Introduction

Atoms, with radii of about angstrom (Å, 10−10 m), are the building blocks of our nature physical world. However, they are too small to be seen with visible light. A conventional optical microscope can only tell spots with a size larger than half of the optical wavelength, which is more than 2,000 Å, limited by the optical diffraction1. A scanning tunneling microscope (STM) is able to reach a subatomic resolution2, but it is expensive and also has some certain limitations. Here we show that atomic sizes can be resolved optically with Raman spectroscopy. We load iodine molecules into nano-channels of a zeolite single crystal and find their Raman spectroscopy is significantly modified by the interaction between the iodine molecules and the rigid frame of the crystal's nano-channels. The number of excitable vibration quantum states of such a diatomic molecule is limited by the size of the nano-channel. We determine the radius of an iodine atom from the data of Raman spectroscopy. Our work not only demonstrates a spectroscopy probe for measuring the atomic sizes with nano-channel, but also may lead to practical applications in nano science and technology, such as designing, fabricating and diagnosing nano materials and structures in the level of single atoms and molecules.

An atom consists of a central nucleus surrounded by a cloud of electrons, which determines the size or the boundary of the atom. Thus the atomic size plays an important role in the physical and chemical properties of the atom, as shown in the periodic table. It is well known that the atomic or molecular size affects the real gas state equation. A deviation from the ideal gas model was first pointed out by Van der Waals3. For this reason, we often call the radius of a spherical atom as the Van der Waals radius. While the electron distribution of the hydrogen atom can be precisely calculated following quantum mechanics, it is extremely difficult to do so for multi-electron atoms. Experimentally, besides the STM, there have been some indirect methods to estimate the scale of atomic sizes, such as X-ray diffraction and neutron diffraction4,5,6,7,8,9.

In this letter, we focus on determining the Van der Waals radius of the atoms in an iodine molecule. As the atoms are in the electronic ground state configuration, we can treat them as two hard spheres with a radius of ra and internuclear separation of r. Our modeled systems are schematically illustrated in Figure 1. In free space, the diatomic interaction can be described by the Morse potential10, which allows a large number of discrete vibration quantum states, as shown in the right panel of Figure 1a. However, the situation is different when the iodine molecule is loaded into a cylindrical cage with an internal diameter 2R (here R is the radius) slightly larger than the size of the ground state molecule. At higher vibration states, the vibration amplitude increases such that at a certain energy level the atoms start to collide with the cage wall (Figure 1b). The vibration states at higher energies are blurred due to the vibration energy dissipation from the molecule to the framework of the wall, as illustrated in the right panel of Figure 1b. As a result, Raman transitions to higher vibration states, which exist in the free space, disappear when the molecule is trapped inside the cage. Thus we may be able to determine the iodine atomic size from the data of Raman spectroscopy if the cage dimension is precisely known.

Figure 1
figure 1

Schematics of diatomic molecules and two-photon Raman transitions between vibration levels at the electronic ground state.

(a) The left panel depicts in free space a molecule at ground vibration energy level and two molecules at higher vibration energy levels and the right panel represents the molecular Raman transitions in the free space. (b) Two molecules confined in a rigid molecular nano channel are illustrated in left panel. The right panel describes molecular Raman transitions in the nano channel. The two atoms of the diatomic molecule at a specific vibrational state would vibrate between the inner turning position (smallest internuclear separation) and outer turning position (largest internuclear separation) of that state. When diatomic molecules are put into a rigid cylindrical cage, if the internuclear separation rt of the outer turning position plus the atomic diameter 2ra is larger than the size of the cage, in other words rt + 2ra ≥ 2R, these vibration states will be blurred due to the vibration energy dissipation from the molecule to the framework of the wall, which will be revealed by Raman spectroscopy. If the internal radius R of the cylindrical cage is well known, the atomic radius ra of the confined molecule can be calculated through ra = R − rtc/2.

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With a given diatomic interaction potential U(r) inside the nano cage, we can determine the outer turning position at rt by U(rt) = E for a specific vibrational state with the energy eigenvalue E which can be obtained from the Raman shift. It is well known that, at a high vibrational energy state, the probability density is concentrated at these classical turning positions, where the atom spends most of its time with minimum speed11. Meanwhile, the atoms at their outer turning positions are much closer to the wall of the nano cage and have higher probability in colliding incoherently with the wall. As the outer turning positions approach to the wall, the two-photon Raman transition probability reduces. Therefore, maintaining a free-state-like vibrational state requires rt + 2ra ≤ 2R. The molecule 5 in Figure 1b represents such a critical case where the sphere of the atom coincides with the wall: rtc + 2ra = 2R, i.e., ra = R − rtc/2.

Results

We use aluminophosphate (AlPO4) porous single crystals as the “molecular ruler”, which have been widely adopted as host template platforms since their ultra-small opening nano-channels are suitable to trap small guest molecules12,13,14,15,16,17,18,19,20,21. AlPO4 single crystals are transparent from ultraviolet to near-infrared, which are appropriate for the optical investigation of the trapped guest molecules18,19,20,21,22,23. We had introduced iodine molecules into the nano-channels of AlPO4-5 (AFI) and AlPO4-11 (AEL) single crystals and investigated the vibrational energy level separations of the confined iodine molecules in our previous work24.

The framework structures of AFI and AEL crystals have been well determined by X-ray diffraction and neutron diffraction experiments25,26,27. The main channel (cylindrical cage) of AFI crystal consists of twelve alternating AlO4 and PO4 tetrahedrons, leading to a circular cross section. The diameter of the space enclosed by channel framework is about 7.3 Å. While the main channel of AEL crystal is composed of ten alternating AlO4 and PO4 tetrahedrons with 7.2 Å-major and 5.3 Å-minor elliptical enclosed space. The iodine molecules are oriented along the major axis not the minor axis in AEL channels. This configuration is confirmed by molecular dynamics simulation as well as polarized Raman spectroscopy20,21,28.

The polarized Raman spectra of iodine molecules in free space, accommodated in AFI (I2@AFI) and in AEL (I2@AEL) crystals are shown in Figure 2a, 2b and 2c, respectively. The Raman scattering is polarization independent for free-standing iodine molecules, reflecting their isotropic characteristics. In contrast, the Raman scattering of the I2@AEL is strongly polarized along the major elliptical axis direction, while the Raman polarization property of the I2@AFI is in between vapor phase iodine and I2@AEL. More than fifty Raman peaks are observable for the free-standing iodine molecules. The intensity of the overtones decreases slowly. In comparison, the intensities of the overtones of the confined iodine molecules decrease rapidly. In the Raman spectra (Figure 2), only fourteen and ten overtones are observable in I2@AFI and I2@AEL crystals. We attribute these phenomena to the confinement effects on the iodine molecules. The vibration movements of the confined molecules are gradually influenced by the framework atoms of the wall. At higher vibration states, the magnitude of vibration decreases dramatically due to the vibration energy dissipation to the framework atoms and the vibration is gradually undetectable in the Raman spectrum.

Figure 2
figure 2

Polarized Raman spectra of iodine molecules in free space, inside an AFI crystal and an AEL crystal for excitation with a 514.5 nm laser line.

(a) In free space, more than 30 peaks are clearly observed with iodine molecules. (b) For iodine molecules trapped inside the AFI crystal, there are about 14 monotonously decreasing peaks. (c) With enclosed molecules in the AEL crystal, only about 10 peaks are identified. It is clearly seen that, in b and c, the intensity of the peaks of confined iodine molecules presented an exponential decay nature. These phenomena are attributed to the different degree of channel framework's confinement on the vibrational motion of the embedded iodine molecules. The configurations of Raman scattering experiment are illustrated in the insets of (b) and (c). The incident excitation laser line is polarized along the y-axis and propagates along the negative z-axis, while the back-scattering Raman signals propagates along the positive z-axis and are polarized along the y-axis. The AFI and AEL crystals are oriented such that their channels' direction parallels the x-axis.

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We define the point of the vibration magnitude down to −20 dB or 0.01 as the maximum reachable boundary of the diatomic vibration. From the Raman spectra shown in Figure 2, this boundary is determined at the vibration state of v = 10 ± 1 for I2@AEL crystals and v = 14 ± 1 for I2@AFI crystals, respectively. The magnitude of the molecular vibration would eventually be damped to zero if the vibration is reaching or overcrossing the vibration boundary.

The vibration potential at electronic ground state can be drawn from the overtones frequencies showed in the Raman spectra (Figure 2). We represent the internuclear potential with a Morse potential

which is determined by three parameters De, a and re, where restands for the equilibrium separation, De denotes the dissociation energy and a determines the “width” of the potential10, as shown in the right panel of Figure 1a. The vibrational energy eigenvalues can be expressed as , where denotes the classical vibrational frequency (μ stands for the reduced mass of the molecule). We determine these parameters by best fitting the numerically calculated vibrational energy spectrum to the Raman shifts shown in Figure 2. For free-standing iodine molecule, it is well determined that ωe = 214.548 cm−1, a = 1.52270 Å−1 and De = 18681.4 cm−1 29. The value re = 2.665 Å can be get from rotational spectrum29. Because the rotational spectrum was not resolved in the confined state, we set re = 2.665 Å in three cases. For I2@AFI, we get ωe = 212.5 cm−1, a = 1.759 Å−1 and De = 1.373*104 cm−1. The vibration boundary at v = 14 ± 1 corresponds to the critical turning position rtc = 3.02 ± 0.01 Å, as illustrated in Figure 3a. In addition, the data of the lattice parameters of the AFI and AEL single crystals were well established by X-ray and neutron diffraction25,26,27. Here we take the radii of framework atoms according to Batsanov's values9: 2.10 Å, 1.95 Å and 1.55 Å for aluminum, phosphorus and oxygen, respectively. From these data, we obtain the effective inner diameter of the enclosed nano channel to be 7.3 Å ± 0.1 Å, as shown in Figure 3b. Then we estimate the radius of the iodine atom to be ra = (2R − rtc)/2 = 2.10 Å ± 0.05 Å.

Figure 3
figure 3

The determination of atomic radius of iodine atoms confined in the AFI crystal.

(a) Vibrational Morse potential curve of iodine molecule in AFI crystal. The energy levels of v = 0, v = 5, v = 10, v = 14 are plotted. The iodine molecule is plotted such that one atom is fixed at zero point, the other one vibrates between the two turning points of that level (where the state's energy coincides with the potential energy). The outer turning point of the last observable vibrational state v = 14 can be identified from the potential curve, which is rtc = 3.02 Å. We model that the iodine atoms would collide with walls at this position. (b) The instant moment of iodine atoms strike the framework of main channel atoms viewed from [001] direction. The internuclear distance of two iodine nuclei is set to 3.02 Å. The diameter of the space enclosed by channel framework is 7.3 Å ± 0.1 Å. Then we estimate the radius of the iodine atom to be ra = (2R − rtc)/2 = 2.10 Å ± 0.05 Å.

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The nano-channels in the AEL crystal are elliptical. The polarization Raman spectrum shows all the iodine molecules in its nano channels are aligned along the major axis not the minor axis. From the Raman shift of the overtones, we determined the parameters of the interatomic potential as: ωe = 217.6 cm−1, a = 1.678 Å−1 and De = 1.582*104 cm−1. The vibration boundary is at v = 10 ± 1, thus we obtain its outer turning position rtc = 2.94 ± 0.01 Å (Figure 4a). The inner major axis of the enclosed nano channel to is 7.2 Å ± 0.1 Å, as shown in Figure 4b. Using the major axis radius Rj = 3.6 Å of the nano channel, we determine the atomic radius ra = Rj − rtc/2 = 2.10 Å ± 0.05 Å. This value is consistent with the one obtained from AFI crystals.

Figure 4
figure 4

The determination of atomic radius of iodine atoms confined in the AEL crystal.

(a) Vibrational Morse potential curve of iodine molecule trapped inside the AEL crystal. The energy levels of v = 0, v = 5, v = 10 are plotted. The iodine molecule is plotted such that one atom is fixed at zero point, the other one vibrates between the two turning points of that level (where the state's energy coincides with the potential energy). The outer turning point of the last observable vibrational state v = 10 can be identified from the potential curve, which is rtc = 2.94 Å. We model that the iodine atoms would collide with walls at this position. (b) The frame of iodine atoms hit the framework of main channel atoms view from [001] direction. The internuclear distance of two iodine nuclei is set to 2.94 Å. The major axis of the elliptical cross section is 7.2 ± 0.1 Å. Using the major axis radius R = 3.6 Å ± 0.05 Å of the nano channel, we determine the atomic radius ra = R − rtc/2 = 2.10 ± 0.05 Å.

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Discussion

Table 1 shows a comparison of the iodine atomic radius determined in this work with those reported in the literature. The radii reported in the literature are mainly determined from X-ray diffraction and neutron diffraction technologies. It can be seen that there is an approximate 10% uncertainty of iodine's radius. This is because the same atom in different molecular crystals would manifest slightly different radii. In fact, the atomic radius just depicts an imaginary sphere to represent the electron density distribution for each atom, the actual distribution may have a rather complicated shape rather than simple sphere. From Table 1 we can see that, generally speaking, the atomic radius of confined iodine atom determined from the Raman spectra are in good agreement with the values got from other methods.

Table 1 Atomic radius of iodine determined by different methods
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In summary, we proposed and demonstrated an optical spectroscopy approach to estimate the atomic radius of iodine atom through a “molecular vibrational confiner”. The confined iodine molecule at a vibrational state higher than a critical value will collide with the channel wall by dissipating vibration energy to the host framework atoms. As a result, the overtones to the higher vibration states disappear in the Raman spectra. With the known size of the nano channels, the Van der Waals radius of the iodine atom can then be determined from the number of excitable vibrational levels. This technique provides a novel method to “see” the size of atoms with a high precision. Our result may lead to practical applications in nano science and technology, as well as fundamental research in atomic, molecular and optical physics.

Methods

The AFI and AEL single crystals were synthesized by hydrothermal method30,31. The as-synthesized AFI and AEL single crystals have typical dimensions of 120 × 120 × 350 μm and 10 × 30 × 70 μm, respectively. Iodine molecules were loaded into the nano-channels of AFI and AEL single crystals by a physical diffusion process. The AFI and AEL crystals were calcined at 580°C and 680°C respectively in O2 atmosphere for about 40 hours to remove the organic template molecules in the channels. Then the calcined crystals were sealed together with solid iodine source (BDH laboratory supplies, > 99%) in a Pyrex tube under a vacuum of 10−2 mbar. Iodine molecules were physically diffused into the nano channels of the AFI and AEL single crystals at temperature 150°C. The loading density of the iodine in the host crystals can be controlled by the weight ratio of the loaded solid iodine to the AFI (AEL) single crystals. We only focus on the low loading density samples in this experiment so as to ignore the intermolecular interaction.

The data were taken at room temperature using Jobin-Yvon T64000 micro-Raman spectrometer equipped with a liquid nitrogen cooled CCD detector in a backscattering configuration. A linearly polarized 514.5 nm beam output from an Innova 70C Ar/Kr ion laser with a line width of 6 GHz and a power of 30 mW was used as the excitation laser line. As shown in the insets of Figure 2b and 2c, the incident excitation laser line is polarized along the y-axis and propagates along the negative z–axis, while the back-scattering Raman signals propagates along the positive z-axis and are polarized along the y-axis.

We use the vibration frequencies got from the first 12 peaks of the Raman spectrum of the iodine molecule confined in AFI crystal for plotting a Morse potential to represent the vibrational potential curve and show it in Figure 3a. And we use the vibration frequencies got from the first 8 Raman peaks for extrapolating a Morse potential to represent the vibrational potential curve for the iodine molecule confined in AEL crystal and show it in Figure 4a. The equilibrium interatomic distance is set to 2.665 Å, which is the same as the one of free iodine molecule determined from rotational spectrum29. The energies of the vibrational states are determined from Raman spectra and then these turning points in potential curve are identified correspondingly.