{"id":129,"date":"2018-11-14T12:19:06","date_gmt":"2018-11-14T12:19:06","guid":{"rendered":"http:\/\/phyp10.epgpbooks.inflibnet.ac.in\/?post_type=chapter&p=129"},"modified":"2019-04-30T06:55:52","modified_gmt":"2019-04-30T06:55:52","slug":"atoms-in-external-fields","status":"publish","type":"chapter","link":"https:\/\/ebooks.inflibnet.ac.in\/phyp10\/chapter\/atoms-in-external-fields\/","title":{"rendered":"Atoms in External fields"},"content":{"raw":"
\"epgp<\/a>\r\n<\/span><\/div>\r\n
\r\n\r\n \r\n\r\n1.\u00a0<\/strong>Zeeman Effect<\/strong>\r\n\r\n \r\n

Zeeman effect is named after the great scientist P.Zeeman who in1876 observed that when a light source is brought into a magnetic field, each spectral line is splitted into number of components. This implies that the energy levels of the atom, those are involved in the transition, in the presence of a magnetic field, must split into several components. Thus the interaction of the electronic magnetic moment of the atom with the magnetic field results in splitting.<\/p>\r\n \r\n

It is known that the ratio of magnetic and mechanical moment of an electron in an orbit is<\/p>\r\n\"\"\r\n

Also, the electron also has orbital angular momentum l and a spin angular momentum s. The ratio of magnetic and mechanical moment for the spinning electron is<\/p>\r\n\"\"\r\n\r\nL-S coupling<\/strong>\r\n\r\n \r\n

It is assumed that the interaction between the spin angular momentum of the electrons on one hand, the interaction between their orbital motions on the other hand is large compared with the interaction between spin and orbital angular momenta of each electron and thus LS coupling holds (in case the atom has more than one valence electron).<\/p>\r\n \r\n\r\nThe magnetic moment \u00b5L is\r\n\r\n\"\"\r\n\r\n<\/div>\r\n

\r\n\r\n \r\n

L is the total orbital angular momentum of all electrons. The magnetic moment \u00b5s then<\/p>\r\n\"\"\r\n\r\n \r\n\r\nS <\/strong>is the total spin angular momentum.\r\n\r\n \r\n

The vector L<\/strong> and S<\/strong> precess together around their resultant J<\/strong> in the absence of a magnetic field.<\/p>\r\n \r\n

When a magnetic field B<\/strong> is applied, L<\/strong> and S<\/strong> couple with it and in the absence of coupling between L<\/strong> and S<\/strong>, the latter precess independently around B<\/strong>.<\/p>\r\n \r\n

However, energy corresponding to coupling of L<\/strong> and S<\/strong> with B<\/strong> is smaller spin-orbit interaction energy, in weak magnetic field. Thus, B<\/strong> does not perturb the coupling between L and S under such condition. The L<\/strong> and S<\/strong> precess about their resultant J<\/strong>. Because of torque, J<\/strong> precesses around B<\/strong> at a rate small compared with the precession rate of L<\/strong> and S<\/strong> about J<\/strong>. The figure depicts the Precession of J about the field direction z in a weak magnetic field B<\/p>\r\n\r\n<\/div>\r\n\"\"\r\n

\r\n

The total electronic magnetic moment \u00b5 = \u00b5<\/strong>L<\/strong> + \u00b5<\/strong>sof the atom is not oriented in the same direction as the total angular momentum J = L + S.<\/strong> This is because of different dependences of \u00b5<\/strong>L and \u00b5<\/strong>s on L<\/strong> and S<\/strong>, respectively. The figure illustrate vectors L,S<\/strong> and J<\/strong> and the associated magnetic moment vectors.<\/p>\r\n \r\n\r\n \r\n\r\n\"\"\r\n\r\n<\/div>\r\n

\r\n

As L<\/strong> and S<\/strong> precess rapidly about J<\/strong>, \u00b5<\/strong>L and \u00b5<\/strong>s precess rapidly as well, resulting \u00b5<\/strong> to zero and the component parallel to \u2013J<\/strong> remains a constant in magnitude \u00b5<\/strong>J<\/strong>. The component of \u00b5 along \u2013J<\/strong> axis from the above figure is<\/p>\r\n \r\n

\u00b5J = \u00b5Lcos\u00a0\u00a0\u00a0 + \u00b5S cos<\/p>\r\n \r\n\r\nwherer J =L + S<\/strong> or S = J<\/strong> \u2013L<\/strong>\r\n\r\n\"\"\r\n\r\n<\/div>\r\n

\r\n\r\nThis is called Lande\u2019s splitting factor\r\n\r\n \r\n\r\njj coupling<\/strong>\r\n\r\n \r\n

In case where interaction between spin and orbital motion of the electron is large compared with the interaction between the spin angular momenta of the electrons on one hand and the interaction between their orbital motions on the other hand, jj coupling holds and hence the s and l of each electron are coupled together to form their own resultant j.<\/p>\r\n \r\n

The figure presents the vector model for jj-coupling in a weak magnetic field. j<\/strong>1 and j<\/strong>2 couples together to give the resultant J<\/strong><\/p>\r\n \r\n\r\n\"\"\r\n\r\n\"\"\r\n\r\n<\/div>\r\n

\r\n\r\nThe angles between j<\/strong>1, j<\/strong>2 and J<\/strong> are constant and J<\/strong> = j<\/strong>1 + j<\/strong>2 or J<\/strong> \u2013 j<\/strong>1 = j<\/strong>2. Taking dot product of j<\/strong>2 with j<\/strong>2\r\n\r\n\"\"\r\n\r\nSimilarly,\r\n\r\n\"\"\r\n\r\n<\/div>\r\n
\r\n\r\nSubstituting\r\n\r\n\"\"\r\n

The magnetic interaction energy resulting from the interaction between the electronic magnetic moment of the atom and an external magnetic field B<\/strong> (directed along the z axis) is<\/p>\r\n

\u2206E = -\u00b5<\/strong>J<\/sub>.<\/strong>B<\/strong><\/p>\r\n \r\n

Here g is chosen depending whether the coupling is LS or jj. J cos (JB) is the projection of J<\/strong> on B<\/strong> that is equal to Jz. The allowed values of Jz are M\u0127 (M being the magnetic quantum number takes values from + J<\/strong> to \u2013 J<\/strong>, a total of 2J +1 value). So,<\/p>\r\n\"\"\r\n

This clearly mentions that the magnetic field lifts the degeneracy giving rise to 2J+\u00a0\u00a0 1 equidistant sublevels those are called Zeeman sublevels. The distance between two consecutive sublevels is gBe<\/sub>B. Let E0<\/sub> is the energy of the atom without magnetic field, and then the energy in the magnetic field is<\/p>\r\n \r\n\r\n\"\"\r\n\r\n \r\n

It is known that there is a relationship between term value T and energy E i.e. T = -E\/hc. The interaction energy in wave-numbers becomes<\/p>\r\n\r\n<\/div>\r\n\"\"\r\n

\r\n

Where L = Be<\/sub> B\/hc is called Lorentz unit and its value is 0.467B cm-1<\/sup> when B<\/strong> is measured in Webers per square metre (tesla). The separation between two neighbouring sublevels is gL and is determined by magnetic field B<\/strong> and g-factor belonging to the energy level.<\/p>\r\n \r\n\r\nExamples\r\n\r\n\"\"\r\n\r\n<\/div>\r\n

\r\n\r\n \r\n

\u2206M = 0 (p component or \u03c0 component)<\/p>\r\n

\u2206M = \u00b11 (s component or component)<\/p>\r\n

\u2206J = 0, with M =0 \u2192 M = 0 is not allowed<\/p>\r\n \r\n\r\nThe 2<\/strong><\/sup>D5\/2<\/strong><\/sub> \u21922<\/strong><\/sup>P3\/2<\/strong><\/sub>transition spits up into twelve components ( four p components and eight s components) .\r\n\r\n \r\n

Example 2. <\/strong>Consider the transition between 1<\/sup>F3<\/sub> \u2013 1<\/sup>D2<\/sub>. As 1<\/sup>F3<\/sub> \u2013 1<\/sup>D2<\/sub>are singlet, therefore their g values are equal to 1. Fig. shows the splitting of the levels in weak magnetic field and allowed transition between them.<\/p>\r\n\r\n<\/div>\r\n\"\"The allowed transitions from <\/span>1<\/sup>F<\/span>3<\/sub> \u2013 <\/span>1<\/sup>D<\/span>2<\/sub>in weak magnetic field are shown in the figure.<\/span>\r\n

\r\n\r\n\"\"\r\n

This indicates that there are only three distinct energies at A+a, A and A-a. Thus the spectral line corresponding to 1<\/sup>F3<\/sub> \u2013 1<\/sup>D2<\/sub>transition splits into three components in the weak magnetic field, one at the same position and other two are displaced on either side of undisplaced line.<\/p>\r\n \r\n

Now it is clear that when levels involved in a transition are singlet, only three lines are observed corresponding to normal Zeeman effect.<\/strong><\/p>\r\n \r\n

If the levels in a transition involved are different from singlet, the anomalous Zeeman effect is observed.<\/p>\r\n \r\n

Conclusevly, when the contribution of spin towards magnetic moment is zero, the normal Zeeman effect is observed and the spin contribution is taken in account, anomalous Zeeman effect is observed.<\/p>\r\n \r\n

Representation of transitions between Zeeman sublevels on a normal energy level diagram can be understood with following arrangement.<\/p>\r\n \r\n

Write in a row all the values of M, for 3<\/sup>P3\/2<\/sub> ---------2<\/sup>S1\/2<\/sub> transition. Below each value of M put the displacement Mg of the magnetic level of the initial spectral term. Similarly write down the displacement of the levels belonging to the final term in the next row. The vertical arrows represent the transitions \u2206M = 0 (\u03c0 or p) Forbidden Transition is represented by arrows of greater inclination. The line positions are expressed as multiples of 1\/r, where r is called Runge denominator and is the least common multiple of the denominator in the values of Mg. The usual notation for a Zeeman pattern consists of a long line with Runge denominator beneath it with integer above it to show at what multiple of 1\/r the components lie. The table demonstrates the procedure for obtaining the Zeeman pattern for 2<\/sup>P3\/2<\/sub>\u21922<\/sup>S1\/2<\/sub> transition.<\/span><\/p>\r\n \r\n

The g values for 2<\/sup>P3\/2<\/sub>and 2<\/sup>S1\/2<\/sub>are 4\/3 and 2, respectively. The vertical differences ( pi or p component) are 3\/6 and -3\/6 while the diagonal difference ( sigma or s components) are 6\/6, 5\/6, -5\/6 and -6\/6. In Runge notation it is written as<\/span><\/p>\r\n\r\n<\/div>\r\n

\r\n\r\n\"\"\r\n\r\n \r\n

With two p components being set in parentheses, followed by the four s components.<\/p>\r\n \r\n\r\nIntensity rules<\/strong>\r\n\r\n \r\n

(i) The intensities of the components of one line are symmetrical relative to the position of the original line.<\/p>\r\n

(ii) The sum of the intensities of combinations of a level characterized by the magnetic quantum number M with level M-1, M and M+1 is independent of M.<\/p>\r\n

(iii)Sum of the intensities of all p components is equal to the sum of the intensities of all s components<\/p>\r\n\r\n<\/div>\r\n\"\"\r\n

\r\n

When Zeeman Effect is observed in a direction perpendicular to the magnetic field, only half of the intensity of the s component is observed. The other half is observed in a direction parallel to the magnetic field. Therefore, in studying intensities s components need to be multiplied by 2. A and B are constants in the above table that need not be determined for relative intensities within each Zeeman pattern.<\/p>\r\n \r\n

The table holds for any coupling scheme. To determine the strongest component the following approximate rule is useful. In case where J1\u2260 J2, the vertical differences in the middle of the scheme and diagonal differences at the ends give, respectively, the strongest p and s components. If J1 = J2,the vertical differences at the end of the scheme and the diagonal differences at the centre give, respectively the strongest p and s components with the restriction that M = 0 to M = 0 is forbidden.<\/p>\r\n\r\n<\/div>\r\n\"\"\r\n\r\n \r\n

\r\n\r\n \r\n\r\n2.\u00a0<\/strong>Paschen Back Effect<\/strong>\r\n\r\n \r\n

In describing the anomalous Zeeman Effect, it is assumed that external magnetic field is weak compared to the internal fields; consequently the interaction between J and H is weak compared to the interaction of orbital and spin magnetic fields. The interaction between l and s starts loosening or gets broken at sufficiently large value if the strength of the external magnetic field is increased to the extent that it starts competing with the internal field.<\/p>\r\n \r\n

Under this condition J starts losing its significance and l<\/em> & s<\/em> interact independently with the external magnetic field resulting in the independent precessions of l<\/em> & s<\/em> about J; their precession is much faster than the precession of residual interaction of l<\/em> & s about j. J processes slowly around H<\/strong> in comparison to l and s about J. This is known as Paschen-Back Effect. The separation between the spin components or between the Zeeman components is a measure of the corresponding processional frequency.<\/p>\r\n \r\n

In the anomalous Zeeman Effect, the precession of l & s<\/em> about j is faster and therefore the average values of their components normal to j are assumed as zero in order to avoid the perturbation of other precessions.<\/p>\r\n \r\n

However, when the external magnetic field is of the same order as that of the internal fields, the case is different and the relation -\u2206T=<\/em> gLm<\/em>j<\/em> no longer holds good. Further, l<\/em> & s will precess independently about H<\/strong> as the field is increased and hence will become quantized independently in the direction of field H. The\u00a0figure depicts the precession of <\/span>l & s<\/em> about external magnetic field H and their space quantization.<\/span><\/p>\r\n\r\n<\/div>\r\n

\r\n\r\n\"\"\r\n\r\n \r\n

As each of the electrons takes each of the two values of m<\/em>s<\/em> there are 2(2l<\/em>+1) different quantum states; each corresponding to different combination of quantum numbers. The figure depicts space quantization for l<\/em>=1 and s=1\/2 . When the ls<\/em>-interaction is significantly loosened or broken, major part of the total energy of the atom consists of the energies due to the precession of l<\/em> around H plus energy due to precession of s about H<\/em>. Thus, the major energy shift is<\/p>\r\n\"\"\r\n\r\n \r\n

Using equations for \u03bcl and \u03bcs respectively and substituting the values of cos(l,H) and cos(s,H) , the equation (1) reduces to<\/p>\r\n \r\n\r\n\"\"\r\n\r\n<\/div>\r\n

In terms of wave numbers, -\u2206T=(ml<\/em><\/sub>+2ms<\/sub>)Lcm-1---------------[3]<\/p>\r\n \r\n

Though <\/span>l<\/em> & s precess independently, but each produces magnetic field on the electron causing some perturbation to other\u2019s motion.<\/span><\/p>\r\n \r\n

Though, the <\/span>l<\/em>s- interaction is small compared to the effect of the external magnetic field but need to be taken into consideration. The separation due to this residual interaction is of the same order of magnitude as the field free fine structure doublet separations. Therefore, using equation 3 and that the j is vector sum of l and s, its contribution can be written as<\/span><\/p>\r\n\"\"\r\n

In field free l,s interaction, l and s rotate together as a rigid system about j thereby the angle between <\/span>l<\/em> and s is constant rendering easy evaluation of cos(l,s). In the present case angle between l and s is varying continuously because of the anomalous behavior of spin (s precess faster than l). This necessitate the use of average value of cos(l,s) that can be evaluated using trigonometry theorem, viz cos(l,s)=cos(l,H)\u00d7cos(s,H). Using this theorem along with the values of cos(l,H) and cos(s,H) from the figure (1), one finds<\/span><\/p>\r\n

\"\"Paschen Back effect of Sodium D<\/strong>1<\/strong> & D<\/strong>2<\/strong> lines<\/strong>: quantum numbers for the states involved in the transitions exhibiting Paschen Back effect are as:<\/span><\/p>\r\n\r\n

\r\n\r\n\"\"\r\n\r\n\"\"\r\n\r\n<\/div>\r\n
\r\n\r\n\"\"\r\n\r\n \r\n

Fig. 2. <\/strong>Transitions showing Paschen Back effect in the absence of ls-interaction.<\/p>\r\n

The effect of the external magnetic field (weak to strong with respect to internal magnetic field) on the spectrum can be summarized as: as long as the external magnetic field is unable to perturb the inner precession, one gets anomalous Zeeman spectrum. On the other side, if the strength of the external field yields the magnetic resolution<\/p>\r\n\"\"\r\n\r\n \r\n

Fig. 3. <\/strong>Transition from anomalous Zeeman effect to Paschen Back effect.<\/p>\r\n \r\n

more than the spin \u2013orbit fine structure, that is normal Zeeman triplet, shown in\u00a0the fig. 3 (transition of interaction for WEAK to STRONG field). In a way, these are the two extreme situations; what about the intermediate fields, i.e. during the transition from relatively weak to strong external field? Usually this transitional zone is referred to as the Paschen Back effect.<\/span><\/p>\r\n \r\n

It is noteworthy that the Paschen Back spectrum is not as simple as discussed; it is somewhat complicated and separation between the various magnetic field components is a function of external magnetic field.<\/span><\/p>\r\n\"\"\r\n\r\n<\/div>\r\n

\r\n\r\n\"\"\r\n\r\n\"\"\r\n\r\n<\/div>\r\n
\r\n\r\n \r\n\r\nNOTE:\r\n\r\n \r\n

It is noteworthy that the weak or Strong external magneticfField is not in absolute term but is relative to the net internal<\/p>\r\n \r\n

A week external field for one atom could be strong for some other atom. Magnetic field of 30k Gauss produces Zeeman separation of 2 cm -1<\/sup> in sodium; while the ls<\/em> interaction yields J1\/2<\/sub> and J3\/2<\/sub> with a separation of 17.8 cm-1<\/sup> (a measure of internal field) indicating that internal field is much stronger than the external field. On the other hand, same magnetic field applied to H-atom, Zeeman separation (measure of external field) is much larger than the ls<\/em> interaction separation (measure of internal\u00a0field) that is less than half a wavenumber. Thus the field which is weak in case of sodium is strong for H-atom.<\/span><\/p>\r\n \r\n

3.<\/strong>The Stark Effect<\/strong><\/p>\r\n \r\n

When a source of light is placed in an electric field, the spectral lines split up into number of components. The Stark effect is due to the interaction between the electric moment of the atom and the external electric field. The term to the interaction energy is,<\/span><\/p>\r\n\r\n<\/div>\r\n

\r\n

W = -p. E,<\/p>\r\n \r\n

Here p is the electric dipole moment and E is the external field.<\/p>\r\n \r\n

Electric dipole moment in atoms arises as a consequence of the way that charge is distributed within atoms. There exist two aspects of the Stark effect:<\/p>\r\n \r\n