Fig. 4: Minimal model and temperature dependent measurements.
From: Magnetic control over the fundamental structure of atomic wires
a Schematic curve of total energy as a function of wire length for parallel and perpendicular magnetic field and magnetization. b Same as (a) for the force. The breaking force (\({{{{{{\rm{F}}}}}}}_{\parallel ,\perp }^{{{{{{\rm{break}}}}}}}\)) is obtained at different wire length d||,⊥ for parallel and perpendicular magnetization and magnetic field orientations. c Experimental length histogram at zero magnetic field (same as Fig. 1c) presented for the sake of comparison to our model. d–f Illustration of the minimal model: (d) At zero magnetic field, two peaks represent the distribution of wire lengths with parallel (gray) and perpendicular (pink) magnetizations. In our model, these distributions are given by similar Gaussians multiplied by Zeeman-based Boltzmann weights that depend on the relative orientations and magnitudes of the magnetization and applied magnetic field. The distributions are centered at \({d}_{\parallel }^{n}\), \({d}_{\perp }^{n}\) for the nth peak, and their summed contribution (black envelope) is adapted by setting the Gaussian widths to σ = 0.5 Å to fit the experimental peaks in (c). e At a finite parallel magnetic field, the two Boltzmann weights are different due to different Zeeman energy for parallel and perpendicular magnetizations. As a result, the heights of the two peaks are different, leading to a down-shift in the position of the maximum of the total distribution (black). f The two distributions at finite perpendicular magnetic field. Here, the different Boltzmann weights for parallel and perpendicular magnetizations lead to an up-shift in the position of the maximum of the total distribution (black). g, h Calculated d2-1 and d3-2 as a function of perpendicular (g) and parallel (h) magnetic field. \({d}_{\parallel }^{n}\), \({d}_{\perp }^{n}\) are determined by the onset of saturation in Fig. 2c, f. This model captures the opposite shifts in the inter-peak distance for parallel and perpendicular magnetic fields for both magnetization directions. i, j Inter-peak distance as a function of perpendicular (i) and parallel (j) magnetic field for different temperatures. k, l Same as in (i), (j), with magnetic field divided by the temperature.
