to find the positions of the peaks corresponding to spin- wave excitations. In order to simplify the identification of the intensity peaks, we performed a post-processing scheme in which the intensity for each q vector is con- voluted with a Gaussian, normalized to unity, and then used in a power function. This allows us to decrease the number of ensemble averages needed for each simulation by a few orders of magnitude. Fig. 2 displays an example of the calculated dynami- cal structure factor S $^{k}$( q , ω ), plotted after the convolu- tion with a Gaussian function, for a selection of differ- ent q vectors along the [001] direction of the Fe/W(110) monolayer. For low q values, the spectral intensity of S $^{k}$( q , ω ) is high, and decreases with increasing q . At the same time, the peak width can be seen to increase with increasing energy. This is made clear by observing the peaks located at q = 0 . 8 ˚ A − 1 and q = 1 . 2 ˚ A − $^{1}$. These are positioned at similar energies, and do not exhibit notice- able broadening as a function of q . The broadening of higher energy excitations can be explained by the damp- ing term in the Landau-Lifshitz-Gilbert equation. In our proposed approach for calculating the magnon spectra, temperature effects enter in two ways, explicitly as the temperature at which the spin dynamics simula- tions are performed, and implicitly by the choice of which magnetic configuration the interatomic exchange interac- tion parameters are calculated from. In order to quantify temperature effects, we have calculated the spin wave stiffness of the Fe monolayer, calculated from LKGM simulations starting from both ferromagnetic and DLM states. The spin wave stiffness was evaluated by a least- squares fit of the magnon curves to a second order poly- nomial, within the range q < 0 . 2 ˚ A − $^{1}$. At T = 0 K we cal- culated the spin-stiffness to be 160 meV ˚ A 2 when the J$_{ij}$ s were obtained from a ferromagnetic configuration, and to be 110 meV ˚ A 2 from the DLM set of exchange interac- tion parameters. When introducing temperature to the spin dynamics simulations, we found that, for both the DLM and the ferromagnetic case, the spin wave stiffness decreased slowly up until the ordering temperature re- gion, whereupon the softening became more pronounced. Around 120 K, the temperature at which the data of Prokop et al. 1 was measured, we find a non-negligible softening of the magnons of approximately 15%, relative to 0 K, for each set of exchange interaction parameters. Thus, a correct description of the system lies somewhere in between the ferromagnetic and the DLM descriptions, which correspond to low- and high-temperature states, respectively. In order to make a direct comparison with the SPEELS data reported for the Fe/W(110) monolayer system $^{1}$, we therefore considered a partially disordered magnetic state with an average magnetization corresponding to 120 K. At this temperature, spin dynamics simulations resulted in an average magnetization of 85% of the saturation magnetization. Consequently, we performed an elec- tronic structure calculation for a partial DLM configu- ration corresponding to this state, and subsequently car- We obtain the spin wave dispersions by identifying the peak positions of the structure factor along particular directions in reciprocal space $^{19,34,35}$. This approach is based on the adiabatic approximation, and since longi- tudinal spin fluctuations are neglected, the interaction between spin waves and the Stoner continuum is not taken into account. However, Stoner excitations are usu- ally present for short wave lengths, so the stiffness of the magnons should be captured by our method. Given the stochastic nature of the simulations, the calculated S $^{k}$( q , ω ) spectra are very diffuse. Unless a considerable averaging of ensembles is performed, it can prove difficult where the angular brackets signify an ensemble average, and k the cartesian component. Using the first principles exchange interactions in the atomistic spin dynamics Hamiltonian allows us to ad- dress the dynamical properties of spin systems at finite temperatures $^{19,34,35}$. Of particular interest to us is the dynamical structure factor S $^{k}$( q , ω ), which is the quan- tity probed in neutron scattering experiments of bulk systems $^{36}$, and can analogously be applied to SPEELS measurements. The dynamical structure factor is readily obtained by a Fourier transform of the space- and time- displaced correlation function, produce the T = 0 K magnon spectra from the frozen magnon calculation with the spin dynamics simulations, at least 20 shells of neighbors had to be included in the Heisenberg Hamiltonian. With the LKGM approach, the convergence was even more sensitive, with up to 80 shells required. Figure 2: (Color online) Left panel: intensity of the dynamical structure factor S $^{z}$( q , ω ) for a selection of q -vectors along the crystallographic [001] direction for a ML Fe on W(110). Up- per right panel: intensity plot of the “raw” S $^{z}$( q , ω ) directly obtained from an ASD simulation. Lower right panel: the same S $^{z}$( q , ω ), obtained by following the peak-finding pro- cess described in the text. C $^{k}$( r − r $^{′}$, t ) = 〈 m k $_{r}$( t ) m k r ′ (0) 〉 − 〈 m k $_{r}$( t ) 〉〈 m k r ′ (0) 〉 , 3