From d0e90bb39dc4b7984ada931415bab24628c91f8c Mon Sep 17 00:00:00 2001 From: Loosetooth Date: Thu, 28 Sep 2023 00:13:02 +0200 Subject: [PATCH] wip: chapter 5 --- components/LatticeGate.js | 1205 +++++++++++++++++++ pages/chapter5-structure-of-the-lattice.mdx | 134 ++- 2 files changed, 1338 insertions(+), 1 deletion(-) create mode 100644 components/LatticeGate.js diff --git a/components/LatticeGate.js b/components/LatticeGate.js new file mode 100644 index 0000000..38b07ca --- /dev/null +++ b/components/LatticeGate.js @@ -0,0 +1,1205 @@ +import React from "react"; +import { useTheme } from 'next-themes' +import styled from "@emotion/styled"; + +const Holder = styled.div` + width: ${props => props.width ? props.width : "100%"}; + display: flex; + flex-direction: column; + align-items: center; + margin: 1rem; +` + +function LatticeGate(props) { + const { resolvedTheme } = useTheme(); + const strokeColor = resolvedTheme === "dark" ? "floralwhite" : "black" + + return ( + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+ {...React.Children.toArray(props.children)} +
+ + ); +} + +export default LatticeGate; \ No newline at end of file diff --git a/pages/chapter5-structure-of-the-lattice.mdx b/pages/chapter5-structure-of-the-lattice.mdx index 09d6bdd..8e40010 100644 --- a/pages/chapter5-structure-of-the-lattice.mdx +++ b/pages/chapter5-structure-of-the-lattice.mdx @@ -2,6 +2,7 @@ import { Callout } from '../components/Callout' import BindingEnergy from '../components/BindingEnergy' import LatticeUnit from '../components/LatticeUnit' +import LatticeGate from '../components/LatticeGate'
@@ -152,4 +153,135 @@ The lattice constant $l_0$ of the epola, calculated in Section 5.6, is $l_0 = \l *Figure 2. Lattice unit of the $\mathrm{NaCl}$ and the $e^- e^+$ lattice* -The central particle is marked $1$. Its six nearest neighbors, belonging only halfwise to the unit, are marked $1 / 2$. Fractions marking other particles show their affiliation with this fcc unit. \ No newline at end of file +The central particle is marked $1$. Its six nearest neighbors, belonging only halfwise to the unit, are marked $1 / 2$. Fractions marking other particles show their affiliation with this fcc unit. + + +### 5.8 The electromagnetic radius of the electron + +There is currently no way to measure the radius of the electron, also not in the foreseeable future. A common approximation is the 'classical' or 'electromagnetic' radius. This value, which is 2.8 fm, was obtained using two wrong assumptions. The first is the 'classical' assumption that the $-e$ charge of the electron is distributed evenly on the outermost surface of the electron, so that there is no charge in its volume. In such case the electrostatic energy of the electron can be calculated as the energy ${}_{es}E$ of a conducting sphere of radius $R$ and charge $e$, which is + +$$ +{}_{es}E = \frac{k e^2}{R} +$$ + +Here, $k = 9 \ \mathrm{Gm / F}$ is and SI units factor. + + +Editorial note: k is "Coulomb's Force Constant". + + +Considering the electron as a conducting sphere means that the charge of the electron is not intrinsically connected with the particle itself. This and the assumed absence of charge in the volume of the electron contradict our knowledge of the structure of the electron. Hence, the classical contribution to the electron radius is incorrect. + +The second incorrect assumption is the relativistic contribution, equating the electrostatic energy of the conducting-sphere electron to $m_e c^2$, the energy-equivalent of the electron mass $m_e$. Then, + +$$ +\frac{k e^2}{R} = m_e c^2 \ , +$$ + +which yields + +$$ +R = \frac{k e^2}{m_e c^2} \ . +$$ + +Substituting all the known values, one obtains $R = 2.82 \ \mathrm{fm}$. + +However, $m_e c^2$ is the radiation energy, appearing in the electromagnetic field when an electron 'disappears' in it. Radiation energy is definitely known to be connected with the vibrational motion of electrons. Therefore, it is also definitely connected with the electrostatic energy of the electron, but this cannot explain why these two energies should be equal. As shown in Section 2.6., equating the two energies is illegal. Therefore, the justly derived part of the expression for R means that if the charge e on a conducting sphere causes it to have an electrostatic energy $m_e c^2 = 511 \ \mathrm{keV}$, then the sphere should have a radius of 2.8 fm. + + +### 5.9 Radii of atomic nuclei + +The radii of nuclei are not precisely determinable quantities, and each type +of experiment serving to determine them yields slightly different values. +Bombardment with fast neutrons yields the 'neutron collision' radius of the +nucleus. The rate of disintegration of $\alpha$-particles and the cross-sections of +nuclear reactions involving other nuclides yield the Coulomb or Gamow 'barrier +radius' of the nucleus. The 'electrostatic radius' is obtained from the analysis of +the binding energy of the nucleus, and the 'electron scattering' radius is obtained +from the scattering of fast electrons from the nucleus (R. Hofstadter). + +In the experiments, especially in the scattering experiments, +a point can be located, at which the charge-density of the nucleus has a maximum. The radius R of +the nucleus is defined as the distance from this point to a point, where the +positive charge-density of the nucleus is decreased to 50 percent of the +maximum. Then all experiments show that the radii of nuclei are proportional +to the cube-root of their mass-number $A$, + +$$ +R = R_1 \cdot A^{1 / 3} +$$ + +The factor $R_1$ which is the radius of a nucleus having $A = 1$, would then be the +radius of a proton or of a neutron. These radii were not yet measured, and in +different experiments the derived values of $R_1$ vary from 1.1 to 1.3 fm. Thus, + +$$ +R = (1.2 \ \mathrm{fm}) \cdot A^{1/3} +$$ + +The so-defined radii of the nuclei vary, therefore, from 1.2 fm in hydrogen $(A = +1)$ to 7.4 fm in uranium $(A = 238)$. + +These results lead to important conclusions. First, that all nuclei have about +the same mass density. Second, if $R_1$ is the radius of protons and neutrons and $A$ +is their number in the nucleus, then $R$ can be proportional to $A^{1/3}$ only if the +protons and neutrons are most closely packed in the nucleus. Therefore, the +electrical charge of the nuclei must be quite evenly distributed in their volumes. + + +### 5.10 The 'nuclear' radius of the electron + +Just how unfit the electromagnetic radius of the electron is can be seen from +the fact that 2.82 fm is more than twice the radius of the proton, or equal to the +size of the nucleus of nitrogen. Hence, the charge density in the proton would be +13 times higher than in the electron. This disagrees with the uniqueness of the +electron charge-to-mass ratio, which is about 1840 times that of a proton. + +Scattering experiments of fast electrons lead to values in +the order of 0.1 fm for the radius $R_e$ of the electron. This value +can also be obtained from the formula for nuclear radii, $R = (1.2 \ \mathrm{fm}) \cdot A^{1/3}$, +by substituting for $A$ the 'atomic mass number' $A_e$ of the electron, $A_e = 1 / 1840$. +The obtained $R_e = 0.1 \ \mathrm{fm}$ value may +be named as the 'nuclear' radius of the electron. + +The 0.1 fm electron radius means that the volumetric charge density +is many times higher in the electron (or positron) than in +the proton, as the volume of the electron is smaller than that +of the proton. On the other hand, the mass-density in the electron +or positron is similar to the mass-density in nucleons and nuclei, +and is, therefore, $10^{15}$ times larger than in atomic bodies. + +The proportionality of the radii of these 'dense' particles +to the cubic root of their masses shows that the subparticles +of which they consist should be closely packed in them. Hence, +the structure of the dense particles is opposed to the planetary +structure in atomic bodies. The dense particles, and nuclear matter +in general, represent therefore a distinct form of matter, which +may not obey laws, established for atomic matter (and vice versa). + + +*Figure 3. Gate to epola unit cube* +*(see text for explanation)* +*scale ~$10^{-13} : 1$* + + +The actual value of the radius of the electron is not important for the epola model, +if only it is less than half the lattice constant $l_0$, i.e., less than 2.2 fm. Figure 3 +shows a 'gate' to an ${l_0}^3$ unit cube of the +epola, with electron and positron radii of 1 +fm. The circle in the center depicts the size +of a proton (or neutron). Black dots +represent the 'nuclear' 0.1 fm radius-size +electrons and positrons, all in one scale. + +The closeness of the densities of nuclei, electrons, positrons and other +elementary particles (of non·zero rest mass) justifies their consideration as +'dense' particles, opposed to the many orders of magnitude lower density of +atomic bodies. As much as the densities of the dense particles may differ from +each other, they still represent a close community, compared with the so much +smaller density of atomic matter. Whenever appropriate, we shall use for +'dense' particles the name 'avotons', from the Hebrew 'avot', meaning dense, +densely interwoven (The Prophet Yehezkiel, 6:13). Thus, *avotons* (or 'densons'?) are elementary particles (of non-zero rest mass) and nuclides. + + +# 5.11 Epola random vibrations and temperature