Subsidiary quantum number

Neben-quantenzahl, /. secondary (or subsidiary) quantum number, -raum, m. additional space anteroom, -reaktlon, /. by-reaction, side reaction, secondary reaction, -rechnung, /. auxiliary calculation, -reihe,/. = Neben-serie. -rohr, n. side tube (or pipe), branch tube (or pipe). -roUe, /. subordinate r61e. -sache, /. secondary matter. 

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. 

The principal quantum number n, the subsidiary quantum number which can take the values 0,1. .. up to — 1), the magnetic quantum number m with the values —l, —(/—1). . . o. . . (1 — 1), /, therefore follow directly from the wave mechanical theory. Each state can further be occupied by two electrons, on account of the spin quantum number 5, which can have the values +V2 an<3 —V2-... 

Bohr theory. The shells are determined by the principal quantum number n, and orbitals in the same shell are distinguished by the subsidiary quantum numbers I and m. (When the orbitals are not strictly hydrogen-like, n can still be defined as the number of nodes in an orbital minus one.) This classification is general, depending only on the spherical symmetry of the... 

This can be shown by writing down only those electronic arrangements of m and ms which do not violate the Pauli exclusion principle. For p electrons, the subsidiary quantum number 1=1, and the magnetic quantum number m may have values from + / 0 /, giving in this case values of m = + 1, 0 and -1. There are 15 possible combinations. 

All but one of the lanthanide ions show absorption in the Visible or near UV region. The exception is Lu3+ which has a full/shell. In the spectra of lanthanides spin orbit coupling is more important than crystal field splitting. The colours are due to Laporte-forbidden f-f transition i. e., transitions between the J states of 4n configuration since the change in the subsidiary quantum number is zero. The forbitals are... 

Certain features of the Bohr theory are still appropriate to the wave-niechanical picture in particular, the orbits can still be grouped in shells characterized by a principal quantum number n, and the maximum permissible number of electrons in any shell is still 2n2. Not all the electrons in one shell, however, are identical, for those in a shell of principal quantum number n are distributed over n sub-shells characterized by an azimuthal or subsidiary quantum number Z, which can assume any of the values o, 1,..., (n— 1). Electrons in sub-shells with l = o, 1, 2, 3 are commonly termed s,p, /electrons, respectively, and the state of an electron in respect of its principal and subsidiary quantum numbers is usually symbolized by a figure representing n followed by a letter representing Z. Thus is is an electron in the K shell with / = 0... 

The subsidiary quantum number, l. This determines the sub-shell within a given shell. For a shell of principal quantum number n there are n sub-shells corresponding to values of l from o to (tz — 1). 

The magnetic quantum number, m. This determines the atomic orbital within a given sub-shell. For a sub-shell of subsidiary quantum number / there are (2Z+1) orbitals. 

Table 2.01. Electron distribution as a function of principal and subsidiary quantum numbers...

We have already explained that the single orbital of the s subshell, of subsidiary quantum number l = 0, corresponds to a distribution of electron density which is spherical in the sense that there is the same probability of finding an electron in any direction in space (fig. 2.01 2). 

The five d and seven/orbitals are also unsymmetrical figures. Their exact form is a matter of less immediate concern to us than the fact that, in common with the p orbitals, they represent a directed distribution of electron density. As we shall shortly see, this spatial direction of the orbitals plays a very important part when we come to consider the interatomic binding forces between atoms, and we must remember that it is determined by the subsidiary quantum number l. 

For a given principal and subsidiary quantum number the energy is the same for all the possible atomic orbitals. Thus the energy of the three p orbitals is the same for any one value of n similarly the energy of the five d orbitals or of the seven / orbitals is the same. (This is implicit in the figure only inasmuch as the different orbitals with the same principal and subsidiary quantum numbers are represented by a single line.)... 

Sommerfeld s work required the introduction of a second quantum number which was denoted by the letter k it was known as the subsidiary quantum number. The value of k is related to the geometry of an ellipse, and the ratio n/k, where n title,.grincipal quantum number, is equal to the ratio of the semi-major to the semi-minor axes of the ellipse. 

Once we accept the possibility of an elliptical orbit, we must take into account a change in the velocity of an electron undertaking such an orbit relative to a circular one. Relativity theory predicts an associated change in mass for the electron, and as a result the energy of an electron describing an ellipse of say, principal quantum number == 3, subsidiary quantum number k— 1, will be different to that of an electron in an orbit for which n—3 and k 2. 

The second quantum number required by the wave theory replaces the subsidiary quantum number of the older theory. It is known as the azi-v muthal quantum number, and it is denoted by the letter /. It can take the values 0,1,2, etc., and they equal the old k values minus one in each case, v The existence of a magnetic quantum number is also required by the wave equation, but again, it arises as a fundamental necessity. It is denoted by the letter m, and it can take the values 0, 1, 2 etc. 

Spin quantum number, 8 Subsidiary quantum number, 7 Sulphur compounds, 48,49... 

In a (non-Coulomb) central field of force the motion depends, according to 21, on the subsidiary quantum number k in addition to the principal quantum number n. k has a simple mechanical significance, being in fact the total angular momentum of the electron measured in units of hfiir. 

In the normal state, hydrogen (1 H) has an electron in an orbit with the principal quantum number 1. As long as the orbit is regarded as an exact Kepler ellipse the subsidiary quantum number is undetermined. We shall see, however, on taking into account the relativity theory in 33, that the total angular momentum is also to be fixed by a quantum condition, without thereby appreciably altering the energy. The normal orbit of the electron is thus a lj-orbit. 

In the absence of external disturbances, only those terms combine, according to the correspondence principle ( 17), for which the subsidiary quantum number k differs by 1. The line series whose limiting term is n = l (in H the Lyman series) consists of single lines the line series having the limiting term n=2 (in H the Balmer series) consists of triplets, the lines of the remaining series show a still more complex character. 

Azimuthal (or Subsidiary) Quantum Number, 1 This quantum number describes the shape of the sub-orbital. Each principal level is divided into sub-shells, which are designated by this number. Magnetic Quantum Number, m This quantum number is used to indicate the direction of the sub-orbital in space. 

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