# 纳米级电子器件&纳米级磁性材料和器件&纳米材料|EEE6215& MAT6390 Nanoscale Electronic Devices&Nanoscale Magnetic Materials and Devices代写

$$V-E+F=2(1-g)$$
For bulk three-dimensional solids, $g$ is equivalent to the number of cuts required to transform a solid structure into a structure topologically equivalent to a sphere (for instance, $g=0$ for a polygonal sphere such as $\mathrm{C}{60}$ or $\mathrm{C}{70}$, and $g=1$ for a torus). Suppose, further, that the object is formed of polygons having different (i) number of sides. The total number of faces $(F)$ is then
$$F=\sum N_{i}$$
where $N_{i}$ is the number of polygons with $i$ sides. Each edge, by definition, is shared between two adjacent faces, and each vertex between three adjacent faces, which is represented as
$$E=(1 / 2) \sum i N_{i}$$
and
$$V=(2 / 3) E$$
By substituting Euler’s postulate for $i \geq 3$ is represented as
$$\begin{array}{r} 3 N_{3}+2 N_{4}+N_{5}-N_{7}-2 N_{8}-3 N_{9}-\cdots-=12(1-g) \ \sum(6-i) N_{i}=12(1-g) \end{array}$$

$$\delta=d A$$
where $d$ is carbide density and $A$ is the mass fraction of carbon in the carbide. Theoretical total pore volume $V_{\Sigma}$ was calculated on the basis of pycnometric density of carbon $\left(d_{C}=2.00 \mathrm{~g} / \mathrm{cm}^{3}\right)$ by the formula
$$V_{\Sigma}=\frac{1}{\delta}-\frac{1}{d_{\mathrm{C}}}$$
It was found that the theoretical $\delta$ and $V_{\Sigma}$ values are generally in satisfactory agreement with experimental data taking into account the porosity of the initial carbide granules.