Abstract
A physical phenomena is always described as a function of time and spatial coordinates \((t,\varvec{{ r}})\). One may Fourier transform the time domain (t) to the frequency domain(\(\omega \)) without losing any information, where \(\omega \) (rad/s) is the angular frequency.
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Notes
- 1.
Where \(\varvec{{ r}}\equiv (x,y, z)\) in Cartesian coordinate system, and \(\varvec{{ r}}\equiv (r,\theta ,\phi )\) in spherical coordinate system.
- 2.
\(\omega =2\pi f\), where \(f\ (Hz)\) is the cyclic frequency.
- 3.
Where \(\hbar \) is the reduced Planck’s constant given as, \(\hbar = h/2\pi \), and h is the Planck’s constant given as, \(h = 6.62 \times 10^{-34}\) Js.
- 4.
\(n=5\) is not allowed since one may not fill a 2D space with pentagons only.
- 5.
A word of caution that the word crystal is loosely defined and even sometimes used for finite structures without any periodicity. Hence, the reader is encouraged to be careful in the way the word “crystal” is used sometimes.
- 6.
For orthogonal basis vectors, \(\hat{a_i}\varvec{\cdot }\hat{a_j}=0\), where \(i\ne j\).
- 7.
For nonorthogonal basis vectors, \(\hat{a_i}\varvec{\cdot }\hat{a_j}\ne 0\), where \(i\ne j\).
- 8.
Not to be confused with the use of ( ) and [ ] brackets for defining the range and the domain of a function.
- 9.
\(1\ \AA \) = 0.1 nm
- 10.
Dirac’s delta function is a different one as discussed later in this book.
- 11.
Note the mathematical notation for the range of an interval. ] and [ brackets mean that the number is included, whereas ) and ( brackets mean that the number is excluded.
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Raza, H. (2019). Atomic Structure. In: Nanoelectronics Fundamentals. NanoScience and Technology. Springer, Cham. https://doi.org/10.1007/978-3-030-32573-2_2
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DOI: https://doi.org/10.1007/978-3-030-32573-2_2
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