# Symmetry Aspects of the Band Structure and Motion Equations Applied in Calculating the Cyclotron Frequency of Electrons in Metals

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## Abstract

**:**

## 1. Introduction. Outline of the Symmetry Properties Concerning the Crystal Electrons Moving in an External Magnetic Field

## 2. The Formalism

## 3. Crystal Hamiltonians as the Oscillator Problems

## 4. Symmetry of Oscillators with Respect to the Dynamic Variables x and ${p}_{x}$ and Separation of Energy into the Kinetic and Potential Parts

## 5. Example 1: The Simple Cubic Lattice

## 6. Example 2: The Body-Centered Cubic Lattice

## 7. Example 3: The Face-Centered Cubic Lattice

## 8. Equations of Motion as a Basis in Calculating the Cyclotron Frequency

## 9. Cyclotron Oscillation Periods and Frequencies

## 10. Extensions of the Method to More Complicated Hamiltonians

## 11. Summary

## References

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**Figure 1.**The $(x,{p}_{x})$ plot of a constant energy ${C}^{\mathrm{sc}}$ of the simple cubic lattice done for two values of $cos{a}_{0}:1/2$ and $-1/3$. Figure 1a is based on the Hamiltonian formula (23), Figure 1b is obtained from the first step of (47), Figure 1c is based on the energy expression given in (48a) and inferences below of it. x and ${p}_{x}$ are expressed in radians.

**Figure 2.**The $(x,{p}_{x})$ plot of a constant energy ${C}^{\mathrm{bcc}}$ of the body centered cubic lattice; two values applied for $cos{a}_{0}$ are: $1/2$ and $1/4$. The plot obtained on the basis of (24) degenerates with the energy plot having ${E}_{\mathrm{kin}}$ taken from the first step in (55) and $V(x)$ taken from the second step in (57), as well as with the plot based on ${E}_{\mathrm{kin}}$ taken from (55a) and $V({p}_{x})$ considered instead of $V(x)$.

**Figure 3.**The $(x,{p}_{x})$ plot of a constant energy ${C}^{\mathrm{fcc}}$ of the face centered cubic lattice done for two values of $cos{a}_{0}:1/2$ and $-1/2$. Figure 3(

**a**) is based on the Hamiltonian formula (25); Figure 3(

**b**) is calculated on the basis of ${E}_{\mathrm{kin}}$ obtained in the first step of (66) and $V(x)$ taken from (67); Figure 3(

**c**) is a plot of the energy formula (37) specified for the case of the fcc lattice.

sc lattice |

$\begin{array}{ccc}\hfill {\omega}^{\mathrm{sc}}& =& 1-\frac{{a}_{0}^{2}}{8}+\frac{5{a}_{0}^{4}}{768}-\frac{17{a}_{0}^{6}}{92160}-\frac{11{a}_{0}^{8}}{16515072}-\frac{5753{a}_{0}^{10}}{29727129600}-\frac{46177{a}_{0}^{12}}{6278369771520}-\dots \hfill \\ & =& 1.-0.125{a}_{0}^{2}+0.00651042{a}_{0}^{4}-0.000184462{a}_{0}^{6}-6.66058\times {10}^{-7}{a}_{0}^{8}\hfill \\ & & -1.93527\times {10}^{-7}{a}_{0}^{10}-7.35493\times {10}^{-9}{a}_{0}^{12}-\dots \hfill \end{array}$ |

bcc lattice |

$\begin{array}{ccc}\hfill {\omega}^{\mathrm{bcc}}& =& 1-\frac{{a}_{0}^{2}}{4}+\frac{{a}_{0}^{4}}{192}-\frac{23{a}_{0}^{6}}{11520}-\frac{2519{a}_{0}^{8}}{5160960}-\frac{140333{a}_{0}^{10}}{928972800}-\frac{23798857{a}_{0}^{12}}{490497638400}-\dots \hfill \\ & =& 1.-0.25{a}_{0}^{2}+0.00520833{a}_{0}^{4}-0.00199653{a}_{0}^{6}-0.000488087{a}_{0}^{8}\hfill \\ & & \phantom{\rule{1.em}{0ex}}-0.000151063{a}_{0}^{10}-0.0000485198{a}_{0}^{12}-\dots \hfill \end{array}$ |

fcc lattice |

$\begin{array}{ccc}\hfill {\omega}^{\mathrm{fcc}}& =& 1-\frac{3{a}_{0}^{2}}{16}+\frac{11{a}_{0}^{4}}{1024}-\frac{21{a}_{0}^{6}}{81920}+\frac{1891{a}_{0}^{8}}{440401920}-\frac{1213{a}_{0}^{10}}{317089382400}+\frac{75113{a}_{0}^{12}}{31890132172800}+\dots \hfill \\ & =& 1.-0.1875{a}_{0}^{2}+0.0107422{a}_{0}^{4}-0.000256348{a}_{0}^{6}+4.29381\times {10}^{-6}{a}_{0}^{8}\hfill \\ & & -3.82542\times {10}^{-9}{a}_{0}^{10}+2.35537\times {10}^{-9}{a}_{0}^{12}+\dots \hfill \end{array}$ |

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**MDPI and ACS Style**

Olszewski, S.; Roliński, T.
Symmetry Aspects of the Band Structure and Motion Equations Applied in Calculating the Cyclotron Frequency of Electrons in Metals. *Symmetry* **2011**, *3*, 541-563.
https://doi.org/10.3390/sym3030541

**AMA Style**

Olszewski S, Roliński T.
Symmetry Aspects of the Band Structure and Motion Equations Applied in Calculating the Cyclotron Frequency of Electrons in Metals. *Symmetry*. 2011; 3(3):541-563.
https://doi.org/10.3390/sym3030541

**Chicago/Turabian Style**

Olszewski, Stanislaw, and Tomasz Roliński.
2011. "Symmetry Aspects of the Band Structure and Motion Equations Applied in Calculating the Cyclotron Frequency of Electrons in Metals" *Symmetry* 3, no. 3: 541-563.
https://doi.org/10.3390/sym3030541