The so-called Bergman representation of effective dielectric functions is the most general form of effective medium approaches:
The function g(n,f) is called spectral density and holds all topological details of the microgeometry. It is a real, non-negative function, normalized in the interval [0,1] for n. In the case of isotropy there is also a condition for the first moment of the spectral density:
Any microgeometry is represented by a certain spectral density. However, in general it cannot be computed exactly for real systems since the topology is too complicated (and in most cases not even known).
To gain some insight into the relation of spectral density and microtopology some research projects in the past had the goal to fit theoretical optical spectra (of systems with known optical constants of the matrix and particle materials) to experimental ones by adjusting g(n,f). This way a lot of experience has been achieved being useful in the treatment of new two-phase composites.
In many cases an important property of the microgeometry is the 'degree of connectivity' of the embedded particles which usually is called percolation strength. Especially metal-insulator composites can switch from metallic to dielectric behaviour when the percolation strength is reduced. It can be shown that this feature shows up in the function g(n,f) as a δ-function at n=0. Therefore, a separation of this diverging term from the continuous part of the spectral density is useful:
The factor g0(f) is called percolation strength.
Implementation
The two parameters volume fraction f and percolation strength g0(f) are specified in the Parameters dialog of Bergman representation objects in SCOUT. Then, to adjust spectral densities to fit experimental data, one has to parameterize the shape of the spectral density in a suitable way. We have chosen to do that in SCOUT by a set of definition points. Each point is defined by a value of n and a corresponding g(n) value. You can use an arbitrary number of definition points. The position of the points can be set 'graphically', i.e. by the mouse.
The shape of the spectral density is then obtained by cubic spline interpolation through the definition points. Usually, any user-defined shape (drawn visually with the mouse) will violate the restrictions mentioned above (normalization, coupling of the first moment of the spectral density to the volume fraction). Therefore, SCOUT adds a smooth peak to the user-defined curve to ensure the proper value of the 1st moment. This is followed by an automatic rescaling which normalizes the spectral density.
The Points submenu contains some commands to control the definition points. Add points brings up a mode where a mouse click creates a new definition point at the mouse cursor position. After the Move points command you can catch an already existing point with the cursor and drag it to a new position. If the Continuous update option is selected the spectral density shape is updated with each mouse move. If the option is switched off, the spectral density is updated when the mouse button is released at the final position of the definition point. Clicking at points after Delete points deletes the points. Delete all points deletes all points and starts with three new definition points at default positions.
A typical situation is this:
Here the user-defined curve had too much weight on the right side, hence SCOUT added the peak to the left. If you move down the point at n=0.72 you reduce the weight on the right side and the peak on the left side is not needed any more:
Note the re-scaling of the height of the definition points. If you reduce the height of that point further on, SCOUT has to add something on the right side to achieve the correct value of the 1st moment: