m. PARAMAGNETIC RESONANCE INTRODUCTORY PAPER BY J. H. E. GRIFFITHS The Clarendon Laboratory, Oxford Received 10th February, 1955 1. INTRODUCTION Paramagnetic resonance is a branch of spectroscopy which is concerned with the investigation of the absorption spectra of paramagnetic substances in a magnetic field. The spectrum often occurs in the microwave region (1-10cm wavelength) for suitable values of the magnetic field. The substances investigated here have been mainly crystalline salts of the transition and rare earth elements, although measurements have also been made on organic paramagnetic substances and on various types of impurities in diamagnetic substances. The purpose of this paper is to present a brief description of this method and of the theory of the spectra observed in the transition and rare earth compounds, with the emphasis on those features which are of intcrest to chemistry.For more detailed treatments two review articles should be consulted. The first by Bleaney and Stevens 1 gives an account of the theoretical background and dis- cusses the experimental results in relation to this and the second by Bowers and Owen 2 collects together all the available experimental results. The paramagnetic substances described here are characterized by two features : (i) they contain ions which possess permanent magnetic morncnts ; this distinguishes them from diamagnetic substances. These ions therefore have a ground statc which can be split by the application of a magnetic field into two or more sub- states and it is the transitions between these which are observed in paramagnetic resonance.(ii) The energy of interaction, whether magnetic or exchange between these magnetic dipoles is very much smaller than kT at ordinary temperatures; this distinguishes them from ferromagnetic or antiferromagnetic substances. This interaction, although small, is still sufficient to cause broadening of the resonance lines; to minimize this effect, considerable use is made of dilute crystals in which the paramagnetic substance is mixed with an isomorphous diamagnetic substance. This reduces the chance that two paramagnetic ions are close together. Again it is most important to use single crystals as the amount of information obtainable by using polycrystalline material is very limited. 2. EXPERIMENTAL METHODS One commonly used experimental arrangement is as folIows.A small crystal (about 1 or 2 mm in size) is placed at the end of a cavity resonator tuned to the appropriate wavelength which is usually about 1% cm or 3 cm. In this position the crystal is in the greatest oscillatory magnetic field which is what is required for the magnetic dipolar transition to be observed. The resonator is placed in the field of an electromagnet, the two magnetic fields being at right-angles to each other. Conventional methods are used to feed the microwave power from the reflex klystron into the resonator and from thc resonator to thc detector which is a silicon-tungsten crystal rectifier. In taking measurements the frequency is kept constant and the magnetic field varied and when absorption occurs the de- tector current decreases.This may be observed with a galvanometer, but a 106J . H . E . GRIFFITHS 107 more convenient and sensitive method is to modulate the magnetic field by about 100 gauss at 50 c/s by passing alternating current through auxiliary coils on the electromagnet. The signal from the detector is then passed through a low frequency amplifier and displayed on a cathode ray tube. This is a simple method, the only complication in dcsign being duc to the fact that it is often necessary or desirable to cool the substancc to the temperature of liquid air, hydrogen or even helium. There arc two main rcasons for this, firstly, that the interactions between the magnetic dipoles and the lattice vibrations are sometimes so large at room temperature that the linc is very broad or unobscrvable.This interaction de- creases with decrease of tempcrature. Sccondly, the intensity of absorption is usually inversely proportional to the absolute temperature, because it is pro- portional to the difference in population of thc levels concerned, which is given by the Boltzmann factor. There are adjustments to tune the resonator and to rotate the crystal which call for careful dcsign, but thc latter can be dispensed with if the electromagnet can bc rotated. Therc are other methods of greater sensitivity, but of greater complication, which need not be considered here. 3. THE IONIC MODEL Most theories of paramagnetic susceptibility and of paramagnetic resonance use the ionic model as their starting point.In this, it is assumed that the substance is purely ionic and that the effect on the ion of the surrounding atoms or ions can be entirely represented by an electric field of appropriate strength and sym- metry. This electric field may be very large and is produced either by the charges on other (diamagnetic) ions or by the dipole moments of such molecules as H20 or NH3. Before considering the effect of the crystalline electric field on the energy levels of an ion, it is useful to discuss two simple cases to which some actual cases approximate. Thc first is that of a system of independent electron spins of magnetic moment /I (the Bohr magneton). In a magnetic field H there are two energy levcls of each spin with energy f PH, corresponding to the two values of the spin angular momentum, Ms = f - - ' Transitions between these levels can be induced if 2 27r' the frequency v of the radiation is given by numerically this gives HA = 10.7, when H is measurcd in kilogauss and A is the wavelength in cm.This rcsult arises as follows. For a system with total spin S, the magictic quantum numbcr M , can have the values S, (S - 1) . . . (- S), i.e. (2s + 1) values and these states have energy 2M3H in the field H. The selection rule is that Ms can only changc by f 1 and therefore in this case all the allowed transitions have the same frcquency which is given by eqn. (1). The sccond case is that of a system of free ions and neglects the effect of the electric field. Hcre the orbital moment has to bc considered as well as the spin moment.If the ion obeys Russell-Saunders' coupling, there is a total orbital quantum number L and a total spin S which combine to give a resultant J. In a magnetic field there are W + 1 levels corresponding to values of MJ of J, (J- 1) . . . (- J ) and these have energies MJgpH. Again, since the selection rule is that MJ can only change by f 1 , there is only one resonance condition such that Here g is the Land6 splitting factor and has the value unity when there is only an orbital moment and 2 when there is only a spin moment (more accurately the spin only value of g is 2.0023). This factor g is of great importance in paramagnetic resonancc. Many results give a value of g near to 2 and the difference from 2 is interpreted as giving a measure of the contribution of the orbital moment.I ~ v == 2pH; (1) hv = gPH. (2)108 INTRODUCTORY PAPER It is now necessary to consider the effect of the crystalline electric field on the ground state of the ion. There is a general theorem due to Kramers which is very useful in this connection and which states that if there is an odd number of electrons in the system, no electric field can remove all the degeneracy of the states. This means that there are at least two states with the same energy in the electric field and these are split by the application of a magnetic field, so that paramagnetic resonance will usually be observed in these systems. The detailed treatment of the effect of the electric field depends on the strength of this field as compared with the other energy terms.Three cases can be distinguished. (i) SMALL CRYSTAL FIELD.-This is the case of the rare earths where the 4f electrons which are responsible for the paramagnetism are we11 shielded by the closed shells of 5s and 5p electrons. This approximates to the second of the two simple cases considered above. L and S combine to form J which is still a " good " quantum number, but the states of different J z , which is the component of J along the axis of the electric Geld, are split by the electric field, often into pairs of f J z with energy separations between the pairs of the order of 100 cm-1. At the low temperature at which it is usually necessary to make measurements on these salts, only the lowest doubIet is occupied and it is the transition between these levels which is observed.The results on the rare earths are not of great interest to chemistry since the electrons concerned are inner electrons and do not seem to form chemical bonds. The other group in which felectrons occur is the uranium group and this is discussed in the paper by Bleaney. (ii) MODERATE CRYSTAL FIELD.-T~~S is the case of the iron group and approxim- ates to the first of the two simple cases, i.e. that of free spins. The reason for this is as follows. In the case of the rare earths, the energy difference between states of different J z but the same J caused by the electric field is small compared with the energy difference in the free ion between states of different J but the same L and S. In other words, the interaction with the electric field is small compared with the spin orbit coupling.In the iron group this is reversed and therefore J loses its meaning and is no longer a " good " quantum number, but L and S are still " good " quantum numbers. The electric field does not have any direct effect on the spin but splits the 2L + 1 orbital levels with splitting2 of the order of 10,000 cm-1 (see Fig. 1). Transitions between these orbital levels are usually responsible for the characteristic colours of these salts. There are still the 2 s + 1 states belonging to each orbital level and if a single orbital lcvel lies lowest, first- order theory shows that these behave like the spin-only case of eqn. (l), i.e. a single line with g = 2. In the more exact theory it is necessary to consider in some detail the sym- metry of the electric field.It is commonly found in iron groups salts that the ion is surrounded by six H2O molecules which are equally spaced along the three cubic axes (at the corners of an octahedron). This gives a field of cubic symmetry. This arrangement is usually slightly distorted and this distortion produces an additional (small) component of lower symmetry (often uniaxial). When this field is taken into account together with the spin orbit coupling, which has so far been neglected, two new features appear. First, the g values are anisotropic and no longer equal to 2, and secondly, if the spin is greater than 1/2, the spin levels are split with an energy difference usually of the order of 0.1-1 cm-1. Two examples may make this clearer : (i) Cu2+ has nine d electrons (d9) and as a free ion has a 2D state lowest.The five orbital levels are split by a cubic field into a lower doublet and a higher triplet and a field of tetragonal or lower symmetry splits the doublet also (see Fig. 1). Since the spin is 1/2, each of thcse levels is a spin doublet and the single line observed in paramagnetic resonance is the transition between the two spin levels of the lowest orbital level which is the only one populated at ordinary temperatures. In the salt K ~ S O A , CuSO4. 6H2O the g values are found to be gll = 2.4, the value when the magnetic field is parallelJ . H . E. GRIFFITHS 109 to the axis of the electric field and g i = 2.1, the perpendicular value. Theory gives approximately gll = 2[1 - (4A/A)] and g l = 2[1 - @/A)], where A is the spin orbit coupling coefficient (in this case negative) which can be found from optical spectra and A is the energy difference between the lowest level and the higher orbital triplet.This gives a connection between the measured g values and the splitting of the orbital levels. (ii) Cr3+ has three d electrons and 4 F is the lowest state as a free ion. A cubic field splits the orbital levels into a lower singlet with two upper triplets (see diagram). Each of these levels has a spin of 20 4F I I.-. (7 'r" MS CU'+ FIG. 1.-Energy level diagrams (not drawn to scale) for the two cases of Cu2+ and Cr3+ : (a) represents the free ion, (6) shows the splitting of the orbital levels in a cubic electric field, and (c) the lower levels in a field of lower symmetry.On the right of each diagram is represented the splitting in a magnetic field which increases from left to right. The number in parenthesis is the number of orbital states of the same energy. The approximate value of the splitting is given in cm-1- The full lines terminated with arrows indicate the microwave transitions observed. 3/2 and therefore is fourfold degenerate. In a salt such as CsCr(SO&. 12H20 the Cr3-1- ion is surrounded by an arrangement of I 3 2 0 molecules which is nearly octahedral but has a small distortion along the body diagonal of the cubic unit ccll. This electric field splits the spin levels into two doublets of spin f 3/2 and & 1/2 with a separation of 0.13 to 0.18 cm-1 depending on the temperature. The allowed transitions in a magnetic field are still AM, = f 1 and this gives rise to threc lines which are not coincident because of the splitting.This is known as the fine structure and from measurements on these lines the splitting may be determined. Another method of measuring the splitting is from the specific heat anomaly at low temperature (0.2"K) to which it gives rise. The exact behaviour of these four levels is complicated and depends on the angle between the applied magnetic field and the axes of the electric field, but can be worked out in detail. (iii) STRONG ELECTRIC FIELD.-This is the case of the palladium and platinum transition groups and here the field is strong enough to break down the Russell- Saundcrs coupling completely and the effect of the electric field has to be con- sidered before the interactions between the electrons.These electrons are d electrons and a cubic field splits the d state into a lower triplet, generally known as de, and an upper doublet dy, the splitting between which is so large that they can be considered as separate sub-groups. As electrons are added they fill first the de states forming a lowest level of maximum spin consistent with Pauli's prin- ciple (Hund's rule). Thus the spin of the first six ions is respectively 1/2, 1, 13, 1, 1/2, 0, which completes the de sub-group. This contrasts with the normal spin of iron group ions for which the first six have values 112, 1, 1+,2,2+, 2, although110 INTRODUCTORY PAPER some iron group complexes such as cyanides belong to the strong field case.The treatment of the strong field case then proceeds in much the same way as that of the moderate field case. Further discussion is left to the paper by Owen. It must be emphasized that the particular arrangement of splittings of the orbital lcvel depends very much on the symmetry of the electric field and the cases discussed here have predominantly cubic symmetry arising from six neighbours. Other arrangements occur such as with four or eight nearest neighbours and these result in different arrangements of the orbital levels and different results. 4. HYPERFINE STRUCTURE There is a hyperfine structure in those cases in which the magnetic electron is associated with a nucleus that has a magnetic moment. Often this takes a simple form in which there are 21 -l- 1 equally spaced lines to each electronic transition, whcre Z is thc spin of thc nuclcus.This arises in the following way. The effect of the magnetic moment of thc nucleus on the electronic system is the same as that produced by a magnetic field AH. In a transition of this kind the nucleus is not affected so that at constant frequency a resonance line is displaced by AH, and sincc there are 21 + 1 orientations of a nucleus of spin 1 which give equally spaced values of AII, there result 21 -1- 1 equally spaced lines. This gives a method of detcrmining the nuclcar spin in suitable cases and some spins have first bcen determined in this way. The magnitude of the separation of the hyperfine lines often varies with the angle between the applied magnetic field and the axis of the electric field, and there are also effects duc to the quadrupole moment of the nucleus.5. INTERPRETATION OF THE RESULTS There are two stages in thc assembling and interpretation of the results. In the fist it is necessary to detcrmine such quantities as the effective spin, the g- values, the electronic splitting and the hyperfine splittings. Some of these quan- titics vary with the oricntation of the crystal in the external magnetic field and from this variation the directions of the axcs and the symmetry of the electric field may be found. All this information is usually collected together in what is known as a “ spin Hamiltonian ” which expresses the results in the smallest number of constants and from which can be worked out the positions and intensities of thc lines for all orientations and strengths of the external magnetic field by straight- forward quantum mechanical mcthods.This procedure is fully described in the review article by Bowers and Owen.2 The second stagc is to interpret the values of thc constants of the spin Hamiltonian in terms of a model, and here it is necessary to consider also any othcr relevant information such as that obtainable from optical absorption spectra. In this papcr the results have bcen interpreted in terms of an ionic model although even hcre assumptions had to be made about the relative size of thc effects of thc electric ficld and the intcrnal interactions between the elcctrons. The effect of assuming a certain amount of covalent bonding is discussed in the paper by Owcn whcre it is shown that the main obscrvable cffccts arc a reduction of the g-valucs and of thc hyperfinc splitting constant, and in cases where thc nuclei OJ the surrounding atoms have n spin, there may be a hyperfine structure duc to these atoms.6. SCOPE AND LIMITATIONS OF THE METHOD Thc limitations arise mainly from the fact that cven if the substance is para- magnetic it may still be difficult or even impossiblc to observe a resonance. This may be duc to thc following reasons. (i) If the number of electrons in the system is cven thcrc is no Kramcrs’ degeneracy, and the splitting between the spin lcvels may be so large that no rcsonance can be obscrved at the frcquencies and magnctic fields availablc in the laboratory.J . H . E. GRIFFITHS 111 (ii) Even if therc is Kramers' degeneracy, the transition between the two lowest levels may be forbidden and no absorption will be observed unless higher levels are occupied at the temperature used.This probably occurs in some of the rare earths. (iii) The exchange of energy between the spin and the lattice may be so rapid that a very broad line results and this may be too broad to be observable. This effect decreases with decrease of temperature and therefore should be negligible at a sufficiently low temperature, c.g. for CsTi(SO&. 12H20 a temperature of less than 8" K had to be uscd. However, if these limitations are absent and a resonance can be observed in a single crystal, precise information can often be obtaincd about a number of features some of which arc listed herc.(i) The electronic state ofthe ion.-This is often known or can be inferred with reasonable certainty from other evidence, but in the case of the uranium group this is not so and the information obtainable by paramagnetic resonance is discussed in the paper by Bleaney. In other cases it may be a useful method of determining the valency state of an ion wherc this is in doubt. (ii) The crystalline electricjield-As stated abovc, the symrnctry of this field can usually be determincd and from this the arrangerncnt of the immediate surroundings of the ion can often be inferred. Sometimes a small distortion from a regular array can be detected by this means more easily than by X-ray crystallography. (iii) Perhaps the most interesting from the chemical point of view is the in- formation that can be obtained about the amount of covalent bonding which occurs in these compounds.In several cases it has been possible to give a numerical estimate and these are discussed in the following papers. (iv) A spccial case of some intercst is that of cupric acetate. Here the reson- ance results showed that the cupric ions are sufficiently closc togcther in pairs to behave somcwhat like a molecule with the two spins of 1/2 forming a lower diamagnetic lcvcl of S = 0 and upper triplet of S = 1 separated by about 300 cm-1. This arrangement of thc cupric ions was later confirmed by X-ray measurement. (v) Paramagnetic resonance has been observed in a number of organic mole- cules which have an unpaired clcctron. The g-value is very close to 2 but usually a little higher.This seems to be a promising field of investigation particularly if a hyperfine structure can be observed, because from this can be obtained an estimate of the time which the unpaired electron spends attached to the particular nucleus which has a magnctic moment. Some examples arc given in the following papers. (vii) Very small concentrations of impurities can be detected and sometimes identified by this mcthod in suitable cases. There are two types of cases. If the impurity is normally diamagnetic it may become paramagnetic by excitation or by attachment or removal of an electron, this new state being produced by irradi- ation by ultra-violet, X-rays, electrons or neutrons. Several cases have been investigated such as colour centres in alkali halides, quartz irradiated by X-rays in which the spectrum is due to A1 impurity, and neutron irradiated diamond. The second type is when the impurity is paramagnetic such as Mn2+ in phosphors and Fe2f in ZnF2. The possibilities are many and various and the sensitivity is such that a total number of impurity ccntres of the order of 1016 in a sample of about 10-2cm3 can be detected with very straightforward apparatus and this numbcr may well be reduced to 1012 with more refined methods. To conclude this vcry brief review, it seems to be the experience so far that although there are some general categories into which substances may be placed because of the similarity of thcir magnetic behaviour, in fact, most substances present their own problems and yield their own interests, because in each there is a different feature which is important. Thus, if a resonance can be found using a single crystal, the problem of interpreting it is one usually well worth pursuing. * Bleariey and Stevens, Reports Prog. Physics, 1953, 16, 108. 2 Bowers and Owen, Reports Prog. Physics (in press).