# Reversal of Photoinduced Bending of Crystals Due to Internal Refraction of Light
###### Abstract  
The well-known effect of bending of acicular crystals undergoing photochemical reactions is associated with the transformation gradient across the crystal thickness caused by the absorption of light. It is believed that the direction of bending is unambiguously dictated by the sign of the axial component of the reaction strain and due to the higher light intensity on the irradiated side of the crystal. In this study, it is shown experimentally and theoretically that thin crystals with a convex irradiated surface bend in the opposite direction if their thickness is less than the light penetration depth. The reversal of the bending direction is due to the lens effect, able to overcome the absorption of light in a thin crystal and provide a higher light intensity on the rear side of the crystal. A crystal of [Co(NO)\({}_{5}\)]NO\({}_{2}\)]Cl(NO\({}_{3}\)) experiencing nitro-nitrito photoisomerization, begins to bend in the opposite direction after it is thinned by etching to 2 \(\upmu\)m when irradiated at the wavelengths of 523 nm, 465 nm, and 403 nm, absorbed at a depth of more than 10 \(\upmu\)m in the crystal, but bends in the normal direction under 350 nm light absorbed at a depth of about 1 \(\upmu\)m. The experimental results are fully confirmed by modeling the interaction of plane EM wave with crystals of various cross sections. photomechanical effect; photoinduced bending of crystals; wavelength dependence of photomechanical effect; absorption of light; refraction of light +
Footnote †: journal: Applied Physics Letters  
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, Berkeley, CA 94720, USA  
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provides an excellent opportunity for their study and quantitative description of the transformation and mechanical response [24], and is also attractive for possible practical applications of the effect. The macroscopic deformation of bending crystals, including longitudinal elongation/shortening and bending of the crystal axis, is completely determined by the distribution of the axial component of lattice deformation in the crystal cross section [1; 24]. The curvature of a bent crystal (the value inverse to the radius of curvature \(R\)) is determined by the moment of distribution of axial strain \(\varepsilon(\mathbf{r})\) over the cross section of the crystal \(S\), according to Equation (1);  
\[\frac{1}{R}\ =\ -\frac{1}{J}\int\limits_{S}x\varepsilon(\mathbf{r})ds, \tag{1}\]  
while the elongation is defined by the average axial strain in the cross section, Equation (2);  
\[\frac{\Delta L}{L}\ =\ \frac{1}{S}\int\limits_{S}\varepsilon(\mathbf{r})ds \tag{2}\]  
Equations (1) and (2) are a generalization of the expressions obtained in [1] for the case of an arbitrary shape of the cross section of crystals. The value of the static moment of inertia of the crystal section \(J\) is determined by Equation (3);  
\[J\ =\ \int\limits_{S}x^{2}ds \tag{3}\]  
where \(x\) is the distance from the neutral plane of the bent crystal. The sign in Equation (1) is taken in such a way that the curvature \(1/R\) is positive with a positive transformation strain, \(\varepsilon>0\), which decreases deeper from the irradiated surface, \(d\varepsilon/dx<0\). The latter condition is a natural consequence of light absorption in the bulk of the crystal, leading to a faster transformation near the irradiated crystal surface and, as a consequence, to the appearance of a negative gradient of the transformation degree in the direction of light propagation, \(dC/dx<0\). This condition is considered to be always satisfied, which leads to an unambiguous direction of crystal bending: in the direction of light propagation if \(\varepsilon>0\), or towards the light source if \(\varepsilon<0\). It should be noted that the strains caused by irradiation can be associated not only with a photochemical transformation, but also, at least, with the direct heating of the crystal, since some part of the energy of the absorbed light is converted into heat.