Electron Diffraction of Molecules and Clusters in Superfluid Helium Droplets

Abstract

In an effort to solve the crystallization problem in crystallography, we have been engaged in developing a method termed “serial single molecule electron diffraction imaging” (SS-EDI). The unique features of SS-EDI are superfluid helium droplet cooling and field-induced orientation. With two features combined, the process constitutes a molecular goniometer. Unfortunately, the helium atoms surrounding the sample molecule also contribute to a diffraction background. In this chapter, we analyze the properties of a superfluid helium droplet beam and its doping statistics, and demonstrate the feasibility of overcoming the background issue by using the velocity slip phenomenon of a pulsed droplet beam. Electron diffraction profiles and pair correlation functions of monomer-doped droplets, small cluster and nanocluster -doped droplets are presented. The timing of the pulsed electron gun and the effective doping efficiency under different dopant pressures can both be controlled for size selection. This work clears any doubt of the effectiveness of superfluid helium droplets in SS-EDI, thereby advancing the effort in demonstrating the “proof-of-concept” one step further.

8.1 Introduction

More than 80% of the atomic-resolution structures can be attributed to single-crystal crystallography in the Protein Data Bank. Still, the difficulty and unpredictability of crystallization remain the nemesis of the technique. Recently, several new ideas have been introduced to solve the crystallization problem in crystallography [1,2,3,4,5]. One of the most successful is termed “diffract and destroy” [1], where ultrashort and ultra-intense x-ray photons are used to diffract from a single particle before the particle is destroyed by the radiation. The fundamental premise is that sufficient sampling of a randomly oriented sample population allows complete sampling of all possible orientations. To date, several dozens of new protein structures have already been solved using this method [6, 7]. The method has been adopted to determine the shape of and detect the vortices in superfluid helium droplets [8]. Another method employs electrons because of their much larger diffraction cross sections [9] and easier accessibility in laboratories than ultra-short x-ray photons. In addition, sample alignment in a laser field prior to diffraction has also been demonstrated, simplifying the data interpretation tremendously [2, 3]. The ease in aligning a molecule embedded in superfluid helium droplets has further prompted the idea of using Coulomb explosion to obtain structures of small molecules [4, 10, 11].

Our group has been developing a method called serial single molecule electron diffraction imaging (SS-EDI) as a potential means to solve structures of large biological molecules and nanomaterials [5]. The procedure starts with electrospray ionization to produce ions for doping into superfluid helium droplets. The cooled ions are aligned by an elliptically polarized laser field and subjected to radiation by high energy electrons. The diffraction patterns from the isolated molecules embedded in superfluid helium droplets are accumulated as the sample is refreshed in repetitive pulses for the desired signal-to-noise (S/N) ratio. Three-dimensional information is obtained from diffraction images collected from different orientations of the sample achieved by different polarizations of the alignment laser beam. So far, we have successfully demonstrated the feasibility of doping proteins such as the green fluorescent protein into superfluid helium droplets; [12,13,14] and performed electron diffraction (ED) of several neutral molecules and clusters embedded in superfluid helium droplets, without laser alignment [15,16,17,18,19].

The role of superfluid helium droplets in SS-EDI is multifaceted. The low temperature of ~ 0.4 K within the droplets is crucial for field-induced alignment and orientation; compared with a room temperature sample, superfluid helium droplet cooling achieves a reduction in the required field strength by nearly four orders of magnitude for the same degree of orientation [20]. In addition, the superfluidity of the droplets allows free rotation of the dopant in response to an external electromagnetic field [21,22,23]. However, the presence of helium atoms surrounding the embedded molecule inevitably contributes to the diffraction background, particularly if the droplet size reaches the order of 10⁶ helium atoms [24]. Here, we demonstrate the feasibility of solving structural information of molecules and clusters embedded in superfluid droplets.

8.2 Theory

8.2.1 Theoretical Concept of Gas-Phase Electron Diffraction

A brief review of the fundamental theory about electron diffraction is presented in this section. More details can be found in the references [25,26,27,28].

The de Broglie-wavelength λ of a high-speed electron including the relativistic correction is calculated as:

$$\lambda = \frac{h}{{\sqrt {2m_{e} {\text{ eV}}(1 + {\text{eV}}/{\text{mc}}^{2} )} }},$$

(8.1)

with the acceleration voltage V, the electron charge e, the electron mass m_e, the speed of light c, and the Planck constant h. For an electron of 40 keV, the de Broglie-wavelength is 6.0 × 10⁻¹² m. For simplicity, the electron beam is treated as a plane wave, and the wavefunction $\Psi_{0}$ of the incident electrons traveling along the z-direction is described as:

$$\Psi_{0} = e^{{i{\mathbf{k}}_{{\mathbf{0}}} z}} ,$$

(8.2)

with a wave vector k₀, where $\left| {{\mathbf{k}}_{{\mathbf{0}}} } \right| = {{2\pi } \mathord{\left/ {\vphantom {{2\pi } \lambda }} \right. \kern-\nulldelimiterspace} \lambda }$.

The independent-atom model treatment, commonly accepted in the community of gas-phase electron diffraction, is used to calculate the molecular diffraction pattern shown in Fig. 8.1. In this approach, the atoms within a molecule are treated as perfect spherical independent diffraction centers. The elastic scattering wavefunction $\Psi_{n}^{^{\prime}}$ of the nth atom can be written as:

$$\Psi_{n}^{^{\prime}} = K\frac{{e^{{ik_{0} R}} }}{R}f_{n} (\theta )e^{{i({\mathbf{k}}_{{\mathbf{0}}} - {\mathbf{k^{\prime}}}){\mathbf{r}}_{{\mathbf{n}}} }} ,$$

(8.3)

Fig. 8.1

Electron diffraction of a pair of atoms

where $K = \frac{{8\pi^{2} me^{2} }}{{h^{2} }}$, R is the distance between scattering center and observation point, the atomic position vector r_n, the scattered wave vector ${\mathbf{k^{\prime}}}$ and the scattering angle θ with respect to z, and the atomic scattering factor $f_{n} (\theta )$, which includes amplitude and a phase shift term. Then the overall scattering wavefunction for a molecule containing N atoms, is the sum of the scattering wavefunctions from individual atoms:

$$\Psi^{^{\prime}} = \sum\limits_{n = 1}^{N} {K\frac{{e^{{ik_{0} R}} }}{R}f_{n} (\theta )e^{{i({\mathbf{k}}_{{\mathbf{0}}} - {\mathbf{k^{\prime}}}){\mathbf{r}}_{{\mathbf{n}}} }} } ,$$

(8.4)

In traditional gas phase electron diffraction (GPED), the magnitude of the momentum transfer s is defined as:

$$\left| {\mathbf{s}} \right| = \left| {{\mathbf{k}}_{{\mathbf{0}}} - {\mathbf{k^{\prime}}}} \right| = 2k_{0} \sin \left( {\frac{\theta }{2}} \right) = \frac{4\pi }{\lambda }\sin \left( {\frac{\theta }{2}} \right),$$

(8.5)

and the total scattering intensity of the molecule can be written as a function of s:

$$I(s) = \frac{{K^{2} I_{0} }}{{R^{2} }}\sum\limits_{n = 1}^{N} {\sum\limits_{m = 1}^{N} {f_{n} (s)f_{m}^{*} (s)e^{{i{\mathbf{sr}}_{nm} }} } } ,$$

(8.6)

with ${\mathbf{r}}_{nm} = {\mathbf{r}}_{n} - {\mathbf{r}}_{m}$, the distance between the nth atom and the mth atom. Here $r_{nm} \ll R$, I₀ is the intensity of the incident electron beam. The scattering factor is rewritten as a function of s with amplitude $\left| {f(s)} \right|$ and the phase term $\eta (s)$ with values tabulated in ref. [29]. The total diffraction intensity I(s) can be split into two parts: a non-structural related contribution from individual atoms, denoted as atomic scattering intensity I_a(s), and a structural related contribution determined by interatomic distances, denote as molecular scattering intensity I_m(s).

$$\begin{aligned} I(s) & = I_{m} (s) + I_{a} (s) \\ & = \frac{{K^{2} I_{0} }}{{R^{2} }}\sum\limits_{n = 1}^{N} {\left| {f_{n} (s)} \right|^{2} } \\ & + \,\frac{{K^{2} I_{0} }}{{R^{2} }}\sum\limits_{n = 1}^{N} {\sum\limits_{\begin{subarray}{l} m = 1 \\ m \ne n \end{subarray} }^{N} {\left| {f_{n} (s)} \right|\left| {f_{m} (s)} \right|\cos \left( {\eta_{n} (s) - \eta_{m} (s)} \right)e^{{i{\mathbf{sr}}_{nm} }} } ,} \\ \end{aligned}$$

(8.7)

For randomly oriented gas-phase molecules, the molecular scattering intensity can be calculated by integrating over all orientations:

$$I_{m} (s) = \frac{{K^{2} I_{0} }}{{R^{2} }}\sum\limits_{n = 1}^{N} {\sum\limits_{\begin{subarray}{l} m = 1 \\ m \ne n \end{subarray} }^{N} {\left| {f_{n} (s)} \right|\left| {f_{m} (s)} \right|\cos \left( {\eta_{n} (s) - \eta_{m} (s)} \right)\frac{{\sin (sr_{nm} )}}{{sr_{nm} }}} } ,$$

(8.8)

In addition to elastic scattering, inelastic scattering should be considered for small angle diffractions. Inelastic scattering is considered to only spreads over a small angle, and the intensity of inelastic scattering decreases rapidly when s is increasing. In practice, inelastic scattering is considered incoherent and is often treated as non-structural related background during data analysis.

8.2.2 Implementation and Challenges

Although the helium atom is light with a small scattering factor, in the case of electron diffraction of a sample-doped helium droplet, hundreds or more helium atoms in one droplet still generate a considerable background. They can easily overshadow the diffraction pattern of the sample. It is a unique feature of the helium droplet environment that requires additional treatment.

The total diffraction intensity $I(s)$ of a doped droplet is shown as the following:

$$I(s) = I_{m} (s) + I_{a} (s) + I_{d} (s),$$

(8.9)

where $I_{d} (s)$ is the contribution from the diffraction of the helium droplets. More discussion of this term will be covered in Sect. 8.4. Each helium droplet may contain a different number of helium atoms, and the analytical expression for this term is not yet clear. Additionally, the observed diffraction patterns show that no structural information of the helium atoms could be detected for droplets containing less than a few thousand helium atoms, so we treat the helium droplet diffraction as a monotonic non-structural contribution. Doping causes a certain amount of helium atoms to evaporate, but helium's contribution to the diffraction profile remains similar to that prior to doping. To account for this effect, we record the diffraction pattern of a pure helium droplet. We then use a scaling factor to model the contribution of helium in the overall diffraction profile.

In GPED, the modified molecular scattering curves sM(s) are used to magnify the contribution from molecular scattering by removing all contributions from atoms and the helium droplet background and are further magnified by the momentum transfer s:

$$sM(s) = \frac{{I_{m} }}{{I_{a} }} \cdot s,$$

(8.10)

The Eq. 8.10 can be further written as:

$$sM(s) = \frac{{\alpha_{s} \cdot I_{total} (s) - \beta \cdot I_{droplet} (s) - \alpha_{b} \cdot I_{background} (s) - I_{T,at} (s)}}{{I_{T,at} (s)}} \cdot s,$$

(8.11)

where $I_{total} (s)$, $I_{droplet} (s)$, and $I_{background} (s)$ are the experimental intensities of the doped droplets, pure droplets, and background; $\alpha_{s}$,$\beta$, and $\alpha_{b}$ are the corresponding fitting parameters; and $I_{T,at} (s)$ is the theoretical atomic diffraction intensity from the dopant molecule. All three fitting parameters contain a common response factor, and the ratio $\frac{\beta }{{\alpha_{s} }}$ indicates the ratio of the number of helium atoms after doping to that without doping.

The above method has been used for doped droplets dominated by single dopant doping. When the dopant number increases following the Poisson distribution, a direct fitting of the total diffraction intensity with all the possible candidate structures can be adopted:

$$I_{total} (s) = \beta \cdot I_{droplet} (s) + \alpha_{1} \cdot I_{1} + \sum\limits_{i} {\alpha_{{2_{i} }} I_{{2_{i} }} } + \sum\limits_{i} {\alpha_{{3_{i} }} I_{{3_{i} }} } + \sum\limits_{i} {\alpha_{{4_{i} }} I_{{4_{i} }} + ...} ,$$

(8.12)

where $I_{{2_{i} }}$,$I_{{3_{i} }}$, and $I_{{4_{i} }}$ are the theoretical diffraction intensities for a given structure i of dimers, trimers and tetramers, and $\beta$,$\alpha_{1}$, $\alpha_{{2_{i} }}$,$\alpha_{{3_{i} }}$, and $\alpha_{{4_{i} }}$ are the fitting parameters. Similar to Eq. 8.11, the value $\beta$ represents the remaining helium’s contribution after doping relative to that of a neat droplet beam. The value $\alpha_{1}$ represents the contribution from monomer and the values of $\alpha_{{2_{i} }}$,$\alpha_{{3_{i} }}$, and $\alpha_{{4_{i} }}$ are the contributions of clusters of dimer, trimer and tetramers with structure i: all contain a common scaling factor. Only the ratios of the $\alpha_{{n_{i} }} /\alpha_{1}$ can be compared with the Poisson statistics. When the signal-to-noise level of the experimental data is limited, additional constraints with the pre-assumption of doping statistics need to be considered. Details of the procedure and the statistical methods will be explained in the case studies.

8.3 Experiment

The apparatus used for electron diffraction in our laboratory to study samples doped in superfluid helium droplets is illustrated in Fig. 8.2. It combines a superfluid helium droplet source with a GPED system. The apparatus can be divided into three regions: a droplet source chamber, a doping chamber, and a diffraction and detection (D&D) chamber.

Fig. 8.2

Experimental setup showing the sample Pulse Valve and in-line Time-of-flight

The droplet source consists of a pulse valve (PV, Cryogenic Copper Even-Lavie Valve) and a skimmer cone of an orifice of 2 mm. High purity helium (99.9995%) is connected to the PV with an effective nozzle diameter of 50 µm and is kept at a temperature between 10 and 20 K with a stagnation pressure between 50 and 70 atm. The skimmer located 11 cm downstream of the PV separates the source chamber from the doping chamber. The droplet source chamber's vacuum level is maintained at 1 × 10^–7 torr when the PV is off and reaches up to 5 × 10^–6 torr when the PV is on.

The sample is introduced into the doping chamber by either a heated pickup cell or a sample pulse valve (SPV). For samples requiring heating temperatures above 200 °C, a heated pickup cell located 5 cm downstream from the skimmer base is used. The vapor pressure is controlled by varying the temperature of the cell. For samples with higher vapor pressures, a SPV is mounted 10 cm away from the skimmer base and 5–10 mm from the droplet beam axis. With the use of the SPV, the amount of introduced sample can be significantly reduced, thus resulting in less diffused sample present in the diffraction chamber. The doping chamber is separated from the diffraction chamber by a home-made conical skimmer with a diameter of 4 mm. The doping chamber’s vacuum level is 1 × 10^–7 torr without the sample and increases to 10^–5 torr when the sample is present.

The GPED system located in the D&D chamber includes the electron source, the diffraction system, and the imaging system. The collimated electron beam is pulsed such that it would spatially and temporally overlap with the doped droplets, and the resulting diffracted electrons are detected by a phosphor screen and a camera. The electrons are emitted from a LaB₆ filament, biased at 40 keV, then focused onto the detector by a magnetic lens included in the electron gun (Kimball Physics, EGH-6210A). An extra magnetic lens can be mounted below the commercial electron gun to reduce the beam size from a diameter of 3 mm to 0.3 mm at the diffraction spot. The diffracted electrons hit the detector while the undiffracted beam is stopped and measured by a Faraday cup that is 4.8 mm in diameter, 50.8 mm in height, and located 2 mm above the detector. The phosphor detector, as well as the imaging system, can be mounted at different distances from the diffraction spot to collect different ranges of diffraction angles. In most of our experiments, the distance is set to cover up to a diffraction angle of 4.4°, equivalent to s = 8 Å⁻¹. The imaging system consists of a scintillation screen with a phosphor coating (Beam Imaging Solutions P43, 40 mm) and an EMCCD camera (Andor Technology, iXON Ultra) to record the image. The camera is synchronized with the pulsed electron beam to capture individual diffraction pattern on the phosphor screen. To dynamically remove the background diffraction, a “toggle” method is used. In addition to images from droplets, extra images are measured between consecutive droplet pulses and are accumulated and saved as the background image. When the doping sample is present, the difference between the two images is the net diffraction from the sample doped droplets I_total. When the sample is not present, the difference corresponds to the net diffraction of pure droplets I_doplet. Hence, the overall experiment's effective repetition rate is equivalent to half of that of the electron beam. To protect the gun filament and reduce the ambient diffraction, a cold trap is connected to the shielding elements of the electron beam. With the cold trap, the vacuum level of the D&D chamber can be maintained at 8 × 10^–9 torr.

To assist in the electron source characterization and optimization of the experimental conditions, a rotatable 6-position wheel holding a variety of components is mounted inside the diffraction chamber as illustrated in Fig. 8.3. Each component on the wheel can be positioned in the diffraction spot at the center of the chamber via a rotating shaft located outside the vacuum chamber. The components’ details are summarized in Table 8.1, and all of them are not necessarily coexist on the wheel.

Fig. 8.3

Rotatable wheel inside D&D chamber

Table 8.1 List of components on the rotatable wheel

Another TOF–MS oriented in-line with the droplet beam consists of a pair of repeller and ground electrodes, and an MCP detector. It is located inside the D&D Chamber for ion detection with high m/z. It can be operated simultaneously with the image collection of electron diffraction. In both TOF–MS setups, the doped droplets are ionized via the 4th harmonic at 266 nm from a pulsed Nd: YAG laser (Quantel, Brilliant). The laser power density on the level of 10⁶ W/cm² is typically high enough to ionize or fragment the dopants inside the superfluid helium droplets via 2 or 3 multiphoton absorptions while leaving the helium droplets mostly intact. When the miniature TOF–MS is in use, the ionization spot is at the same location as the diffraction spot, whereas in the case of the in-line TOF–MS, the ionization region is 5 cm upstream. In this latter case, the laser timing needs to be adjusted accordingly.

8.4 Characterization of Droplet Sizes

Benzene doped droplet was used for the characterization of droplet sizes for different pre-cooled temperatures. The ionization threshold of benzene is only 9.24 eV [30], and 2 photons at 266 nm, with a total energy of 9.35 eV is sufficient for ionization. The 2-photon process is achievable at a power density of 10⁵ to 10⁶ W/cm² without any focusing lens. The mass spectrum of benzene doped droplets contains no fragments, with only parent ions and clusters of parent ions with helium, ideal for size measurement. Since most of the excess energy (0.1 eV) released upon ionization is taken away by the departing photoelectron, size measurements from benzene doped droplets should be reliable. One example of droplets forming with a stagnation temperature of 20 K is shown in Fig. 8.4. The detected current contains contributions of bare ions and ion-doped droplets, and the latter could contain multiple groups with different sizes, as revealed in our previous measurements [12, 31].

Fig. 8.4

Arrival time profiles of benzene doped helium droplets recorded at a source temperature of 20 K and a doping pressure of 2.5 × 10^–4 Torr. The acceleration voltages on the repeller electrode are labeled in the figure

The time profiles are fitted using the exponentially modified Gaussian probability distribution function [32], i.e., a convolution of a Gaussian function with an exponential decay function, and used the nested F-test to avoid overfitting. The center of the fitted Gaussian function is then used to represent the group arriving time. Combined with the acceleration voltage and travel distance, each group's size can then be obtained.

8.5 Image and Data Processing

The procedure of image treatment and data process for structure analysis is discussed in this section. One of the examples used here is the diffraction image of CBr₄ doped droplets. As mentioned in Sect. 8.2, each unique pair of atoms generates a set of rings, and the diffraction image is highly symmetric with the superposition of all the rings from each pair. However, the Faraday cup and its supporting arm cast a shadow on the phosphor screen as shown in the raw image of CBr₄ doped droplet in Fig. 8.5a. The portion inside the red dash box is then removed and replaced by the content in the opposite direction. In this experiment, the diffraction signal is relatively weak and a Wiener filter [33] is used in the image treatment to reduce the noise level of the image. A radial profile of the treated image can be calculated by a MATLAB script and then used in structure fitting as I_total(s). For a better visualization effect, the image is then averaged over all four quadrants to smooth the image further. With the fitting parameters, the helium droplet contribution part is then removed to emphasize the dopant contribution as in Fig. 8.5c. Contrasting features can be seen by the naked eye only in this figure in the case of the weak signal.

Fig. 8.5

Diffraction images from CBr₄ doped helium droplets. Left: Raw image from CBr₄ doped helium droplets, Middle: Wiener filtered regenerated image from left one by refilling the portion inside the dashed box with data from the opposite side of the box and averaging over all four quadrants, Right: image after droplet background removal. Reprinted from “Electron diffraction of CBr4 in superfluid helium droplets: A step towards single molecule diffraction”, J. Chem. Phys. 145, 034,307 (2016), with the permission of AIP Publishing

8.6 Case Study

8.6.1 Electron Diffraction of Pure Droplets at Different Temperatures

The electron diffraction pattern of pure helium droplets under different temperatures of the helium source was recorded. The diffraction intensity I_total is shown in Fig. 8.6 as a function of momentum transfer s. The diffraction intensity increases with the decrease in temperature of the droplet source, and the decay rate also exhibits a dependence on the source temperature. A single exponential function can be used to fit the decay of the diffraction profiles, and the fitted width parameters (w) are listed in Table 8.2.

Fig. 8.6

Total electron diffraction intensity from pure superfluid helium droplets. The smooth line shows single exponential fits to the experimental curve. Reprinted from “Electron Diffraction of Superfluid Helium Droplets” https://pubs.acs.org/doi/abs/10.1021/jz5006829. J. Phys. Chem. Lett. 2014, 5, 11, 1801–1805 with the permission from ACS publishing. Further permission related to the material excerpted should contact ACS

Table 8.2 Width parameters (w) from electron diffraction profiles of superfluid helium droplets^a

The curves of different temperatures can be categorized into two groups: the curves of 8 and 12 K with higher diffraction intensities and narrower distributions and 18 and 40 K with weaker intensities and slower decays. The diffraction profiles of the latter group are essentially identical to that of pure helium gas at 298 K. We have then attempted a biexponential fitting for the lower temperature group (≤12 K) with one of the exponents fixed at 1.48 Å⁻¹ (exponent of the higher temperature group), and the resulting second exponential function has a much faster decay, on the order of 0.3 Å ⁻¹. We have also performed statistical analysis (F-test) and confirmed the biexponential nature of the decay profiles. For further evidence, we have performed a biexponential analysis for the higher temperature group and confirmed the experimental data's single exponential nature.

The amplitude of the exponential fitting is affected by many factors. Under constant electron fluxes and constant gas fluxes from the nozzle, the amplitude from atomic diffraction should be similar. The amplitude from molecular diffraction, that is, coherent diffraction from a correlated atom pair, should be related to the number of helium pairs within each droplet. For the higher temperature group with negligible coherent molecular diffraction, we therefore expect similar amplitudes of atomic diffraction because the total number of atoms arriving at the diffraction region is similar. This expectation is qualitatively confirmed in Fig. 8.6. On the other hand, from 12 to 8 K, we do expect a rise in diffraction amplitude due to the presence of correlated atom pairs in large droplets, again as evidenced from Fig. 8.6.

Compared with previous reports of neutron diffraction of bulk superfluid helium [34,35,36], our monotonic decay profiles lack the weak oscillatory portion of the pair correlation function at large s values. We attribute this difference to two reasons: one is the polydispersity in the size of the droplets and the other is the variation in density from the core to the surface of a helium droplet. Theoretical simulations of superfluid helium droplets have revealed a diffuse surface layer and a bulklike interior for a droplet of over 100 atoms.

Although the droplet size distribution varies from setup to setup for pulsed droplet sources, we tentatively use the average droplet size from ref. [37] as a general guide. If we further assume a surface layer of 6 Å, the resulting percent of surface atoms under each source temperature is listed in Table 8.2. Below 12 K, most of the atoms in a droplet are considered interior atoms, whereas above 18 K, diffuse surface atoms dominate. In this sense, biexponential functions should be better representations of the experimental data at the two lowest source temperatures, a point confirmed from our statistical analysis. When the source temperature is above 18 K, diffraction profiles of the droplet beam are essentially the same as that of gas-phase helium atoms, consisting of only incoherent atomic scattering. The negligible pair correlation between atoms of small droplets is essential for analyzing the diffraction pattern of doped droplets: all recorded coherent molecular scattering should be from the doped molecules, whereas coherence of the surrounding helium atoms can be neglected.

One thing to note here is that this work is done at the early stage of our study; thus, the whole apparatus used is a prototype with low sensitivity and limited signal to noise ratio. With this work, we have demonstrated electron diffraction of pure superfluid helium droplets. The diffraction profile of pure helium droplets is affected by the polydispersity of the droplet beam, but it is in qualitative agreement with the size variation of the droplets. Larger droplets with a substantial compact interior component demonstrate stronger diffraction and faster decay with momentum transfer, whereas smaller droplets converge to gas phase isolated molecules when the droplet source temperature reaches 18 K.

8.6.2 Single Dopant Case: Ferrocene

Velocity slip [38] can beexploited in order to achieve extensive single molecule doping of small droplets in some favorable cases. Each collision between a dopant molecule and a droplet can result in the evaporation of a certain amount of helium atoms, cool the translational, rotational, and vibrational degrees of freedom, and eliminate the binding energy between the dopant and the droplet. When a large amount of helium atoms are required in this process, for example in the case of large molecules with many low frequency vibrational modes and high binding energy with the helium environment, a large size separation is created between those that can only accommodate one dopant molecule and those that are large enough to accommodate two dopant molecules. By timing the pulsed electron gun at the leading edge of the doped droplets, we can sample only singly doped small droplets, while larger droplets containing more dopant molecules arrive at the diffraction region after the termination of the electron pulse.

Ferrocene powder was placed in a SPV heated to 346 K to provide sufficient vapor pressure for doping. In the toggle mode, the sample PV and the electron gun were operated at 14 Hz, while the droplet PV kept at 18 K ran at half the frequency, and the difference image obtained when the droplet PV was on and off was the net image from the droplets, with or without dopant. A successful example of this approach is illustrated in Fig. 8.6 from the diffraction of ferrocene-doped droplets [19]. Fig. 8.7b shows the radial distributions directly obtained from an image accumulated from 200,000 shots. The electron gun's timing was set to sample droplets with about 2000 atoms/droplet. With these profiles as the intensities of Eq. 8.11 and based on comparisons with the theoretical sM(s) [39], multilinear regression is performed to obtain the coefficients of each component α_s, β, and α_b. The left half of the image shown in Fig. 8.7a is the difference image after removing pure droplets’ contribution based on the obtained coefficients. The right side of the image is the theoretical calculation.

Fig. 8.7

Electron diffraction of ferrocene doped droplets. a The left half of the image is the averaged experimental data after removing the contribution of helium, and right half is the simulation result. b is the radial profile with and without dopant. Reprinted from “Communication: Electron diffraction of ferrocene in superfluid helium droplets” J. Chem. Phys. 144, 221,101 (2016), with the permission of AIP Publishing

Due to the transient nature of the doping process, we could not measure the actual pressure in the doping region. However, basedon the Poisson statistics [40] and the fitting results β/α_s, we can determine the average number of effective collisions between ferrocene and droplets, assuming that trapping of one ferrocene molecule results in a droplet size change from 2000 to 800. We calculate the equivalent pressure for doping based on the experiment setup and the probability of picking zero (P₀) and one (P₁) ferrocene molecules. Further assuming that no droplets contain two or more ferrocene molecules in the diffraction region, we can then calculate the fraction of singly doped droplets.

Table 8.3 shows that at delays of 1000 and 1200 μs, more than 80% of the droplets are singly doped. It is worth noting that under these doping conditions, for a droplet of size 2000, only 7% are doped with one ferrocene molecule and 2% are undoped, while the remaining 91% are destroyed because of further collisions with dopant molecules. The above results also imply that regardless of the doping pressure, as long as helium atoms’ contribution can be effectively removed according to Eq. 8.11, there is no essential difference between the resulting sM(s) profile. Hence the images obtained from the last three columns were added to improve the signal-to-noise ratio of the final result shown in Fig. 8.7. The data from the first column were not used because of its low ferrocene content. The excellent quality of the fitting procedure also confirms the hypothesis on the sampling condition: in all delay conditions, only singly doped droplets are sampled in the experiment, with no detectable contributions from ferrocene dimers.

Table 8.3 Fitting results at different delay times between the two PVs^a

Inverse Fourier transform of the modified molecular scattering intensity results in the pair correlation function and gives the distances of atomic pairs in the molecule. Figure 8.8a shows the experimental and theoretical sM(s) profiles and the comparison between the pair correlation functions obtained from Fig. 8.8a and from the calculation of known gas phase structure is shown in Fig. 8.8b. There is a good agreement between the ferrocene from the doped droplet and the simulation from gas phase ferrocene. However, the high damping factor used in the transform introduces artificial wiggles in the simulation. Gas phase ferrocene is known to be in the eclipse conformation for the two pentacene rings [41]. The resolution is directly related to momentum transfer, which is limited by the detector's size. Unfortunately, with our current setup, we cannot resolve between the eclipse and the staggered structure. To resolve such a difference from the diffraction pattern, the range of s values needs to exceed 12 Å⁻¹.

Fig. 8.8

Modified molecular scattering intensities (a) and pair correlation profiles (b) of doped ferrocene. The residue is shown in the lower panel of (b). Reprinted from “Communication: Electron diffraction of ferrocene in superfluid helium droplets” J. Chem. Phys. 144, 221,101 (2016), with the permission of AIP Publishing

This work demonstrates the feasibility of sampling only singly doped droplets. Several factors determine the degree of size selection via velocity slip, including the physical and chemical properties of the dopant and the dimension and performance of the experimental apparatus. The current approach hence is not “one-size-fits-all”. In many cases, statistical analysis and deconvolution procedures are necessary.

8.6.3 Small Cluster of the Simple Molecules: CBr₄

With the aid of electron impact ionization (EI) and non-resonant multiphoton ionization (MPI), the most probable size of the droplet beam is determined to be ∼2000 helium atoms/droplet in the study of CBr₄ doping experiment, in agreement with the general result by Gomez et al. [37]. Doping one CBr₄ molecule requires evaporation of ∼600 helium atoms [40]. The MPI experiment also confirmed the velocity slip feature in the pulsed droplet beam: the overall droplet beam spans over 200 μs in the diffraction region, but depending on the ionization laser's timing, different sized droplets are sampled in the experiment.

Different from the ferrocene case, the droplets contain not only singly doped but also multiple doped CBr₄. There are many possible structures of the CBr₄ clusters. To modeling all of the structures are extremely difficult with the limited data resolution. Among contributions of intermolecular atomic pairs from clusters, only the shortest and most prominent Br···Br intermolecular pairs are considered in the treatment of the cluster to simplify the fitting process. The Eq. 8.11 is modified to add terms to represent the cluster contribution as:

$$sM(s) = \frac{{\alpha_{s} \cdot I_{total} (s) - \beta \cdot I_{droplet} (s) - \alpha_{b} \cdot I_{background} (s) - I_{T,at} (s) - C_{c} \cdot I_{T,c} (s)}}{{I_{T,at} (s)}} \cdot s,$$

(8.13)

I_T,c(s) is the theoretical diffraction profile of the shortest Br···Br pair with a separation of 4 Å [42, 43] and C_c is the weight of the theoretical contribution of the cluster in the overall diffraction. The radial profiles of different components are shown in the main part of Fig. 8.9. There the total experimental radial distribution (“Total Exp”) is compared with the scaled theoretical contribution of clusters in the form of Br···Br interference denoted “Br ···Br Calculation,” and the difference between the experimental data and this theoretical component is labeled as “Exp w/o Br···Br.” The latter profile should be compared with the theoretical diffraction profile labeled as “CBr₄ Calculation.” The halo of the Faraday cup is not included in the profiles. The experimental corrected radial profile contains one clearly observable ring, corresponding to the feature at s = 2.5 Å⁻¹ and a weaker feature at about 4.3 Å ⁻¹.

Fig. 8.9

Radial profiles relevant to diffraction from CBr₄ doped superfluid helium droplets. Reprinted from “Electron diffraction of CBr₄ in superfluid helium droplets: A step towards single molecule diffraction”, J. Chem. Phys. 145, 034,307 (2016), with the permission of AIP Publishing

The gas phase electron diffraction images of CBr₄ are also recorded by injecting the gaseous sample through a needle positioned directly in place of the droplet beam. The radial profiles and the corresponding pair correlation functions are shown in Fig. 8.10, and they are similar to those from a previous report [44] (not shown in Fig. 8.10). It is important to note that the “doped” trace in Fig. 8.9 is the net difference recorded with and without the droplet beam, while the doping chamber was maintained at a constant pressure. Contributions from the diffused gas in the trace of the “doped” sample are therefore removed. Nevertheless, the doped sample shows a similar diffraction profile as the gaseous sample, and both experimental results are similar to the theoretical calculation. Although the gaseous sample was at room temperature while the doped sample was at 0.38 K, the predominant factor determining the width of each interference ring is the wave physics, not the vibrational movement of the atoms in a molecule [9]. Evidently, the superfluid helium environment exerts negligible perturbation to the enclosed molecular structure. The weak features near the major features centered at 4.5 and 6.5 Å⁻¹ in the sM(s) profile from the doped sample could be due to residual contributions from clusters of CBr₄. The same argument applies to the pair correlation profiles, where the profile from the doped sample also contains an extra shoulder near 4 Å.

Fig. 8.10

Comparisons of results from gaseous and doped CBr₄ with theoretical calculations: a molecular diffraction profiles sM(s), and b pair correlation functions. Reprinted from “Electron diffraction of CBr₄ in superfluid helium droplets: A step towards single molecule diffraction”, J. Chem. Phys. 145, 034,307 (2016), with the permission of AIP Publishing

The diffraction of the droplet beam contains contributions from dopant clusters as well as pure droplets. The contribution of pure droplets from the fitting of Eq. 8.13, β/α_s, should be proportional to P₀, the probability of not picking up any dopant. The contribution of dopant molecular clusters C_c should be proportional to the ratio between the number of intermolecular Br···Br pairs with a separation of 4 Å and the average number of dopant molecules in a droplet $\left\langle k \right\rangle$

$$\left\langle k \right\rangle = \sum\nolimits_{k} {k \cdot P_{k} } ,$$

(8.14)

where P_k is the probability of picking up k molecules by a droplet. If we assume that the electron gun samples the most probable size of the droplet beam of 2000, then there should be at maximum 3 dopant molecules in a droplet. Since there are 3 equivalent Br···Br pairs with a separation of 4 Å in a dimer and 4 such pairs in a trimer [45], the value of C_c should be proportional to $(3P_{2} + 4P_{3} )/\left\langle k \right\rangle$.

Figure 8.11 compares the experimental fitting results with the Poisson statistics. Figure 8.11a shows the contribution of pure undoped droplets relative to that of doped droplets, and the ratio (vertical axis) is expected to be proportional to the probability of undoped droplets. At a doping pressure of 1 × 10^–5 Torr, less than 15% of the droplets are undoped, and this value is reproduced from the experimental fitting value of β/α_s. More and more droplets are doped with increasing doping pressure, and the portion of undoped pure droplets decreases accordingly. At the lowest doping pressure, the error bar is considerably large and a disagreement exists. We attribute this point to the residual gas in the doping chamber since the base pressure in the doping chamber without any doping gas was 1 × 10^–6 Torr. Figure 8.11b compares the contribution of clusters calculated using $(3P_{2} + 4P_{3} )/\left\langle k \right\rangle$ (designated as “Cluster” in the figure) based on the Poisson statistics [31, 46] with the experimental fitting value for clusters C_c. At a doping pressure of 1 × 10^–5 Torr, contributions of clusters relative to monomers are slightly more than 1:1, and this ratio increases to 1.3:1 at higher doping pressures. This upper limit is attributed to the maximum number of molecules that an average-sized cluster can pick up before it is destroyed completely. The coefficients of P₂ and P₃ in the expression ${{\left( {3P_{2} + 4P_{3} } \right)} \mathord{\left/ {\vphantom {{\left( {3P_{2} + 4P_{3} } \right)} {\left\langle k \right\rangle }}} \right. \kern-\nulldelimiterspace} {\left\langle k \right\rangle }}$$(3P_{2} + 4P_{3} )/\left\langle k \right\rangle$ do not play a critical role when varied between 3 and 6: they only affect the scaling of the profile, not the general trend.

Fig. 8.11

Comparisons between Poisson statistics and experimental results. a Calculated probabilities of undoped droplets P₀ in a droplet beam and fitting results β/α_s from experiment. b Contributions of CBr₄ clusters C_c from experiment and calculated ratios of cluster and monomer contributions (designated as “Cluster”). Reprinted from “Electron diffraction of CBr4 in superfluid helium droplets: A step towards single molecule diffraction”, J. Chem. Phys. 145, 034,307 (2016), with the permission of AIP Publishing

The above comparison highlights the crucial issue in using superfluid helium droplets as an ultra-cold gentle matrix for electron diffraction from field aligned and/or oriented molecules, i.e., the helium background. The large unwanted diffraction intensity from undoped droplet can potentially overwhelm the detector. This issue should be much more severe for an ion doped droplet beam since the equivalent vapor pressure of ions at the space charge limit is only 10^–9 Torr [13]; in this case, more than 99% of the droplets contain no protein ions at all even in the unrealistically favorable assumption of Poisson statistics. Under low doping conditions, it is therefore the presence of undoped droplets that dominate the background, and elimination of undoped droplets is essential for reducing the background of helium. The agreement between experiment and analysis alludes to the background problem’s solution in electron diffraction of embedded molecules in superfluid helium droplets. By reducing the background from pure undoped droplets via multiple doping, with small corrections for dimers and trimers, clearly resolved diffraction rings of CBr₄ similar to those of gas phase molecules can be observed. This condition is achievable for neutral molecules by heavy doping via increased doping pressure or path length. For charged species from an ESI source, fortunately, eliminating neutral undoped droplets from the charged doped droplets is straightforward using electric fields (magnetic fields are generally avoided because of the difficulty in field containment for electron diffraction). An ion-doped droplet beam can be bent from the initial path via an electric field generated by a stack of electrodes in, for example, a reflectron type of design, which also has the benefit of compressing the droplet pulse spatially and temporally. Alternatively, charged droplets can be accelerated or decelerated relative to the neutral undoped droplets, while a pulsed laser and/or electron gun can be synchronized to interact only with the doped beam. Electrostatic steering has the additional benefit of size selection, which could be used to eliminate excessively large droplets due to their high helium content and correspondingly large background contribution.

8.6.4 Halogen Bond Case in the Case of I₂

By taking advantage of the velocity slip in the pulsed droplet beam, shifting the electron gun's timing relative to the droplet pulse offers a limited degree of control over droplet size and dopant cluster size. We have explored this option in the study of iodine clusters doped in superfluid helium droplets [16]. In this effort, we have chosen two experimental conditions: different source temperatures, doping conditions, and time delays for the electron gun, which resulted in very different diffraction patterns. Figure 8.12 shows the diffraction profiles and the pair correlation functions under the two different conditions: the top panel was obtained under a lower effective doping pressure and by sampling the leading edge of the droplet beam, while the bottom panel was obtained under opposite conditions. The experimental radial profiles (red dots) in the two panels are quite different, indicating significantly different structures sampled under the two different experimental conditions.

Fig. 8.12

Modified molecular scattering intensities of smaller iodine clusters (a) and larger iodine clusters (b). The solid black lines are calculated profiles based on proposed structures. Reprint from “Self-Assembly of Iodine in Superfluid Helium Droplets: Halogen Bonds and Nanocrystals” Angew. Chem. Int. Ed. Engl. 2017 Mar 20; 56(13): 3541–3545 with the permission of WILEY

The upper panel corresponds to diffraction from smaller sized iodine cluster. With the comparison between the theoretical sM(s) of I₂ monomer and the experimental result, the conclusion is that there must be iodine clusters or mixtures of monomers and clusters dominant the diffraction with the possibility of only monomers be eliminated. Based on the timing of the electron gun and the size distribution of the droplet beam, as well as the calibration from previous experiment, the general size of the sampled droplets should be smaller than 1500 atoms/droplet [15, 31]. In general, pickup of one iodine molecule requires removal of 400–600 helium atoms for cooling, and to ensure that the doped droplets continue to travel into the diffraction region, at least 500 helium atoms need to remain with the droplet after doping [31, 40]. With these into consideration, the sampled iodine cluster in the top panel of Fig. 8.12 should contain no more than three iodine molecules.

We were then surprised to discover that no structural information was available in the literature on iodine dimers or larger clusters. We then fixed the intramolecular bond length at 2.67 Å, manually varied the relative distance and angle between the two iodine molecules, calculated the theoretical sM(s), and compared the resulting diffraction profiles with the experiment by relying on the fitting result of multilinear regressions of the diffraction profiles from Eq. 8.11. The best fit is a possible “L” shape structure with an adjusted intermolecular distance of 3.65 Å between the two nearest iodine atoms. The calculation is not a perfect reproduction of the experimental result, but it has sufficient merit in reproducing the general trend. We have also calculated the sM(s) profiles of trimers based on several possible structures, a one-layer structure from several different cuts of crystalline iodine, and a bi-layer structure with a dimer and a 3rd molecule in a different plane. None of the profiles can be considered qualitatively acceptable. We therefore conclude that the diffraction profile is predominantly due to iodine dimers.

The lower panel in Fig. 8.12 corresponds to diffraction from larger sized iodine clusters, such as tetramers, pentamers or even hexamers. Among those, the pentamers is the statistically most likely cluster. We tried to place all five iodine molecules in one layer according to the crystalline structure [47] (light-blue dashed line), and the result is qualitatively unacceptable. Realizing that the most salient feature in the diffraction profile centered at 3.5 Å⁻¹, corresponding to a distance of ~ 4.3 Å, we then considered pseudo-double layer structures with two iodine molecules in each plane and the fifth adjustable out of either plane. We cut fragments from an iodine single crystal, calculated the diffraction profiles, and compared with the experimental result. The black line shows the best result from this adjustment. We can confidently state that the larger iodine clusters sampled in the bottom panel have a bi-layer feature. Unlike the case of the smaller clusters, the calculated sM(s) profiles for tetramers, pentamers, and hexamers are all similar, as long as the iodine molecules form bi-layer structures. The diffraction technique is insensitive to the actual size of the iodine cluster under the current conditions. The pairings from atoms on the outer edges of each cluster do not have repeats, hence their contribution in the overall diffraction profile is overshadowed by those that have many repeats, such as the interlayer distances between corresponding atoms. We choose pentamer as a representation of this cluster group, partly because we have the best success in reproducing the experimental diffraction profile with one particular pentamer structure. We do acknowledge that the experimental data could well be a mixture of clusters with sizes from tetramer to hexamer but with similar structural motifs.

Figure 8.13 shows the pair correlation functions of the two diffraction profiles obtained from Fig. 8.12. The limited range of s values from our image detector requires a large damping factor in the calculation, which not only broadens the profile but also introduces extraneous oscillations in large distances. The estimated uncertainty in the resulting distance is on the order of 0.1 Å. The inset of each panel shows our proposed structures and the numbering schemes. To avoid clumsiness in labeling, all intramolecular distances between the two covalently bonded iodine atoms are labeled “Intra”, while only a few intermolecular distances are labeled. The shaded region represents intermolecular pairs such as 3···8, 4···9 and 5···10.

Fig. 8.13

Pair correlation profiles of doped iodine clusters and their proposed structures. The insets show the structures of the clusters and the numbering of the atoms. Reprint from “Self-Assembly of Iodine in Superfluid Helium Droplets: Halogen Bonds and Nanocrystals” Angew. Chem. Int. Ed. Engl. 2017 Mar 20; 56(13): 3541–3545 with the permission of WILEY

For the case of dimer dominated diffraction, each unique interatomic distance can be more or less resolved under the current conditions, although the fitting is still imperfect. The proposed structure has a distance of 3.65 Å between atoms 1 and 3, and this value is substantially shorter than the sum of the van der Waals distances of two iodine atoms (4.3 Å), but are similar to the in-plane intermolecular distance of 3.5 Å in crystalline iodine [47]. Moreover, all four iodine atoms are in the same plane in the proposed dimer structure. It is the first time that halogen bonds are observed from iodine clusters.

Our experiment is incapable of resolving several unique intermolecular distances for the larger clusters, but a few features are still identifiable. The intramolecular distance at 2.67 Å, which has five repeats, is clearly resolved. The current pentamer structure contains two halogen bonds between atoms 2···3 and 2···5 at ~3.5 Å, while other distances of the parallelepiped of atoms 3···5 and 3···7 are van der Waals in nature. Attempts to rearrange the 1–2 atoms into a more symmetric position stapling the two layers failed to match the current fitting result. An interesting result is the structure of the larger clusters where instead of a single layer, a bi-layer structure seems to dominate when more than three iodine molecules are present. It is possible that different from bulk crystals, small clusters are more stable in a bi-layer structure, particularly in a superfluid helium environment. Alternatively, the missing one-layer structure might be related to the limited size range sampled by the electron beam. It is possible that within the size range of the droplet beam sampled by the electron gun, only bi-layer structures can survive the evaporative cooling process and maintain the traveling momentum to the diffraction region. Unfortunately, given the insensitivity of the diffraction technique, further experimental confirmation of this speculation is difficult, if possible at all.

8.6.5 CS₂

The experimental results of electron diffraction of CS₂ are presented in this section [48]. It demonstrates the feasibility of structural determination of dimers, trimers, tetramers, and clusters containing a large number of monomers embedded in superfluid helium droplets. We can narrow down the range of parameters of the least squares fitting procedure of the diffraction patterns from detailed droplet size measurements and modeling of the doping statistics.

The experiment setup is the same as explained in the previous ferrocene case. Room temperature CS₂ with a vapor pressure of 344 Torr is routed to the sample PV via a vacuum feedthrough. Figure 8.14 shows the mass spectrum of CS₂ doped in superfluid helium droplets at a stagnation temperature of 10 K. At least three photons at 266 nm are needed for ionization, and the resulting mass spectrum shows extensive fragmentation. The presence of attached helium with the sample ions C⁺ and S⁺ manifests the presence of the helium droplet. Although not seen explicitly in the mass spectrum, the presence of CS₂ clusters can be deduced from the presence of CS₂C⁺.

Fig. 8.14

Mass spectrum of CS₂ doped helium droplets recorded at a source temperature of 10 K and a doping pressure of 5 × 10^–4 Torr. Reprint from “Electron diffraction of CS2 nanoclusters embedded in superfluid helium droplets” J. Chem. Phys. 152, 224,306 (2020) with permission from AIP publishing

Based on the size analysis mentioned in Sect. 8.4, two groups of doped droplets at the source temperature of 20 K can carry CS₂ to the diffraction region. The predominant one has an average size of 800 atoms/droplet and a less abundant group has a much larger average size, with 5000 atoms/droplet. When the source temperature is reduced to 10 K, the two size groups are 2.2 × 10⁴ atoms/droplet and 1.4 × 10⁵ atoms/droplet with similar abundance. It is also worth noting that each size group travels at a unique speed, regardless of the source temperature, and the effect of the source temperature is primarily manifested in the relative abundances of the different groups, while the average size of each group is only mildly dependent on the source temperature [12]. The overall average size of the droplets at a fixed source pressure and temperature, weighted by each group's abundance, is in agreement with literature reports [37, 49]. Due to limited velocity slip [38], however, there is more than one group of droplets with different sizes and velocities at any sampling time within the droplet pulse. Hence, the pulsed electron gun's time setting selects the size composition of the sampled droplets during the diffraction experiment. We note here that these sizes are measured after ionization; hence, the corresponding size of the droplet prior to doping should be larger by a few hundred—on the order of 500—after picking up each dopant molecule [31, 40].

Figure 8.15 shows two scaled radial profiles of the diffraction patterns obtained at two different time settings of the electron gun: the solid squares were recorded at a delay time of 1289 μs after the droplet PV, and the open squares were at 1269 μs. Both profiles result from the diffraction of neutral samples without ionization and without sample orientation, and the diffraction profiles are concentric rings due to the orientation average. To emphasize the contribution of molecular interference and to contrast the difference between the two results, the radial profiles are normalized by each profile's exposure time, and the intensity is multiplied by s. The difference in the scaled profiles is due to the different doping conditions of the droplet beam: at the earlier timing of 1269 μs, the droplets are smaller and contain fewer sample molecules.

Fig. 8.15

Scaled radial profiles of diffraction patterns from CS₂ doped droplets. The experimental profiles were recorded at a source temperature of 20 K, two different time settings of the electron gun: solid symbols at 1289 μs and open symbols at 1269 μs. All profiles are results of multiplying the radial intensity by s to emphasize the contribution of molecular interference. The insets show the resulting structures for the dimer, trimer, and tetramer. Reprint from “Electron diffraction of CS2 nanoclusters embedded in superfluid helium droplets” J. Chem. Phys. 152, 224,306 (2020) with permission from AIP publishing

Figure 8.16 shows the diffraction profile after further shifting the electron gun's timing to 1581 μs, lowering the source temperature to 10 K, and increasing the effective doping pressure from 2 × 10^–4 Torr to 5 × 10^–4 Torr (estimated doping pressure). Compared with Fig. 8.14, the overall diffraction intensity has decreased, but the feature of molecular interference is more prominent, implying more contribution from the sample than from the helium atoms.

Fig. 8.16

Scaled radial profile of the diffraction pattern from CS2 doped droplets recorded at a source temperature of 10 K. The profile is scaled by s to emphasize the contribution of molecular interference. The inset shows the structure of the crystal and the axes of the unit cell. Reprint from “Electron diffraction of CS2 nanoclusters embedded in superfluid helium droplets” J. Chem. Phys. 152, 224,306 (2020) with permission from AIP publishing

Structures of gas phase clusters and single crystals of CS₂ have been reported in the literature [50,51,52,53,54]. Experiments in high resolution spectroscopy have concluded that gas phase clusters prefer highly symmetric shapes: the dimer is cross-shaped with a D_2d symmetry, [52] as shown in the inset of Fig. 8.14, the trimer forms a pinwheel with a D₃ symmetry [50], and the tetramer forms a barrel with a D_2d symmetry [51]. A few theoretical calculations have proposed various geometries for small clusters, but the most stable isomer for each cluster size agrees with the experimental results [55, 56]. For clusters containing more than four monomers, convergence to the crystal structure has been predicted [54]. The crystal structure of CS₂ at temperatures below 150 K is known to be orthorhombic, containing pairs of CS₂. A unit cell contains three monomers at the edges in addition to a central monomer slanted at 55.56° to the c axis, as shown in the inset of Fig. 8.16. The molecular arrangement in crystals is substantially different from those of reported gas phase clusters.

Equation 8.12 is used for fitting the diffraction profiles, including different cuts from the crystals for dimers, trimers, and tetramers, in addition to their gas phase structures. In this case, α_ig (2 ≤ i ≤ 4) as contributions of gas phase clusters containing i monomers, and α_ic (2 ≤ i ≤ 4) represents all possible cuts of clusters from the crystal structure. A modified Poisson model using Markovian arrival process with non-identical exponential interarrival times [57] to accommodate the change of cross section reduction with sequential pick up of CS₂. The details of the treatment is explained in the reference [48]. The calculated probabilities P_n of pickup n CS₂ molecules with a source temperature of 20 K are listed as the first number in each cell of Table 8.4. Even for the larger size group at 1289 µs, the probability of P₄/P₁ is only 0.04. About 50% of the droplets contain no sample at all for both delay times, and about 35% of the droplets contain just one CS₂. The second number in each cell of the above table shows the fitting results of the diffraction profiles based on Eq. 2.4. The values of β represent the degree of helium loss due to the pickup process. In both delay times, nearly 50% of helium is lost before the droplets can reach the diffraction region. The absolute values of α_i have no significance since it is affected by the detector's sensitivity, while the ratio of α_i/α₁ shows the relative contributions of clusters to monomers.

Table 8.4 Doping statistics and fitting results of diffraction images from Fig. 15^a

With the statistical analysis of the fitting result, only the gas phase structure for dimers α_2g, the closely packed crystal structures for trimers α_3c and the tetramer α_4c are statistically significant in the diffraction profiles. Furthermore, the contribution of trimers is lower than the calculation results based on the Markovian arrival process. The discrepancy could be related to trimers_’ rlative instability compared to the other clusters since crystalline CS₂ essentially is a composition of dimers [53]. The values on the relative contributions of the other clusters to that of monomers are very consistent between the diffraction fitting results and the calculation based on doping statistics.

The fitting for the diffraction profile of Fig. 8.16 is simpler because we only considered crystalline structures, as suggested in the literature [54]. Both groups of droplets contain at least 2 × 10⁴ atoms/droplet, and after picking up one CS₂, the droplet size changes by less than 3%. Standard Poisson statistics is therefore applicable. At the experiment doping pressure, the most probable number of CS₂ monomers for the smaller droplet group is 12 and for the larger group is 40. The continuous red line results from fitting the linear regression, with only α_c·I_c term other than the droplet contributing in Eq. 8.12, with I_c is the theoretical diffraction profile containing numerous monomers based on the crystalline structure of CS₂, and α_c is the fitting coefficient. When more than four monomers form a nanocrystal within our detection range, the resulting diffraction profile becomes insensitive to the cluster's size. The calculated diffraction profile results from two unit cells along the a axis, one unit along the b axis, and one unit along the c axis with an overall monomer count of 18. The profile is insensitive to the number of unit cells along each direction, as long as there is more than one monomer (not unit cell) along each direction. However, limiting the structure to two-dimensions spanning only the a and c axis resulted in a substantially lower quality of fitting. It is highly plausible that the nanocrystal is most likely of the same 3D structure as that of a single crystal of CS₂.

It is interesting to see that, similar to the iodine case, clusters containing more than two monomeric units deviate from their corresponding gas phase structures and seem to adopt structures of their bulk crystals. The preference for bulk structures in helium droplets has been attributed to the droplet environment's fast cooling effect, even though the helium environment should be more similar to that of vacuum than to that of typical crystal formation. A newly captured dopant can be trapped in a metastable configuration upon entering the droplet, incapable of finding the global minimum.

8.6.6 Diffraction of Molecules Only with Light Atoms: Pyrene

All the previous cases mentioned above involve molecular species that contain at least one heavy atom (with an atomic number larger than 20) to help with the contrast between the molecular diffraction and the atomic diffraction from helium. However, biological samples contain mostly carbon atoms, and the contrast issue due to similar diffraction cross sections of carbon [58] and helium has to be addressed. The work with all-light-atom-containing species, pyrene (Py, C₁₆H₁₀) demonstrates the feasibility of extracting structural information from the helium background for molecular systems that do not contain any contrasting element [17].

Pyrene's doping condition is characterized by a mass spectrometer via MPI with a 266 nm pulsed laser at a power density of 10⁶ W/cm² (without focus lens). The miniature TOF located on the wheel shown in Fig. 8.3 resolves the ionized parent, fragment and cluster ions. Figure 8.17 presents the TOF mass spectra of gaseous pyrene, pyrene-doped droplets, and the difference. No fragmentation is observed under this laser power level. The gaseous sample only contains monomeric parent ions, while doped droplets show cluster ions Py_n⁺(n = 2–4). Since the degree of fragmentation after ejection from the doped droplet is unknown. the presence of pyrene clusters should be treated as evidence of existence and cannot be used for quantitative analysis of the clusters. The energy of two photons at 266 nm (total energy: 9.35 eV) is more than sufficient to both ionize [59, 60] and dissociate (or dissociate and ionize) a pyrene dimer to produce Py⁺  + Py [61, 62].

Fig. 8.17

Copyright 2020 American Chemical Society

TOF mass spectra of pyrene-related species. Reprinted with permission from “Electron Diffraction of Pyrene Nanoclusters Embedded in Superfluid Helium Droplets” J. Phys. Chem. Lett. 2020, 11, 3, 724–729.

Figure 8.18 shows the scaled radial profiles of the experimental diffraction patterns, which are obtaioned with an accumulation of 232,559 shots (12.92 h at a repetition rate of 5 Hz). The insert shows the unscaled radial distribution from the raw experimental data. After a scaling factor of 7.5 for the doped droplet, no difference can be seen between the doped and neat droplets on the linear scale. To contrast the difference between the two results, the radial profiles are scaled by s², where s is the momentum transfer.

Fig. 8.18

Copyright 2020 American Chemical Society

Radial profiles of diffraction patterns from neat and pyrene doped droplets. The inset shows the relative intensities of the radial profiles. Reprinted with permission from “Electron Diffraction of Pyrene Nanoclusters Embedded in Superfluid Helium Droplets” J. Phys. Chem. Lett. 2020, 11, 3, 724–729.

To derive structural information from the diffraction profiles, contributions from the helium background and from all possible pyrene clusters need to be included. Figure 8.19 shows the theoretical diffraction profiles of pyrene clusters based on a few theoretical calculations [61, 63,64,65,66] and some representative cuts from crystalline pyrene [67, 68]. The crystal structure and designations of molecular axes are shown in the inset of (a).

Fig. 8.19

Copyright 2020 American Chemical Society

Theoretical diffraction profiles from selected structures of pyrene clusters. The diffraction profile of each structure is color-coded within each panel. Reprinted with permission from “Electron Diffraction of Pyrene Nanoclusters Embedded in Superfluid Helium Droplets” J. Phys. Chem. Lett. 2020, 11, 3, 724–729.

Several theoretical calculations on the structures of pyrene clusters have been reported in the literature. The most recent is by Dontot, Spiegelman, and Rapacioli (DSR), reporting a rotation angle of 67° but a slightly nonparallel arrangement between the two molecular planes [61]. However, the authors reported a shallow minimum, with four other structures competitive within 20 meV: they all have parallel molecular planes but are shifted or rotated by different angles, as shown in the inset of Fig. 18b. The structure labeled SPL is the global minimum by Gonzales and Lim [66], and it involves a parallel slip between the two monomers along the long axis and an interplanar distance of 3.51 Å, in agreement with the distance in the dimeric unit of crystals. The other three parallel dimers include SPS—slip along the short axis, GR—slip along a C–C bond, and cross—a rotation of 90° [62]. All four structures have very similar diffraction profiles and hence are referred to as the Para dimer in the following discussion. The trimer structure from the DSR calculation is stacked but slightly nonparallel, quite different from a trimeric cut of crystalline pyrene, while the tetramer structure is a 3 + 1 construct, with the fourth pyrene nearly perpendicular to the stacked trimer. An earlier report by Takeuchi (HT structure) contains a parallel dimer [65], a parallel trimer, and a near-cyclic tetramer. Although slightly different from the four parallel dimers, the HT dimer has a very similar diffraction profile to those of the parallel dimers.

If we include all the possible clusters for a global fit, there will be too many independent parameters with limited data points from the diffraction profile. To alleviate model complexity, we chose to fit four sets of structures independently, including the DSR and the HT set, a mix set containing the parallel dimer and the DSR trimer, and a mix set containing the HT dimer and the DSR trimer. The structures of trimers and tetramers derived from crystalline pyrene are eliminated because when added to any one of the sets the resulting coefficients for these structures are essentially zero. The Alkaike information criterion and bootstrap resampling method through balanced variable selection is used to confirm the significance of the regression coefficients, and the details are explained in the reference paper. [17] Table 8.5 summarizes the best fitting coefficients and their ratios and uncertainties.

Table 8.5 Constrained least-squares fitting result of embedded Py_n (n = 1–3) in superfluid helium droplets from the best model^a

Figure 8.20 compares the experimental data with the fitting results, and the residue is shown in the bottom panel. Similar to Fig. 8.17, both the radial profiles and the residues are scaled with s². The small value of β signifies that more than 90% of the helium atoms could not reach the diffraction region. This level of elimination is on par with our previous work on ferrocene and iodine [16, 19]. The effective high vapor pressure in the doping region destroys most of the small droplets with or without a dopant monomer.

Fig. 8.20

Copyright 2020 American Chemical Society

Comparison of scaled experimental and fitting results. The residue is the difference between the scaled radial profiles. Reprinted with permission from “Electron Diffraction of Pyrene Nanoclusters Embedded in Superfluid Helium Droplets” J. Phys. Chem. Lett. 2020, 11, 3, 724–729.

The doping process is modeled using Poisson statistics to further understand the contribution of Py_n in the diffraction pattern. We estimate the number of evaporated helium atoms (2000) upon cooling a pyrene molecule to 0.4 K based on the heat capacity of solid pyrene (229 J/K·mol) [69] and the binding energy of helium (0.6 meV) [70]. After the first collision, 4% of the helium atoms is lost in a droplet of 5 × 10⁴ atoms/droplet. This size change is negligible and standard Poisson distribution can be used to calculate the probability of doping [46]. The doping pressure is estimated with the empirical formula of supersonic expansion as 1.3 × 10^–5 Torr. With a doping distance of 7 mm, the probability of doping 0–4 pyrene is 0.71:0.24:0.04:0.004:0.0005 (the ratios of the corresponding α_i values are listed in the last column of Table 8.5). The relative abundance of Py₂ and Py₃ is in qualitative agreement with that from the fitting. The much larger contribution of monomers from the doping statistics than that from fitting of the diffraction pattern is attributed to contamination in the neat droplet diffraction profile. The doping statistics and the fitting results of the diffraction profile are on par with the abundance of Py_n⁺ in the mass spectra of Fig. 8.16. We have limited information on the ionization mechanism of Py_n⁺. However, we speculate that the abundant Py⁺ is most likely a result of dissociation of Py_n or Py_n⁺ after desorption from the droplet. The missing contribution from Py₄ in the diffraction profile should be a result of low concentration.

In conclusion, the diffraction profile from this experiment of pyrene doped droplets contains mostly contributions from Py and Py₂, with indications of a ~10% contribution from Py₃. The structure of Py₂ contains two parallel pyrene molecules, and that of Py₃ appears to be stacked but not completely parallel. This structure of Py₃, in our best fitting model, is different from that of the crystalline structure, demonstrating that at least in superfluid helium droplets, the stacking force prevails against the tendency of forming a 3-D closely packed structure. Unlike our previous work, pyrene contains no heavy atoms, and the success of this work offers promise in obtaining molecular parameters from all-light-atom containing species in superfluid helium droplets.

8.7 Conclusion

The extreme cooling effect of superfluid helium droplets is attractive in the structural solving of unstable species under normal conditions. Still, the associated background issue in electron diffraction is of practical concern. By taking advantage of the velocity slip of pulsed droplet beams and using Poisson statistics with the Markovian arrival process in sample doping, the background problem for neutral molecule-doped in droplets can be largely minimized. The success of the above variety cases shows the feasibility of this approach even with an all-light-atom system. As for the electron diffraction of protein ions in superfluid helium droplets, the background problem will be easied with manipulating charged particles using electric fields. The space charge limit gives a low charge density of ions will result in a limited diffraction strength. However, the complete separation of undoped droplets and the ability to select small droplets are both beneficial to the ultimate diffraction signal. The next step is to perform diffraction of the ion-doped droplets. The “proof-of-principle” experiment involving the laser-induced alignment of ions embedded in superfluid helium droplets will ensue thereafter.