We propose a new method to probe for variations in the fine structure constant alpha using clusters of galaxies, opening up a window on a new redshift range for such constraints. Hot clusters shine in the X-ray mainly due to bremsstrahlung, while they leave an imprint on the CMB frequency spectrum through the Sunyaev-Zeldovich effect. These two physical processes can be characterized by the integrated Comptonization parameter Y_SZ DA^2 and its X-ray counterpart, the Y_X parameter. The ratio of these two quantities is expected to be constant from numerical simulations and current observations. We show that this fact can be exploited to constrain alpha, as the ratio of the two parameters depends on the fine structure constant as alpha^{3.5}. We determine current constraints from a combination of Planck SZ and XMM-Newton data, testing different models of variation of alpha. When fitting for a constant value of alpha, we find that current constraints are at the 1% level, comparable with current CMB constraints. We discuss strategies for further improving these constraints by almost an order of magnitude.
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