Fig. 1: Schematic for multi-atom phase analysis.

a Simulation of density of states perturbation δN(r) in amplitude units I₀ due to two-dimensional Friedel oscillations surrounding 100 impurity atoms at random locations R_i. b Real part of Fourier transform ReδN(q) from δN(r) in a. We use an integer grid, hence the units of Fourier transform are also I₀. c Imaginary part of Fourier transform ReδN(q) from δN(r) in a. d Real part of Fourier transform ReδN_MA(q) calculated using multi-atom technique of Eq. (7). e Imaginary part of Fourier transform ReδN_MA(q) calculated using multi-atom technique of Eq. (7). f ReδN(q) from δN(r) in a for \(\vartheta = 0\) and \(\vartheta = \pi\), integrated azimuthally from b. Its strong random fluctuations versus |q| are due to summing the Friedel oscillations in δN(r) of a with random phases due to the random locations R_i. g ReδN_MA(q) from δN(r) in a integrated azimuthally from d. ReδN_MA(q) is now orders of magnitude more intense than in f, and the phase of the Friedel oscillations in δN(r) of a is now very well defined because the effects of random locations R_i are removed by using Eq. (7). Note that, now, changing the oscillation phase \(\vartheta = 0\) to \(\vartheta = \pi\) surrounding all R_i in δN(r) produces the correct evolution of ReδN_MA(q).