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We study the local vortex-vortex interaction force $\textbf{f}_i$ of thestructure frozen during a field-cooling process in an electron-irradiatedBi$_2$Sr$_2$CaCu$_2$O$_{8+\delta}$ sample. We compute this magnitude fromsnapshots of the vortex structure obtained via magnetic decoration experimentsat various fields H in the same sample. Since the observed structurescorrespond to the equilibrium ones frozen at T$\sim$T$_{irr}$(H), at thistemperature the local modulus of $\textbf{f}_i$ roughly equals the localpinning force at the decorated surface of the sample. We estimate the mostprobable local pinning force from the mode value of the f$_i$(r) distribution,f $_{p}^{m}$. We found that f $_{p}^{m}$ grows algebraically with H and inelectron-irradiated samples is 50-20% smaller than for samples with columnardefects.

Direct visualization of local interaction forces in Bi$_2$Sr$_2$CaCu$_2$O$_{8+\delta}$ vortex matter