Post Graduate, ORCID: 0009-0009-6370-7496
A.I.Akhmetova1, 2, Cand. of Sci. (Physics and Mathematics), Senior Researcher, Leading Specialist,
ORCID: 0000-0002-5115-8030
A.I.Fedoseev1, Doct. of Sci. (Physics and Mathematics), Prof., ORCID: 0009-0007-7282-1093
I.V.Yaminsky1, 2, Doct. of Sci. (Physics and Mathematics), Prof., Director General,
ORCID: 0000-0001-8731-3947 / yaminsky@nanoscopy.ru
Abstract. In modern metrology, the size of an atom is determined statistically as half the interatomic distance in a crystal lattice, rather than through direct measurement of an isolated atom. Scanning tunneling microscopy (STM) and atomic force microscopy (AFM) are key tools for directly observing and measuring the geometry of individual atoms weakly bound to a substrate. The image obtained in STM is the spatial distribution of the electron density of the outer shell of the atom. Although STM is traditionally considered a more detailed method, AFM with modified probes can in some cases provide greater resolution and the ability to chemically identify atoms. In this article, we have collected data on the observation of atoms and their size measurements using scanning tunneling and atomic force microscopy.
Keywords: scanning tunneling microscopy, atomic force microscopy, atoms, nanometer standard
For citation: D.I. Yaminsky, A.I. Akhmetova, A.I. Fedoseev, I.V. Yaminsky. About dimension of the atom. NANOINDUSTRY. 2025. Vol. 18. No. 6. PP. 346–354. https://doi.org/10.22184/1993-8578.2025.18.6.346.354.
INTRODUCTION
In modern metrological measurements, the radius of an atom is defined as half of the interatomic distance in the crystal lattice. For example, this is how the radius of carbon and silicon atoms forming crystals with covalent bonding is defined. Accordingly, the observed radius is called the covalent radius. In the case of carbon, it is 0.076±0.001 nm for diamond, 0.073±0.002 nm for graphite and 0.069±0.001 for linear carbon. Silicon has one crystal shape and therefore only one value of covalent radius, equal to 0.111±0.002 nm. For a number of other elements, the metallic radius of the atom (metals), ionic radius (ionic crystals), van der Waals radii (molecular crystals and noble gases) are determined.
These data are based on statistical analyses of a large number of experimental measurements and are given in the Cambridge Structural Database [1].
RESEARCH METHODS
The determination of the size of a single atom is undoubtedly of great interest. Significant progress in this field became possible after invention of scanning probe microscopy tools - scanning tunnelling and atomic force microscopes. Strictly speaking, these methods do not provide an opportunity to measure a completely isolated atom, but they allow us to determine the atom geometry weakly coupled to its environment. A successful model solution is an atom on a substrate with weak interaction with it. A series of works on visualisation of Xe and Fe atoms was carried out by D.Eigler’s research group at IBM using a high-vacuum scanning tunnelling microscope [2]. The scanning tunnelling microscope was invented by IBM employees H.Rohrer and G.Binnig in 1981, for which they were awarded the 1986 Nobel Prize in Physics. When xenon atoms were observed on the surface of a nickel crystal, the nickel atoms could move freely when interacting with the probe, indicating that they were weakly bound to the substrate [3].
The image of a single copper atom obtained using a scanning tunnelling microscope [4] is shown in Fig.1.
When imaging an atom, a scanning tunnelling microscope obtains a spatial image of the occupied electron level of the outer shell at a positive potential on the probe of the tunnelling microscope. At negative potential on the probe, the image of the unoccupied electron level of the atom is obtained. In this case, the observed sizes of the obtained images should be slightly different. The size of the atom should be considered the first case, since it corresponds to the ground state of the atom.
If an atom is in an excited state, its radius increases accordingly due to transition of an electron to an electron orbital more distant from the atomic nucleus [5]. This was experimentally demonstrated for the gold atom in the ground and excited states (STM conference in Japan, 2005).
Another type of probe microscopy, the atomic force microscope, is also used to determine the size of an atom. The probe of the atomic force microscope lightly touches the atom and traces its topography. The principle of a gramophone player works here - a needle slides over a surface and traces its topography. The interaction force between the probe and the surface has an electromagnetic component, although the main contribution to the interaction is determined by the principles of relativistic quantum mechanics - Pauli’s rule (or prohibition). Pauli’s principle forbids the finding of two identical electrons in one point of space. The same principle does not allow a person to fall through the floor and in general provides the whole spatial architecture of the world we observe.
In atomic force microscopy, chemical identification of individual atoms can be realised by the nature of force interaction, in particular, it is possible to distinguish individual atoms of silicon, tin and lead [6]. Force measurements have recently become popular in various fields from materials science to living systems [7].
To visualise a single atom on a substrate, the magnitude of the contact force on the probe side should be minimised. In this case, the atomic force microscope will obtain images in many respects similar to the scanning tunnelling microscope images. At the same time, certain differences in the images may be observed due to the different nature of the probe-sample interaction.
The scanning tunnelling microscope is often considered to give a more detailed image of the atomic structure of the sample. However, there are also opposite results. A good example is the comparative observation of the internal structure of the pentacene molecule by two methods - tunnelling and atomic force microscopy [8]. The atomic force microscope allows you to look inside the molecule with a greater degree of detail. It should be noted that an unusual probe was used in this experiment. One molecule of carbon monoxide CO was attached to its tip. That is, scanning of one molecule took place with the help of another molecule. Scanning tunnelling microscopy also uses probes with CO molecules at the tip. In this work, visualisation of copper atoms depending on orientation (rotation angle) of the CO molecule was carried out [9].
As can be easily understood, the best spatial resolution of atomic force microscopy can be obtained if the smallest atom, a hydrogen atom, is placed at the tip of the probe. By the way, such an experiment was carried out [10]. And its result was also a measurement of the size of the hydrogen atom itself.
Strictly speaking, from the very beginning, the tunnelling and atomic force microscope were instruments with unique capabilities, but they were not metrological tools. In the following years, many efforts were made to bring them closer to the modern metrology requirements. In 2019, the ISO 11952:201 international standard - Surface chemical analysis - Scanning probe microscopy - Determination of geometric quantities by SPM: calibration of measuring systems was approved. Additions were made in 2024.
If we consider small sizes - units of nanometres - probe microscopes carry out measurements with high accuracy. Certain errors caused by the piezoceramics properties, the material of the moving parts, appear in the range of tens to hundreds of microns. To eliminate them, precise displacement meters based on capacitive, strain-gauge and interference measuring systems are used.
In the case of small fields of observation, such as single atoms, it is necessary to ensure that the displacement systems are correctly calibrated. For example, if a displacement of 1 nm is required, the displacement and is carried out exactly one nanometre. To calibrate microscopes directly during measurements, we have previously proposed a nanometre standard.
Unlike fixed-magnification optical microscopes, SPMs create images by sliding a sharp probe across a surface. The size of the image is directly dependent on the voltage applied to the scanners (piezoelectric elements) and the resulting physical motion. Piezo scanner motion is inherently non-linear, subject to hysteresis, creep and cross-coupling between axes. These effects lead to distortions and scaling errors that can be on the order of 10% or more without correction.
Calibration of SPM is usually carried out in three axes: X, Y (lateral) and Z (vertical). Calibration consists in comparing measured parameters on the AFM with passport values of standards in X, Y and Z axes, in correction and determination of correct coefficients to account for the piezoscanner nonlinearity. For high-precision calibration, software methods of data processing and distortion compensation are used (Fig.2) [11].
Arrays of isolated, well-defined objects (e.g., nanoparticles or fabricated columns) with known spacing between them are useful for evaluating scanner linearity at various points in the scanning range.
Vertical standards are used to calibrate the scanner’s Z-axis response, which is critical for height, depth and roughness measurements.
Atomic force microscope (AFM) calibration uses calibration gratings: silicon TGZ gratings (Fig.3) and diffraction gratings [12], as well as test samples with known surface topography: for example, freshly chipped mica, highly oriented pyrolytic graphite (HOPG) chromium columns [13], blu-ray disc (Fig.4), and others.
Atoms, with their blurred electron clouds, are often thought to be vague and somewhat abstract. However, scanning probe microscopy provides precise information that atoms are rather solid entities with well-defined contours. It is these data that allow us to speak about the exact dimensions of individual atoms. To a certain extent, the question remains open as to what, if I may say so, is the mechanical rigidity of the atom. In other words, it is interesting to know how the size of an atom depends on the magnitude of the force applied to it. The dependence of energy and force of interaction between atoms is determined by pair potentials. Among them are empirical dependences - Lennard Jones, Buckingham, Morse, etc. potentials. Accordingly, when atoms approach each other, repulsive forces appear, which can change the observed atom size. It seems achievable in principle that such experiments can be carried out using an atomic force microscope. At present, we have not found such published data.
The size of an atom can be interpreted in terms of the Heisenberg uncertainty principle [14]. The standard deviation Δr of the electron shell radius and the standard deviation Δpr of the electron momentum projection satisfy the relation:
Δr Δpr ≥ ħ/2,
where ħ is the reduced Planck constant. If at measurements in a scanning probe microscope the standard deviation Δr of the radius is 10–3 nm, then the standard deviation of the momentum is 0.5 · 10–22 J. At the same time, the electron velocity is approximately three times less than the speed of light. In a scanning tunnelling microscope, an electron tunnels from the filled shell of an atom to the unfilled electronic state of the probe, or in the opposite direction from the filled electronic level of the probe to the free electronic level of the atom. In the quantum theory of tunnelling transition (in certain approximations) probability of tunnelling depends on the spatial arrangement of electronic levels and the distance between them.
To image an atom in atomic force microscopy, one should take into account the Pauli principle, which forbids two electrons with the same set of quantum numbers to be in the same place in space. This rule leads to spatial separation of electrons of the probe and the atom under study. Figuratively speaking, the probe only touches the atom. At that, repulsive forces and their corresponding energy arise. For description of forces and interaction energy there are empirical potentials mentioned earlier. Whether it is possible to derive a rigorous conclusion of this potential theoretically from the relativistic quantum theory is an open question for us at present. It is clear that in order to obtain by atomic force microscopy the size of an undisturbed atom, its measurement should be carried out at minimum force of influence.
Whether the Pauli principle allows us to determine the atom size with such a high, achievable in experiment accuracy, given that only the coordinate is measured without measuring the momentum, remains open. However, this does not prevent us from admiring the beauty of individual atoms and molecules observed in scanning tunnelling and atomic force microscopes.
Probe microscopy is not the only method that allows visualising the atom. The first image of an atom was obtained using an ion projector by Erwin Muller [15]. The peculiarity of this method is that the atom itself must be at the top of the metal tip during observations.
Fifteen years later, the image of the atom was obtained using a transmission electron microscope [16]. Taking into account the peculiarities of this method, an image of a heavy atom, uranium, was obtained. The atom was located on a graphite substrate. For the first time it was possible to visualise individual heavy atoms of mercury in a transmission electron microscope with a high probability [17]. In the ion projector and electron microscope, the image of an atom is usually obtained on a phosphor screen on which ions or electrons fall after interaction with the atom under study.
DETERMINATION OF ATOM SIZE BY AFM AND STM METHODS
In an atomic force microscope, the situation is different. The tip of the cantilever (atom in the tip) touches the observed atom, and the tip of the cantilever describes a trajectory, repeating the contour of this atom. The resulting images of the atom are striking in their beauty. In the atomic force microscope, atoms look like billiard balls. They have a certain size and shape. In this case, the probe-atom interaction, due to which the image is obtained, is conditioned by the Pauli prohibition. This interaction does not belong to any of the interactions of the standard model of physics - strong, weak, electromagnetic and gravitational. The force and interaction in atomic force microscopy is a consequence of Pauli’s prohibition. The atomic force microscope measures this force and interaction. However, we have not found a theoretical description of the nature of this interaction involving relativistic quantum mechanics in the scientific literature.
In scanning tunnelling microscopy, individual images of atoms also have clear contours. The repeat accuracy of the contours is at the level of thousandths of a nanometre. These images give information about the size of the atom. Whether these images provide information about the size of an electron is an open question. In the standard model, the electron is considered to be an elementary particle that has no size. In school physics, too, there is a concept of point charge, the use of which allows many problems from electricity to be solved accurately. The electron has measured the exact values of mass, charge and spin. The question of size remains open. So far there is no experimental data indicating the size of the electron.
ACKNOWLEDGEMENTS
This work was partially performed under the state order of the Lomonosov Moscow State University. FemtoScan Online software is provided by Advanced Technologies Center, www.femtoscan.ru
PEER REVIEW INFO
Editorial board thanks the anonymous reviewer(s) for their contribution to the peer review of this work. It is also grateful for their consent to publish papers on the journal’s website and SEL eLibrary eLIBRARY.RU.
Declaration of Competing Interest. The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Original paper
ABOUT DIMENSION OF THE ATOM
D.I.Yaminsky1, Post Graduate, ORCID: 0009-0009-6370-7496
A.I.Akhmetova1, 2, Cand. of Sci. (Physics and Mathematics), Senior Researcher, Leading Specialist,
ORCID: 0000-0002-5115-8030
A.I.Fedoseev1, Doct. of Sci. (Physics and Mathematics), Prof., ORCID: 0009-0007-7282-1093
I.V.Yaminsky1, 2, Doct. of Sci. (Physics and Mathematics), Prof., Director General,
ORCID: 0000-0001-8731-3947 / yaminsky@nanoscopy.ru
Abstract. In modern metrology, the size of an atom is determined statistically as half the interatomic distance in a crystal lattice, rather than through direct measurement of an isolated atom. Scanning tunneling microscopy (STM) and atomic force microscopy (AFM) are key tools for directly observing and measuring the geometry of individual atoms weakly bound to a substrate. The image obtained in STM is the spatial distribution of the electron density of the outer shell of the atom. Although STM is traditionally considered a more detailed method, AFM with modified probes can in some cases provide greater resolution and the ability to chemically identify atoms. In this article, we have collected data on the observation of atoms and their size measurements using scanning tunneling and atomic force microscopy.
Keywords: scanning tunneling microscopy, atomic force microscopy, atoms, nanometer standard
For citation: D.I. Yaminsky, A.I. Akhmetova, A.I. Fedoseev, I.V. Yaminsky. About dimension of the atom. NANOINDUSTRY. 2025. Vol. 18. No. 6. PP. 346–354. https://doi.org/10.22184/1993-8578.2025.18.6.346.354.
INTRODUCTION
In modern metrological measurements, the radius of an atom is defined as half of the interatomic distance in the crystal lattice. For example, this is how the radius of carbon and silicon atoms forming crystals with covalent bonding is defined. Accordingly, the observed radius is called the covalent radius. In the case of carbon, it is 0.076±0.001 nm for diamond, 0.073±0.002 nm for graphite and 0.069±0.001 for linear carbon. Silicon has one crystal shape and therefore only one value of covalent radius, equal to 0.111±0.002 nm. For a number of other elements, the metallic radius of the atom (metals), ionic radius (ionic crystals), van der Waals radii (molecular crystals and noble gases) are determined.
These data are based on statistical analyses of a large number of experimental measurements and are given in the Cambridge Structural Database [1].
RESEARCH METHODS
The determination of the size of a single atom is undoubtedly of great interest. Significant progress in this field became possible after invention of scanning probe microscopy tools - scanning tunnelling and atomic force microscopes. Strictly speaking, these methods do not provide an opportunity to measure a completely isolated atom, but they allow us to determine the atom geometry weakly coupled to its environment. A successful model solution is an atom on a substrate with weak interaction with it. A series of works on visualisation of Xe and Fe atoms was carried out by D.Eigler’s research group at IBM using a high-vacuum scanning tunnelling microscope [2]. The scanning tunnelling microscope was invented by IBM employees H.Rohrer and G.Binnig in 1981, for which they were awarded the 1986 Nobel Prize in Physics. When xenon atoms were observed on the surface of a nickel crystal, the nickel atoms could move freely when interacting with the probe, indicating that they were weakly bound to the substrate [3].
The image of a single copper atom obtained using a scanning tunnelling microscope [4] is shown in Fig.1.
When imaging an atom, a scanning tunnelling microscope obtains a spatial image of the occupied electron level of the outer shell at a positive potential on the probe of the tunnelling microscope. At negative potential on the probe, the image of the unoccupied electron level of the atom is obtained. In this case, the observed sizes of the obtained images should be slightly different. The size of the atom should be considered the first case, since it corresponds to the ground state of the atom.
If an atom is in an excited state, its radius increases accordingly due to transition of an electron to an electron orbital more distant from the atomic nucleus [5]. This was experimentally demonstrated for the gold atom in the ground and excited states (STM conference in Japan, 2005).
Another type of probe microscopy, the atomic force microscope, is also used to determine the size of an atom. The probe of the atomic force microscope lightly touches the atom and traces its topography. The principle of a gramophone player works here - a needle slides over a surface and traces its topography. The interaction force between the probe and the surface has an electromagnetic component, although the main contribution to the interaction is determined by the principles of relativistic quantum mechanics - Pauli’s rule (or prohibition). Pauli’s principle forbids the finding of two identical electrons in one point of space. The same principle does not allow a person to fall through the floor and in general provides the whole spatial architecture of the world we observe.
In atomic force microscopy, chemical identification of individual atoms can be realised by the nature of force interaction, in particular, it is possible to distinguish individual atoms of silicon, tin and lead [6]. Force measurements have recently become popular in various fields from materials science to living systems [7].
To visualise a single atom on a substrate, the magnitude of the contact force on the probe side should be minimised. In this case, the atomic force microscope will obtain images in many respects similar to the scanning tunnelling microscope images. At the same time, certain differences in the images may be observed due to the different nature of the probe-sample interaction.
The scanning tunnelling microscope is often considered to give a more detailed image of the atomic structure of the sample. However, there are also opposite results. A good example is the comparative observation of the internal structure of the pentacene molecule by two methods - tunnelling and atomic force microscopy [8]. The atomic force microscope allows you to look inside the molecule with a greater degree of detail. It should be noted that an unusual probe was used in this experiment. One molecule of carbon monoxide CO was attached to its tip. That is, scanning of one molecule took place with the help of another molecule. Scanning tunnelling microscopy also uses probes with CO molecules at the tip. In this work, visualisation of copper atoms depending on orientation (rotation angle) of the CO molecule was carried out [9].
As can be easily understood, the best spatial resolution of atomic force microscopy can be obtained if the smallest atom, a hydrogen atom, is placed at the tip of the probe. By the way, such an experiment was carried out [10]. And its result was also a measurement of the size of the hydrogen atom itself.
Strictly speaking, from the very beginning, the tunnelling and atomic force microscope were instruments with unique capabilities, but they were not metrological tools. In the following years, many efforts were made to bring them closer to the modern metrology requirements. In 2019, the ISO 11952:201 international standard - Surface chemical analysis - Scanning probe microscopy - Determination of geometric quantities by SPM: calibration of measuring systems was approved. Additions were made in 2024.
If we consider small sizes - units of nanometres - probe microscopes carry out measurements with high accuracy. Certain errors caused by the piezoceramics properties, the material of the moving parts, appear in the range of tens to hundreds of microns. To eliminate them, precise displacement meters based on capacitive, strain-gauge and interference measuring systems are used.
In the case of small fields of observation, such as single atoms, it is necessary to ensure that the displacement systems are correctly calibrated. For example, if a displacement of 1 nm is required, the displacement and is carried out exactly one nanometre. To calibrate microscopes directly during measurements, we have previously proposed a nanometre standard.
Unlike fixed-magnification optical microscopes, SPMs create images by sliding a sharp probe across a surface. The size of the image is directly dependent on the voltage applied to the scanners (piezoelectric elements) and the resulting physical motion. Piezo scanner motion is inherently non-linear, subject to hysteresis, creep and cross-coupling between axes. These effects lead to distortions and scaling errors that can be on the order of 10% or more without correction.
Calibration of SPM is usually carried out in three axes: X, Y (lateral) and Z (vertical). Calibration consists in comparing measured parameters on the AFM with passport values of standards in X, Y and Z axes, in correction and determination of correct coefficients to account for the piezoscanner nonlinearity. For high-precision calibration, software methods of data processing and distortion compensation are used (Fig.2) [11].
Arrays of isolated, well-defined objects (e.g., nanoparticles or fabricated columns) with known spacing between them are useful for evaluating scanner linearity at various points in the scanning range.
Vertical standards are used to calibrate the scanner’s Z-axis response, which is critical for height, depth and roughness measurements.
Atomic force microscope (AFM) calibration uses calibration gratings: silicon TGZ gratings (Fig.3) and diffraction gratings [12], as well as test samples with known surface topography: for example, freshly chipped mica, highly oriented pyrolytic graphite (HOPG) chromium columns [13], blu-ray disc (Fig.4), and others.
Atoms, with their blurred electron clouds, are often thought to be vague and somewhat abstract. However, scanning probe microscopy provides precise information that atoms are rather solid entities with well-defined contours. It is these data that allow us to speak about the exact dimensions of individual atoms. To a certain extent, the question remains open as to what, if I may say so, is the mechanical rigidity of the atom. In other words, it is interesting to know how the size of an atom depends on the magnitude of the force applied to it. The dependence of energy and force of interaction between atoms is determined by pair potentials. Among them are empirical dependences - Lennard Jones, Buckingham, Morse, etc. potentials. Accordingly, when atoms approach each other, repulsive forces appear, which can change the observed atom size. It seems achievable in principle that such experiments can be carried out using an atomic force microscope. At present, we have not found such published data.
The size of an atom can be interpreted in terms of the Heisenberg uncertainty principle [14]. The standard deviation Δr of the electron shell radius and the standard deviation Δpr of the electron momentum projection satisfy the relation:
Δr Δpr ≥ ħ/2,
where ħ is the reduced Planck constant. If at measurements in a scanning probe microscope the standard deviation Δr of the radius is 10–3 nm, then the standard deviation of the momentum is 0.5 · 10–22 J. At the same time, the electron velocity is approximately three times less than the speed of light. In a scanning tunnelling microscope, an electron tunnels from the filled shell of an atom to the unfilled electronic state of the probe, or in the opposite direction from the filled electronic level of the probe to the free electronic level of the atom. In the quantum theory of tunnelling transition (in certain approximations) probability of tunnelling depends on the spatial arrangement of electronic levels and the distance between them.
To image an atom in atomic force microscopy, one should take into account the Pauli principle, which forbids two electrons with the same set of quantum numbers to be in the same place in space. This rule leads to spatial separation of electrons of the probe and the atom under study. Figuratively speaking, the probe only touches the atom. At that, repulsive forces and their corresponding energy arise. For description of forces and interaction energy there are empirical potentials mentioned earlier. Whether it is possible to derive a rigorous conclusion of this potential theoretically from the relativistic quantum theory is an open question for us at present. It is clear that in order to obtain by atomic force microscopy the size of an undisturbed atom, its measurement should be carried out at minimum force of influence.
Whether the Pauli principle allows us to determine the atom size with such a high, achievable in experiment accuracy, given that only the coordinate is measured without measuring the momentum, remains open. However, this does not prevent us from admiring the beauty of individual atoms and molecules observed in scanning tunnelling and atomic force microscopes.
Probe microscopy is not the only method that allows visualising the atom. The first image of an atom was obtained using an ion projector by Erwin Muller [15]. The peculiarity of this method is that the atom itself must be at the top of the metal tip during observations.
Fifteen years later, the image of the atom was obtained using a transmission electron microscope [16]. Taking into account the peculiarities of this method, an image of a heavy atom, uranium, was obtained. The atom was located on a graphite substrate. For the first time it was possible to visualise individual heavy atoms of mercury in a transmission electron microscope with a high probability [17]. In the ion projector and electron microscope, the image of an atom is usually obtained on a phosphor screen on which ions or electrons fall after interaction with the atom under study.
DETERMINATION OF ATOM SIZE BY AFM AND STM METHODS
In an atomic force microscope, the situation is different. The tip of the cantilever (atom in the tip) touches the observed atom, and the tip of the cantilever describes a trajectory, repeating the contour of this atom. The resulting images of the atom are striking in their beauty. In the atomic force microscope, atoms look like billiard balls. They have a certain size and shape. In this case, the probe-atom interaction, due to which the image is obtained, is conditioned by the Pauli prohibition. This interaction does not belong to any of the interactions of the standard model of physics - strong, weak, electromagnetic and gravitational. The force and interaction in atomic force microscopy is a consequence of Pauli’s prohibition. The atomic force microscope measures this force and interaction. However, we have not found a theoretical description of the nature of this interaction involving relativistic quantum mechanics in the scientific literature.
In scanning tunnelling microscopy, individual images of atoms also have clear contours. The repeat accuracy of the contours is at the level of thousandths of a nanometre. These images give information about the size of the atom. Whether these images provide information about the size of an electron is an open question. In the standard model, the electron is considered to be an elementary particle that has no size. In school physics, too, there is a concept of point charge, the use of which allows many problems from electricity to be solved accurately. The electron has measured the exact values of mass, charge and spin. The question of size remains open. So far there is no experimental data indicating the size of the electron.
ACKNOWLEDGEMENTS
This work was partially performed under the state order of the Lomonosov Moscow State University. FemtoScan Online software is provided by Advanced Technologies Center, www.femtoscan.ru
PEER REVIEW INFO
Editorial board thanks the anonymous reviewer(s) for their contribution to the peer review of this work. It is also grateful for their consent to publish papers on the journal’s website and SEL eLibrary eLIBRARY.RU.
Declaration of Competing Interest. The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
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