Electron Spin A Fourth Quantum Number

The relative phases in these orbitals are shown by the colors red and blue. The radial nodes are represented by the dashed circles. 

What type of orbital has three angular nodes and one radial node  

Two possibilities for electron spin are shown with their associated magnetic fields. Two electrons with opposing spins have opposing magnetic fields that cancel, leaving no net magnetic field for the pair. 

Ag atoms vaporized in the oven are collimated into a beam by the slit, and the beam is passed through a nonuniform magnetic field. The beam splits in two. The beam of atoms would not experience a force if the magnetic field were uniform. The field strength must be stronger in certain directions than in others. 

Actually, electron spin is characterized by using two quantum numbers, s and m. The s quantum number determines the magnitude of the magnetic field produced and m, the orientation of this field. For an electron, s is always equal to 5, and we say that an electron is a spin particle. For other particles, s can have other values. For example, s = 1 for a photon. For a given value of s, the allowed values of are -s, -s +1, -s + 2. s. For s =, the possible values for are - and As long as we keep in mind that s = for all electrons, we can safely omit explicit reference to the quantum number s when characterizing an electron s spin. 

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Energy Levels, Spectrum, and Ionization Energy of the Hydrogen 8-8 Electron Spin A Fourth Quantum Number... 

In addition, a relativistic treatment of the electron introduces a fourth quantum number, the spin, m, with ms = j. This is because every electron has associated with it a magnetic moment which it quantized in one of two possible orientations parallel with or opposite to an applied magnetic field. 

Dirac showed in 1928 dial a fourth quantum number associated with intrinsic angidar momentum appears in a relativistic treatment of the free electron, it is customary to treat spin heiiristically. In general, the wavefimction of an electron is written as the product of the usual spatial part (which corresponds to a solution of the non-relativistic Sclnodinger equation and involves oidy the Cartesian coordinates of the particle) and a spin part a, where a is either a or p. A connnon shorthand notation is often used, whereby... 

The spins of two electrons are said to be paired if one is T and the other 1 (Fig. 1.43). Paired spins are denoted Tl, and electrons with paired spins have spin magnetic quantum numbers of opposite sign. Because an atomic orbital is designated by three quantum numbers (n, /, and mt) and the two spin states are specified by a fourth quantum number, ms, another way of expressing the Pauli exclusion principle for atoms is... 

In late fall 1925, the Dutch physicists G. Uhlenbeck and Samuel Goudsmit gave a physical interpretation to Pauli s postulate of a fourth quantum number. The electron, they proposed, may spin in one of two directions. In a given atom, a pair of electrons having three identical quantum-number values must have their spin axes oriented in opposite directions, and if paired oppositely in a single orbital, they neutralize each other magnetically. 22... 

As you learned from the previous section, three quantum numbers—n, 1, and mi—describe the energy, size, shape, and spatial orientation of an orbital. A fourth quantum number describes a property of the electron that results from its particle-like nature. Experimental evidence suggests that electrons spin about their axes as they move throughout the volume of their atoms. Like a tiny top, an electron can spin in one of two directions, each direction generating a magnetic field. The spin quantum number (mj specifies the direction in which the electron is spinning. This quantum number has only two possible values or —... 

The principal quantum number, n, is related to the size of the orbital. A second quantum number, the angular momentum quantum number, I, is used to represent different shapes of orbital. The orientation of any non-spherical orbital is indicated by a third quantum number, the magnetic quantum number, m. A fourth quantum number, the spin quantum number, s, indicates the spin of an electron within an orbital. 

When dealing with atoms possessing more than one electron it is necessary to introduce a fourth quantum number s, the spin quantum number. However, to take into account the intrinsic energy of an electron, the value of 5 is taken to be A. Essentially the intrinsic energy of the electron may interact in a quantized manner with that associated with the angular momentum represented by /, such that the only permitted interactions are l + s and / - s. For atoms possessing more than one electron it is necessary to specify the values of s with respect to an applied magnetic field these are expressed as values of ra of + A or - A. 

According to quantum mechanics, an electron has two spin states, represented by the arrows t and 1 or the Greek letters a and (3. We can think of an electron as being able to spin counterclockwise at a certain rate (the T state) or clockwise at exactly the same rate (the i state). These two spin states are distinguished by a fourth quantum number, the spin magnetic quantum number, ms. This quantum number can have only two values Tj indicates an t electron and —j indicates a l electron (Fig. f.30). Box 1.1 describes how spin explains the results of an important experiment. 

A FIGURE 5.15 Electrons behave in some respects as if they were tiny charged spheres spinning around an axis. This spin (blue arrow) gives rise to a tiny magnetic field (green arrow) and to a fourth quantum number, ms, which can have a value of either +1/2 or—1/2. 

The three quantum numbers n, l, and wi/ discussed in Section 5.7 define the energy, shape, and spatial orientation of orbitals, but they don t quite tell the whole story. When the line spectra of many multielectron atoms are studied in detail, it turns out that some lines actually occur as very closely spaced pairs. (You can see this pairing if you look closely at the visible spectrum of sodium in Figure 5.6.) Thus, there are more energy levels than simple quantum mechanics predicts, and a fourth quantum number is required. Denoted ms, this fourth quantum number is related to a property called electron spin. 

For atoms with more than one electron, we must take account of a fourth quantum number, ms, the electron spin quantum number, which has only two values, ms = 1/2. An electron has a magnetic moment which can be rationalized by imagining that electrons spin about an... 

FJectron correlations are intimately associated with two assumptions (1) a fourth quantum number, the electron-spin quantum number s, and ( ) the Pauli exclusion principle. In order to account for spectral data, it is necessary to postulate that electrons spin about their own axis to create a magnetic moment (G2o). Whereas the magnetic moment associated with the angular momentum may have (2Z + 1) components mi in the direction of an external magnetic field H, the spin moment may have only two components corresponding to s = ms = 1/2. Classically the magnitude of the moment fia associated with an angular momentum p is... 

So far we have three quantum numbers. However, we know from the relativistic treatment of the electron due to Dirac, which we described in chapter 3, that there is a fourth quantum number, called electron spin. In the first instance the need for a fourth quantum number became evident from experiment in the Dirac theory it is a consequence of introducing time as the fourth dimension. The spin angular momentum % s has the value (1/2)15, so that the magnitude of the spin angular momentum is [(1 /2)(3/2) l/ 2 ti. It can be oriented in two possible directions, with the fourth quantum number ms taking the values +1/2 or —1/2. Conventionally, these two orientations are described as a or f) respectively. 

As already pointed out, terms such as wave function, electron orbit, resonance, etc., with which we describe the formulations and results of wave mechanics, are borrowed from classical mechanics of matter in which concepts occur which, in certain respects at least, show a correspondence to the wave mechanical concepts in question. The same is the case with the electron spin. In Bohr s quantum theory, Uhlenbeck and Goudsmit s hypothesis meant the introduction of a fourth quantum number j, which can only take on the values +1/2 and —1/2- In wave mechanics it means that the total wave function, besides the orbital function, contains another factor, the spin function. This spin function can be represented by a or (3, whereby, for example, a describes the state j = +1/2 and P that with s = —1/2. The correspondence with the mechanical analogy, the top, from which the name spin has been borrowed, is appropriate in so far that the laevo and dextro rotatory character, or the pointing of the top in the + or — direction, can be connected with it. A magnetic moment and a... 

The concept of electron spin was developed by Samuel Goudsmit and George Uhlenbeck in 1925 while they were graduate students at the University of Leyden in the Netherlands. They found that a fourth quantum number (in addition to n, t, and me) was necessary to account for the details of the emission spectra of atoms. The new quantum number adopted to describe this phenomenon, called the electron spin quantum number (ms), can have only one of two values, + and — j. 

In the above treatment of the hydrogen molecule ion and the hydrogen molecule, the effect of electronic spin has been excluded, but the complete wave function of an electron must include not only the orbital motion, with which we have been concerned so far, but also a contribution for the spin. With single atoms it was possible to introduce a fourth quantum number s in addition to the three quantum numbers , I and m in order to account for the spin of the electron, and for polyelectronic molecules it is possible to proceed in an analogous manner. The complete wave function of an electron is considered to be the product of the orbital wave function, i,e. the wave function that we have been considering so far, and a wave function representing the orientation of the spin axis of the electron. 

There is also a fourth quantum number, the spin angular momentum quantum number, ttig, which can take values of + 1 or -1. The spin is not a property of orbitals but of the electrons that we put in the orbitals... 

There is a fourth quantum number that is necessary but does not result from the solution to the Schrodinger equation as we have written it. Rather, it results from a relativistic form of the equation. This is the spin quantum number, ms, which is needed for many-electron atoms and has values... 

In quantum mechanics, three quantum numbers are required to describe the distribution of electrons in hydrogen and other atoms. These numbers are derived from the mathematical solution of the Schrodinger equation for the hydrogen atom. They are called the principal quantum number, the angular momentum quantum number, and the magnetic quantum number. These quantum numbers will be used to describe atomic orbitals and to label electrons that reside in them. A fourth quantum number—the spin quantum number—describes the behavior of a specific electron and completes the description of electrons in atoms. 

In 1926 it was realized that there is a property of the electron other than the charge which must be taken into account, namely, the magnetic moment associated with intrinsic spin. It was shown by Goudsmit and Uhlenbeck [53] that this property, which represents an extra degree of freedom and therefore demands a fourth quantum number, could account for the doublet structure of the alkali spectra and the anomalous Zeeman effect. If was necessary and sufficient that the extra quantum number he two-valued. 

Although it does not follow from the Schrodinger equation, there is a fourth quantum number, m that describes the spin of the electron. It can assume two values, -I-1/2 and —1/2. According to the Paufi Exclusion Principle no two electrons in an atom can have the same set of four quantum numbers. If two electrons have the same values for n (main shell), / (subshell), and (orbital), they must differ in spin. Each orbital in an atom can hold no more than two electrons, and they must be opposed in spin. Such a couple of electrons, opposite in spin, constitutes an electron pair. 

One final note The conclusions arrived at so far tend to indicate that all sublevels with the same n have exactly the same energy, when in reality they have slightly different energies. Also a fourth quantum number, the spin quantum number m, which denotes the direction of electron spin, was not mentioned. Both of these omissions are a direct result of ignoring relativistic effects which, when taken into account, are fully accounted for. 

A fourth quantum number, s, is used to define the spin of the electron about its own axis and the orientation of the magnetic field produced by the motion of an electron. It has two possible values, + 112, corresponding with two opposite direc-... 

Each orbital has the energy shown in Figure 115 and two electrons can be housed in each orbital if their values of a fourth quantum number are different. This fourth quantum number, the spin quantum number s, has only two possible values—-1-1/2 and -1/2. The spin quantum number is interpreted as indicating that the electron also has properties associated with spin and the two values of s indicate either clockwise or counterclockwise spin. Thus, if two electrons are to be described by the same orbital, their s values must be different one electron must have a value of -i-l/2, the other must have an s value of -1/2. Another way to phrase this is that no two electrons in an atom can have the same set of four quantum numbers. This is known as the Pauli Exclusion principle (Austrian physicist, Wolfgang Pauli). 

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