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4. La Soberanía Imperial y el régimen de verdad americano
All matter is composed of atoms. The atoms,in turn,consist of electrons,protons,and neutrons. The properties of an atom are determined by many factors including: (1) the atomic number Z that corresponds to the number of electrons or protons in a neutral atom, (2) the mass of the atom,(3) the spatial distribution of the electrons in orbits around the nucleus,(4) the energy of the electrons in the atom,and (5) the ease of adding or removing one or more electrons from the atom to create a charged ion. The latter three factors, involving electrons,can be influenced by external conditions such as mechanical forces, electromagnetic fields,and temperature. Since we are interested in the influence of exactly these variables on the properties of solids,the bulk of our discussion will focus on the characteristics of electrons in atoms. An understanding of electron behavior,however, requires some understanding of quantum mechanics. Quantum mechanics theory,or QMT,is a mathematical framework developed by physicists in the early part of the 20th century to describe the interaction of electrons,protons,and neutrons in atoms and molecules.
One of the key features of QMT is the recognition that an electron exhibits both particle and wave characteristics. The properties of an electron in an atom can be best modeled by treating the electron as an energy wave. The equation describing the electron wave motion was developed by Erwin Schro¨dinger in 1925 and is known as the Schro¨dinger equation. Both this equation and its solution are beyond the level of this text.
However,we will state without derivation some results of Schro¨dinger’s equation,and of QMT,as they relate to the atomic structure of materials.
Only certain types of electron wave motion can satisfy the constraints of Schro¨dinger’s equation. The valid solutions to this equation can be numbered for identification purposes.
That is,each solution is identified by a set of three integer values (n, l,and m) known as
Bohr model for a Zn atom
Electrons
Nucleus 1s 2s
2p 3s3p4s 3d
quantum numbers.1 While the wave equation is extremely successful in explaining a great many aspects of the properties of electrons, the physicist Wolfgang Pauli found that a complete description of an electron required the specification of an additional value associated with electron “spin.” This fourth quantum number, ms, can only tak e on values of⫾12. Another outcome of Pauli’s work, known as thePauli exclusion principle, states that no two interacting electrons may have the same four values for their quantum numbers. Thus, each electron in an atom has a unique set of four quantum numbers that completely describe its characteristics.
One of the main reasons for introducing quantum numbers in this text is that they can be used to characterize the energy levels and arrangement of the electrons in an atom. The energy of an electron is primarily a function of its n and l quantum numbers (with m having a weaker influence). As shown in Figure 2.2–1, in the planetary or Bohr model of an atom the electrons are arranged in subshells in which all electrons have the same n and l values and, therefore, approximately the same energy.
The electron subshells are identified using an alphanumeric code in which the number represents the value of n and the letter gives the value of l using the following convention:
s for l⫽ 0, p for l ⫽ 1, d for l ⫽ 2, and f for l ⫽ 3. The Pauli exclusion principle can be used to show that the maximum number of electrons permitted in any subshell is
1For a potential energy function with spherical symmetry, as in the case of the hydrogen atom, in which the negatively charged electron is electrostatically attracted to the positively charged nucleus, the wave equation is solved more easily in spherical共r,,兲 coordinates. The principal quantum number n is associated with the boundary conditions on r; the angular momentum quantum number l is derived from the boundary conditions on ; and the magnetic quantum number m is associated with the coordinate. More detailed quantum calculations show that while n can assume any integer value greater than or equal to 1, the values of the quantum numbers l and m are restricted as follows: l can only have integer values from 0 to n⫺ 1, and m can only assume the integer values from⫺l to ⫹l.
FIGURE 2.2–1 A sche-matic illustration of the Bohr model of a Zn atom showing the arrangement of electrons in the sub-shells.
determined by the value of the quantum number l and is given by 2共2l ⫹ 1兲. Thus, the maxi-imum numbers of electrons in an s, p, d, and f subshell are respectively 2, 6, 10, and 14.
Theelectron configurationrepresents the distribution of electrons within the permis-sible energy levels. In theground state, an atom’s electrons occupy the lowest-energy subshells consistent with the Pauli exclusion principle. The subshells can be arranged in order of increasing energy as follows:
1s, 2s, 2p, 3s, 3p, 4s, 3d, 4p, 5s, 4d, 5p, 6s, 4f, 5d, 6p, 7s, 5f, 6d . . .
In this notation, the number of electrons in each subshell is indicated using an integer superscript on the corresponding letter. For example, a half-filled subshell with quantum numbers n⫽ 3 and l ⫽ 2 would be designated as 3d5.
Howcan we use this notation to describe the ground-state electron configuration for an oxygen atom that contains eight electrons? In the ground state the subshells will “fill”
in the order 1s, 2s, 2p . . . and the maximum number of electrons in s and p subshells will be two and six, respectively. Thus, the ground-state electron configuration for oxygen is 1s22s22p4, indicating two electrons in each of the (filled) 1s and 2s subshells and four electrons in the (partially filled) 2p subshell.
In addition to the quantization of energy, another key result of the wave model is that the exact position of an electron within an atom can never be known. Instead, probability density functions (PDFs) are used to describe the spatial location of electrons. As shown in Figure 2.2–2, the shape of the PDF depends on the value of the quantum number l.
Note that not all the distribution functions are radially symmetric. The consequence of a nonsymmetric PDF is that definite bond angles can be found in structures such as dia-mond, organic molecules, and polymeric chains. We will see that these specific bond angles influence the macroscopic engineering properties of the corresponding materials.
FIGURE 2.2–2 A highly schematic illustra-tion of the probability den-sity functions for electrons in certain subshells of an atom. Note that the s sub-shells are radially symmet-ric while the p subshells (and all other subshells) are highly directional.
s orbitals are spherically symmetric p orbitals have a dumbbell shape
H
Another less abstract consequence of QMT is a rational explanation of the periodic table of the elements, which was originally developed on the basis of experimental observations (see Figure 2.2–3). The elements were placed in order of increasing atomic number and arranged in a series of vertical columns, or groups, so that all the elements in a group display similar chemical properties.An explanation for the regularity of atomic properties within a group is obtained from the electron configurations of the elements.
Elements within a group have the same number of electrons in their outer, orvalence, shells that participate most strongly in chemical reactions.
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EXAMPLE 2.2–1
Determine the electron configurations for a silicon atom 共Z ⫽ 14兲 and a germanium atom 共Z ⫽ 32兲.Explain why these two elements display similar characteristics.
Solution
Since the maximum number of electrons in a subshell is given by the equation 2共2l ⫹ 1兲, the corresponding numbers for an s shell共l ⫽ 0兲, a p shell 共l ⫽ 1兲, and a d shell 共l ⫽ 2兲 are respec-tively 2, 6, and 10.Combining this result with the observation that the order of the subshells is given by 1s, 2s, 2p, 3s, 3p, 4s, 3d, 4p allows one to determine that the electron configurations for silicon and germanium are关1s22s22p6兴3s23p2and关1s22s22p63s23p63d10兴4s24p2.Both elements have a va-lence electron structure of the form x s2x p2where x is 3 for Si and 4 for Ge.Since the valence electron distributions are similar for these elements, we should expect them to exhibit similar properties.
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FIGURE 2.2–3 The periodic table of the elements.