fix units

pages/chapter4-bound-electrons.mdx

@@ -12,7 +12,7 @@ The positron was discovered in 1932 by Carl David Anderson (born 1905), who obse
 
 Anderson also discovered that when a $\gamma$-photon of at least 1.02 MeV is absorbed in the vacuum, an electron and a positron may appear. He also observed reverse processes, occuring when an electron and a positron approach each other. Then it could happen that these particles disappear, and, simultaneously, two $\gamma$-photons of combined energy 1.02 MeV are emitted.
 
-Anderson's discovery was in agreement with some theoretical predictions made by P.A.M. Dirac (1902-1984) in 1928. Dirac developed a quantum theory of the electron, which tried to unify the quantum approach with relativity. He assumed that there is an 'electron sea' with quantized negative energy levels filled with electrons. Electrons on neqative energy levels are not detectable. The highest negative energy level is 511 keV below the zero energy of an electron, at which the electron becomes observable. Here 511 keV is the energy equivalent of the mass $m_e$ of the electron, $m_e c^2 = 511 \mathrm{keV}$. For consistency, Dirac had to assume, that negative energy levels, which are not occupied by electrons, or "holes" in the electron sea, should appear and behave as particles having the mass of the electron and a positive charge, equal and opposite to the charge of the electron. An electron which falls into such a hole, or "recombines" with it, becomes undetectable, as are the electrons of the "electron sea".
+Anderson's discovery was in agreement with some theoretical predictions made by P.A.M. Dirac (1902-1984) in 1928. Dirac developed a quantum theory of the electron, which tried to unify the quantum approach with relativity. He assumed that there is an 'electron sea' with quantized negative energy levels filled with electrons. Electrons on neqative energy levels are not detectable. The highest negative energy level is 511 keV below the zero energy of an electron, at which the electron becomes observable. Here 511 keV is the energy equivalent of the mass $m_e$ of the electron, $m_e c^2 = 511 \ \mathrm{keV}$. For consistency, Dirac had to assume, that negative energy levels, which are not occupied by electrons, or "holes" in the electron sea, should appear and behave as particles having the mass of the electron and a positive charge, equal and opposite to the charge of the electron. An electron which falls into such a hole, or "recombines" with it, becomes undetectable, as are the electrons of the "electron sea".
 
 Dirac's 'hole' could be considered as a prediction of the positron. The appearance of the electron and positron in Anderson's experiment with the absorption of 1.02 MeV could be interpreted as the pulling of an electron out from the highest negative energy level in the 'electron sea'; 511 keV are then needed to "produce" the mass of the electron, and 511 keV for the mass of the appearing hole. Therefore, Anderson's discovery was initially considered as having been predicted by Dirac, thus proving the theory. However, Dirac's theory did not fit the developing theories and was abandoned in particle physics. It inspired, though, the electron-hole generation and recombination interpretations in solid state physics.
 
@@ -38,7 +38,7 @@ We interpreted the disappearance of masses in a way similar to the commonly acce
 
 The law that electric charges cannot be created or destroyed was introduced in the middle of the XVII century without any elaborate argumentation. It was not challenged, and stands firm in modern science. Relativity, which reduced the mass-conservation law to a second-rate status, has raised charge conservation to become an absolute law.
 
-Unlike the "poor childhood" of the charge conservation law, the law that masses cannot be created or destroyed was subject to elaborate investigations by the most prominent physicists and chemists, and was a most important pillar of science until 1905. It was then challenged by A. Einstein with the introduction of the formula for the 'equivalence' of mass and energy. As we saw, this formula leads to the wrong conclusion that 511 keV is the creation energy of an electron. With our conclusion that $m_e c^2 = 511 \mathrm{keV}$ is the binding energy of the electron to the bound positrons and electrons existing in the vacuum space, the mass-conservation law is restored to its full glory, while the Einstein mass-energy relation remains perfectly valid.
+Unlike the "poor childhood" of the charge conservation law, the law that masses cannot be created or destroyed was subject to elaborate investigations by the most prominent physicists and chemists, and was a most important pillar of science until 1905. It was then challenged by A. Einstein with the introduction of the formula for the 'equivalence' of mass and energy. As we saw, this formula leads to the wrong conclusion that 511 keV is the creation energy of an electron. With our conclusion that $m_e c^2 = 511 \ \mathrm{keV}$ is the binding energy of the electron to the bound positrons and electrons existing in the vacuum space, the mass-conservation law is restored to its full glory, while the Einstein mass-energy relation remains perfectly valid.
 
 
 ### 4.4 Efficiency of e<sup>&ndash;</sup>e<sup>+</sup> pair production

pages/chapter5-structure-of-the-lattice.mdx

@@ -14,7 +14,7 @@ import LatticeUnit from '../components/LatticeUnit'
 
 Our acceptance of the $\mathrm{NaCl}$ crystal lattice as the closest solid-state analog of the electron-positron lattice is based on several reasons. First, the epola analog must be an alkali-halide crystal because of the equality of the electrical charges of their positive ions (e.g., $\mathrm{Na^+}$) to the charge $+e$ of the positron, and of their negative ions (e.g., $\mathrm{Cl^-}$) to the charge $-e$ of the electron. Also, in terms of the types of crystal bonds, the epola bonds correspond to 'fully ionic'. Therefore, the crystal bonding of the analog must have the highest possible rate of ionicity, which again leads to the alkali halides.
 
-Second, the equality of the masses $m_e$ of the electron and the positron requires that the ion masses of the analog be as close as possible to one another. Therefore, the best choice would be $\mathrm{KCl}$, with $m_{\mathrm{K}} = 39 \mathrm{AMU}$ and $m_{\mathrm{Cl}} = 35 \mathrm{AMU}$. However, the symmetry in the inner structure of the electron and positron suggests that the electronic shell structure of the alkali and the halide atoms should be similar, i.e., they should belong to the same period in the periodic table of elements. Such are the $\mathrm{Na}$ and $\mathrm{Cl}$ atoms, $\left( m_{\mathrm{Na}} = 23 \mathrm{AMU},\  m_{\mathrm{Cl}} = 35 \mathrm{AMU} \right)$. Both belong to the third period, have the same filled inner shells (two electrons on the $s$-shell and eight electrons on the $d$-shell), and the same unfilled outer shell (one $p$-electron in $\mathrm{Na}$, seven $p$-electrons in $\mathrm{Cl}$). The atomic masses of other alkali and halide atom pairs from a common period have more diverse masses $\left( m_{\mathrm{Li}} = 7, m_{\mathrm{F}} = 19; \ m_{\mathrm{K}} = 39, m_{\mathrm{Br}} = 80 \mathrm{AMU} \right)$.
+Second, the equality of the masses $m_e$ of the electron and the positron requires that the ion masses of the analog be as close as possible to one another. Therefore, the best choice would be $\mathrm{KCl}$, with $m_{\mathrm{K}} = 39 \ \mathrm{AMU}$ and $m_{\mathrm{Cl}} = 35 \ \mathrm{AMU}$. However, the symmetry in the inner structure of the electron and positron suggests that the electronic shell structure of the alkali and the halide atoms should be similar, i.e., they should belong to the same period in the periodic table of elements. Such are the $\mathrm{Na}$ and $\mathrm{Cl}$ atoms, $\left( m_{\mathrm{Na}} = 23 \ \mathrm{AMU},\  m_{\mathrm{Cl}} = 35 \ \mathrm{AMU} \right)$. Both belong to the third period, have the same filled inner shells (two electrons on the $s$-shell and eight electrons on the $d$-shell), and the same unfilled outer shell (one $p$-electron in $\mathrm{Na}$, seven $p$-electrons in $\mathrm{Cl}$). The atomic masses of other alkali and halide atom pairs from a common period have more diverse masses $\left( m_{\mathrm{Li}} = 7, m_{\mathrm{F}} = 19; \ m_{\mathrm{K}} = 39, m_{\mathrm{Br}} = 80 \ \mathrm{AMU} \right)$.
 
 Third, the binding energy in the $\mathrm{NaCl}$ crystal is the largest among the alkali-halide crystals, which fits the strength of the bond between electrons and positrons. Then, the face-centered cubic (fcc) lattice structure of the $\mathrm{NaCl}$ crystal is most common: it is represented more or less exactly in 17 out of the 20 alkali halide crystals. It should be noted., however, that our calculations based on the analogy with the $\mathrm{NaCl}$ crystal would not be significantly affected by the use of some other alkali halide crystal as analog.