The Electron-Spin Quantum Number

Recall from Chapter 7 that the three quantum numbers n, I, and m, describe the size (energy), shape, and orientation, respectively, of an atomic orbital. However, an additional quantum number is needed to describe a property of the electron itself, called spin, which is not a property of the orbital. Electron spin becomes important when more than one electron is present. 

Like its charge, spin is an intrinsic property of the electron, and the spin quantum number (fn ) has values of either or -j. Thus, each electron in an atom is described completely by a set of four quantum numbers the first three describe its orbital, and the fourth describes its spin. The quantum numbers are summarized in Table 8.1. 

Now we can write a set of four quantum numbers for any electron in the ground state of any atom. For example, the set of quantum numbers for the lone electron in hydrogen (H Z = 1) is n = 1, / = 0, m/ = 0, and m = +5. (The spin quantum number for this electron could just as well have been —5, but by convention, we assign +5 for the first electron in an orbital.) 

Orbital energy (size) Orbital shape (The / values 0, 1,2, and 3 correspond to s, p, d, and / orbitals, respectively.) 

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MOs around them - rather as we construct atomic orbitals (AOs) around a single bare nucleus. Electrons are then fed into the MOs in pairs (with the electron spin quantum number = 5) in order of increasing energy using the aufbau principle, just as for atoms (Section 7.1.1), to give the ground configuration of the molecule. 

The valence electron for the cesium atom is in the 6s orbital. In assigning quantum numbers, n = principal energy level = 6. The quantum number l represents the angular momentum (type of orbital) with s orbitals = 0, p orbitals = 1, d orbitals = 2, and so forth. In this case, l = 0. The quantum number m is known as the magnetic quantum number and describes the orientation of the orbital in space. For, v orbitals (as in this case), mt always equals 0. For p orbitals, mt can take on the values of -1, 0, and +1. For d orbitals, can take on the values -2, -1, 0, +1, and +2. The quantum number ms is known as the electron spin quantum number and can take only two values, +1/2 and -1/2, depending on the spin of the electron. 

In Eqs. (4)-(7) S is the electron spin quantum number, jh the proton nuclear magnetogyric ratio, g and p the electronic g factor and Bohr magneton, respectively. r//is the distance between the metal ion and the protons of the coordinated water molecules, (Oh and cos the proton and electron Larmor frequencies, respectively, and Xr is the reorientational correlation time. The longitudinal and transverse electron spin relaxation times, Tig and T2g, are frequency dependent according to Eqs. (6) and (7), and characterized by the correlation time of the modulation of the zero-field splitting (x ) and the mean-square zero-field-splitting energy (A. The limits and the approximations inherent to the equations above are discussed in detail in the previous two chapters. 

S is the electron spin quantum number and E = gfiSiH. Nuclear resonance field shifts AH then are observed which are given by the expression... 

The electron spin quantum number, m t dctfcnniijcs the magnetic field generated by the electron and has a value of... 

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... 

For isolated electron spins the magnetic moment, p, is equal to —gHoS, where Po is the Bohr magneton, the electron spin quantum number, S, is equal to %, and g is a constant (known as the g-value). The y-value is characteristic of the... 

A. Mn(II) EPR. The five unpaired 3d electrons and the relatively long electron spin relaxation time of the divalent manganese ion result in readily observable EPR spectra for Mn2+ solutions at room temperature. The Mn2+ (S = 5/2) ion exhibits six possible spin-energy levels when placed in an external magnetic field. These six levels correspond to the six values of the electron spin quantum number, Ms, which has the values 5/2, 3/2, 1/2, -1/2, -3/2 and -5/2. The manganese nucleus has a nuclear spin quantum number of 5/2, which splits each electronic fine structure transition into six components. Under these conditions the selection rules for allowed EPR transitions are AMS = + 1, Amj = 0 (where Ms and mj are the electron and nuclear spin quantum numbers) resulting in 30 allowed transitions. The spin Hamiltonian describing such a system is... 

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... 

What do quantum numbers mean As we shall see in Sections 3.5, the three spatial quantum numbers (n, l, m ) for the H atom identify the allowed eigenstates for the solution of the Schrodinger89 equation, with certain characteristic energies and spatial features (e.g., the angular momentum quantum number describes how much angular momentum the atom has). But then, what is the meaning of the "electron spin" quantum number ... 

Before we delve further into the properties of the nucleus, let us momentarily shift our attention back to one of the electrons zooming around the nucleus. Just like photons, electrons exhibit both wave and particle properties. Each electron wave in an atom is characterized by four quantum numbers. The first three of these numbers can be taken as the electron s address and describe the energy, shape, and orientation of the volume the electron occupies in the atom. This volume is called an orbital. The fourth quantum number is the electron spin quantum number s, which can assume only two values, or - f. (Why J was selected rather than, say, 1 will be described a little later.) The Pauli exclusion principle tells us that no two electrons in an atom can have exactly the same set of four quantum numbers. Therefore, if two electrons occupy the same orbital (and thus possess the same first three quantum numbers), they must have different spin quantum numbers. Therefore, no orbital can possess more than two electrons, and then only if their spins are paired (opposite). 

Four quantum numbers describe the position and behavior of an electron in an atom the principal quantum number, the angular momentum quantum number, the magnetic quantum number, and the electron spin quantum number. A branch of physics called quantum mechanics mathematically derives these numbers through the Schrodinger equation. 

Electron spin quantum number (mg) The electron spin quantum number describes the spin of an electron. Magnetic fields have shown that the two electrons in an orbital have equal and opposite spins. The m values for these spins are - and... 

Electron spin quantum number The electron spin quantum number describes the spin of an electron. 

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. 

For our purposes the main significance of the electron spin quantum number is connected with the postulate of Austrian physicist Wolfgang Pauli (1900-1958), which is often stated as follows In a given atom no two electrons can have the same set of four quantum numbers (n, , me, and ms). This is called the Pauli exclusion principle. Since electrons in the same orbital have the same values of n, i, and mc, this postulate requires that they have different values of ms. Since only two values of ms are allowed, we might paraphrase the Pauli principle as follows An orbital can hold only two electrons, and they must have opposite spins. This principle will have important consequences when we use the atomic model to relate the electron arrangement of an atom to its position in the periodic table. 

The fourth quantum number is the electron spin quantum number, tns. The elec-, tron spin quantum number can have values of -V2 or +1/2. Any orbital can hold up to two electrons and no more. If two electrons occupy the same orbital, they have the same first three quantum numbers. The Pauli exclusion principle says that no,... 

Answer To start with, we know that the principal quantum number n is 3 and the angular momentum quantum number must be 1 (because we are deahng with a p orbital). For = 1, there are three values of given by 1, 0, 1. Since the electron spin quantum number can be either or we conclude that there are six possible ways to designate the electron ... 

The Electron-Spin Quantum Number The Exclusion Principle Electrostatic Effects and Energy-Level Splitting... 

The electron spin quantum number also defines the number of allowed spin states that may exist. This number is equivalent to 27e + 1- In the case of the electron, is 1/2 therefore there are only two allowed spin states that are initially degenerate. Since spin angular momentum is quantised in terms of magnitude and orientation according to the electron spin quantum number, 7e> then z-axis components of angular momentum // are similarly quantised. Each allowed z-axis component is represented by an individual magnetic spin quantum number, ms, according to... 

The quantity is the spin quantum number it may have only the values or —j. The Schrodinger equation in its usual form gives no indication of the existence of the electron spin. However, Dirac has shown that if the Schrodinger equation is cast into a form that satisfies certain requirements of relativity theory, then four quantum numbers, the fourth being the electron spin quantum number, appear in the solution for the hydrogen atom. Thus the spin is a coherent part of the fundamental theory and is not tacked on just to patch things up. 

Let a and / be the two spin wave functions corresponding to the two possible values of the electron spin quantum number then a(l) indicates that electron 1 has spin a. The possible spin functions for two electrons are cTj = a(l)a(2) <72 = a(l) (2) cr3 = / (l)a(2) 0-4 = P(T)P(2). By making linear combinations where necessary, show that three functions are symmetric (triplet state) and one is antisymmetric (singlet state) under the interchange of the two electrons. 

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