Spin, electrons

Spin refers to rotation that is a linear function of time. A spherically rotating object the initial configuration of which is given by a spinor  

The three spatially linked matrices are identified as the Pauli matrices 7j. Choosing the matrices 

According to general properties of Pauli matrices (a p)2 = p2 hence (9) is recognized as Schrodinger s equation, with E and p in operator form. On defining the electronic wave functions as spinors both Dirac s and Schrodinger s equations are therefore obtained as the differential equation describing respectively non-relativistic and relativistic motion of an electron with spin, which appears naturally. 

The nature of spin as revealed with a model of spherical rotation is fully consistent with all known attributes of electron spin [97]. It represents an intrinsic magnetic moment of one bohr magneton8 

8This assumption was made by Uhlenbeck and Goudsmit, the discoverers of electron spin, and later shown [98] to be the correct value for an electron viewed as a rapidly rotating body, not anywhere exceeding c in tangential velocity. 

Spin-up and spin-down efer to electron spins oriented in opposite directions. The sign given to a particular spin is arbitrary. 

The wavefunction that we have just derived for the helium atom is incomplete because it does not include the spins of the two electrons. The occupation of atomic oritals in many-electron atoms is controlled by the Pauli exclusion principle, which states that  

Two electrons cannot have the same set of quantum numbers. 

This set includes not only the three orbital quantum numbers, , / and ntp but also the spin quantum number m, which was discussed in Chapter 5. The spin quantum number is restricted to one of two values +J, corresponding to spin-up, and -4, corresponding to spin-down. This leads to an alternative formulation of the Pauli exclusion principle as  

A maximum of two electrons can occupy the same. spatial orbital, and then only if their spins are paired. 

2 Electron Spin. - With electron spin resonance (ESR) experiments one studies the spin properties of the electrons. The hyperfine coupling constants describe the interactions between the total spin of the electrons and those of the nuclei and they are only then non-vanishing when none of the two spins vanishes. This means that most often such systems have an odd number of electrons, although deviations from this exist. These coupling constants depend essentially on the difference in the spin-up and spin-down electron densities at 

Also Cohen and Chong75 examined the basis-set dependence of these hyperfine coupling parameters by studying a larger set of smaller first-row molecules. They found that larger basis sets do not necessarily lead to more accurate coupling constants, which might be understood from the discussion 

Barone et al.76 studied the transition-metal complex CUC2H2 using different density-functional methods. They found that the experimental hyperfine coupling constants were well reproduced by GGA calculations with fairly large basis sets. But all studies indicate that these constants can be calculated accurately only with great care. 

Electrons behave as if they are spinning on an axis much as the earth spins on its axis. A spinning electron acts like a very small bar magnet with north and south poles. Small arrows pointing upward, T, or downward, -l, are used to indicate the two orientations of spin. Electron spin is important because two electrons in the same orbited must spin in opposite directions, T 4, a fact stated below in the Pauli principle. 

Two electrons spinning in opposite direction act like two small bar magnets with opposite poles aligned. Upward and downward arrows are used to show the two directions of spin. 

In the following orbital diagram, a total of six electrons is added, one at a time, to the three orbitals in a p-subshell. Each orbital is represented as a box, and each electron as an arrow. Notice how the Pauli principle and Hund s rule are followed. 

Electrons will fill the orbitals in a p-subshell singly before any electrons are paired. 

The same holds true for the five orbitals in a d-subshell. Adding three electrons to a d-subshell will place one electron in three of the five orbitals, leaving two orbitals empty. No pairing would take place until more than five electrons are added to the d-subshell. 

The three quantum numbers n, /, and mi are all associated with the movement of the electron around the nucleus of the hydrogen atom. In order to explain certain precise spectral observations, Goudsmit 

In our development of quantum mechanics to this point, the behavior of a particle, usually an electron, is governed by a wave function that is dependent only on the cartesian coordinates x, y, z or, equivalently, on the spherical coordinates r, 6, cp. There are, however, experimental observations that cannot be explained by a wave function which depends on cartesian coordinates alone. 

Uhlenbeck and S. Goudsmit (1925) explained the splitting of atomic spectral lines by postulating that the electron possesses an intrinsic angular momentum, which is called spin. The component of the spin angular momen- 

Following the hypothesis of electron spin by Uhlenbeck and Goudsmit, P. A. M. Dirac (1928) developed a quantum mechanics based on the theory of relativity rather than on Newtonian mechanics and applied it to the electron. He found that the spin angular momentum and the spin magnetic moment of the electron are obtained automatically from the solution of his relativistic wave equation without any further postulates. Thus, spin angular momentum is an intrinsic property of an electron (and of other elementary particles as well) just as are the charge and rest mass. 

Prior to Dirac s relativistic quantum theory, W. Pauli (1927) showed how spin could be incorporated into non-relativistic quantum mechanics. Since the subject of relativistic quantum mechanics is beyond the scope of this book, we present in this chapter Pauli s modification of the wave-function description so 

are again observed to be composed of several very closely spaced lines, with equation (6.83) giving the average wave number of each grouping. The splitting of the spectral lines in the alkali and alkaline-earth metal atoms and in hydrogen cannot be explained in terms of the quantum-mechanical postulates that are presented in Section 3.7, i.e., they cannot be explained in terms of a wave function that is dependent only on cartesian coordinates. 

Following the hypothesis of electron spin by Uhlenbeck and Goudsmit, P. A. 

The electrons in an atom also have an intrinsic angular momentum in addition to their orbital angular momentum about the nucleus. This is called the electron spin or sometimes just referred to as spin. Even an electron in the / = 0 orbital that has zero angular momentum will have an intrinsic spin. The intrinsic spin of the electron is not a classical mechanical effect hence, it is not a correct picture to view the electron spinning about one of its axes, as the classical mechanical picture would indicate. The term spin is more of a name for this phenomenon rather than an actual description of the electron. Though the intrinsic spin of the electron is real, there is no example in the macroscopic world to form a visual model. The electron spin arises naturally when relativistic mechanics is combined with quantum mechanics. Since this text is confined to quantum mechanics, the concept of electron spin must be introduced as a hypothesis. 

Since an electron has an intrinsic spin, there must be a corresponding operator for the overall intrinsic spin angular momentum squared,. It is expected that the intrinsic spin eigenfunctions, Xsm, are analogous to the spatial spherical harmonic wavefunctions, Yi (6, ). The operators S, and 5 will be the only operators for which the intrinsic spin functions are eigenfunctions just like YUd. ) are only eigenfunctions of i) and operators. 

Equation 8-43 is the analog of the equation for overall orbital angular momentum squared. 

Equation 844 is the analog to the equation for the z-component of orbital angular momentum. 

There are only two possible values for the M and S quantum numbers for a single electron (such as in a hydrogen atom) +V2 or -V2. The eigenfunction forthe M= +M eigenstate is given the symbol a and is called spin up . The M=-% eigenstate is given the symbol 3 and is called spin down . 

So far, I have ignored the existence of spin. Spin is an internal angular momentum that some particles have and others do not. Electron spin is a two-valued quantity we denote the spin variable for a single electron s, and the spin states are written a(s) and /J(s), or just a and /I for short when the meaning is obvious. The notation I am going to use is that a(si) means electron 1 in spin state a. With an eye to the discussion above about indistinguishability, we consider the following four combinations of spin states for two electrons  

All four of these combinations allow for the indistinguishability of the electrons. 

All electrons, protons and neutrons, the elementary constituents of atoms, are fermions and therefore intrinsically endowed with an amount h/2 of angular momentum, known as spin. Like mass and charge, the other properties of fermions, the nature of spin is poorly understood. In quantum theory spin is treated purely mathematically in terms of operators and spinors, without physical connotation. 

In practice, spin is as real as mass and charge, and routinely measured spectroscopically by the techniques of electron and nuclear magnetic resonance. These measurements are done under widely different conditions and with minor interference between the two phenomena. As implied by the terminology, atomic spin is of two different types - separately associated with the nucleus and extranuclear electrons respectively. The theoretical challenge is how to describe these two independent rotations within the same body. 

It is necessary to first understand the spin of a free fermion. Considered as an isolated dimensionless point particle, no conceivable mechanism can explain the physical origin of its magnetic moment. Even the rotation of a spherically symmetrical indivisible unit charge, associated with a wave structure in the aether, cannot have an intrinsic magnetic moment. 

The electron distribution in an atom or molecule containing more than one electron is determined by the electrostatic repulsion between the electrons and the attraction of the nuclei for the electrons. But there is another property of electrons that influences the electron density substantially, albeit in an indirect way. This property is called electron spin. 

All chemists are familiar with the yellow color imparted to a flame by sodium atoms. The strongest yellow line (the D line) in the sodium spectrum is actually two closely spaced lines. The sodium D line arises from a transition from the excited configuration ls 2s 2p 3p to the ground state. The doublet nature of this and other lines in the Na spectrum indicates a doubling of the expected number of states available to the valence electron. 

To explain this/me structure of atomic spectra, Uhlenbeck and Goudsmit proposed 

In 1928, Dirac developed the relativistic quantum mechanics of an electron, and in his treatment electron spin arises naturally. 

In the nonrelativistic quantum mechanics to which we are confining ourselves, electron spin must be introduced as an additional hypothesis. We have learned that each physical property has its corresponding linear Hermitian operator in quantum mechanics. For such properties as orbital angular momentum, we can construct the quantum-mechanical operator from the classical expression by replacing p, py, p by the appropriate operators. The inherent spin angular momentum of a microscopic particle has no analog in classical mechanics, so we cannot use this method to construct operators for spin. For our purposes, we shall simply use symbols for the spin operators, without giving an explicit form for them. 

Analogous to the orbital angular-momentum operators L, L, Ly, L, we have the spin angular-momentum operators S, S, Sy, S, which are postulated to be linear and Hermitian. is the operator for the square of the magnitude of the total spin angular 

We postulate that the spin angular-momentum operators obey the same commutation relations as the orbital angular-momentum operators. Analogous to [Lj, Ly = ihL [Ly, LJ = ihL [Lj, Lj,] = ihLy [Eqs. (5.46) and (5.48)], we have 

When a beam of He atoms similarly undergoes a Stem-Gerlach experiment, the beam passes through without being deflected. This implies that there is no magnetic field associated with the He atoms, even though there are two electrons present. Thus, the two electrons in the atom must have opposite spins—one up and one down — which cancel each other out and provide no overall magnetic moment. 

Model 1. The Electron Configurations of the Ground States (lowest energy states) of Several Elements. 

What generalization can be made about 2 electrons in a filled s subshell  

For each case, predict the results of a Stern-Gerlach experiment on a beam of atoms. That is, predict whether the atoms will pass through undeflected or will be split into different components. 

Here are some simulated data concerning this phenomenon  

All of our orbitals have disappeared. How do we escape this terrible dilemma We insist that no two elections may have the same wave function. In the case of elections in spatially different orbitals, say. Is and 2s orbitals, there is no problem, but for the two elechons in the 1 s orbital of the helium atom, the space orbital is the same for both. Here we must recognize an extr a dimension of relativistic space-time 

If two electrons occupy the same space orbital but have different spins, we can write 

These equations are legitimate spinorbitals, but neither is acceptable because they both imply that we can somehow label elections, ot for one and p for the other. This violates the principle of indistinguishability, but there is an easy way out of the problem we simply write the orbitals as linear combinations 

Neither of these equations tells us which spin is on which electron. They merely say that there are two spins and the probability that the 1, 2 spin combination is ot, p is equal to the probability that the 2, 1 spin combination is ot, p. The two linear combinations i i(l,2) v /(2,1) are perfectly legitimate wave functions (sums and differences of solutions of linear differential equations with constant coefficients are also solutions), but neither implies that we know which electron has the label ot or p. 

Now that we have selected two wave functions that do not violate the principle of indistinguishability, let us look at their exchange properties. The linear combinations are 

J2 and the components of J obey similar CRs to those of L2 and the components of L, so that results which follow from the CRs for L therefore also hold for S and for J. In 

The quantum number j is an integer or half-integer. The eigenvalues of j2 are j(j +1) and the eigenvalues ofjz are m = —j, /I 1,. .., j (in atomic units). These results, which are 

Warning Dirac notation is used in eq. (8) matrix notation is used in eq. (9). 

The MRs of the spin operators are readily obtained using their known properties (as given above) and the MRs of the spin eigenvectors. 

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In an electron spin resonance spectrometer, transitions between the two states are brought about by the application of the quantum of energy hv which is equal to g H. The resonance condition is defined when hv = g H and this is achieved experimentally by varying H keeping the frequency (v) constant. Esr spectroscopy is used extensively in chemistry in the identification and elucidation of structures of radicals. 

In addition to this electron spin fine structure there are often still finer lines present. These are known as the hyperfine structure, which arises from the dilTerent weights of the isotopes of an element or from the spin of the nucleus. 

ESR Electron spin (paramag- Chemical shift of splitting of Chemical state of adsorbed... 

EPR) netic) resonance [218-222] electron spin states in a magnetic field species... 

The polymer concentration profile has been measured by small-angle neutron scattering from polymers adsorbed onto colloidal particles [70,71] or porous media [72] and from flat surfaces with neutron reflectivity [73] and optical reflectometry [74]. The fraction of segments bound to the solid surface is nicely revealed in NMR studies [75], infrared spectroscopy [76], and electron spin resonance [77]. An example of the concentration profile obtained by inverting neutron scattering measurements appears in Fig. XI-7, showing a typical surface volume fraction of 0.25 and layer thickness of 10-15 nm. The profile decays rapidly and monotonically but does not exhibit power-law scaling [70]. 

Electron Spin Resonance Spectroscopy. Several ESR studies have been reported for adsorption systems [85-90]. ESR signals are strong enough to allow the detection of quite small amounts of unpaired electrons, and the shape of the signal can, in the case of adsorbed transition metal ions, give an indication of the geometry of the adsorption site. Ref. 91 provides a contemporary example of the use of ESR and of electron spin echo modulation (ESEM) to locate the environment of Cu(II) relative to in a microporous aluminophosphate molecular sieve. 

Initially, we neglect tenns depending on the electron spin and the nuclear spin / in the molecular Hamiltonian //. In this approximation, we can take the total angular momentum to be N(see (equation Al.4.1)) which results from the rotational motion of the nuclei and the orbital motion of the electrons. The components of. m the (X, Y, Z) axis system are given by ... 

If we allow for the tenns in the molecular Hamiltonian depending on the electron spin - (see chapter 7 of [1]), the resulting Hamiltonian no longer connnutes with the components of fVas given in (equation Al.4.125), but with the components of... 

Finally, we consider the complete molecular Hamiltonian which contains not only temis depending on the electron spin, but also temis depending on the nuclear spin / (see chapter 7 of [1]). This Hamiltonian conmiutes with the components of Pgiven in (equation Al.4,1). The diagonalization of the matrix representation of the complete molecular Hamiltonian proceeds as described in section Al.4,1.1. The theory of rotational synnnetry is an extensive subject and we have only scratched the surface here. A relatively new book, which is concemed with molecules, is by Zare [6] (see [7] for the solutions to all the problems in [6] and a list of the errors). This book describes, for example, the method for obtaining the fimctioiis ... 

These limitations lead to electron spin multiplicity restrictions and to differing nuclear spin statistical weights for the rotational levels. Writing the electronic wavefunction as the product of an orbital fiinction and a spin fiinction there are restrictions on how these functions can be combined. The restrictions are imposed by the fact that the complete function has to be of synnnetry... 

Karplus M 1959 Contact electron spin coupling of nuclear magnetic moments J. Chem. Phys. 30 11-15... 

S spin remains in tliennal equilibrium on die time scale of the /-spin relaxation. This situation occurs in paramagnetic systems, where S is an electron spin. The spin-lattice relaxation rate for the / spin is then given by ... 

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