Interpolation between spectra satisfying sum rules

A method is presented which is able to interpolate between spectra depending parametrically on one variable and obeying a sum rule. This enables the description of experiments with a finite resolution in that parameter because integrals over certain parameter ranges are easily obtained, as for instance in the case of inelastic x-ray scattering with finite resolution in momentum transfer. Beyond the sum rule, the method does not use further assumptions about the physics of the system. It is applicable to a wide range of spectra as for instance the dynamic structure factor or the dielectric function for different moduli or directions of momentum transfer, absorption spectra for different alloy compositions or for a range of nanocrystal sizes, probability distributions, etc. The method is expected to be useful not only in the simulation of experimental spectra but also in calculations where the determination of certain spectral quantities is numerically cumbersome. A code carrying out the interpolation is provided.

Figure 1

(Color online) Illustration of the method—the integrals Eq. (7) for two different input curves (thick lines) and indication of the interpolation direction. Also shown are a number of interpolated curves (linearly between the two inputs) for the integrals and the resulting spectra.

Figure 2

(Color online) Interpretation of the interpolation method within the picture of a function $f$ of the two variables $q$ and $\omega$. Shown are the input spectra $f(q,\omega )$ for three values of $q$ (thick red lines), as well as two interpolated spectra (thin black lines), along with the integrals given with the corresponding line types. The blue dotted line shows the (here quadratic) interpolation between the intersections of the integrals with a plane $F(q,\omega )=\text{const.}$ within this plane. The curves used for this schematic figure are the EEL spectra shown in Fig. 10. We show as $f(q,\omega )$ the spectra multiplied by $\omega$, as well as the integrals $F(q,\omega )$ of this quantity. The intermediate spectra have been calculated using the present method.

Figure 3

(Color online) Interpolation (linear) between input Lorentz curves (thick black and green) with parameter $\gamma$ and center ${\omega }_0$ as indicated. The interpolated curve in the middle coincides precisely with the Lorentz curve (thick red) calculated for comparison with these intermediate parameters. (The interpolated result is given by the crosses to make the equality apparent.) Also shown are a number of interpolated curves (thin lines). The simple average of the input curves (blue dashed) gives a double-peak structure far from the desired result.

Figure 4

(Color online) Schematic of the $\omega \text{−}q$ plane showing the two possibilities to interpolate between two spectra exhibiting two peaks each, indicated by the asterisks (red). On the one hand, the peaks can move without crossing, as indicated by the green solid arrows. At all spectra for intermediate $q$, there will be two peaks. On the other hand, the peaks may cross according to the blue dotted arrows.

Figure 5

(Color online) Spectrum made of two peaks which change their broadening. The dashed green and the thick black lines are the inputs, and the thin lines are the curves obtained for intermediate $q$ using the interpolation method (linear interpolation). ${\sigma }_i$ and ${\omega }_{0i}$ refer to the first and the second peaks, respectively. The arrows are a guide for the eyes to clarify the direction of the change in the peaks $\left(\text{black}\to \text{green}\text{dashed}\right)$.

Figure 6

(Color online) Same situation as in Fig. 5 for the two outer input spectra (thick black and red) but with a third input spectrum (thick green) created using the intermediate parameters which forces a crossing of the peaks. ${\sigma }_i$ and ${\omega }_{0i}$ refer to the first peak and the second peak, respectively. Panel (a): second-order polynomial interpolation using the two outer (black and red) and the middle (green) peaks as inputs. The interpolated curves (dashed) for two $q$ are compared with the result created for these values (solid). Panel (b): second-order polynomial interpolation between the black and the middle green inputs as in (a), with an additional input between the two (red).

Figure 7

(Color online) (a) Input spectra composed of two Gaussians with parameters as indicated, ${\sigma }_i$ and ${\omega }_{0i}$ referring to the first peak and the second peak, respectively. The two outer curves (black and green) are taken as input for the (linear) interpolation, and the result (red dashed) is compared with a curve created for comparison using the corresponding parameters (red solid). Also shown are the integrals according to Eq. (7) showing the crossing (corresponding colors, thin lines). (b) Blowup of the region of the crossing showing the integrals of panel (a). Colors as in (a).

Figure 8

(Color online) Input spectra (thick green and black) as sums of peaks which give different weights to the sum rule integrals. While the two parts of the peaks which do have the same weight behave roughly like in Fig. 5, a third peak carries the weight from one side to the other (red dashed).

Figure 9

(Color online) Interpolation (linear) between the loss function of the homogeneous electron gas for $r_s=4.0$ and momentum transfers as indicated.

Figure 10

(Color online) Interpolation (second-order polynomial) between loss spectra of bulk silicon calculated within TDLDA with inclusion of lifetimes (Ref. 5) for three $\mathbf{q}$ values (thick black, red, and green lines). The intermediate calculations (thin red and blue) are compared with the interpolated curves (dashed red and blue).

Figure 11

(Color online) Interpolation (linear) between the imaginary part of the normalized dielectric function of Ge nanocrystals (Ref. 15) for two sizes, Ge 83 and Ge 239 (solid lines as indicated). The numbers indicate the number of Ge atoms in the nanocrystal. The calculation of the intermediate size Ge 147 (thin blue) is compared with the interpolated curve (dashed red) for the same size.

Figure 12

(Color online) Interpolation (linear) between the imaginary parts of the dielectric function of the ${\text{Ge}}_{1-x}{\text{Si}}_x$ bulk alloy of different composition $x$. The measurements used for input and comparison are from Ref. 16.