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Transcript
Chapter 7 Atoms in a magnetic field In chapter 6, we have seen that the angular momentum or the associated magnetic moment of atoms precess in the presence of magnetic field, and it is directional quantised in z direction. The interaction energy between the magnetic field and the magnetic moment of the electrons in an atom leads to a splitting of the energy terms, which is described by the different possible values of the magnetic quantum number. This energy splitting can be determined in the treatment of the Stern-Gerlach experiments, and other types of experiments: electron spin resonance (ESR), nuclear magnetic resonance (NMR), the Zeeman effect, and the Paschen-Back effect. Electron spin resonance Electron spin resonance abbreviated ESR, and sometimes EPR for electron paramagnetic resonance. The method involves the production of transitions between energy states of the electrons which are characterised by different values of the magnetic quantum number ms. The spin of an electron has two B0 possible orientations in an applied ms = 1/2 magnetic field. They correspond to two values of the potential energy. 0 s s( s 1) B g s s , z ms B g s 12 g s B ms = -1/2 The difference of the potential energy of these two orientations: E g s B B0 The transition frequencies are usually in the range of microwave frequencies depending on the strength of the applied magnetic field. If a sinusoidally varying magnetic field B1 = B1sint is applied in a direction perpendicular to B0, transitions between the two states are induced if the frequency = /2 fulfils the condition: E h g s B B0 Where = 2.8026 × 1010 B0 Hz(tesla)-1 The frequency depends on the choice of the applied magnetic field B0. For reasons of sensitivity, usually the highest possible frequencies are used, corresponding to the highest possible magnetic fields. The fields and frequencies used in practice are limited by questions of technical feasibility. Usually, fields in the range 0.1 to 1T are chosen. This leads to frequencies in the GHz region (centimeter waves). Electron spin resonance Schematic representation of the experimental setup. The sample is located in a resonant cavity between the pole pieces of an electromagnet. The microwaves are generated by a klystron and detected by a diode. To increase the sensitivity of detection, the field B0 is modulated. Below-left: energy states of a free electron as functions of the applied magnetic field. Below-right: signal U from the diode as a function of Bo for resonance. ESR spectrometers count as standard spectroscopic accessories in many physical and chemical laboratories. For technical reasons, usually a fixed frequency is used in the spectrometers. The magnetic field is varied to fulfil the resonance condition and obtain ESR transitions. A frequently used wavelength is 3cm (the so-called X-band). ESR is utilised for: Precision determinations of the gyromagnetic ratio and the g factor of the electron; Measurement of the g factor of atoms in the ground state and in excited states for the purpose of analysing the term diagram; The study of various kinds of paramagnetic states and centers in solid state physics and in chemistry: molecular radicals, conduction electrons, paramagnetic ions in ionic and metallic crystals, colour centers. Electron spin resonance was observed for the first time in 1944 by the Russian physicist Zavoisky. The analogous spin resonance of paramagnetic atomic nuclei is seen under otherwise identical conditions at a frequency which is 3 orders of magnitude smaller, due to the fact that nuclear moments are about a factor of 1000 smaller than atomic magnetic moments; the corresponding frequencies are in the radio frequency region. This nuclear magnetic resonance (NMR) was observed in the solid state for the first time in 1946 by Bloch and Purcell, nearly 10 years after it had first been used by Rabi to measure the gyromagnetic ratio of nuclei in gas atoms. NMR (Nuclear Magnetic Resonance) and its applications in medical diagnosis, atomic clock and chemical shift Hyperfine structure The presence of the hyperfine structure was first postulated by Pauli in 1924 as means of explaining spectroscopic observations. In the year 1934, Schüller further postulated the existence of electric quadrupole moments in nuclei. Similar to electron, the atomic nuclei also exists spins I and magnetic moments µI. The interactions of these nuclear moments with the electron leads to an additional splitting of the spectral lines, which is the hyperfine structure. Comparing to fine structure, the splitting resulting from hyperfine structure is much smaller, the measurement of which generally requires an especially high resolution. The angular momentum and magnetic moment of the atomic nuclei: I I ( I 1), gI N I I Where the quantum number I may be integral or half-integral; the nuclear magneton µN = eħ/2mp is the unit of the nuclear magnetic moment. The spin of atomic nuclei will directional quantised in the presence of external magnetic field. The z component of the nuclear magnetic moment: I , z mI g I N Where the quantum number mI = -I, -I+1, …, I, has 2I+1 possibility. The maximum observable value of µI is: I max Ig I N For example: for the hydrogen nucleus, the proton: µI(1H) = +2.79µN; I = ½; gI = 5.58; For the potassium nucleus with mass number 40: µI(40K) = -1.29µN; I = 4; gI = -0.32 There are also numberous nuclei with vanishing spins I = 0. These nuclei do not contribute to the hyperfine structure. Examples of this type of nucleus are: 4 2 40 56 88 188 208 238 He,126C ,168O, 20 Ca, 26 Fe,38 Sr ,144 Cd , Hf , Pb , 48 72 82 92 U Nuclear magnetic resonance Energy levels for a nucleus with spin quantum number I No field Applied magnetic field mI = -1/2 energy Resonance frequency: = E / h E mI = 1/2 gI N hB E , 2 e N , 2 mP Nuclear magneton B mP 1836 N me A magnetic dipole moment (usually just called “magnetic moment”) in a Magnetic field will have a potential energy related to its orientation with Respect to that field. 2e B 2 2 12 (5.79 105 eV / T )(1T ) electron spin 1.7608 1011 s 1 16 6.58 10 eV s 28.025GHz Larmor frequency 2 2 p B 2 2.79 (3.15 108 eV / T )(1T ) 8 1 proton spin 2 . 6753 10 s 16 6.58 10 eV s 42.5781MHz Larmor frequency 2 The Larmor frequency Larmor/B Particle Spin /B s-1T-1 Electron 1/2 1.7608 x 1011 28.025 GHz/T Proton 1/2 2.6753 x 108 42.5781 MHz/T Deuteron 1 0.4107 x 108 6.5357 MHz/T 1/2 3/2 1/2 1 1.8326 x 108 29.1667 MHz/T 0.7076 x 108 11.2618 MHz/T 1.0829 x 108 17.2349 MHz/T 0.1935 x 108 3.08 MHz/T 13C 1/2 0.6729 x 108 10.71 MHz/T 19F 1/2 2.518 x 108 40.08 MHz/T Neutron 23Na 31P 14N The Larmor frequency of the electron spin is in the microwave region, the Larmor frequency of proton or other nucleus is three orders smaller. Diagram of a simple nuclear spin resonance apparatus Fixed magnetic field B0 The radio-frequency field B1 In 1946, Purcell and Bloch showed both experimentally and theoretically that the precessional motion of the nuclear spin is largely independent of the translational and rotational motion of the nucleus, and that the method of NMR can be applied not only to free atoms, but to atomic nuclei in liquids and solids. The medical application — Magnetic Resonance Imaging (MRI) Proton nuclear magnetic resonance detects the presence of Hydrogens (protons) by subjecting them to a large magnetic field to partially polarize the nuclear spins, then exciting the spins with properly tuned radio frequency (RF) radiation, and then detecting weak RF radiation from them as they “relax” from this magnetic interaction. The frequency of this proton “signal” is proportional to the magnetic field to which they are subjected during this relaxation process. an MRI image of a cross-section of tissue can be made by producing a wellcalibrated magnetic field gradient across the tissue so that a certain value of magnetic field can be associated with a given location in the tissue. Since the proton signal frequency is proportional to that magnetic field, a given proton signal frequency can be assigned to a location in the tissue. This provides the information to map the tissue in terms of the protons present there. Since the proton density varies with the type of tissue, a certain amount of contrast is achieved to image the organs and other tissue variations in the subject tissue. MRI image Since the MRI uses proton NMR, it images the concentration of protons. Many of those protons are the protons in water, so MRI is particularly well suited for the imaging of soft tissue, like the brain, eyes, and other soft tissue structures in the head as shown above. The bone of the skull doesn't have many protons, so it shows up dark. Also the sinus cavities image as a dark region. Bushong's assessment is that about 80% of the body's atoms are hydrogen atoms, so most parts of the body have an abundance of sources for the hydrogen NMR signals which make up the magnetic resonance image. The setup of MRI Be possible to image soft tissues: Joint, brain, and spinal cord normal knub The gradient magnetic field Two-dimensional map of the proton density A rotating field gradient is used, linear positioning information is collected along a number of different directions. That information can be combined to produce a two-dimensional map of the proton densities. The proton NMR signals are quite sensitive to differences in proton content that are characteristic of different kinds of tissue. Even though the spatial resolution of MRI is not as great as a conventional x-ray film, its contrast resolution is much better for tissue. Rapid scanning and computer reconstruction give well-resolved images of organs. the applications of MRI: In 1999, In 2002, 2,170 in all of the world 22,300 and 500 in china (also several in Jinan) Usage, ~ 60 million times in 2000 Cost: ~ 160$ Advantage: non-invasive, non-ionising radiation, and a high soft-tissue resolution and discrimination in any imaging plane. Chemical applications of NMR The frequency detected in NMR spectroscopy is proportional to the magnetic field applied to the nucleus. This would be a precisely determined frequency if the only magnetic field acting on the nucleus was the externally applied field. But the response of the atomic electrons to that externally applied magnetic field is such that their motions produce a small magnetic field at the nucleus which usually acts in opposition to the externally applied field. This change in the effective field on the nuclear spin causes the NMR signal frequency to shift. The magnitude of the shift depends upon the type of nucleus and the details of the electron motion in the nearby atoms and molecules. It is called a "chemical shift". The precision of NMR spectroscopy allows this chemical shift to be measured, and the study of chemical shifts has produced a large store of information about the chemical bonds and the structure of molecules. Chemical shift in NMR spectra The effective magnetic field at the nucleus can be expressed in terms of the externally applied field B0 by the expression: B B0 (1 s ) where s is called the shielding factor or screening factor. The factor s is small - typically 10-5 for protons and <10-3 for other nuclei (Becker). In practice the chemical shift is usually indicated by a symbol d which is defined in terms of a standard reference. (vS vR ) 10 d vR 6 quoted as ppm Chemical shift The signal shift is very small, parts per million, but the great precision with which frequencies can be measured permits the determination of chemical shift to three or more significant figures. The reference material is often tetramethylsilane, Si(CH3)4, abbreviated TMS. Since the signal frequency is related to the shielding by B0 (1 s ) 2 gyromagnetic ratio the chemical shift can also be expressed as: (s R s S ) d 106 (s R s S ) 106 1s R To determine the chemical bonding A sample of a chemical Shift spectrum which is a proton spectrum. The high-resolution peaks Can be identified with the functional groups in the radicals: d=1.23, (CH3)2; 2.16, CH3C=O; 2.62, CH2; 4.12, OH The Caesium(Cs) atomic clock — as a time and frequency standard Cs has a nuclear spin I=7/2, and in the atomic ground state, the angular momenta of the electrons J=1/2; The total angular momenta F=4 and F=3 The transition frequency used for the Cs atomic clock corresponds to the transition between the states: F=3, mF=0, and F=4, mF=0 A portion of the term scheme of the Cs atom in the ground state as a function of a weak applied magnetic field B0. The atomic beam resonance method of Rabi (1937) Data curve from the Rabi atomic beam resonance experiment. The intensity at the detector is at a minimum when the homogeneous field B0 of magnet C fulfils the resonance condition. Zeeman effect A splitting of the energy terms of atoms in a magnetic field can be observed as a splitting of the frequencies of transitions in the optical spectra (or as a shift). A splitting of this type of spectral lines in a magnetic field was observed for the first time in 1896 by Zeeman. The effect is small. Spectral apparatus of very high resolution is required. These are either diffraction grating spectrometers with long focal lengths and a large number of lines per cm in the grating, or else interference spectrometers, mainly Fabry-Perot interferometers. With a Fabry-Perot interferometers or with a grating spectrometer of sufficient resolution, the splitting in magnetic fields may be quantitatively measured. Fabry-Perot Interferometer This interferometer makes use of multiple reflections between two closely spaced partially silvered surfaces. Part of the light is transmitted each time the light reaches the second surface, resulting in multiple offset beams which can interfere with each other. The large number of interfering rays produces an interferometer with extremely high resolution (106), somewhat like the multiple slits of a diffraction grating increase its resolution. Ordinary Zeeman effect Without Magnetic field With magnetic field Transverse observation EB0 E⁄⁄B0 EB0 With magnetic field Longitudinal observation EB0 , circular Ordinary Zeeman effect for the atomic Cd line at λ = 6438. With transverse observation the original line and two symmetrically shifted components are seen. Under longitudinal observation, only the split components are seen. Transverse and longitudinal observation of emission spectral lines in a magnetic field. S is the entrance slit of the spectrometer. Anomalous Zeeman effect D1 D2 Without magnetic field With magnetic field The D lines of sodium. The D1 line splits into four components, the D2 line into six in a magnetic field. The wavelengths of the D1 and D2 lines are 5896 and 5889 ; the quantum energy increases to the right in the diagram. The Zeeman effect results from the splitting of energy states with the interaction of the resultant angular momentum and external magnetic fields. If the resultant angular momentum is composed of both spin and orbital angular momentum, one speaks of the anomalous Zeeman effect. The normal Zeeman effect describes states in which no spin magnetism occurs, therefore with pure orbital angular momentum. In these states, at least two electrons contribute in such a way that their spins are coupled to zero. Therefore, the normal Zeeman effect is found only for states involving several (at least two) electrons. Explanation of the Zeeman effect from the standpoint of classical electron theory The ordinary Zeeman effect may be understood to a large extent using classical electron theory, as it was shown by Lorentz shortly after its discovery. In the model, the emission of light by an electron whose motion about the nucleus is interpreted as an oscillation. The radiation electron is treated as the electron by three component oscillators according to the rules of vector addition: component oscillator 1 oscillators linearly, parallel to the direction of B0; oscillators 2 and 3 oscillate circularly in opposite senses and in a plane perpendicular to the direction of B0. This resolution into components is allowed, since any linear oscillation may be represented by the addition of two counterrotating circular ones. An oscillating electron is resolved into three component oscillators e- oscillator Component 2 and 3 Component 1 B0 2 3 Without the magnetic field B0, the frequency of all the component oscillators is equal to that of the original electron, namely 0. With the field B0: component 1, parallel to B0, experiences no force. Its frequency remains unchanged. It emits light which is linearly polarised with its E vector parallel to the vector B0. The circularly oscillating components 2 and 3 are accelerated or slowed down by the effect of magnetic induction, depending on their direction of motion. Their circular frequencies are increased or decreased by an amount: e B d B0 B0 2m0 Calculation of the frequency shift for the component oscillators Without the applied magnetic field, the circular frequency of the component electrons is 0. The Coulomb force and the centrifugal force are in balance. In a homogeneous magnetic field B0 applied in the z direction, the Lorentz force acts in addition. In Cartesian coordinates, the following equations of motion are then valid: mx m02 x ey B0 0 my m02 y exB0 0 mz m02 z 0 For component 1, z = z0exp(i0t), the frequency remains unchanged. For component 2 and 3, we substitute u = x + iy and v = x – iy. The equations have the following solutions: u = u0exp[i(0 – eB0/2m)t] and v = v0exp[i(0 + eB0/2m)t] The component electron oscillators 2 and 3 thus emit or absorb circularly polarised light with the frequency 0 ± d. The frequency change has the magnitude: d e d B0 2 4m0 For a magnetic field strength B0 = 1T, this yield the value: d 1.4 1010 s 1 0.465cm 1 For each spectral line with a given magnetic field B0, the frequency shift d is the same. Theory and experiment agree completely. For the polarization of the Zeeman components, we find the following predictions: component electron oscillator 1 has the radiation characteristics of a Hertzian dipole oscillator, oscillating in a direction parallel to B0. In particular, the E vector of the emitted radiation oscillates, and the intensity of the radiation is zero in the emitted radiation oscillates parallel to B0. This corresponds exactly to the experimental results for the unshifted Zeeman component. It is also called the component ( for parallel). If the radiation from the component electron oscillators 2 and 3 is observed in the direction of B0, it is found to be circularly polarised; observed in the direction perpendicular to B0, it is linearly polarised. This is also in agreement with the results of the experiment. This radiation is called s+ and s– light, where s stands for perpendicular and the + and – signs for an increase and decrease of the frequency. The s+ light is right-circular polarised, the s– light is left-circular polarised. The direction is defined relative to the lines of the B0 field, not relative to the propagation direction of the light. Description of the ordinary Zeeman effect by the vector model Both ordinary and anomalous can be described by a complete quantum mechanical treatment, which we will not discuss here. For simplicity, we employ the vector model. The angular momentum vector j, and the magnetic moment µj, precess together around the field axis B0. The additional energy of the atom due to the magnetic field is then: Vm j ( j ) z B0 m j g j B B0 with m j j , j 1, j B0, z jz = mjħ µj j µj,z = mjgjµB Precession of j and µj about the direction of the applied field B0, j = l. The (2j+1)-fold directional degeneracy is lifted in the presence of the magnetic field, and then the term is split into 2j+1 components. These are energetically equidistant. The distance between two components with mj = 1 is E g j B B0 For the ordinary Zeeman effect, the spin S = 0 and consider only orbital magnetism. gj has a numerical value of 1. The frequency shift: e d B0 2m0 The magnitude of the splitting is thus the same as in classical theory. For optical transitions, the selection rule: mj = 0, ±1. From quantum theory one also obtains the result that the number of lines is always three: the ordinary Zeeman triplet. The splitting diagram for a cadmium line mj 2 1 0 -1 -2 1D 2 =6438 1P 1 0 -1 1 mj -1 0 1 s- s+ Splitting of the = 6438 line of the neutral Cd atom, transition 1P1 – 1D2, into three components. The spins of the two electrons are antiparallel and thus compensate each other, giving a total spin S = 0. The splitting is equal in each case because only orbital magnetism is involved. R. A. Beth in 1936 found that the circular polarised light quanta has not only the energy but also the angular momentum. s-, circular polarised photon l propagation s+, circular polarised photon l propagation Based on the conservation of the angular momentum for the system of electrons and light quanta: For mj = 0, the angular momentum of the system was not changed after the transition, the emitting light has no angular momentum, and it is thus linearly polarised, which is light. For mj = -1, the angular momentum of the system was changed -ħ after the transition, the emitting light has angular momentum -ħ, and it is thus circular polarised, which is s- light. For mj = +1, the angular momentum of the system was changed +ħ after the transition, the emitting light has angular momentum +ħ, and it is thus circular polarised, which is s+ light. The anomalous Zeeman effect In general case, the atomic magnetism is due to the superposition of spin and orbital magnetism, which results the anomalous Zeeman effect. The term “anomalous” Zeeman effect is historical, and is actually contradictory, because this is the normal case. In cases of the anomalous Zeeman effect, the two terms involved in the optical transition have different g factors, because the relative contributions of spin and orbital magnetism to the two states are different. The g factors are determined by the total angular momentum j and are therefore called gj factors. The splitting of the terms in the ground and excited states is therefore different, in contrast to the situation in the normal Zeeman effect. This produces a larger number of spectral lines. The relation between the angular momentum J, the magnetic moment µJ and their orientation with respect to the magnetic field B0 for strong spin-orbit coupling. The angular momentum vectors S and L combine to form J. J and uJ are not coincide. For the transitions of the Na D lines, three terms involved, namely the 2S1/2, the 2P1/2 and the 2P3/2, the magnetic moments in the direction of the field are ( j ) j , z m j g j B The magnetic energy is Vm j ( j ) j , z B0 The number of splitting components in the field is given by mj and is again 2j+1. The distance between the components with different values of mj – the so-called Zeeman components – is no longer the same for all terms, but depends on the quantum numbers l, s, and j: Em j ,m j1 g j B B0 Experimentally, it is found that gj = 2 for the ground state 2S1/2, 2/3 for the state 2P1/2 and 4/3 for the state 2P . For optical transitions, the selection rule is 3/2 again mj = 0, ±1. It yields 10 lines. D1 line 2P 2S 1/2 mj mjgj +1/2 -1/2 +1/3 -1/3 +1/2 +1 1/2 -1/2 s s D2 line 2P 2S 3/2 mj mjgj +3/2 +6/3 +1/2 -1/2 -3/2 +2/3 -2/3 -6/3 +1/2 +1 -1/2 -1 1/2 -1 ss ss Magnetic moments with spin-orbit coupling In anomalous Zeeman splitting, other values of gj than 1 or 2 are found. The gj factor links the magnitude of the magnetic moment of an atom to its total angular momentum. The magnetic moment is the vector sum of the orbital and spin magnetic moments j s l B ( gl l g s s ) The directions of the vectors µl and l are antiparallel, as are those of the vectors µs and s. In contrast, the directions of j and µj do not in general coincide. This is a result of the difference in the g factors for spin and orbital magnetism. The magnetic moment µj resulting from vector addition of µl and µs precesses around the total angular momentum vector j, the direction of which is fixed in space. Due to the strong coupling of the angular momenta, the precession is rapid. Therefore only the time average of its projection on j can be observed, since the other components cancel each other in time. This projection (µj)j precesses in turn around the B0 axis of the applied magnetic field B0. In the calculation of the magnetic contribution to the energy Vmj, the projection of µj on the j axis (µj): ( j ) j l cos(l , j ) s cos( s , j ) B l (l 1) cos(l , j ) 2 s( s 1) cos( s , j ) Vector model S 3 j ( j 1) s ( s 1) l (l 1) ( j ) j B 2 j ( j 1) L J gj L 1 2 J S 1 2 S ( J ) J j ( j 1) B The magnetic moment projected in j direction: ( j ) j g j B j / j ( j 1) s( s 1) l (l 1) g j 1 2 j ( j 1) The component of magnetic moment in z direction: ( j ) j , z m j g j B The paschen-Back effect For the Zeeman effect, the splitting of spectral lines in a magnetic field hold for “weak” magnetic fields. “weak” means that the splitting of energy levels in the magnetic field is small compared to fine structure splitting; or in other words, the spin-orbit coupling is stronger than the coupling of either the spin or the orbital moment alone to the external magnetic field. When the magnetic field B0 is strong enough so that the above condition is no longer fulfilled, the splitting picture is simplified. The magnetic field dissolves the fine structure coupling. L and s are, to a first approximation, uncoupled, and process independently around B0. The quantum number for the total angular momentum j, thus loses its meaning. This limiting case is called the Paschen-Back effect. The Pachen-Back effect The components of the orbital (µl)z and spin (µs)z moments in the field direction are now individually quantised. The corresponding magnetic energy is Vms ,ml (ml 2ms ) B B0 The splitting of the spectral lines: In a strong magnetic field B0, the spin S and orbital L angular momenta align independently with the field B0. A total angular momentum J is not defined. E (ml 2ms ) B B0 Term diagram and optical transitions of Na atoms glml+gsms 2 1 0 -1 -2 1 -1 (a) D1 and D2 lines of the neutral Na atom; (b) the anomalous Zeeman effect; (c) Pachen-Back effect. Question 1: Why is the 4D1/2 term not split in a magnetic field? Explain this in terms of the vector model. Question 2: Calculate the angle between the total and the orbital angular momenta in a 4D3/2 state. homework Pp220, 13.1, 13.3, 13.5, 13.8 Many-electron atoms Possible electronic configuration Angular momentum coupling Magnetic moments of many-electron atoms Electronic configuration and atomic term scheme: ground state, excited states Angular momentum coupling In the one-electron system, the individual angular momenta l and s combine to give a resultant angular momentum j. In many-electron atoms, there is a similar coupling between the angular momenta of different electrons in the same atom. These angular momenta are coupled by means of magnetic and electric interactions between electrons in the atom. They combine according to specific quantum mechanical rules to produce the total angular momentum J of the atom. The vector model provides insight into the composition of the angular momentum. Since the total angular momentum of an atom is equal to zero in closed shell, in calculating the total angular momentum of an atom, it is therefore necessary to consider only the angular momenta of the valence electrons, i.e. the electrons in non-filled shells. There are two limiting cases in angular momentum coupling: the LS coupling, and jj coupling. LS coupling (Russell-Saunders coupling) For many-electron atoms if the spin-orbit interactions (si · li) between the spin and orbital angular momenta of the individual electrons i are smaller than the mutual interactions of the orbital or spin angular momenta of different electrons coupling (li · lj) or (si · sj), the orbital angular momenta li combine vectorially to a total orbital angular momentum L, and the spins combine to a total spin S. L couples with S to form the total angular momentum J. L li , i S si , i J SL LS coupling gives a good agreement with the observed spectral details for many light atoms. For heavier atoms, another coupling scheme called j-j coupling provides better agreement with experiment. The vector model: For example for a two-electron system like the He atom The orbital angular momentum L of the atom: L l1 l2 , L L( L 1) L l1 l2 , l1 l2 1,, l1 l2 The quantum number L determines the term characteristics: L = 0, 1, 2, … indicates S, P, D, … terms. It should be noted here that a term with L = 1 is called a P term but this does not necessarily mean that in this configuration one of the electrons is individually in a p state. For the total spin angular momentum S: S s1 s2 with S S ( S 1) The spin quantum number: S = ½ + ½ = 1 or S = ½ - ½ = 0 The interaction between S and the magnetic field BL, which arises from the total orbital angular momentum L, results in a coupling of the two angular momenta L and S to the total angular momentum J: J L S, The quantum number J: For S = 0, J = L; For S = 1, J = L +1, L, L – 1 J J ( J 1) singlet; triplet In the general case of a many-electron system, there are 2S + 1 possible orientations of S with respect to L, i.e. the multiplicity of the terms is 2S + 1. The complete nomenclature for terms or energy states of atoms: n 2 S 1 LJ For many-electron systems, the possible multiplicities: For two electrons: S=0 S=1 singlet triplet For three electrons: S = ½ S = 3/2 doublet Quartet For four electrons: S=0 S=1 S=2 singlet triplet Quintet For five electrons: S=½ S = 3/2 S = 5/2 doublet Quartet Sextet Atomic terms of He atom If both electrons are in the lowest shell 1s2, they have the following quantum numbers: n1 = n2 = 1, l1 = l2 = 0, s1 = s 2 = ½ The resulting quantum numbers for the atom: L = 0, S = 0, ms1 = -ms2, J = 0, the singlet ground state 1S0; Or L = 0, S = 1, ms1 = ms2, J = 1, the triplet state 3S1, which is forbidden by the Pauli principle. If the atom in the electron configuration 1s2s, we have the following quantum numbers: n1 = 1, n2 = 2, l1 = l2 = 0, s1 = s2 = ½ , The resulting quantum numbers: L = 0, S = 0, J = 0, the singlet state 1S0; Or L = 0, S = 1, J = 1, the triplet state 3S1 In the same way, the states and term symbols can be derived for all electron configurations: 1s2p, 1s3d, 2p3d, … The selection rule: L = 0, 1; S = 0; J = 0, 1. Term scheme of the He atom. Some of the allowed transitions are indicated. There are two term system, between which radiative transitions are forbidden. Term diagram for the nitrogen. Nitrogen has a doublet and a quartet systems. The electronic configuration of the valence electrons is given at the top. Term diagram for the carbon. Carbon has a singlet and a triplet systems. The electronic configuration of the valence electrons is given at the top. jj coupling jj coupling is the case for coupling of electron spin and orbital angular momenta is larger compared to the interactions (li · lj) and (si · sj) between different electrons. It occurs mostly in heavy atoms, because the spin-orbit coupling for each individual electron increases rapidly with the nuclear charge Z. j1 l1 s1 ; j2 l 2 s 2 ; J ji with J J ( J 1) In jj coupling, a resultant orbital angular momentum L is not defined. There are therefore no term symbols S, P, D, etc. one has to use the term notation (j1, j2) etc.. The number of possible states and the J values are the same as in LS coupling. A selection rule for optical transitions: J = 0, 1, and a transition from J = 0 to J = 0 is forbidden. Purely jj coupling is only found in very heavy atoms. In most cases there are intermediate forms of coupling (intermediary coupling), which the intercombination between terms of different multiplicity is not so strictly forbidden. Transition from LS coupling in light atoms to jj coupling in heavy atoms in the series C – Si – Ge – Sn – Pb. Magnetic moments of many-electron atom In the case of LS coupling, the magnetic moment: J L S The total moment µJ precesses around the direction of J, and the observable magnetic moment is only that component of µJ which is parallel to J: 3 J ( J 1) S ( S 1) L( L 1) ( J ) J B g J J ( J 1) B 2 J ( J 1) gJ 1 J ( J 1) S ( S 1) L( L 1) 2 J ( J 1) In one of chosen direction z, the only possible orientations are quantised and they are described by the quantum number mJ, depending on the magnitude of J. ( J ) J , z mJ g J B With mJ = J, J - 1, … , -J Atomic ground states The possible electronic configurations of the atoms, concerning to the quantum numbers n and l, are governed by Pauli principle. The atomic term scheme, including of the ground state and the excited states, related to the energetic order of the states with different values of ml and ms and the combination of the angular momenta of individual electrons to form the total angular momentum of the atom. There are several rules for the energetic ordering of the electrons within the subshells in addition to the Pauli principle. In LS coupling, the angular momenta are governed by Hund’s rules. Hund’s rules Rule #1: Full shells and subshells contribute nothing to the total angular momenta L and S. Rule #2: The term with maximum multiplicity lies lowest in energy. Rule #3: For a given multiplicity, the term with the largest value of L lies lowest in energy. Rule #4: For atoms with less than half-filled shells, the level with the lowest value of J lies lowest in energy. Rule #2: The term with maximum multiplicity lies lowest in energy. For example: in the electronic configuration p2, we expect the order 3P < (1D, 1S) The explanation of the rule lies in the effects of the spinspin interaction. Though often called by the name spinspin interaction, the origin of the energy difference is in the coulomb repulsion of the electrons. The Pauli principle requires that the total wavefunction be antisymmetric. A symmetric spin state forces an antisymmetric spatial state where the electrons are on average further apart and provide less shielding for each other, yielding a lower energy. 2 Space wavefunction Rule #3: For a given multiplicity, the term with the largest value of L lies lowest in energy. For example: in the configuration p2, we expect the order 3P < 1D < 1S. The basis for this rule is essentially that if the electrons are orbiting in the same direction (and so have a large total angular momentum) they meet less often than when they orbit in opposite directions. Hence their repulsion is less on average when L is large. These influences on the atomic electron energy levels is sometimes called the orbit-orbit interaction. The origin of the energy difference lies with differences in the coulomb repulsive energies between the electrons. 3 For large L value, some or all of the electrons are orbiting in the same direction. That implies that they can stay a larger distance apart on the average since they could conceivably always be on the opposite side of the nucleus. For low L value, some electrons must orbit in the opposite direction and therefore pass close to each other once per orbit, leading to a smaller average separation of electrons and therefore a higher energy. Rule #4: For atoms with less than half-filled shells, the level with the lowest value of J lies lowest in energy. For example: since p2 is less than half-filled, the three states of 3P are expected to lie in the order 3P0 < 3P1 < 3P2. When the shell is more than half full, the opposite rule holds (highest J lies lowest). The basis for the rule is the spin-orbit coupling. The scalar product S · L is negative if the spin and orbital angular momentum are in opposite directions. Since the coefficient of S · L is positive, lower J is lower in energy. Influence on the atomic energy levels Hund’s rule #2 Hund’s rule #4 Hund’s rule #3 Identical particle: the electrons have the same rest mass, charge and spin, and can not be identified in quantum mechanics. Equivalent electrons: electrons with the same quantum numbers n and l, or the electrons in the same shell and subshell. Non-equivalent electrons The complete schemes for atoms correspond to a particular electron configuration and to a certain type of coupling of the electrons in non-filled shells. The energetic positions of these terms are uniquely determined by the energies of interaction between the nucleus and electrons and between the electrons themselves. Quantitative calculations are extremely difficult, because atoms with more than one electron are complicated. The possible atomic terms for a given electron configuration: 1) only the electrons in open shells must be considered; 2) each electron is characterised by the four quantum numbers n, l, ml and ms (a set of quantum numbers); To derive all the possible terms (LS coupling for example), all the possible variations of the couplings have to be considered: 1) For each value of S, MS = mSi have the possible values S, S-1, …, -S; 2) For each value of L, ML = mli have the possible values L, L-1, …, -L; 3) When the electrons are completely decoupled by a strong magnetic field (according to Ehrenfest), the individual electrons are quantised according to ml = l, l-1, …, -l and ms = ±½. The complete term scheme For non-equivalent electrons (LS coupling): ss: sp: sd: pp: pd: dd: 1S, 3S 1P, 3P 1D, 3D 1S, 1P, 1P, 3S, 3P, 3D 1P, 1D, 1F, 3P, 3D, 3F 1S, 1P, 1D, 1F, 1G, 3S, 3P, 3D, 3F, 3G For equivalent electrons, less terms: p2, p4: P3: d2, d8: d3, d7: d4, d6: d5: 1S, 1D, 3P 4S, 2P, 2D 1S, 1D, 1G, 3P, 3F 2P, 2D, 2F, 2G, 2H, 4P, 4F 1S, 1D, 1F, 1G, 1I, 3P, 3D, 3F, 3G, 3H, 5D 2S, 2P, 2D, 2F, 2G, 2H, 2I, 4P, 4F, 4D, 6S How to determine the shell structures and terms in experiments? ----- X-ray spectrum homework pp344 19.1, 19.4, 19.6, 19.7