The structure of amorphous materials can be described by their correlation functions gn(r₁,…,r_n) where g_n(r₁,…,r_n) d³r₁···d³r_n gives the probability of finding particle i in the volume d³r_i and so on [1]. The pair correlation function g₂(r₁, r₂) is well known experimentally from scattering experiments. However, it describes only the distribution of atomic distances. Information about bond angles is lost. This would be accessible if the triple correlation function g₃ (r₁,r₂,r₃) could be measured.
The Fourier transform of the correlation functions are the so-called structure factors. These can be directly obtained from experimental images [2]. Previous attempts to implement this approach, however, have failed [3, 4].
A few years ago Huang et al. [5] made incredibly well defined images of a layer of amorphous silica where the resolution was high enough to resolve atomic spacings. As the atomic positions are directly visible in these images, we used them to obtain the three particle structure factor.

In a first approximation we assumed ideal imaging and determined the structure factor S⁽¹⁾ simply as the Fourier transform of the measured intensity. The two particle structure factor S⁽²⁾ is then calculated as the square of its absolute value and finally the three particle structure factor is obtained as S⁽³⁾ (q₁,q₂) = S⁽¹⁾(q₁) S⁽¹⁾(q₂) S⁽¹⁾(−q₁−q₂) where q₁ , q₂are spatial frequencies in two dimensions [2].

Figure 1 shows the two particle structure factor of amorphous silica. There are two peaks. The first one at q ≈ 0.3 Å^-1 is broader and we expect it to represent the atomic distances in the specimen. The second peak at q ≈ 0.5 Å^-1 is sharper and due to the graphene substrate. Since amorphous matter is expected to be isotropic we average over one spatial angle and consider S⁽³⁾ as a function of only three variables |q₁|, |q₂| and the angle φ between q₁ and q₂. To reduce the number of degrees of freedom even more, we took q₁ = q₂ =: q and chose q to be at the first peak in S⁽²⁾(q) and only varied φ. A first result is shown in figure 2.
In a crystal one has well-defined binding angles and thus expects S⁽³⁾(φ) to have sharp peaks at those angles. In our case we find peaks around 60° and 120° stemming from the approximate 6-fold symmetry of silica but compared to the case of crystals they are smeared out. For the first time we have been able to determine the three particle structure factor from TEM images. We are now undertaking a systematic study to obtain further insights into the amorphous structure of two-dimensional glasses.

Acknowledgements: We are grateful to Prof. Dr. Ute Kaiser (University of Ulm) for providing her image data to us.

[1] J.M. Ziman. Models of Disorder: The Theoretical Physics of Homogeneously Disordered Systems. Cambridge University Press, 1979.
[2] M Hammel and H Kohl. Determination of the triple correlation-function of amorphous specimens from em micrographs. In Inst. Phys. Conf. Ser., number 93, pages 209–210, 1988.

[3] Michael Hammel. Bestimmung der abbildungsparameter und der korrelationsfunktion dritter ordnung amorpher objekte aus elektronenmikroskopischen phasenkontrastaufnahmen. Master’s thesis, TH Darmstadt, 1988.
[4] Ansgar Haking. Bestimmung des drei-teilchen-strukturfaktors amorpher stoffe aus hochaufgelösten elektronenmikroskopischen aufnahmen. Master’s thesis, WWU Münster, 1995.
[5] Pinshane Y. Huang, Simon Kurasch, Anchal Srivastava, Viera Skakalova, Jani Kotakoski, Arkady V. Krasheninnikov, Robert Hovden, Qingyun Mao, Jannik C. Meyer, Jurgen Smet, David A. Muller, and Ute Kaiser. Direct imaging of a two-dimensional silica glass on graphene. Nano Letters, 12(2):1081–1086, 2012.

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