Magnetic Quantum Number m

The magnetic quantum number (nt() describes the orientation of the orbital in space (see Section 6.7). Within a subshell, the value of nt( depends on the value of t. For a certain value of , there are (2 + 1) integral values of mi as follows  

If = 0. there is only one possible value of Wf 0. If = 1. then there are three values of ni( — 1, 0. and +1. If = 2, there are Jive values of mi, namely. -2. —1.0. +1, and +2. and so on. The number of ni( values indicates the number of orbitals In a subshell with a particular t value that is, each ntf value refers to a different orbital. 

Sample Problem 6.7 gives you some practice with the allowed values of quantum numbers. 

CHAPTER 6 Quantum Theory and the Electronic Structure of Atoms 

Think About It Consult Table 6.2 to make sure your answer is conect. Table 6.2 confirms that it is the value of , not the value of n, that determines the possible values of me. 

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According to quantum mechanics, the maximum observable component of the angular momentum is Ih/lir, where h is Planck s constant. A nucleus can assume only 21+1 energy states. Associated with each of these states is a magnetic quantum number m. where m has the values I, I — I, I —2,, —1+ 1, —I. 

The Hamiltonian operator for the electric quadrupole interaction, 7/q, given in (4.29), coimects the spin of the nucleus with quantum number I with the EFG. In the simplest case, when the EFG is axial (y = Vyy, i.e. rf = 0), the Schrddinger equation can be solved on the basis of the spin functions I,mi), with magnetic quantum numbers m/ = 7, 7—1,. .., —7. The Hamilton matrix is diagonal, because... 

Depending on the permitted values of the magnetic quantum number m, each subshell is further broken down into units called orbitals. The number of orbitals per subshell depends on the type of subshell but not on the value of n. Each orbital can hold a maximum of two electrons hence, the maximum number of electrons that can occupy a given subshell is determined by the number of orbitals available. These relationships are presented in Table 17-5. The maximum number of electrons in any given energy level is thus determined by the subshells it contains. The first shell can contain 2 electrons the second, 8 electrons the third, 18 electrons the fourth, 32 electrons and so on. 

A The magnetic quantum number, m, is not reflected in the orbital designation. 

Figure 10.5 Energy levels of atomic orbitals, n is the principal quantum number, and the 5, p, d notation indicates the azimuthal quantum number (/). For / = 1 and above the orbital is split into multiple suborbitals (indicated by the number of lines), corresponding to the values of the magnetic quantum number m Each of these lines can hold two electrons (corresponding to spin up and spin down ), giving rise to the rules for filling up the orbitals.

The magnetic quantum number m is related to the fact that only in an applied magnetic field it is possible to define a direction within the atom with respect to which the orbital can be directed. In general for a value of the orbital quantum number we have 21 + 1 possible values of the magnetic quantum number (which are 0, 1, 2,... up to ). To an s orbital, for instance, for which = 0 and is spherically symmetrical, only one value corresponds for the magnetic quantum number (m = 0). For p orbitals = 1) we have three possibilities (m = —1,0,+1) corresponding to three orientations (generally assumed as the x, y, z directions in Cartesian coordinates). Similarly we have five possibilities for d orbitals ( = 2) (that is m = —2, — 1,0, +1, +2), seven for/orbitals ( = 3), etc. 

All electrons in an atom can be defined in terms of four quantum numbers. The four quantum numbers are the principal quantum number, n, the angular momentum quantum number, /, the magnetic quantum number, m, and the spin quantum number, s. 

N2 = 2. For carbon, the subshells are expansions of 6, respectively, 4, Slater-type functions, that is, V = 6, v2 = 4. Because of the spherical averaging of pc and pv, the occupancies of orbitals with the same n and l values are the same, regardless of their m values. In other words, the electrons in a subshell are evenly distributed among the orbitals with different values of the magnetic quantum number m. 

But the difficulty can be overcome if we observe that Il0, 11° and H(i) are all invariant under the rotation about the z-axis and that in consequence we can consider the problem in each of the subspaces of where the z-component of the angular momentum takes on definite values. Considered in any one fm) of such subspaces belonging to the magnetic quantum number m, Il0) reduces to a constant, so that we have only to take the term k II into consideration. Then, since H0 and If"J are both bounded below, we can apply the Case ii) of 7. 3 and conclude that the condition C) is also satisfied in... 

In the presence of magnetic field a further splitting into (2J+l) equispaced energy levels occurs. These correspond to the number of values that can be assumed by the magnetic quantum number M ranging from +J>M>—J (Zeeman effect). 

For the case of a central field, the energy of an atom does not depend on magnetic quantum number m/. This means that the energy level, characterized by quantum numbers n and /, is degenerated 2/+1 times. For a pure Coulomb field there exists additional (hydrogenic) degeneration the energy of such an atom does not depend on /. Wave function (1.14) may... 

The fine-structure constant a indicates that first-order perturbation theory has been applied the linear dependence on the photon energy Eph is due to the length form of the dipole operator used in equ. (2.1), and the wavenumber k compensates the 1 /k which appears if the absolute squared value of the continuum wavefunction is used (see equ. (7.29)). The summations over the magnetic quantum numbers M, of the photoion and ms of the photoelectron s spin are necessary because no observation is made with respect to these substates. Due to the closed-shell structure of the initial state with f — 0 and M = 0, the averaging over the magnetic quantum numbers M simply yields unity and is omitted. 

In a next step the absolute squared value of the matrix element has to be evaluated together with the summation over the magnetic quantum numbers M, and ms. This gives... 

The magnetic quantum number, m, describes the orientation of orbitals in space, m may have any integral value between -/ and +/. 

The coupled equations do not depend on the magnetic quantum number M therefore, the radial functions Xjii R Jp) need not to be labeled by M. 

From the WET, Eq. [166], it is obvious that the reduced matrix element (RME) depends on the specific wave functions and the operator, whereas it is independent of magnetic quantum numbers m. The 3/ symbol depends only on rotational symmetry properties. It is related to the corresponding vector... 

Magnetic Quantum Number (m) It refers to different orientations of electron cloud in a particular subshell. These different orientations are called orbitals. The number of orbitals in a particular sub-shell with in a principal energy level is given by the values allowed to m which in turn depends on values of 7. The possible values of m range from l through 0 to + /, thus making a total of (21+ 1) values. 

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