Spin quantum values

The simplest case arises when the electronic motion can be considered in temis of just one electron for example, in hydrogen or alkali metal atoms. That electron will have various values of orbital angular momentum described by a quantum number /. It also has a spin angular momentum described by a spin quantum number s of d, and a total angular momentum which is the vector sum of orbital and spin parts with... 

Figure 3. Low-energy vibronic spectrum in a. 11 electronic state of a linear triatomic molecule, computed for various values of the Renner parameter e and spin-orbit constant Aso (in cm ). The spectrum shown in the center of figure (e = —0.17, A o = —37cm ) corresponds to the A TT state of NCN [28,29]. The zero on the energy scale represents the minimum of the potential energy surface. Solid lines A = 0 vibronic levels dashed lines K = levels dash-dotted lines K = 1 levels dotted lines = 3 levels. Spin-vibronic levels are denoted by the value of the corresponding quantum number P P = Af - - E note that E is in this case spin quantum number),...

As was shown in the preceding discussion (see also Sections Vin and IX), the rovibronic wave functions for a homonuclear diatomic molecule under the permutation of identical nuclei are symmetric for even J rotational quantum numbers in and E electronic states antisymmeUic for odd J values in and E elecbonic states symmetric for odd J values in E and E electronic states and antisymmeteic for even J values in Ej and E+ electeonic states. Note that the vibrational ground state is symmetric under pemrutation of the two nuclei. The most restrictive result arises therefore when the nuclear spin quantum number of the individual nuclei is 0. In this case, the nuclear spin function is always symmetric with respect to interchange of the identical nuclei, and hence only totally symmeUic rovibronic states are allowed since the total wave function must be symmetric for bosonic systems. For example, the nucleus has zero nuclear spin, and hence the rotational levels with odd values of J do not exist for the ground electronic state f EJ") of Cr. 

One identifies the highest Ms value (this gives a value of the total spin quantum number that arises, S) in the box. For the above example, the answer is S = 1. 

In addition to being negatively charged electrons possess the property of spin The spin quantum number of an electron can have a value of either +5 or According to the Pauli exclusion principle, two electrons may occupy the same orbital only when... 

THE SCHRODINGER EQUATION AND SOME OF ITS SOLUTIONS Table 1.3 Some values of the nuclear spin quantum number / 19... 

We often say that an electron is a spin-1/2 particle. Many nuclei also have a corresponding internal angular momentum which we refer to as nuclear spin, and we use the symbol I to represent the vector. The nuclear spin quantum number I is not restricted to the value of 1/2 it can have both integral and halfintegral values depending on the particular isotope of a particular element. All nuclei for which 7 1 also posses a nuclear quadrupole moment. It is usually given the symbol Qn and it is related to the nuclear charge density Pn(t) in much the same way as the electric quadrupole discussed earlier ... 

Suppose that the spin quantum number could have the values, 0, and —j. Assuming that the rules governing the values of the other quantum numbers and the order of filling sublevels were unchanged,... 

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. 

S is the spin quantum number. The expected magnetic moments for a sextet and a doublet state are 5.91 and 1.73 respectively, measured in Bohr magnetons. The values calculated from the observed paramagnetic susceptibilities of the crystals are 5.88 for (NH aFeFg and 2.0 for... 

The multiplicity can be determined from the experimental values of the magnetic susceptibility, the magnetic moment in Bohr magnetons being equal to 2 VS(S + l), in which S is the spin quantum number. (The multiplicity is 2S + 1.) The moments for 22 and 62 are 1.73 and 5.91, respectively. The experimental values for K3Fe(CN)6 and (NH jFeF are 2.0 and 5.88, respectively, so that the bonds in the [FefCN ] ion are electron-pair bonds, and those in [FeFe]a are ionic. 

The calculated energy of interaction of an atomic moment and the Weiss field (0.26 uncoupled conduction electrons per atom) for magnetic saturation is 0.135 ev, or 3070 cal. mole-1. According to the Weiss theory the Curie temperature is equal to this energy of interaction divided by 3k, where k is Boltzmann s constant. The effect of spatial quantization of the atomic moment, with spin quantum number S, is to introduce the factor (S + 1)/S that is, the Curie temperature is equal to nt S + l)/3Sk. For iron, with 5 = 1, the predicted value for the Curie constant is 1350°K, in rough agreement with the experimental value, 1043°K. 

An atomic orbital is designated by its and / values, such as Is, 4p, 3d, and so on. When / > 0, there is more than one orbital of each designation three p orbitals, five d orbitals, and so on. When an electron occupies any orbital, its spin quantum number, lit, can be either + or -. Thus, there are many sets of valid quantum numbers. An electron in a 3p orbital, for example, has six valid sets of quantum numbers n = 3, / = 1, m =+1, j — -2 / —i — — 1... 

A very convenient feature of this formalism is that the values of the A and B coefficients only depend on three numbers the number of electrons N), the number of orbitals of the basis K), and the Spin quantum number (S). 

The last quantum number was proposed to solve a mystery. Some spectral lines split into two lines when theory predicted that only one should exist. Several physicists had a hand in trying to solve this problem. By 1924, a consensus was reached. A new quantum property and number were needed to explain spectral splitting. At the time, the electron was considered to be a particle, and scientists called this new property spin, usually designated as mg. The spin quantum number has only two possible values -1-1/2 or -1/2. It is usually depicted as an arrow pointing up or down. 

A particle possesses an intrinsic angular momentum S and an associated magnetic moment Mg. This spin angular momentum is represented by a hermitian operator S which obeys the relation S X S = i S. Each type of partiele has a fixed spin quantum number or spin s from the set of values 5 = 0, i, 1,, 2,. .. The spin s for the electron, the proton, or the neutron has a value The spin magnetie moment for the electron is given by Mg = —eS/ nie. 

In general, the spin quantum numbers s and Ws can have integer and halfinteger values. Although the corresponding orbital angular-momentum quantum numbers / and m are restricted to integer values, there is no reason for such a restriction on s and mg. 

Fig. 1.1 Schematic representation of the population difference of spins at different magnetic field strengths. The two different spin quantum number values of the ]H spin, +34 and -34, are indicated by arrows. Spins assume the lower energy state preferentially, the ratio bet-ween upper and lower energy level being given by the Boltzmann distribution.

The spin quantum number s is used to characterize the spin. It can have only the one numerical value of x = h = 6.6262 10—34 J s = Planck s constant. 

The spin quantum number, denoted s, is related to the spin of the electron on its "axis. It ordinarily does not affect the energy of the electron. Its possible values are -1 and + T The value of s does not depend on the value of any other quantum number. 

Ans. The letter s represents the fourth quantum number—the spin quantum number. It also represents the subshell with an / value of 0. 

The relative size of atomic orbitals, which is found to increase as their energy level rises, is defined by the principal quantum number, n, their shape and spatial orientation (with respect to the nucleus and each other) by the subsidiary quantum numbers, Z and m, respectively. Electrons in orbitals also have a further designation in terms of the spin quantum number, which can have the values +j or — j. One limitation that theory imposes on such orbitals is that each may accommodate not more than two electrons, these electrons being distinguished from each other by having opposed (paired) spins, t This follows from the Pauli exclusion principle, which states that no two electrons in any atom may have exactly the same set of quantum numbers. 

Spin quantum number Number indicating the number of allowed orientations of a particular nucleus in a magnetic field. For example, H has an I value of lk, allowing for two possible orientations, whereas 14N has an I of 1, allowing three possible orientations. 

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