Electron Spin Quantum Number ms

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... 

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 ms = j) 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 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. 

The interaction between the electric quadrupole moment of a nucleus with I > Vi and the electric field gradient from the surrounding medium is usually difficult to extract from ESR spectra. By contrast, the influence of quadrupole couplings is clearly shown by the appearance of more than one ENDOR line for each electron spin quantum number ms of a species with electron spin S and nuclear spin I > Vi. In the common case of an 5 = Vi species with an nqc that is small compared to the hfc the frequencies are given by an equation of the type (see Chapter 3) ... 

The subscripts correspond to the two electron spin quantum numbers, ms = Vi. For each ms value 21 lines with different frequencies occur depending on the value of the nuclear quantum number mi for the transition (ms, mi) (ms, mi -1). 

The field Ba can be oriented two ways depending on the value of the electron spin quantum number (ms = 1/2). The field is according to the direct field model [36] given by (4.2) when the electronic -factor is isotropic ... 

For the electrons on a carbon atom in the ground state, decide which of the following statements are true. If false, explain why. (a) Zeff for an electron in a ls-orbital is the same as Zcff for an electron in a 2s-orbital. (b) Z,.fl for an electron in a 2s-orbital is the same as Zcff for an electron in a 2/ -orbital. (c) An electron in the 2s-orbital has the same energy as an electron in the 2p-orbital. (d) The electrons in the 2p-orbitals have spin quantum numbers ms of opposite sign, (e) The electrons in the 2s-orbital have the same value for the quantum number ms. 

The eigenfunctions of the spin Hamiltonian [eqn (1.7)] are expressed in terms of an electron- and nuclear-spin basis set ms, mr), corresponding to the electron and nuclear spin quantum numbers ms and mr, respectively. The energy eigenvalues of eqn (1.7) are ... 

Given the Hamiltonian eqn (3.1), it is reasonable to express the eigenfunctions in terms of the electron and nuclear spin quantum numbers ms,mi). Applying to this function only the two terms in the Hamiltonian operator that involve the -direction of the field B we get ... 

Before going on to calculate the energy levels it is necessary to digress and briefly describe the wavefunction. The spin Hamiltonian only operates on the spin part of the wavefunction. Every unpaired electron has a spin vector /S = with spin quantum numbers ms = + and mB = — f. The wavefunctions for these two spin states are denoted by ae) and d ), respectively. The proton likewise has I = with spin wavefunctions an) and dn)- In the present example these will be used as the basis functions in our calculation of energy levels, although it is sometimes convenient to use a linear combination of these spin states. 

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. 

The fourth quantum number is the spin quantum number (ms) and indicates the direction the electron is spinning. There are only two possible values for ms, + V2 and —V2. When two electrons are to occupy the same orbital, then one must have an ms = +V2 and the other electron must have an ms = -V2. These are spin-paired electrons. 

The spin quantum number, ms, describes the spin of the electron and can only have values of +/ and —A. 

In certain respects, electrons behave as if they were spinning around an axis, much as the earth spins daily. Unlike the earth, though, electrons are free to spin in either a clockwise or a counterclockwise direction. This spinning charge gives rise to a tiny magnetic field and to a spin quantum number (ms), which can have either of two values, +1/2 or -1 /2 (Figure 5.15). A spin of +1/2 is usually represented by an up arrow ( ), and a spin of -1/2 is represented by a down arrow (l). Note that the value of ms is independent of the other three quantum numbers, unlike the values of n, /, and mJr which are interrelated. 

The positions of electrons around the nucleus are determined with the help of four quantum numbers. There is the principal quantum number (n), secondary quantum number (1), magnetic quantum number (m,) and spin quantum number (ms). Two electrons in an atom never have identical sets of the four quantum numbers. At least one of the four quantum numbers must be different. This is known as Pauli s principle. 

Magnetic spin quantum number, ms describes the direction of the rotation of the electron around its own axis and it is denoted by ms. Since there are two possible opposite spin directions for an electron, the values of the magnetic spin quantum number, ms, can be +1/2 and -1/2. 

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... 

The only quantum number that flows naturally from the Bohr approach is the principal quantum number, n the azimuthal quantum number Z (a modified k), the spin quantum number ms and the magnetic quantum number mm are all ad hoc, improvised to meet an experimental reality. Why should electrons move in elliptical orbits that depend on the principal quantum number n Why should electrons spin, with only two values for this spin Why should the orbital plane of the electron take up with respect to an external magnetic field only certain orientations, which depend on the azimuthal quantum number All four quantum numbers should follow naturally from a satisfying theory of the behaviour of electrons in atoms. 

Slater determinants enforce the Pauli exclusion principle, which forbids any two electrons in a system to have all quantum numbers the same. This is readily seen for an atom if the three quantum numbers n, l and mm of ij/(x, y, z) (Section 4.2.6) and the spin quantum number ms of a or /i were all the same for any electron, two rows (or columns, in the alternative formulation) would be identical and the determinant, hence the wavefunction, would vanish (Section 4.3.3). 

The spin quantum number ms or s has one of two possible values -1/2 or +1/2. ms differentiates between the two possible electrons occupying an orbital. Electrons moving through a magnet behave as if they were tiny magnets themselves spinning on their axis in either a clockwise or counterclockwise direction. These two spins may be described as ms = -1/2 and +1/2 or as down and up. 

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... 

In addition to the three quantum numbers used to describe the one-electron wave function, the electron has also a fourth, the spin quantum number, ms. It is related to the intrinsic angular momentum of the electron, called spin. This quantum number may assume the values of + /2 or -V2. Usually the sign of ms is represented by arrows, (t and ), or by the Greek letters a and (3. Thus, the wave function of an orbital is expressed as... 

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. 

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... 

Shell Principal quantum number n Angular momentum quantum number X Orbital designation Magnetic quantum number Spin quantum number ms Total number of electrons per orbital... 

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... 

A Figure 6.24 Illustration of the Stem-Gerlach expeiimenL Atoms in which the electron spin quantum number (uij) of the unpaired electron is are deflected in one direction, and those in which ms is —j are deflected in the other. 

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