=2 ) ,4L* EXCHANGE-COUPLED CHROMIUM I0N PAIRS IN RUBY by L. F. Mollenauer M. L. Report No. 1329 May 1965 SPECTROSCOPY AND QUANTUM / VARIAN LABORATORY ELECTRONICS OF PHYSICS GROUP AND W. W. MICROWAVE LABORATORY HANSEN LABORATORIES OF GPO @ PHYSICS PRICE $ CFSTI PRICE(S) $ Hard copy (HC) Microfiche (MF) ff e53 July 05 STANFORD STANFORD, UNIVERSITY CALIFORNIA https://ntrs.nasa.gov/search.jsp?R=19660030055 2020-03-16T17:57:27+00:00Z , ° . EXCHANGE-COUPLED CHROMIUM ION PAIRS IN RUBY by L o Fo Mollenauer Mo L_ Report NOn 1325 Interoal Memorandum NASA Grant NsG-331 ;.965 Spectroscopy and Quan_umElectronics Group Varian Laboratory of-Physics and Microwave Laboratory W_ Wo Hansen L_borator_es of Physics Stanford University Stanford_ California _4 ABSTRACT This investigation has as its primary objective measurementof the exchange coupling between the various near neighbor types of Cr>+ ion pairs in the antiferromagnetic compound Cr203. The exchange coupling is expressed by the hamiltonian (Hex)i : - Ji S1 $2 .th where J. is the exchange integral of the l nearest neighbors, and i S I and S2 are the total spin on each of the interacting ions. Advantage is taken of the fact that the homologous lattice AI203 will accept Cr203 in solid solution, and that for solute concentrations on the order of 0.i_ to 1.0_, the resultant dark ruby lattice contains a sufficient number of near neighbor Cr ion pairs that the sharp line fluorescence spectrum produced by these can be investigated. At low temperatures, this spectrum consists of a quartet of lines for each pair type whose spacings yield a very accurate measure of Ji The problem then consists of separating the various and sometimes overlapping ex- change multiplets and identifying each with a particular near neighbor type. When uniaxial stress is applied, differently oriented pairs of a given type are made inequivalent and an apparent splitting of the lines is produced. A constant magnitude of stress is applied to each of approxi- mately eighteen samples whose axes span a full 180 ° range in the crystal mirror plane; the resultant pattern of splitting versus angle, unique to each pair type, allows the desired identifications to be made. 0nly the first four nearest neighbor types were found to have signi- ficantly large exchange interaction; the corresponding exchange integrals -i -I were found to have the values Jl = - 183.5 cm ' J2 = -11.56 cm , J3 = - 46 cm -I , and J4 = 6"94 cm-I The set of four J values can then be inserted into a rather well established expression for @ , the - iii - Curie Weiss constant; the value obtained, @= 565°K , is in excellent agreement with the experimental value of @= 550 ± 20°K The full set of J values offer a unique opportunity to test certain u_u_ u± _b±±_±±__ u_±ng. First_ in _he special case dis- covered here, where Jl and J3 predominate, it can be shownthat at least one other exchange coupling is required to complete long range ordering, and that in fact a change in the sign of J4 between the corundumcompoundsCr203 and _-Fe203 is required to explain the observed difference in their respective antiferromagnetic spin arrangements. Attempts to calculate TN , the temperature below which antiferromagnetic ordering sets in, or the ground state exchange splitting observed in the optical spectrum of pure Cr203, yield wildly wrong answers when the measured J values are inserted into expressions derived from the molecu- lar field theory; yet more realistic models, such as a clustering theory which calculates effects of the molecular field on the first neighbor pair, are indicated to produce muchbetter agreementbetween the J values measuredin dark ruby and the magnetic and optical properties of Cr203• An incidental consequenceof these studies is a rather complete empir- ical knowledge of the large shifts which can be produced in the frequencies of the various pair lines by the application of mechanical stress. This effect suggests the possibility of making a readily and highly tunable laser. - iv - hO ¸ ACKNG_LEDGEMENTS This investigation was carried out under the supervision of Professor A. L. Schawlow. Not only is the somewhat unorthodox and imaginative technique used here the result of his suggestion, but the fact of a successful and completed work is due in no small measure to his unflagging will to see his ideas given complete and fair trial. It has indeed been a privilege and an honor to work with Professor Schawlow. Many thanks are due to Professor P. L. Scott, whose strong interest in this work and whose willingness to discuss the many theoretical questions which arose during the writing of this report have made him an invaluable counselor and friend. A similar note of thanks is due to Professor R. L. White for his accepting to proofread this report on very short notice and for making a number of valuable suggestions. A large part of the rather grueling task of cutting and polishing of samples was p_rfonned by Susan Broadston Nichparenko; both the skill and the cheerfulness with which she carried out this task made it a joy to work with her. Thanks again, Sue. I am indebted to my colleagu_sS. E. Stokowski and W. C. Scott for taking the helium temperature spectrogram and for providing the excellent cryogenic apparatus used in its production, respectively. A special word of thanks is due to Mr,. A. S. Braun and the efficient staffs of the Reports and Drafting Offices for their expert and invaluable help in the preparation of this report. Finally, I would like t0_pay tribute to two persons whose influeneee on this work, though indirect, is of great importance: to my wife, Marjorie, for her constant encouragement and moral support, and to my father, whose own love of learning first set me on the path of scholarship. - V - mI, II. III. IFVo TABLE OF CONTENTS Page Abstract .......................... iii Acknowledgements ...................... v List of Figures ..................... viii Introduction ....................... i i.i Historical note .................... 1 1.2 Modern theory of the C3+ ion in A1203 ........ 7 1.3 The pair spectrum ................... lO The piezospectroscopic method ............... 31 2.1 Introduction ..................... 31 2.2 Discussion ...................... 31 2.3 Symmetries of A1203 and location of the pairs .... 34 2.4 Identification of the pair lines ........... 38 2.5 Orientation and cutting of the samples ........ 46 2.6 The pressure apparatus ................ 50 Results of the piezospectroscopic studies ......... 56 3.1 Introduction ..................... 56 3.2 Splitting of the N1 and N 2 lines ......... 56 3°3 Splitting of the lines at 7452_ ........... 62 3°4 Assignment of pair types to the NO and N2 lines 66 3°5 Assignment of a pair type to the N1 line ...... 73 3.6 Verification of the known exchange multiplets .... 76 3.7 Discovery of the first nearest neighbor lines .... 78 3.8 Accuracy of the piezospectroscopic measurements . . 80 Absorption measurement s ................ 84 84 84 9o 4.1 Introduction ..................... 4.2 Equipment for detecting weak absorptions ....... 4.3 Determination of J for the first nearest neighbors. 4.4 Determination of the third nearest neighbor exchange multiplet ...................... i01 - vi - Vo Conclusions ......................... 108 5.1 Summaryof exchangevalues for near neighbor pairs. • 108 ._ J values In __j and the theory of antiferromagnetism in Cr203 ..... 109 5.3 Correspondencebetween the measured J values in ruby and the absorption spectrum of Cr203 ......... 119 5.4 The piezospectroscopic sensitivity of the ruby pair lines ......................... 122 5.5 Application of the piezospectroscopic effect to laser technology ....................... 125 References............................. 127 - vii - l_lo 1.2. 1.3. 1.5. 1.6_ 1o7o i_8. 1.9. I. i0_ _l_llo I. 12o 1.13. 2.2_ 2. 3 • 2.4_ 2.5° LIST OF FIGURES Edmond Becquerel's Phosphoroscope ............. Portion of the A1203 lattice = 4.3 ° ........... Effects of the cubic field on the energy levels of the Cr 3+ ion ......................... Fluorescence intensity ratio N]R 1 as a function of chromium concentration ................... Fluorescence intensity ratio N/R for various satellite lines ........................... Exchange splitting of a Cr 3+ ion pair ground state .... Exchange splitting of Crj+ ion pair excited state .... Emission and absorption in dark ruby at 4o2°K ....... N line laser action in dark ruby ............. Temperature dependence of N line absorption intensities Energy level scheme of pair type associated with N I Energy level scheme of pair type associated with N 2 Antiferromagnetic spin arrangement in Cr203 as known from neutron diffraction studies ................ Ground state splitting of N I and N2 lines at 4.2°K Symmetry elements of the point group D3d and their stereographic projection ................. Locations of the second, third and fourth neighbor pair axes ........................... Location of sixth nearest neighbor pair axes ....... Mechanics of sample under stress ............. Diagram showing angles used in computing splitting pattern for third neighbors .................... Predicted third neighbor splitting pattern ........ Predicted fourth neighbor splitting pattern ........ - viii - Page 2 8 9 12 13 16 m8 2O 22 23 24 25 27 29 35 36 37 4o 42 43 44 Page 2_ 7 . 2.10. 2. ii. 2.12. 2.13. 3.1. 3.2. 3.3. 3.6. 3.7. 3.8. 3.9. 3.i0. 3.11. 3.12. 4.2. (a) Predicted second neighbor splitting pattern. (b) Pre- dicted sixth neighbor splitting pattern ........... 45 Elements of Laue camera ................... 47 Back reflection Laue photographs of A1203: (a) X-ray beam parallel to C] ; (b) X-ray beam parallel to C2 ..... 48 The mechanical press .................... 51 Photograph of the mechanical press ............. 52 Detail of press; showing sample in holder .......... 53 Mercury arc lamp and housing ................ 54 Recorder traces of NI line splitting ........... 57 Splitting pattern of NI Magnitude of stress is i00 Kg/mm 2 59 Recorder traces of N splitting for two different sample orientations ......................... 60 Splitting pattern of N2 . Magnitude of stress is i00 Kg/mm 2 61 (a) Fourth N.N. Cr-0-Cr complez (b) Third N.N. Cr-O-Cr complex ........................... 63 " "b" . . . .Stress equivalent sets "a and of Cr-O-Cr complex 64 N O splitting as it appeared on the original plates ..... 67 Splitting pattern of NO . Magnitude of applied stress is 2 i00 Kg/ mm ......................... 68 and N O splitting patterns and splitting curves shown side by 69 N2 side for easy comparison .................. Definition of axes relevant to the stress tensor given in Section 3.4 ......................... 72 Stress equivalence of second neighbor pairs lla_ llb_ llla 3 for stress axis lying in mirror plane I ........... 75 Illustration of the splitting pattern common to all members of a given exchange multiplet ................ 77 NI splitting pattern and splitting curve .......... 82 Schematic representation of arrangement used to detect weak absorptions ......................... 86 Housing for sample and photomultiplier ........... 89 - ix - 4V 4.5. 4.8. Page 91First nearest neighbor energy level diagram ......... Increase in absorption strength with temperature of the first nearest neighbor line at 7302 _ ........... 93 Enhancetron output showing first neighbor absorption lines at 20°K .......................... 94 Behavior of the cluster of lines near 7300 _ with temperature 96 Resolution of line at 7302 _ into two components when the temperature is lowered to 77°K ............... 97 Microphotometer trace of fluorescence plate of l_ Cr203 ruby at 4.2°K .......................... 99 Third neighbor energy level diagram ............. 103 Possible third neighbor energy level schemes not entirely ruled out by present data .................. 106 Antiferromagnetic spin ordering for -Fe203 and Cr203 .... ll2 Schematic illustration of Osmund's theory of super-exchange in the third and fourth neighbor pairs ........... ll5 Variations of I/X _ (reference 78) ............ 117 Energy level diagram for Cr203 ............... 120 - x - I. INTRODUCTION io Io HISTORICALNOTE Solid state spectroscopy is so strongly associated with the recent invention of the optical maser that it may comeas a surprise to learn that the first studies of the spectrum of ruby were madearound the middle of the last century_ predating not only the quantumtheory but Maxwell's classical theory of radiation as well: In a treatise on light published in 1867, EdmondBecquerelI devoted nearly a full chapter to "Alum/ne et ses Combinaisons." Nearly all the observations reported there were madewith the aid of his "phosphoroscope," a simple but clever device in which the sample, located between a pair of rotating, segmented disks 9 would be excited by a shaft of sunlight, then viewed through a prism spectrometer a controllable time after the sunlight had been shut- tered off. Becquerel's apparatus is shownin Fig. 1.1. Hewas thus able to determine that the fluorescent lifetime in ruby was of the order of a milliseconds, in excellent agreementwith modern observations; he was also able to note the unusual sharpness of the lines and the great simi- larity of the spectra produced by rubies; sapphires; and samples of alumina powder prepared in various ways. Although he correctly observed that the fluorescence was greatly enhancedby the addition of small amounts of Cr203 to his alumina preparations, he was misled by the persistence of the red fluorescence in so-called "pure" samples of alumina to theorize that the sharp fluorescence lines were the result of someinherent property of the A1203 lattice, and that chromiummerely played the role of an activator, - 1 - -uS ............................... R_II- "............................ I-,,,,,,f_0,,,_ FIG. l.l--Edmond Becquerel's Phosphoroscope. Sunlight enters at S , passes through a water solution filter at E , and excites fluorescence in the sample located in the upper half of the enclosure AB. The rotating, segmented disks are contained in AB and are seen edge on; as the disks rotate_ they alter- nately block off the sunlight and the line of sight to the spectrometer HDG. (From La Lumiere, ses Causes _t ses Effets') -2 - This theory evidently remained unchallenged until 1886, whenthe 2following short notice appeared in Comptes Rendues: "M. Lecoq de Boisbaudran annonce, pour prendre date, que l'a!umine calcin@e et soumise _ l'effluve @lectrique darts le vide ne lui a pas donn6 trace de fluorescence rouge. Cette fluorescence [ainsi que son spectre sp@cial(1)] se montre bril!amment quandl'alumine contient !/i00 et mSme I/ii00 de Cr203 ,. Avec I/i00000 de Cr203 , on obtient encore du rose tres visible .... .o.D'apr_s ees observations, la presence du chrome para_t Stre indispensable _ la production de la fluorescence rouge de l_alumine... '' The aging Becquerel, revered member of the Academic, made immediate and rather haughty reply 3 to the upstart Boisbaudran, the beginning of a long and sometimes amusing fight, which is well chronicled in Comptes 4-8 Rendues. The argument seemed to reach an impasse with Boisbaudran going to ever greater lengths to produce "pure" alumina and Becquerel attacking what we now know to be irrelevant side issues. From a modern viewpoint, both sides of the controversy are unsupportable; in the first place, the chromium fluorescence has such a high quantum efficiency, the lines produced are so sharp, and modern light detectors are so sensitive, that even the most highly purified sapphires produced to date show the R lines of ruby with good signal to noise ratio. Even more important is the consideration that a statement such as "chromium is indispensable in the production of the red fluorescence in alumina" or "chromium is only an activator in ruby" is meaningless outside the context of a complete theory based on the statement in question, such that esentially all predictions of the theory can be checked quanti- tatively against the results of experiment. On the other hand, it would not be fair to judge Becquerel and Boisbaudran by this modern standard, but rather, in view of the nearly complete theoretical vacuum in which they were forced to work, we should be amazed by their wil- lingness to tackle a seemingly insoluble mystery, and should credit them with bringing one of the most beautiful and useful phenomena of nature to the world's attention. L {_Ipspectre si bein d@crit dans l'Ouvrage classique de M. Becquerel: La Lumiere, t. I, p. 340. -3- 01d controversies die hard• Thus it is not too surprising to find, as late as 1904, an article in Annalen der Physik 9 by G. C. Schmidt with the subtitle "Antwort an Hrn. Jo _f_l, ._..... " w c_ t_ns out to be nothing more than a re-run of the Becquerel-Boisbaudran argument. It is surpris- ing_ however, that just a few years later, when the Boisbaudran-Schmidt hypothesis seems to have acquired general acceptance among solid-state spectroscopists, Schmidt is given credit for having finally established i0 bhe correctness of Boisbaudran's contention, although the modern reader will find it difficult to see any essential differences between their methods and arg_mments. Although brief mention of the ruby spectrum by J. Becquerel II and a not very accurate description by A. Miethe 12 appear earlier, the first thorough study of ruby by means of a high-resolution spectrometer was made by H. du Bois and G. J. Elias in 1908.10 Du Bois and Elias were able to observe the absorption spectrum as well as the fluorescent emission, since large single crystals of synthetic ruby had become available by that time, and they gave the principal absorption lines their presently used designations, the R (rot) and B (bla_u) lines• They were able to produce a resolved Zeeman splitting of the R lines for the first time by immersing their samples in liquid air and by using magnetic fields as high as 42 kilogauss. In reading the du Bois and Elias paper, J. Becquerel realized that some of the observations of the Zeeman effect reported there, such as supposed polarization differences for the same line in absorption as opposed to fluorescence, would lead to strange theoretical consequences, and he soon published a series of corrections backed by his own experi- mental observations. 13_14 D_ Bois and Elias were then prompted to • 15 publish a revised and extended verslon of their original paper• Here they were able to explain their former errors in terms of a badly wander- ing optic axis in their long absorption sample, along with visible strains and fractures in their crystals--possibly the earliest recogni- tion of the important role played by crystalline defects. They also recognized the rather rare opportunity presented by the R lines to allow an accurate study of the temperature shift of absorption lines in -4- crystals, and carried out such measurements over a temperature range beginning with liquid air and extending to 845°C. Finally, they report the first measurements of the weak lines and bands seen in fluorescence to the red of the R lines, which they call the "Nebenlinien." Spectro- scopists were somewhat handicapped in those days by the lack of red- sensitive p_ates, and du Bois and Elias were forced to make their measurements directly through a magnifing glass, for the observations would otherwise have used up "too much time and liquid air." Hence, to see the weak neighbor lines, nothing but concentrated sunlight would do as exciter of the fluorescence, and it is amusing to read their apology that "...these observations, which would otherwise have been completed by the end of 1910, had to be delayed until the present spring." The above work was extended somewhat by Mendenhall and Wood in 1915.16 They observed resonance radiation with the R lines, and also, by using photographic plates, were able to observe the "most curious" Zeeman effects in the neighbor lines. An attempt to induce Stark effect in the R lines failed, either because of improper alignment of the crystal axis with the field direction or because of the rather large residual line width due to microscopic strain in the rather poor quality crystals available at that time, or perhaps both at once. The failure is unfortunate, since a positive result may have given just a hint of the crystal field effects so important in solid-state spectra. So far there had been little or no attempt at theoretical inter- pretation of the relatively well observed spectrum, and the few pronounce- ments made sound highly cryptic to the modern reader. Then the new quantum theory of the late 20's began to explain the spectra of excited gases in an avalanche of ever-mounting detail, and it was only a question of time until someone would attempt to apply this new theory to the more complex problem of solids. The foundation was laid by Hans Bethe in what is now a classic paper, published in 1929.17 Here was outlined the modern crystal field theory, which views the electrostatic field imposed by the host lattice as the fundamental perturbation to be applied to the known spectral terms of the free atom or ion_ and classifies the new levels thus determined by the representations of the symmetry group -5- of the crystal field to which the new levels belong. It is therefore 18-22 disappointing to find no mention of Bethe's theory in a series of papers by Otto Deutchbein published_ in the early _h_,_.______ _n_ _._ __÷ is" given of his extensive experimental investigations of chromium phospho- re_ence in a series of host lattices. In observing the relative invar- lance of the sharp-line spectra to the particular host lattice type, he was led to believe a suggestion of Saha 23 that such spectra were the result of an electron spin-flip. Thus he was able to correctly surmise that the R lines, were essentially a 4F-to-2G transition by noticing that the energy differences between the various components of these terms in the spectrum of CrlV (measured by White in 1929) 24 clustered about the R line frequencies. From this point on, however, his interpreta- tion goes astray; for example, his assignment of definite term values resulting from spin-orbit splitting in CrlV to the various R line components shows that he was unaware of the quenching_ so well described in Bethe's paper, of orbital angular momenta by crystalline fields. Incidentally_ Deutchbein was the first to make a distinction between true "neighbor lines" and the vibrational sidebands of the R lines. That is, he observed that the latter obeyed Stokes' law, while the former did not. (Stokes' law states that the vibrational sidebands of an electronic transition will be mirror-symmetric in energy about the zero phonon, or "principal" line, and that at low temperatures only the long- wave side will appear in emission, while only the short-wave side will appear in absorption at those same low temperatures. In modern terms, Stokes _ law is a simple consequence of the population of phonon states according to Bose-Einstein statistics.) With the exception of the relatively minor error that he attributed Maxwell-Boltzman statistics to the population density of phonon states, he understood his observations of the vibrational sidebands in an essentially modern way. With the dawning of World War II, the attention of most physicists was diverted to nuclear physics, microwaves, and other technical projects related to the war effort, with the consequent near abandonment of the optical spectroscopy that Just a few years before had been at the very center of physics, both as source and as testing ground for the new -6- quantum mechanics. Thus it is not too surprising that the careful working out of the implications of Bethe's theory for the spectrum of ruby was delayed nearly a quarter of a century, until the appearance, between 1954 and 1958_ of a series of papers by Satoru Sugano, Yukito Tanabe, and Hiroshi Kamimura. 25-29 Their patient and thorough work has provided the basis for almost all modern studies of ions with d electrons in crystal f_elds of cubic and lower symmetries. (The rather precise predictions of the Sugano and Tanabe theory, especially those concerning anisotropy and Zeeman effect of the R and B lines, were given immediate experimental check by I_ Tsujikawa. 50) 1.2. MODERN THEORY OF THE Cr 3+ ION IN A1203 Since the Sugano and Tanabe theory of the "isolated" Cr 3+ ions in ruby has been recapitulated elsewhere, 31'32 only the briefest description possible will be given here, mainly to introduce notation. Figure 1.2 shows the AI 0 host lattice; the Cr 3+ ion is substituted directly 23 for an AI 3+ ion in the formation of ruby. Strictly speaking, the metal ion site symmetry is only trigonal. However, since the surrounding six oxygen ions lie on the corners of a not too badly distorted oetahedron, the crystal field seen by the metal ion can be analyzed as the sum of a large cubic field and a small trigonal one. Figure I. 3 shows the effects of the cubic field on the energy levels of the Cr 3+ ion. The old energy levels of the free ion, for which the total angular momentum i had been a good quantum number, are broken up into new levels, whose eigenfunetions are now appropriate basis functions for the various representations of the cubic group. The new levels thus formed are simply named after the 33 representation of the cubic group to which they belong. Thus, for example, the 4F level of the free ion is broken up into '_A2 , _4TI , and 4T 2 levels, the A 2 being a one-dimensional representation, the T I and T 2 being the two three-dimensional representations. (Here the superscript retains its usual spectroscopic meaning of spin degeneracy, i.e., it equals 2S + i .) Similarly, the 2G level of the free ion is split by the cubic field into 2T 2 , 2T 1 , and 2E levels. In ruby, - 7 - i2 3 02- / a [zoYo] FIG. 1.2--Portion of the AI 0 lattice. = 4.3 ° Remeika, Phys. Re_.3122, 757). (from Geschwind and -8- 2F 2G 4p 2p % 2G 4p Cr IV: 3d2(3F)4s, 4F~113 1(=4"50 B = 918 4F I 4"F2(d_2d If ) 2F2 4A_d g3)4 F 4'Fo Obs_ Cr IV) I 2 3 Dq/B FIG. 1.3--Effects of the cubic field on the energy levels of the Cr 3+ ion_ (from Sugano and Tanabe27) o -9- h the cubic field parameter Dq is sufficiently strong that the T I and 4T 2 levels lie above the 2E level, and since the splitting between J, these two T levels s_d t_ _A_ l....l is ............. _7_±_ proportional to the cubic field strength, the transitions_ 4A 2 -+2T I and 4A2 -+2T 2 -i form two bands, each about 2000 cm wide. By contrast, the energy separation between 2E and 4A 2 is nearly independent of the cubic field strength, and hence the transition 4A 2 -+2E can be quite sharp -i at low temperatures, in the best quality ruby less than 0.i cm In a purely cubic field, the '_A2 _2E transition is magnetic dipole, electric dipole transitions being both spin and parity forbidden. The above picture of the behavior of Cr 3+ ions in a purely cubic field has been well verified experimentally by studies of Cr 3+ in MgO 34-36 In ruby, the small trigonal field (in combination with spin orbit effects) introduces two important perturbations. First, it splits the 2E level into two levels separated by 28 cm -I Sugano and TsuJikawa were ablej.to make positive identification of the RI and R2 lines with the two _A 2 _2E transitions by comparing observed and predicted Zeeman effect and anisotropy of these lines. Second, it induces an electric dipole oscillator transition (oscillator strength f on the order of 10 -6 ) between the _A 2 and 2E levels. (The mechanism by which the trigonal field induces the electric dipole oscillator will be discussed in another chapter.) Although the oscillator strength of the R lines is very small compared to the usual allowed transitions encountered in gaseous spectroscopy, it is sufficiently great to allow R line fluo_- rescence to compete favorably with vibrational-sideband emission and with nonradiative processes; i.e., the R line fluorescence in ruby has a very high quantum efficiency. 37'38 It is no wonder, then, that these lines were among the first to be seen by early observers, even with the most insensitive equipment. 1.3. THE PAIR SPECTRUM Although the R lines and absorption bands were considered rather well understood after Sugano and Tanabe's work, the complex fluorescence and absorption spectrum of the "neighbor lines" remained to be explained. - i0 - o_ S. Jacobs, in a thesis written in 1956, demonstrated that this spectrum is very complex indeed_ as he had tabulated there accurate wavelength measurements of over ll0 lines. 39 Because they usually had only one or two samples of ruby on hand, the earlier workers, including Jacobs, missed the most obvious clue to the origin of the neighbor lines: their increase in intensity, relative to that of the R lines, as the Cr con- centration is increased. This increase in relative fluorescent intensity was first observed 4o and measured by A. L. Schawlow in 1959, and later by Tolstoi and 41 Abramov. Their findings are reproduced in Figs. 1.4 and 1.5, respec- tively. Both found the same general behavior: the ratio of fluorescent intensity of the various prominent neighbor lines to that of the R lines was directly proportional to the Cr concentration 1or all but the higher concentrations studied, where the ratio began to increase even faster than linearly. The above result is most readily explainable by the hypothesis that the neighbor lines are due to interacting, near- neighbor Cr ion pairs. Under the assumption that the Cr ions enter specific A1 sites in a purely random way, the number of near-neighbor pairs of a given type should increase with the square of the Cr con- centration. Hence the first part of the curves of Figs. 1.4 and 1.5 merely represent the increasing proportion of pairs as the Cr con- centration increases. The nonlinear part of the curve was explained in terms of nonradiative energy transfer from the "isolated" ions to the pairs_ whenever the average distance of the isolated ions from the pairs diminished beyond a certain critical size. Incidentally, a certain tendency for the Cr ions to cluster together would not invalidate the above argument_ although no evidence of such clustering has been revealed to date. The most important clue as to the nature of the interaction between the ions of a Cr pair comes from the fact that pure Cr203 is known to be antiferromagnetic below 308°K and to have a Curie-Weiss constant of 550°K; 42'78 this, coupled with the fact that the Cr203 and AI20_ lattices are homologous with nearly identical unit cell dimensions,43 leads one to expect sizable exchange coupling for at least one of the near neighbor types. - ll - J_ 40 o G) .H O .H 40 40 .H © 40 H 40, 30- 20- i0- °o o.2 o.4 o.6 o.8 1.0 Concentration of Chromium in WT_ FIG. 1.4--Fluorescence intensity ratio as a function of chromium concentration, (from Schawlow et al., ). - 12 - o, _.. Iz' .4...,2¸ l.q OG i i i •0 Ol 02 a3 O# I2 Iz / O_ 08 QD _ QG O# Oq J" O_ az c i J o o, a_ o'3o_ o.5 | i . i 0 0.! 02 Q3 Ok Q5 0 Ol Q2 03 _# 0.5C (%) e "08 io.s "0_ Jl / ' " '3 • "_c0 01 O2 0 0._ f. (%) FIG. !o5--Fluorescence intensity ratio N/R for various satellite lines, (from Tolstoi and Abramov 41) . 13- The exchange interaction is a consequence of the Pauli exclusion principle and the overlap of the wave functions of the interacting electrons. For electrons of ion i in orbitals _i and electrons,, of ion 2 in orbitals _j , the exchange interaction can be written 4_ H : -,_; ji_jsi s. , (1.1)ex j i,j where J.. is the exchange integral: ij Jij i 2= _i(rl) 9j(rl) Ir+l - r%I V(r) _j(r2) _i(r2) dr I dr 2 J (1.2) where V(r) means V(rl) + V(_2) The general tensor relation (i.i) can often be simplified to either the isotropic form Hex = - J Sl s2 ' (1.]) or the slightly more general biquadratic form 45'46 + + + ÷ 2 (1._)H -J sI s2 + j I s21 ,ex = Sl where sI and s2 are the total electron spins for each ion. The use of either (1.3) or (1.4) in the interpretation of the ruby spectrum will be Justified by the observed spectrum itself] hence (1.3) or (1.4) will be understood in all future reference to exchange coupling. - 14 - The question of the sign of J is rather complex, and the following is offered merely to suggest someof the considerations involved. If only the first term in expression (1.2) is of importance, then Jij is positive, since it then merely represents the self-energy of the complex * The coupling (1.3) is then called fer-charge distribution _i _j romagnetic, since the lowest energy occurs when the spins sI and s2 are parallel. On the other hand, when the attractive potential of the positive ion cores involved is stronger than the electron-electron repulsions, the second term in (1.2) can predominate, making J negative, and the coupling is then antiferromagnetic. As pointed out by Anderson in reference 44, the antiferromagnetic coupling is by far more common in insulators, and the ferromagnetic coupling described above an es- sentially hypothetical case. The explanation of exchange coupling in real ferromagnetic materials is much more involved. We are now in a position to calculate the effects of exchange coupling upon the spectrum of a Cr ion pair in ruby. Consider first the ground state. Since, for our purposes, the crystal field splitting of the ground state is essentially negligible compared to the optical linewidths of the pair lines, the total spin S = sI + s2 will be a good quantum number for the pair. Hence, we can insert the identity ÷ + 1 _ + 12 ÷sI s2 = _,[1_12 IsI - ls21e] (1._) into (1.3) to obtain H _t[1_12 I_*112 -,- 2..... Is21] (l.61ex 2 from which it can be seen that the ground state exchange splitting is determined by S alone, according to the well known Lande interval role. I _ I I÷s2 I = 3/2 1or the ground state, yielding S = 0,i_2, orSince ,s] = 3 , the resulting exchange multiplet is a quartet. The splittings are illustrated in Fig. 1.6. - 15 - 15 J _yj s =i 2J 3 -gJ 0 S =2 3J _9# ' s=3 4 Ferromagnetic Coupling J > 0 9 IJI s--3 S3 I 0 3 IJI S =2 21JI I ll IJI I s --i -'_F- T IJl S =0 Antiferromagnetic Coupling J<0 Hex = _ J sI s2 j-> -_sll = 1_21 -_ 3/2 .-). -). -._ S = sI + s2 FIG. 1.6--Exchange splitting of a Cr 3+ ion pair ground state. - 16 - The optically excited state of a Cr pair presents a more complex problem. The basic theory has been worked out by A. M. Clogston 47 and can be summarized as follows: The excited state of a pair will be formed from the excited state of one ion and the ground state of its neighbor. Energy transfer between the two ions would split the excited state of the pair into an odd and an even state. But it can be argued that the two possibilities for such energy transfer, the exchange coupling itself and direct dipole interaction, would have negligible effect. Thus the excited state in which the excitation resides on one ion will be con- sidered degenerate with the state in which the excitation resides on the other ion. As for the energy level splittings induced by the ex- change term, the situation is simple when the excited state exchange integral K is much larger than the crystal field splitting: the two spins sI = i/2 and s2 = 3/2 combine to make just two levels of total spin S = 1 and S = 2 , separated in energy by 2]( (The letter K has been substituted for J to indicate that the exchange couplings may not be exactly the same in the excited and ground states.) That K is much larger than the crystal field splitting seems to be true for the nearest neighbors, as the experiments of this study will show. But for the other near-neighbor pairs, the exchange and crystal field splittings are of the same order of magnitude. In this latter case, since the energy levels cannot be simultaneous eigenfunctions of the exchange and crystal field operators, we no longer have to deal with pure S states, and the level splitting becomes rather complex. Figure 1.7 shows the level splitting E as a function of K , in units of the crystal field splitting parameter A , as reproduced from Clogston's paper. With essentially the above theoretical picture in mind, Schawlow began to search the pair line spectrum for evidence of exchange multiplets 48 in the ground state. His method was gmply to compare fluorescence and absorption spectra of a dark ruby sB_ple, both taken at a sufficiently low temperature. (The temperature actually used was that of liquid He 4 , 4.2°K.) If the exchange energies involved could be assumed large compared to kT at 4.2°K, the two spectra should have revealed (for each pair type) - 17 - E,4 / f _J J J o _J1_J f8 Iv5 -B -3 --Z 0 -0.8 -0.6" -0..4 -0.2 0 0.2. K O..4 O.G O.8 /o0 FIG, 1.7--Exchange splitting of Cr 3+ ion pair excited state. K is the upper state exchange i_tegral, and A the crystal field splitting._ (from Clogston47). - 18 - _Q one line as a common turning point, the transition from the lowest-lying of the excited states to the lowest-lying ground state. The fluorescence photograph would reveal only three other lines, and these would be found to the long-wavelength side of the common line; in this manner the full ground state exchange multiplet would be displayed. On the other hand, the absorption photograph would reveal lines only to the short-wavelength side of the common line_ representing transitions from the lowest of the ground state levels to the excited state complex. Figure 1.8 reproduces what Schawlow actually fonnd. A moment's study of the photograph will reveal some of the difficulties encountered in attempting to see the above-mentioned pattern in it: First, there is almost certainly more than one turning point, indicating that onel must separate the overlapping spectra of at least two different pair types. Second, the selection rules are so strange, with vast intensity differences among the various members of a multiplet, that one can never be sure what lines are lost on that account. Finally, some of the pair lines may lie buried in the vibra- tional sidebands, making it difficult to distinguish true pair lines from the numerous sharp features of the vibrational sidebands. Although the above-mentioned difficulties prevented Schawlow from making immediate positive identification of the various ground state exchange multiplets, he was able to determine that two of the more prominent pair lines, at 7009 and 7041 _ (respectively, the N 2 and N1 lines of du Bois and Elias), terminated on levels which would be empty at low temperatures. He was thus led to suggest, 49 shortly before the existence of the first working laser, that if ruby were usable as a laser material, laser action might be more readily obtainable in the N lines than in lhe R lines on account of the greater ease in attaining population inversion in the former. Although the vastly better optical quality obtainable in pink ruby dictated that the first laser would use the R1 line, Schawlow was able to prove his point late in 1960 by observing simultaneous laser action in the N lines of a cooled, dark ruby laser rod (Cr203 concentration approximately 0.5_). 50 As the energy supplied to the flash lamp was raised, first the N2 line alone, then both N 2 and N1 l_nes simultaneously displayed laser action, and - 19 - FYG. 1.8--Emission _nd absorption in dark ruby at 4.2°K, (from Schawlow_°) . - 20 - tjo finally, at the very highest pump powers, these two were joined by the R1 line. (In some instances the order of appearance of N1 and R1 was reversed.) Figure 1.9 shows the spectra obtained from the first N-line lasers. 51 The experiment offered dramatic proof that the two N lines stemmed from distinctly different pair types, as laser action from one excited level of a given physical system almost invariably pre- empts the build-up of excess population in the other excited states of the same system. Final determination of the energy level schemes associated with the two N lines represents culmination of the effort of several workers. From the selective excitation data of Dal_2- and measurements of the absorption strengths of the two N lines as a function of temperature made by Kisliuk, Schawlow, and Sturge, the ground state exchange multiplet and two excited levels of the N2-1ine pair were worked out. 53 Tentative assignment of the ground state exchange multiplet associated with the N 1 line was also made in reference 53, but subsequent measurements by Kisliuk proved this assignment wrong. 54 Kisliuk's temperature data is reproduced in Fig. i.i0, and the complete energy level schemes worked out by him are illustrated in Figs. i.ii and 1.12. (The piezospectroscopic measurements of this thesis have provided strong confirmation of Kisliuk's work, as will be seen later_) Kisliuk was able to fit the two ground state exchange multiplets found in the above-mentioned study to a biquadratic expression of the form (1.4); for the multiplet associated with the N2 (7009 _) line -1 -1 he found J = 6.94 cm and j = 0.14 cm ; for the N I (7040 _) -1 -1 mu_Itiplet he found J = -11.56 cm and J = 0o19 ca" (The signs of both J and j have been altered from those listed in Kisliuk's original paper to make them conform to the opposite sign convention used here.) Thus two exchange couplings of moderate strength had been found so far, one ferromagnetic, the other antiferromagnetic. But since the antiferromagnetic coupling was too small, by more than an order of magnitude, to explain the Curie-Weiss constant of Cr203 , at least one more antiferromagnetic coupling remained to be discovered from the dark ruby spectrum. - 21 - 6,9i9A. 6,934A 7,009A 7041 6,900 7000 7 00 FIGo !og-LN Line laser action in dark ruby. Top spectrum is ordinary fluorescence. Lower three spectra show laser action at increasing pump levels, (f'rom SchawlowSl). - 22 - e" 4O i I I S 20 I0 8 6 4 2 7041 LINE 7009 LI NE FIG. I I I 8 I0 -2 2 4 I/T 6 8 I0 -_ 1.10--Temperature dependence of N line absorption intensities_ (_rom Kisliuk54). - 23 - 14,387.5 361 o 344.2 ¢OF) 0000 2 TE tO0 32.7 7= o ,og:_' rI.IJ 1 1 Rz 1 R I i 1 S=3 2 I 0 b 6 FIG. l.ll--Energy level s_heme of pair type associated with NI , (from Kisliuk5 ). - 24 - i, 324.3"_ 323.8% 299.3_/j 14,298.6 _JL_ h- ODOOIZ) i .J _0-- _00 I i l I _mON _mN t111 --; II i ' II I'-- u') I_.a) I I I I I I I I , I 3 NOTE" UPPER LEVEL IS NOT TO SCALE FIG. 1.12--Energy level s_.heme of pair type associated with N2 (from Kisli_k_ _). - 25 - In addition, the newly determined energy level schemes had yet to be assigned to specific near-neighbor pair types. The problem is very difficult in ruby, s_nce the separation between ion centers increases s_owly for the first through fifth nearest neighbors, as showm in T_oie i.i, making it impossible to exclude any of these on the basis of a too large ion separation. 55 8 TABLE i. I Near neighbor pair type Separation between ion centers, ist 2.75_ 2nd 2.809 3rd 3. 185 4th 3.504 5th 3.774 6th 4. 766 It was thought that the known spin arrangement in Cr203 , as determined from neutron diffraction studies, 56'57 might provide a clue. The ar- rangement is shown in Fig. 1.13, as reproduced from a paper of Tachiki and Nagamiya. 58 From the figure it can be seen that in pure Cr203 the spin arrangement is antiferromagnetic for the first and second and third nearest neighbors_ and ferromagnetic for the fourth. Of course, the complete spin arrangement in Cr203 may be dictated by the two strongest exchange couplings, say that of the first nearest and one other pair type; then there is no necessary relation between that spin arrangement and the signs of all the smaller exchange couplings. In the meantime, Statz et al., thought that their paramagnetic resonance studies of Cr ion pairs in ruby 59'60 had provided the fol- lowing information: (i) J values for all but the first, second, and -i fourth nearest neighbors could be declared less than or equal to i cm ; (2) J was of the order of 390 cm -I for the nearest neighbors; (3) no - 26 - t(_ = REFERENCE ION FIG. 1.13--Antiferromagnetic spin arrangement in Cr203 as known from neutron diffraction studies. The reference ion is shown cross-hatched; only Cr ions are shown. The figure shows antiferromagnetic spin arrangements for pair types i_ 2 and 3, and a ferromagnetic arrangement for fourth neighbors_ (from Nagamiya5_). - 27 - signal could be found corresponding to the second nearest neighbors; (4) the assignment of a resonance signal to the fourth nearest neighbors was considered highly tentative. Doubt was firs_...... _+_ upon +_ v_d±_N 61 of Statz's work by J. Owen, who pointed out the extreme technical dif- ficulties involved in making unambiguous identification of the first nearest neighbor pair resonance signal. But the optical studies m_ntioned above undermine Statz's most fundamental assumption about the various near neighbor pairs -- that the splitting of the ground state of a pair -i will be exactly the same as that observed for the isolated ions, 0.38 cm or 11.5 kMc. On the contrary, the observed splittings for at least the first through fourth nearest neighbors are often several times that value. -i For example, see the beautiful i cm splitting of the 7041 _ line, or the complex fine structure of the 7009 _ line reproduced in Fig. 1.14, from reference 31. Since Statz' method for identifying resonance signals with pair types depends critically on the value assumed for the ground state splitting, those identifications would seem rather doubtful, at least for the first four or five nearest neighbor types. It is no wonder, then, that the conclusions of this report will often deviate from those of Statz's paper. Thus one is led to conclude that any attempt to assign the known exchange multiplets to specific pair types on consideration of the above two studies-- neutron diffraction and previous paramagnetic resonance studies -- would amount to little more than guesswork. But this negative conclusion is not without benefit, since it has provided at least part of the impetus for the piezospectroscopic studies to be described here. Although the original intent was merely to make positive identification of the particular near neighbor types associated with the two N lines possible, the piezospectroscopic studies have paid a rich harvest of dividends over and above that first object. The accomplishments of this work can be listed as follows: (i) Positive identification has been made of the N 2 and N I multiplets with the second and fourth nearest neighbors, respectively. I m - 28 - a= ! CM -I _. 7041 A i ,_. I CM -I = 1 FIG 1.14--Ground state splitting of N and N lines at 4 2°K Exactly the same relative intensities of the various components is seen at !.4°K _ (from Schawlow31)o 29 - (2) The previously missing first and third nearest neighbor spectra have been found and the corresponding ground state exchange multiplets determined. A relatively large antiferromagnetic coupling (J N -46 cm -I) was found for tile third neighbors, in contrast to previous conclusions that the J value for this -i neighbor type was less than i cm in absolute value. An even stronger exch_ng_ _ f_,1_a ?_ _ _-_ __ ...... ( .o_. 5 -i_ (3) The four J values determined experimentally in dark ruby agree rather well with all that is known about the spectrum and magnetic properties of pure Cr203 ; at least to within the limitations on accuracy imposed by the theories used in making the correspondencesj for details: see Chapter V. It is hoped that the J values given by this report will stimulate the generation of better theories of magnetic ordering in Cr203 and related structures. (4) Positions and piezospectroscopic sensitivities of the newly dis- covered lines should pave the way for a valid theory of the opti- cally excited state of ion pairs. (5) The rather high sensitivity to mechanical stress of the frequencies of the lines investigated; especially that of the strongest line of the third neighbor complex; offer the possibility of making a readily and highly tunable laser. - 30 - _Q II. THE PIEZOSPECTROSCOPIC METHOD 2.1. INTRODUCTION This chapter describes the piezospectroscopic experiments with the object of identifiying the N lines and other pair lines with specific near neighbor Cr ion pair types. Background information relating to the symmetries of the host lattice and the locations of the various pairs relative to the symmetry axes of that lattice are given, leading to development of a scheme for interpreting the piezospectroscopic be- havior of the pairs. In addition, a brief description of the techniques used in the orientation and cutting of the ruby samples is given, and, finally, the piezospectroscopic apparatus itself is described, along with techniques related to its use. Results of the piezospectroscopic experiments are given in Chapter III. 2.2. DISCUSSION Experimental physics can be thought of as consisting of two phases -- one passive, the other active. The passive phase was the more common mode of earlier times, as exemplified by the astronomical observa- tions that led to Newtonian mechanics, or the early spectroscopic work that led to the quantum mechanics. More recently, the bulk of our in- formation about the atomic world has come from observations in response to active prodding. Often the desired perturbation is produced by an externally-applied, static, magnetic, or electric field. But in solid state physics the effects of externally applied fields sometimes are masked by the effects of the usually much larger internal crystalline field. For example, Zeeman studies of line spectra in solids are often precluded by too-large residual linewidths due to internal strain° When such a situation arises_ it is found that mechanical stress, unsophisticated - 31 - as it may be_ often provides the only effective "handle" on the atomic system umder study. This success can be explained by the fact that mechanical stress almost guarantees a certain percentage modulation of the internal field in question, no matter how large that field maybe. And even whenother probes can be used_ the added information of a stress study maybe just that necessary to allow a complete solution. Thus_ for example_ in order better to understand the crystal field splittings of the 2E level of Cr3+ in various noncubic host lattices, such as AI203 or Ga20] _ A. L. Schawlowwas led to try to induce such a splitting in the normally degenerate 2E level of Cr3+ in MgO by the application of uniaxial stress along the (i00), (ii0) or (iii) 35 axes. A well resolved splitting was in fact produced_ and was observed to increase in direct proportion to the applied stress. Stress sensi- -i tivity was such as to allow a splitting of approximately 0.6 cm for a stress of I0 Kg/mm_- (approximately i0 × 108 dynes/cm 2) applied along the (i00) direction. (Although dynes/cm 2 are the truly correct units, the more practical laboratory units of Kg weight/mm _ will be used here to avoid an unnecessary conversion, and by reason of the better "feeling" most physicists seem to have for the latter.) In a theoretical treatment by S. Sugano (also in reference 35)5 a calculation of the splittings, in which the crystal field was based on a point charge model, produced ex- cellent agreement with experiment_ However_ the same model predicted a shift to the blue in the average position of the two components of the split line, by reason of the increased cubic field strength with applica- tion of pressure; the experimentally measured shift was to the red. The only plausible explanation, according to Sugano, was that the Racah parameter B (describing the coulomb interaction between the t2 electrons of the Cr ion) was made to decrease faster with pressure than the cor- responding increase of the cubic field° Nearly the same red shift was found for each of the two R lines of ruby (approximately -0.3 cm -I for a stress of I0 Kg/mm_), again roughly independent of the direction of applied stress. In addition to their intrinsic theoretical interest, the above-mentioned studies were valuable_ at the time they were made, for suggesting that the limit to line narrowing at low temperatures can - 32 - Bbe attributed to a residual microscopic strain field. Now they are valuable for purposes of comparison with the stress induced shifts of the N lines. Piezospectrosnopic studies of the ruby N lines were first made by Kaplyanskii and Przhevuskii, who concentrated their efforts on the N I 62 line. They observed an apparent splitting which was highly dependent upon the direction, relative to the crystallographic axes, of the applied stress. Now in ruby, some of the near neighbor pair types can lie in any one of three possible spatial orientations. In an unstressed crystal, these three orientations are energetically equivalent, by virtue of the three-fold rotation symmetry operator (C3) of the host lattice, and only one line is produced° But C 3 will be destroyed by application of stress to the crystal along any direction, save that of the C3 axis itself, with the result that the different pair orientations are no longer equivalent. Again here, as in the study of the RI and R2 lines, there is no longer a question of splitting the energy levels of one particular pair, since all orbital degeneracies already have been lifted. By observations on the splitting patterns produced for a few different sample orientations (P IIc2 , P IIc3 , P I c2 and C 3 , _L_C 2 = _/4) , and on the rather complete polarizations of some of the individual stress- produced components, Kaplyanskii was l_d to assign the N1 line to the sixth nearest neighbors, although he admitted that his data could be fit almost equally as well by the second neighbors. Since the crystals available to us were of much higher quality than those used by Kaplyanskii, and since we intended to extend his observa- tions anyway, especially to the other prominent pair lines, we have con- sidered it worthwhile to repeat Kaplyanskii's experiments. Hence his results will not be reproduced here, but instead we merely note that those results (as far as they went) are in good agreement with our own. But we strongly prefer the second of the two assignments suggested by Kaplyanskii for the N 1 line, both on grounds of our extended stress data, and for purely theoretical reasons related to the observed strength and sign of the exchange coupling associated with that line. - 33 2.3, SYMMETRIES OF AI203 AND LOCATION OF THE PAIRS The AI203 host lattice belongs to the point symmetry group D3d (Schoenflies notation), alternately known as 3--m (Hermann-Mauguin symbol). A picture of tlhe symmetry elements and the corresponding _tsreographic projection are shown in Fig. 2 io The group D3d formally 2 C2contains five classes in addition to the identity operator_ C 3 _ C3 , _three of these); m(mirror plane - three of' these also); and an inversion center In sapphire (pure AI203) or ruby, the C 3 axis is also known _,_t the optic axis, since the index of refraction for light polarized parallel to this axis is different from the index for light polarized perpendicular to it; this fact allows the C 3 axis to be located with tlhe aid of a pair of polarizing filters The other axes can be located only with the aid of x-ray techniques; more about orientation later The location of a Cr ion pair will be described in terms of the line joining the centers of the two ions, or the "pair-axis." Among the pair types under consideration here, only two -- the first and fifth nearest neighbors, have axes parallel to the C3 axis. Hence these two are only one of a kind_ and their lines should never split under application of stress,, But the axes of all the others seco_,i_ Ithir°d._ fourth, and sixth _ ,_ _'. s_e-__e to ; in this latter situation the C, operator produces two additional equivalent pairs from any given one. Thus_ as mentioned previously, the lines produced by any member of this second group should undergo a _wo_ or three-fold splitting for all but a few special directions of the applied stress° Figures 2.2 and 2 3 show the l.ocations of the various pair-axes. It should be noticed from Fig. 2_.2 that the second_ third and fourth nearest neighbor pairs all lie in a common mirror plane. To specify completely the locations of these three, it is necessary to establish a convention for measuring angles in the mirror plane, relative to the C 3 axis. In the one chosen, the zero angle is perpendicular to C3 , and the quadrant containing the second and third neighbor axes arbi- trarily made positive. Thus the second and third neighbors lie at +11.7 ° and +30.4 °, respectively, while the fourth lies at -_8.3 °. oC Dsd - 3m FIG. 2.1--Symmetry elements of the point group D3d and their stereographic projection. (from Wood 64) - 37 - _earest neighbors _" _\ plane \ = L!. 7° Fourth \ nearest neighbors q ° ., o_'O _' r _ _ 120- d b) Lookir_ down the C3 axis c 3 ± 90 ° -38o3 ° a) ]_cation of pairs e) convention definir_ angle @ FTGo 2°2 ......Locations o:f the seeond_ third and fourth neighbor pair axes. - 36 - J 0o% 0 ivo 0 0 rio ,,e.I 0 0 4-_ N g-t 0 0 .r-I o i I d I--4 O_ rJ - 37- Use of the pairs themselves in the definition of the angular convention allows the most direct relation possible between the actual pair location in the mirror-plane slab from which the samples are cut and certain features of the back-reflection Laue photograph used in its orientation. The above should not be constructed to imply that plus and minus ends can be defined for the C 3 axis; on the contrary, the +90o and -90 ° directions are identical, or to put it another way, all possible direc- tions are encompassed in traversing from -90 ° through 0° to +90 ° (see Fig. 2.2c). Finally, it should be pointed out that the sixth nearest neighbors are in a class by themselves, since their axes lie parallel to the C2 axes (see Fig. 2.3). Thus they lie at right angles to the other pairs, and this fact will be useful in developing a scheme to differentiate between lines due to them and lines due to the second nearest neighbors. 2.4. IDENTIFICATION OF THE PAIR LINES From the above it is obvious that lines due to the first and fifth nearest neighbors can be readily identified by their refusal to split under stress. Furthermore, since the exchange coupling is certainly known a priori to be much larger for the first than for the fifth nearest neighbors_ knowledge of the exchange multiplet should allow the identifica- tion to be complete. It is not nearly as easy to make positive identifica- tion of the others; one must somehow take advantage of the relatively large angular separations between their various axes. The idea used here is that the energy levels of a pair should be more sensitive to stress applied along the pair axis than to stress applied at right angles to it. Although the above idea may seem a bit arbitrary, originally it was not without a rather clear rationale. That is, it was thought that the observed shifts would be due in the main to induced changes in the ex- change coupling. Pushing the ions together would increase the overlap of their electronic wavefunctions, thereby increasing the exchange coupling; whereas stress applied at right angles, in tending to push the ions apart_ would have an opposite effect. On the other hand, ex- periment has shown that modulations of the exchange coupling can account - 38 - tfor only a small fraction of the observed shifts. But those same experi- ments have shown also that the more general notion that the energy levels of a pair should be most sensitive to stress applied along the pair axis is indeed correct. The above can be made less vague by postulating that the induced shift should be directly proportional to the component of strain in the direction of the apir axis. Figure 2.4 illustrates the basic mechanics involved. The sample is a long rectangular rod of square cross-section, with a uni-axial stress SO applied parallel to its long dimension, the "sample-axis." The axis of the ion pair in question lies cocked at an angle _ to the sample axis. The stress parallel to the pair axis can be computed from the simple relation 63 So = -- • (l+ cos (2ol) 2 Assuming an elastically isotropic material with Young's modulus Y and Poi_son's ratio p , the strain e(_) along the pair axis can be written so{ }e(_) = -- (1 - p) + (1 + p) cos 2J_ (2.2)2Y Now consider a fan of samples cut from a common slab containing the pair axis and spanning a range of 180 °. With the exception of a term constant in _ which will be of no real interest here, the frequency shift Av for a constant magnitude of stress will undergo a very simple variation as the samples are tested in order, that is Z_(_) = Z_ 0 cos 2_ (2.3) - 39 - So II11] s(_) s(_) Pair axis Sample FIG. 2.4--Mechanics of sample under stress. -4o- We are now ready to compute the splitting pattern expected for the pair types that are three of a kind. The situation is illustrated in Fig. 2.5. Here YZ is the mirror plane from which the samples are cut; the Z axis represents the CB axis° Inclination of the sample axis is described by the angle @ For purposes of illustration, three pairs are shown with axes tilted up 30° from the plane normal to C3 ; one of the three lies in the YZ plane (hereafter known as I), and the other two out of it (pairs II and III). For pair I, we have _ = @ - 30 ° For pairs II and III, several intervening steps of trigonometry are required to derive the relation cos 2_ = 0.439 cos 2(9!+ 49 °) - 0.561 The result of plugging these two relations into expression (2.3) is illustrated graphically in Fig. 2.6a. The above example corresponds to the third nearest neighbors. Figure 2.6b shows the results of a -similar computation for the fourth nearest neighbors. Several qualitative features of the curves of Fig. 2.9 should be pointed out here. Notice that in each case the curve corresponding to pairs II and III has about half the peak to peak amplitude of the curve corresponding to pair I, and that the smaller amplitude curve is shifted about 90 ° from the larger one. The latter fact can be readily understood by observing that for the third and fourth neighbors the three pair axes of each t2pe form an approximately mutually orthogonal set of directions. Thus, for these two pair types, when stress is applied along pair I, it is approximately at right angles to II and III. Furthermore, the two sets of curves form a complementary set -- the one being very approximately the mirror image of the other across the 0 ° line. By way of contrast, Fig. 2.7a shows the splitting curve calculated for the second neighbors -- here the amplitude ratio between the curve for _air I and that for pairs II and III is greater, but the angular shift between them Ts cohslderablysmaller. Figure 2._o shows the curve calculated for a sixth neighbor pair; now the II and III curve has the larger amplitude, the curve for I being a constant, since the stress is always applied at right angles to it, regardless of the value of @ - hl - i i i i I I I t i z_ C3 x Mirror plane \ X \ \ Y I FIG. 2o_--Diagram showing angles used in computing splitting pattern for for third neighbors. - 42 - o0 0 0 ÷ 0 0 I 0 ! 0 +1 I1) -0 t_ -0 4_ 0 4_ P-t I ! H - 43 - 0I 00 I ! o 0 0"_ 0 0 cY_ 0 0 ! 0 ! OJ 0 OJ _ _ CO 0 • • • • • e I ! ! I ! 0 4_ 4_ ,_ 4_ 4_ _4 O t_ q-t I ! d d I-4 t - 1,4 - 11 ! I I (_) 1 0 FIG. (b) 2.7--(a) predicted secor_d _i_'::bor splitting pattern. sixth neir;hbor splitti> _ 9_ ........ - 45 - (b) Pre6ict ._,l Thus, the schemefor identifying pair types consists simply in trying to match up the observed splitting patterns with the patterns computed above. Although the above is obviously an over-simplification, the numberof choices possible in the assignment of a given line to a pair type is so limited (only three or possibly four) that no more complicated theory has been found necessary. 2.5. ORIENTATIONANDCUTTINGOFTHESAMPLES Except for an initial rough location, by optical means, of the C_ axis of the boule from which the samples were cut, all orientation was done with the aid of back-reflection Laue photographs. The essential elements of a Laue camera are shownin Fig. 2.8. A beamof white x-rays passes through a hole in the film and strikes the sample. Usually, the minimumwavelength available is sufficiently short to allow satisfaction of the Bragg condition for all atomic plane spacings and angles. Hence the various atomic planes are able to reflect the beamback onto the film, muchas a set of plane mirrors would reflect a beamof light. Since the AI203 lattice contains a center of inversion, there is a one-to-one correspondence between the symmetry of the Laue spot pattern 64 and the crystal symmetry. This fact makes identification of the various symmetry axes very easy. Figure 2.9a shows the beautiful spot pattern produced whenthe x-ray beami_ parallel to the C_ axis; three mirror planes can be identified in it in addition to the three-fold rotation symmetry. By contrast, the pattern shownin Fig. 2.9b has but one symmetry -- the two-fold rotation of the C2 axis to which the x-ray beamwas parallel. After a pattern such as Figs. 2.9a or 2.9b had been obtained, small angular corrections could be madeon the crystal holder, such that the crystal axis in question would be parallel to the x-ray beamdirection to within about 0.2 ° or better. The holder was so arranged that the information thus obtained could be tra_sferre_ accurately to the diamond sawused for sample cutting; whenever a crystalline axis was located it was permanently marked by cutting a plane normal to it. Thus, for 0_.) 0 O4 _4 - _7 - ' I M 0 Q 4 \ FJ,g, 2,9a -_ac., r¢i'l_c_io_ Lsuc _- -_,. o _ ..... AI_O,,: (a) X-ray -!_:_m _," uallol . _._ot_,_r_pn_ oi _i _X-ray ' ..... " I •to C..; M M indicates edge of mirador plane, (b oeam po_'.__Lei .,o Cp_ C C_>J!lc]Jeatos row of Laue spots produced by C< zone axis; thus C 5 is at ri:77£: a:,-]les to C C'; 4 4 f indicates row of Lade ::pobs produced Ly zo_c d]zecuzol , heG(_(: ]_ands pazalle: to fourth neighbor axis "" _" _- 4' is perpendicular to oct<:ai ±'our%]i-_"_'"m_z_**oor axis diroctio_ - 48 - example_ a cut made normal to a C2 axis would _ _Yaetlv_ oarallel_ to a mirror plane; a second cut parallel to the first would then produce the desired mirror plane slab. Location of the pair axes in the mirror plane slab was accomplished on consideration of the following. In a crystal, all the atomic planes parallel to a single direction are known as a zone; that direction it- self is the zone axis. All the Laue spots produced by a given zone lie on an hyperbola; in the special case where the zone axis is normal to the x-ray beam, the spots lie on a straight line running through the center of the photograph. Now the fourth nearest neighbor pair cor- responds to such a zone axis in a very simple way: it merely represents two successive sites in an infinite row of AI ions whose direction defines that particular zone axis. In a '!C2" photograph, the fourth neighbor zone axis produces a very prominent row of spots. The identity of that row can then be checked by measurement on the photograph of the unique angle between it and the equally prominent row of spots produced by the zone axis parallel to C3 (There is no question of confusing the two, since the location of C_ is already known.) Orientation of the crystal is then complete, but as a double check similar measurements can be made on the less prominent rows of spots produced by the zone axes associated with the other near neighbor pairs. Finally, in transferring information from the photograph to the crystal, it must be kept in mind that the row of Laue spots is located at right angles to the zone axis that produced it. After the mirror plane slab had been oriented in the above manner, the various samples were cut from it and individually boxed and labeled. Final sample dimensions were usually about 1.5 mm square in cross section and about i0 mm long. Before each sample was used in piezospectroscopic measurements, its cross-section was measured with a micrometer, and the e_act dimensions recorded. " - 49 - 2.6. THE PRESSURE APPARATUS The simple mechanical press used in the piezospectroscopic studies described here is shown schematically in Fig. 2.10_ and Fig. 2.11 is a photograph of it. The lever arm assembly multiplies by a factor of 60 times the force supplied by a set of adjustable weights; the multiplied force is then transmitted by a stainless steel push rod to the sample i_elf. Thus a set of weights of convenie_ size(providing up to 6 Kg) could be used to apply stresses in excess of i00 Kg/mm _ to samples whose 2 cross-section was of the order of 2 to 3 mm . The assembly containing the push rod was made entirely of stainless steel in order to minimize thermal conduction from the top end, at room temperature, to the end containing the sample, which was immersed in liquid N 2 Two sets of windows, cut into the sample holder at right angles to each other, allowed the sample to be exposed to pumping light and permitted the resulting fluorescence to be transmitted to a spectrometer. In order not to over-constrain the ends of the sample, such that a nonuniform stress might result over the sample cross section, force was transmitted to the samples through two "pillow%" consisting of 1/4" dia. copper disks punched from 1/32" thick sheet. The copper was able to undergo plastic flow, even at the working temperature of 77°K, allowing the sample to form a deep impression in it after pressures of the order of 50 to i00 Kg/mm _ had been applied. Of course, new copper disks were supplied for each sample tested. That little or no broadening of the line components could be detected, even at the highest values of stress used, attributes to the effectiveness of the above scheme. Figure 2.12 is a photograph of the holder with a sample mounted. It was discovere_ through the rather bitter experience of losing one or two samples that had cost a great deal of labor to produce before any data had been gathered from them, that the samples would not take very much pressure while they were still at room temperature. Evidently the various microscopic crystalline defects are able to propagate and grow much more rapidly and to a greater extent at room temperature than at 77°K, since most samples were able to withstand pressures up to - 50 - x0 T •f-.t Q_ r_ E_ t_ .r-I c_ ,-.4 _ I i r-4 r_ _o r.-I c_ t_ _rd a_ 0 © _H 0 © c_ cH 0 \ © c_ or-I c_ E_ I ! 0 c_ H ?.IC. 2.11--Photograph of the mechanical press. The arc lamp can be seen behind the dewar tip; part of the photographic spectrometer can be seen to the left. - 52 - . - . . FIG. 2.12--Detail of press, s h o d 3 s-,rqle ir, holder, - 33 - Cooling water for lamp __Ip-_.out_ / I Cooling water tubing . / I -_/_ r-'- tl-- Standard Go E. water _ _ Jacket assembly _ _ t _°u_°4 _°_" _!_ J _i-L_'/////-Yl _ing'c°aled by conduction_'_' _ _ through heavy Cu ITr Z _ HI I "///// *==._.._,. "////////_/i]/] * __ G.Eo type AH=6 lamp arc 25 mm x 1.5 mm DIA. IKW electrical i_put r i / Sample Plastic fresnel lenses condense arc light onto sample FIG. 2.13--Mercury arc lamp and housing. - 54 - lOOKg/mm _ for an indefinite length of time at the _"_+_c_ temperate[re, while they would crumble under a pressure only 1/3 or 1/4 as large at the former. Sample fluorescence was excited by a 1 kW, high pressure, water cooled mercury arc lamp (G. E. type A-H6). The high pressure mercury arc makes a nearly ideal source for pumping ruby, as a very large frac- tion of its electrical input (nearly 20_) appears as light coincident with the pump bands, and only a small fraction appears in the red and infrared. All trace of the latter was eliminated by filtering the light through a water solution of copper sulfate. Arc light was imaged on to the sample by two plastic Fresnel lenses of low f number, placed with their flat surfaces back to back, thereby creating a space for the copper sulfate solution. Lamp, housing and filter were all cooled by the same stream of tap water. The lamp assembly is shown in Fig. 2.13. Piezospectroscopic studies of the 7009 _ and 7041 _ lines'were performed with a large (1.8 m focal length) scanning spectrometer made by the Jarrell-Ash company. To take best advantage of. the grating blaze angle, the instrument was used in eighth order. In that order, instru- mental resolution was nearly an order of magnitude greater than the linewidths involved, even when the slits were opened up to 50 _ to allow a large light gathering power. The detector used was an R.C.A. type 7265 photomultiplier with a cathode of S-20 response. Studies of lines further to the red were made with a dual grating, photographic instrument produced by Bausch and Lomb. The two gratings permit choice of either 4 X/ram or 8 X/ram dispersion. Kodak type I-N plates were used in conjunction with this instrument because of their o nearly uniform response from the visible red to 8000 A. - 55 - III. RESULTS OF THE PIEZOSPECTROSCOPIC STUDIES 3.i INTRODUCTION This chapter presents detailed information relating to the production of "splitting patterns" - curves of line component shift under a standard stress as a function of sample axis angle @ The splitting pattern of a newly discovered line of high piezospectroscopic sensitivity, occurring at 7452 _ and here named the NO line, is given in addition to the split- ting patterns of the well known N I and N2 lines. By matching those patterns to the theoretical patterns predicted in Chapter II_ the N O is identified with third nearest neighbors, the N I with second nearest neighbors_ and the N2 with fourth nearest neighbors. In addition, the principle that all the members of a given exchange multiplet exhibit a common splitting pattern unique to the particular pair type involved is discovered and used to identify part of the first nearest neighbor exchange multiplet. Finally, the various sources of error and probable accuracy of the piezospectroscopic data are discussed. 3.2 SPLITTING OF THE NI AND N 2 LINES The N I and N2 lines were selected for the first studies, since these two were the strongest representatives (in fluorescence at low temperatures) af the only two exchange multiplets known at that time. Since these two are so easily seen in fluorescence, the samples used in their study were cut from a boule of approximately 0.15_ Cr203 by weight, in order to take advantage of the higher crystal quality avail- able at that relatively low concentration. The raw piezospectroscopic data obtained on the N I (7041 _) line were the easiest to reduce, since the splittings Gotained with the higher values of stress were usually well resolved for all but a few angles. Figure 3.1 shows several typical recorder traces of the splitting of the N I line. For each sample, the shifts of the various line components tO SPLITTING OF 7040 A LINE _ II cm-I b I \- FIG. 3.1--Recorder traces of NI line splitting. - 57 - away from the zero pressure position were measured carefully_ and a plot of shift versus pressure was made. In every case, the data points were found to make a close fit to a straight line passing through the origin. V o]_.... o _ _o_+_ ......_,_zed to a pressure of i00 Kg/mm 2 could then be read from the shift versus pressure plots, and a graph of shift versus angle made. Figure 3.2 shows the resultant splitting pattern for the N line: where ...........A m__ a_!es in +_^_ mirror _ _v_±_ to l the convention established in Chapter II. Splittings of the N 2 (7009 _) line were not always as well resolved, and as a consequence the only data points used were those obtained with the highest stress the sample would withstand_ on the order of i00 Kg/mm 2 or slightly higher. Even then, in many instances the exact positions of some of the line components had to be determined by fitting a pair of hand drawn curves to the actual recorder trace, such that their sum would reproduce the unresolved trace. The curve fitting is fundamentally an iterative process; and in the more difficult cases several successive approximations were made before a good fit was obtained. Figure 3.3 shows several recorder traces of some of the better- resolved splittings of the N 2 line, and Fig. 3.4 shows the mirror plane splitting pattern produced by that line; here, again, all values of line shift were normalized to a standard pressure of i00 Kg/mm 2. Except for the extra splitting of the lower amplitude curve, Fig. _.4 strongly suggests the theoretical splitting pattern predicted for the fourth nearest neighbors. However, the extra splitting has a rather simple and obvious expla- nation on the basis of a fourth neighbor assignment. Until now, we have talked only in terms of the symmetries of the lattice, thereby giving the impression that the two pairs lying out of the mirror plane slab from which the samples were cut were true mirror images of each other. (From now on, that pair of any given near neighbor type whose axis lies in the particular mirror plane slab from which the samples were cut will be referred to as I , whereas the two lying out of that mirror plane will be referred to as II and III.) That picture is indeed correct when only the metal ions are taken into account. However, when the arrangement of all the ions in a unit cell is considered, the mirror symmetry - 58 - -1 12 cm K Angle of applied stress, e I0 20 30 50 60 70 80 90 FIG. 3.2--Splitting pattern of N I • Magnitude of stress is i00 Kg/mm 2. - 59 - I I I I I -90 -80 -70 -60 -50 -40 -30 -20 -i0 +I0 Angle of applied stress, I i t I I ' I +20 +30 +40 +50 +60 +70 +80 +90 FIG. 3.4--Splittirg pattern of N 2 Magnitude of stress is i00 Kg/mm 2 . " - 61 - degenerates into a glide plane_ and pairs II and III are no longer equivalent• Most probably_ the single oxygen ion that almost intervenes between the two Cr ions of the fourth nearest neighbor pair is of _o+_ _u_ _u_ than any other The Cr ion pair might then be more properly thought of in terms of a Cr-0-Cr complex_ having a flattened v shape. The Cr-O-Cr complexes and their relation to each other are illustr_a _ Fig. _ _ Perhaps it should be pointed out here, if only parenthetically, that Fig. 3.5 really shows only half the number of possible Cr-O-Cr complexes for a given pair type. However_ the other three complexes can be shown to be stress equivalent to the first set. In Fig. 3.6, the two sets are labeled "a" and "b" hence the individual complexes are labeled la, Ib, lla, etc. The "a" set is related to the "b" set through the inversion center of the AI203 lattice, and since the inversion operation produces no change in an even perturbation such as stress_ the complexes la and Ib are stress equivalents; in like manner_ lla is stress equivalent to lib, and llla is stress equivalent to llIb. Although the above has been stated in terms of the Cr-0-Cr complex, the argument holds equally for the pair taken together with its complete environment. Tentative assignment of the N I and N 2 lines could be made at this point, but that assignment will be delayed until the rather startling piezospectroscopic behavior of a pair line, discovered far to the red of the N lines_ has been described. 3.3 SPLITTING OF THE LINE AT 7452 After the above studies of the N lines had been completed, it was thought that the pressure technique could be used in a search for the missing exchange multiplet of the first nearest neighbors• Discovery of that multiplet would provide one of the most significant pieces of information to be obtained from the spectrum of dark ruby, since it was realized that the first nearest neighbor exchange was almost certainly much greater than the two known so far, thereby making the first neighbor exchange the one most significant in the determination of the antiferromagnetic arrangement of Cr203 . Although the lines of the first neighbors would never split under stress, their piezospectroscopic - 62 - 0- 63 - IIIb Looking along C3 Ia Ib IIa i ! Inversion _ _ Center _ay_ /l _-. I I / Mirror plane is plane of paper FIG, 3,6--Stress equivalent sets "a" and "b" of Cr-O-Cr complex. - 64 - sensitivity would surely be considerably greater than that of the R lines, thereby allowing them to be distinguished from the various slmrp features of the vibronics of those lines, through comparison of a survey plate of the ruby spectrum taken at zero stress with one taken with stress applied to the sample. Since the lines in question, if at all detectable, would be much weaker than the N 1 and N2 lines, two new samples were cut from a boule of 0.29% Cr203 by weight, with sample axes parallel to C3 and C2 , respectively. But before the first neighbor lines could be ferreted out with the aid of these two samples, the very first plate taken with pressure applied parallel to C2 showed a large, three-fold splitting of a line at 7452 _, and the first neighbor search was temporarily sidetracked. Intensity of the new line was also known to increase steadily from zero relative to that of the R lines as the Cr concentration was increased° Could the new line possibly be attributed to the previously m_ssing third nearest neighbor pair? To find the answer, a new mirror- plane fan of samples was cut from the 0.29% Cr203 boule mentioned above, and a piezospectroscopic study of the new line carried out. (To facili- tate constant reference to it, the new line at 7452 _ was named by us as the "No" line, consistent with the pattern established by DuBois and Elias. Although their numbering scheme has lost its original meaning_ now the decreasing subscripts 2, l, 0 can be taken to indicate increasing piezospectroscopic sensitivity. ) The only difference between the studies on the N] and N 2 lines described above and that made on the N O line lay in the fact that the latter was carried out on a photographic instru- ment, rather than on the scanning spectrometer. Although marginal for piezospectroscopic study of the N2 line, resolution of the photographic instrument was more than sufficient for study of the NO line, since the width of that line and its piezospectroscopic sensitivity were, respectively, about four and ten times as great as those of the N 2 line. Choice of the photographic instrument was then a matter of convenience, the most important consideration being that the piezospectroscopic behavior of the whole ruby spectrum would be recorded simultaneously with that of the N O line under study at the time. As will be revealed later, important additional infor- mation was gleaned from a re-examination of the plates used in the study of N O - 67 - Positions of the stress produced components of the NO line were measured with reference to the line at 7438.90 _ of a superimposed neon spectrum. Figure 3-7 shows a few of the plates. The actual measurements were ÷_ir_ from ....u_ trace produced by a recording microphotometer, rather than directly from observations of the plates themselves through a magni- fying glass, in order to facilitate accurate location of the line centers. The _hifts thus determined were then awl normalized to the standard pressure of I00 Kg/mm2; the mirror-plane splitting pattern plotted from the normal- ized shifts is shown in Fig. 3.8. 3.4 ASSIGNMENT OF PAIR TYPES TO THE N O AND N 2 LINES Figure 3.9 reproduces the splitting patterns of the N2 and NO lines side by side to facilitate comparison; the striking similarity of the two patterns_ when the one is seen as an approximate mirror image of the other across the 0° line, leaves little room for doubt that these two represent the complementary set of splitting patterns expected for the fourth and third nearest neighbors_ respectively. Note that the similarity in shape exists in spite of a difference in scale (of the shifts) of approximately one order of magnitude; the maximum splitting between components is on the -i order of 5 cm for the N 2 line_ while the corresponding splitting is approximately 50 cm -I for the NO line. The extra splitting (between components II and III) should have exactly the same explanation for the N O line as was given in Section 3.2 for the splitting pattern of the N 2 line. That is_ just as for the fourth nearest neighbors, the third also have one oxygen ion that almost intervenes between the two Cr ions_ forming a v-shaped Cr-0-Cr complex. Figure 3.5b shows the Cr-0-Cr complexes of the third nearest neighbor pair. In addition to the splitting pattern itself_ the difference between the shift of component I and the average of components II and III can also be plotted as a function of e ; that plot will be called the "splitting curve_" in order to distinguish it from the splitting pattern. In Fig. 3.9 the splitting curves are shown below the corresponding split- ting patterns. Note that the maxima of the splitting curves make much better fit to the actual pair axis directions than do the maxima of the large ampli- tude curves of the original splitting pattern. The explanation for the - 66 - PJ s p l i t t i n g as it appeared on the o r ig ina l p la tes . The two s k r p neon l ines a t 7438.9a ( l e f t ) and a t 7488.98 ( r i g h t ) define t:ie wavelength scale . The t o p spectrum i s of the l i n e a t zero pressure; t h e other th ree correspond t o a stress magnitude of about 80 Kg/m2. bottom, t h e angles of applied s t r e s s a r e +30°, Oo, and -l5O. From next t o the top t o the 4O 2O II - 3O Angle of applied stress, III FIGo 3.8--Splitting pattern of N O Magnitude of applied stress is i00 Kg/mm 2. - 68 - °o I_ ,,t g -o -_- - -a__ ...1_ /,-uJ:)°/JIHS-- _ _l-W_0:::m • __0 0 --'- 0 ++++o.o.z + -o0+_+ _0: 7 m -+-+ , .0 0 I-- :D I-- .-"-'. .I('P "%'0 _ gz _ _, I-W3 0 I 0 O_ 0 c_ 0 .H _O (1) -H o _r_ b.0 °_ 4._ -r-t m m i1) 4._ b.0 -,-t ,--t m O (',1 I I Cc_ H - 69 - improved fit may be that when the difference between curve I and the average of II and III is taken, a term common to all three components, but not accounted for in the simple uniaxial stress theory given in _i_± II_ is s_Dtracted out Such a term may come from response of the Cr-O-Cr complexes to shear strains. Even if the postulate that the shift should be directly proportional ÷_ +_ strain ..... the pair axis is entirely correct, there are a number of reasons why the observed splitting patterns would not be expected to fit precisely the theoretical curves calculated in Chapter II. In the first place, the macroscopic strain represents some sort of average of all the atomic displacements within a unit cell, whereas the true relative displacement of the Cr ions of a given pair may deviate somewhat from that average. The proper relation between such microscopic strains and the macroscopic strain is undoubtedly rather complex, and is certainly not known at this time. Secondly, the elastic isotropy assumed previously was only a rough approximation; the correct relation between macroscopic stress and strain in non-isotropic materials is determined by a nine-by- nine tensor of second rank_ containing six independent constants. The six elastic constants for sapphire have been measured accurately by J. B. Wachtman et al., at the National Bureau of Standards.65 The following table of elastic compliances is taken from their paper. TAB_ 3.1 EIASTIC COMPLIANCES OF SAPPHIRE (Inunitsof lO-13cm2/dyne) SII 2.353 S12 -0.716 s33 2.170 s13 -o.364 $44 6.940 S14 0.489 - 70 - The tensor in question is written: SII SI2 S13 Sl4 0 0 S12 SII SI3 -S14 0 0 S13 S13 $33 0 0 0 SI_ -SI4 0 S44 0 0 0 0 0 0 $44 2S14 o o o o 2s14 2(Sll- s12) Figure 3.10 shows the relation of the axes i, 2 and 3 to the crystal axes, and the way in which the subscripts 4, 5 and 6 describe shear strain and stress. A computer calculation of the predicted splitting patterns might improve the fit to the observed piezospectroscopic behavior, but such a calculation has not yet been attempted. However, the first consideration mentioned above may tend to blur any improvement gained in an exact cal- culation of macroscopic strain. The third and final consideration is that the Cr ions, especially those of a pair, may not choose the exact same locations that were occupied by the AI ions they have replaced. There are two pieces of evidence for such a shift in location of the isolated Cr ions, the one unfortunately contradicting the other on the indicated direction of that shift. A crystal-field calculation by D. S. McClure 66 predicted a shift of about 0.i _ away from the nearest neighbor site; X-ray studies by S. C. Moss and R. E. Newh 67 on ruby of 4% Cr203 concentration indicated a shift of 0.06 _ in the opposite direction. An improved calculation by J. 0. Artman and J. C. Murphy 68 could be said to be in partial agreement with both McClure's calculation and the x-ray studies, since it predicted an abso- lute shift away from the nearest neighbor, but since their calculation took a more general distortion of the lattice into account, the shift, expressed as a fraction of the separation between oxygen layers, was toward the nearest neighbor site. But no matter which of the above pre- dictions is correct, such shifts in location of the two Cr ions of a pair would rotate the second and third nearest neighbor axes by a few - 71 - /X 2 x 3 I' I I I J -7 I I i s 4 x 3 s 4 X2 _- X2 s4 "4" component of shear strain ,i_,,component of shear stress FIG. 3.10--Definition of axes relevant to the stress tensor given in Section 3.4. - 72 - ot degrees in one or the other direction. (Such a shift in the Cr ion loca- tion would not be expected to produce a rotation of the _ourth nearest neighbor pair axis, since for that pair both Cr ions are shifted in the same direction.) 3.5 ASSIGNMENT OF A PAIR TYPE TO THE N I LINE Positive identification of the NO and N2 lines with the comple- mentary set of pairs, the third and fourth nearest neighbors, leaves only one or possibly two choices, either the second or sixth nearest neighbors, for assignment to the NI line. Discrimination between these two alterna- tives could probably be left to the simple consideration that the observed -i exchange -- with a J value of -i0 cm (antiferromagnetic) -- would be extremely difficult to explain on the basis of a sixth nearest neighbor assignment. However, the stress data on the N I line provide good con- firmation of the second neighbor assignment. The sixth nearest neighbor assignment is ruled out by the fact that both the large amplitude curve of the splitting pattern and the splitting curve for the N I line (see Fig. 3.2) are anything but symmetric about the 0° direction_ instead, their peaks lie well into the quadrant containing the second neighbor axis. In fact, by any reasonable criterion for establishing a center of symmetry of the splitting curve, that center lies _o 70within to of the second nearest neighbor pair axis direction. And despite the rather strange shape of the large amplitude curve of the split- ting pattern, if the fundamental sinusoidal variation is extrapolated from it, it will be seen to make fairly good fit to the pattern predicted for second neighbors. Note that in the splitting pattern of the NI line, the curve attributed to pairs II and III (the two lying out of the particular mirror plane slab from which the samples were cut) does not display the extra splitting observed for the third and fourth nearest neighbor pat- terns. This splitting, exactly two-fold, should be unique to the second nearest neighbors within the set comprised of first through fifth nearest neighbors. Of course, pairs II and III of the second neighbors are not mirror images of each other_ when the intervening oxygen ions are taken into account, just as pairs II and III are not mirror images of each other for the third and fourth nearest neighbors. However, unlike the situation for third and fourth neighbors, pairs II and III of the second neighbors can be shownto be stress equivalents for stress applied _.__ _ mn" the I mirror plane. For thesecond neighbors, two nearly intervening oxygen ions are equally involved in the formation of a Cr-O2-Cr complex. The complex is di_mnn_ _p_ { _--............ _near± square, in fact)_ with the two Cr ions on diagonally opposite corners, and the 0 ions on the remaining corners. Again, as already indicated in Section 3.2 with respect to the third and fourth nearest neighbors, there are "a" and "b" sets of complexes, which can be related to each other; through an inversion center. The six complexes la, Ib, lla, etc. are illustrated in Fig. P.ll. From the figure, it can be seen that the lla and lllb (also the llla and llb) complexes are (very nearly, at least) exact images of each other across the I mirror plane; hence these two should be stress equivalent for stress applied anywhere in the I plane. By the same argument given in Section 3.2 with respect to the third and fourth neighbors, lla and lib are general stress equivalents, as are llla and llIb. Therefore, all four complexes -- lla, lib, llla, llIb -- are stress equivalents for stress applied in the I mirror plane. Since the two remaining complexes, la and Ib, are stress equivalents, the mirror plane splitting pattern for second nearest neighbors will show a precisely two-fold splitting. Incidentally, the above argument cannot be applied to the third or fourth neighbors, since, for example, complexes lla and lllb are not images of each other across mirror plane I for either type of pair. The strange shape of the large amplitude curve of the splitting pat- tern is itself a kind of evidence for the second neighbor assignment. The deviation from the predicted pattern probably has as its major reason the difference between microscopic and macroscopic strain discussed in Section 3.4 with reference to the third and fourth nearest neighbor patterns. Furthermore, the degree to which the relative displacement of the ions of a pair would conform to macroscopic strain should decrease markedly with the size of the pair separation. Thus the second neighbor pair is expected to produce a more badly distorted pattern than more distantly separated pairs. - 74 - °III -a FIG. 3.11--Stress e_ Looking along C3 c IIa plane I seen edge on m ealent of sec_>nd neighbor lla, llb, Ilia, 1117o for stress axls lying in _or plane Io Unlike the situation found for third and fourth neighbors, sets "a" and "b" are not related to each other through an actual inversion center of the lattice, butthe Cr-O2-Cr complexes can _evertheless be related to each other through an inversion center by _r_nslation of one of the complexes parallel to itself. Since the stress is unaffected by translation parallel to itself, the stress equivalence of sets "a" and "b" is still maintained. - 7_ - .J Finally, the relative intensities of the stress-produced components agree with the second neighbor assigmment and contradict a sixth neighbor assignment. That is, when polarization effects are taken into account, the low amplitude curve of the NI splitting_ pattern corresponds to a component having twice the intensity of the component from which the large amplitude curve was obtained, in agreement with assignment of the low amplitude curve to the two pairs II and III, and the high amplitude curve to just one pair. For the sixth nearest neighbors_ the high amplitude curve would be produced by pairs II and III, giving just the reverse of the observed intensity ratio. 3.6 VERIFICATION OF TKE KNOWNEXCHANGEMULTIPLETS When the second and fourth nearest neighbor exchange multiplets were examined on the plates originally used in the piezospectroscopic study of the NO line, the various members of a given multiplet were found to produce nearly identical splitting patterns. Figure 3.12 shows one example of the common pattern produced for each of the two known multiplets. The most important implication of the above is that stress-induced variation of the ground state exchange coupling can provide at most only a tiny fraction of the observed piezospectroscopic effect. That is, if modulation of the exchange coupling were largely responsible for production of the observed pressure effect, the magnitude of splitting would be expected to vary greatly from one member of the multiplet to another, since the exchange energy itself represents a radically different multiple of J for each member of the multiplet. The relative insignificance of modulation of the exchange coupling terms by stress can probably be understood in terms of the following simple dimensional argument. Since the exchange energies in question are -I of the order of a few tens of cm , those energies would have to be mod- ulated by nearly 30 to i00 percent in order to produce the observed shifts. In the meantime, the standard stress of i00 Kg/mm 2 produces strains on the order of 2 × i0-3. It is unreasonable to expect the degree of overlap of the wavefunctions in question to be so sensitive to changes in the Cr ion separation that a change of 0.i_ in that separation could modulate the overlap by nearly i00 percent. However, under the assumption that the - 76 - . - e . FIG. 3 .12 - - I l l u s t r a t ion of t h e s p l i t t i n g p a t t e r n common t c a l l nnembers of a given exchange m u l t i p l e t . Above, second neighbor s p l i t t i n g ; beiow, i'ourCh neighbor s p i i t t i n g s . Each stress s p l i t s p e c t r m has a zero s t r e s s spectrum above it f o r comparison. - 77 - stress somehow is able to modulate the over-all energy separation between L the _E and *A 2 states, multiplying that energy of approximately -i 14,000 cm by the strain of 10 -3 yields shifts more nearly commensurate with __.... _........ The above consideration makes the stress effect a powerful tool for the identification of the various members of a given exchange multiplet wi_......._ . _+_. m_ is, one can formulate the principle that all the members of a given exchange multiplet undergo a common splitting under stress, where the common pattern is unique to the particular pair type involved. When that principle is applied to the second and fourth nearest neighbors, it merely provides verification of exchange multiplets previously determined by other means. However, as will be revealed shortly, the same principle has allowed discovery and identification of the first nearest neighbor exchange multiplet. 3.7 DISCOVERY OF TW0 FIRST NEAREST NEIGHBOR LINES While measurements of the NO splitting were being taken from the plates intended primarily for that purpose, a not too distant line, centered at 7540 _ (13,259.0 cm-l), was observed to undergo a rather large shift to the red (-22.8 cm -I for the standard pressure of i00 Kg/mm 2) on the plate corresponding to stress applied along the C3 axis, yet shift of the same line for stress applied at right angles to C3 was very small by comparison. Furthermore, the line at 7540 _ was never observed to split or broaden at any angle of applied stress. Finally, examination of spectra of rubies of different Cr20 n_ concentration showed that the intensity of the line at 75_0 _, relative to that of the R lines, increased steadily from zero as the Cr203 concentration increased. Thus, behavior of the new line was exactly that expected for a line stemming from first nearest neighbors. Without the piezospectroscopic evidence, the new line would have been dismissed as being some sort of vibronic of the R lines or of the N I or N 2 lines, on account of its strange flat-topped shape and its -I rather unusual width, which was about 16 cm A careful search of the plates finally revealed a second line, at 7302 (13,691.1 cm-l), whose piezospectroscopic behavior was identical to that of the line at 7540 _. The line at 7302 _ was very difficult to spot_ since at - 78 - zero pressure it lay atop the edge of a broad feature of the long wave, length tail of the R line vibrational sidebands. But for a pressure of about 50 Kg/mm 2 applied parallel to C 3 3 the line was pushed into a clear space and was readily identifiable as a rather sharp line. At a pressure of 80 Kg/mm 2, the line had been pushed out of the clear space and could be seen intensifying the edge of another vibronic feature farther to the red. Again 3 without the piezospectroscopic technique, the line at 7302 _ would never have been discovered in the fluorescence spectrum. As pointed out earlier, at least one of the nearest neighbor types -1 must have an exchange coupling of several hundred cm in order to explain the Curie-Weiss constant of Cr203 Since the other near neighbor types (second, third_ fourth)_ for which such a strong coupling might have been plausible a priori_ had been shown experimentally to have exchange couplings an order of magnitude or more too small, the first nearest neighbor pair was expected to have a J value in the range -1 of -200 cm -1 to -400 cm . (Although the third nearest neighbor multiplet was not yet known_ it was thought to have an exchange somewhat like that of the fourth nearest neighbors, by virtue of the similar atomic arrange- -1 ment of these two pair types.) Thus the approximately 430 cm difference between the two lines at 7302 _ and 7540 _ appeared to be a likely candi- date for the exchange splitting of. J or 2J found between the S = 0 and S = 1 , or the S = 1 and S = 2 levels of the first nearest neighbor pair type. Furthermore, the two new lines were known to rep- resent transitions from the lowest lying levels of the excited stat% since their strength in fluorescence remained unchanged as the sample temperature was lowered from 77 ° K to 20 ° K. -1 If the 430 cm difference between the lines at 7302 _ and 7540 represented splitting between the S = 1 and S = 2 levels, then the line representing the transition from the lowest excited state to the S = 0 level would lie hidden (in a fluorescence photograph) in the most intense part of the vibrational sidebands of the R lines. Hence the only way to decide between assignment of the line at 7302 _ to a trans- ition terminating on the S = 1 level or terminating on the S = 0 level was to study its behavior in absorption. If it were a transition - 79 - to S = i , absorption by the line should increase rapidly as the sample temperature was raised from very low temperatures, but if it were a transition to S = 0 , the absorption should remain strong even at zero temperature. The absorption studies are the subject of the next chapter. Incidentally, two other features were finally discovered to have the same piezospectroscopic behavior as the lines at 7302 _ and at 7540 _: -i i)first, a line of a few cm width centered at 7497 _ (13,335.0 cm- , whose intensity was much less than that of the near by line at 7540 _ and second, a very diffuse feature centered at 7744 _ (12,909.7 cm-l). The very diffuse nature of the line at 7744 _, when combined with its -i approximately 350 cm spacing from the line at 7540 _, would make it appear to be a vibronic of that line. The line at 7497 _ might also be thought of as being a vibronic, in this instance of the line at 7302 _, but its relative sharpness also makes it seem to be the equally likely result of a splitting of the level upon which the line at 7540 terminates. 3.8 ACCURACY OF THE PIEZOSPECTROSCOPIC MEAS_NTS Discussion of the various sources of error and the over-all accuracy of the piezospectroscopic measurements has been delayed until now in order not to interrupt the logical sequence in the assignment of pair types. Such a discussion would not have been pertinent to the above, since the accuracy of the measurements generally exceeds the degree to which the experimental curves could be expected to fit the simple theory used in interpreting them. However, for future reference, if it is ever desired to extract more from the piezospectroscopic measurements than has been obtained above, the following should indicate the most important sources of error and their probable effect on the accuracy of the measurements. The measurements connect three variables -- sample orientation, magnitude of applied stress, and displacement in frequency of the components of a line away from the zero pressure position. Orientation of the sample axis can probably be guaranteed to lie within a cone of half angle 2° from the supposed direction. The figure given is simply an estimate of the error involved in the transfer of information from the Laue camera to the cutting operation; a slight tendency for the plane of the saw blade to - 80 - 4 wander produces the greatest part of tk_t error. Determination of the orientation of a crystallographic axis relative to the x-ray beam direction was relatively more accurate, since the position of a Laue spot could be read to within 0.2 ° or better. Positions of the rotation axes and pressure points of the lever arm assembly were sharply defined and known to within a small fraction of one percent; the weights themselves were guaranteed accurate to better than + 0.13 by the manufacturer. The sample cross sections were rather precisely rectangular and the two nearly e_ual dimensions could be read with a good micrometer to within a small fraction of one percent. Hence the magnitude of the applied stress could certainly be guaranteed to be within + l_ of the supposed value. The grating drive mechanism on the large scanning spectrometer used in the study of the N1 and N2 lines was known from past experience to be precise to much better than one _ unit on its digital wavelength indicator_ implying a precision of wavelength determination for the order -1 used (eighth) of better than +_0.05 _ or, equivalently, about + 0.i cm Study of e_ch sample was begun with determination of the line position at zero stress; after the run was completed, the zero pressure line position was measured once more; no drift in that position could be detected. Hence the shift of a well resolved component could be determined to within -i better than + O.1 cm . By far the most uncertain process was that of determining the centers of lines involved in an unresolved or only partially resolved splitting. It would be rather foolish to try to attach a number to the accuracy of that process, since its value depends so critically on the degree of resolution already present. When the curves of the splitting patterns were drawn_ the data points from well resolved components were taken most seriously, and the shape of the curves in the vicinity of badly resolved splittings were based on considerations of simplicity of shape and derivative continuity as well as on consideration of the data points themselves° For example, splitting in the vicinity of @ = -70 ° for the splitting pattern of Figs. 3.2 and 3.13 is large,ely the product of i_,_--j t_tion. _nd erroi's s.s hi_ 41 cm-I In addition, considerations of antiferromagnetic ordering in Cr203 exclude the possibility of the large positive value (ferromagnetic coupling) suggested above for J3 , when the more certain experimentally determined values for Jl ' '/2 , and J4 are taken into account° (For a discussion of the ordering, see Section 5.2 of Chapter Vo) Thus, we can probably also say that it is now knownthat -i J3 < -41 cm In addition, it has been demonstrated that the tools necessary to determine experimentally the true J value do indeed exist, and that a more thorough application of them should allow a de- finite choice to be madeamongthe various possibilities. - - lO7 - V. CONCLUSIONS 5.1 SUMMARYOFEXCHANGEVALUESFORNEARNEIGHBORPAIRS The principal results of this paper are summarized_n Table 5.1; only the exchange integrals listed for the second and fourth nearest neighbors were knownpreviously. The other two exchange couplings and all the identifications of the exchange couplings with particular near neighbor types constitute the major newresults of this work. The J value listed for the third neighbors is to be considered tentatlve_ for details and reservations concerning this ass_gnment_see Chapter IV_ Section 4.4. TABLE5. i Neighbor Type First Second Third Fourth -1 -i -iJ_ cm j, cm E (S = i) - E(S = 0)j cm - 183.5 - 11..56 - 46 + 6.94 -6.7 + 0.19 + 0.2 + 0.14 227 lO.3 41 - 7.9 The rather precise values of J and j listed for the first and third neighbors should be qualified in the following way: fairly large aniso- tropic splitting within the various S states makes assignment of an average location of the various components of a given S state to be somewhat uncertain; the values listed were based on the median position. If _t ever w_ll be possible to take a more appropriate average, or if other components are discovered_ the values listed above for J may be changed by a few percent_ and those listed for j drastically altered. - 108 - 5 o2 CORRESPONDENCE BETWEEN THE MEASURED J VALUES IN RUBY AND TEE THEORY OF A/ITIFERBOMAGNETISM IN Cr203 The theory of antiferromagnetism is concerned with three kinds of problems (1) the calculation of J values for the important near neighbor types; (2) the determination of an ordered spin arrangement at low temp_ eratures by a given set of exchange forces; and (3) the generation of theories connecting the measured or calculated set of J values to the macroscopic magnetic constants TN (Neel temperature)_ XN (magnetic susceptibility at the Neel point)_ and @ (%he Curie_Weiss constant)° Here TN is the temperature below which an%iferromagnetic ordering sets in; well above TN the behavior is essentially paramagnetic, as described by the law c (5ol)(T) = T The various aspects of the problem are sufficiently formidable that much of the pr_*. theory is capable of making only qualitative predictions_ in spite of the great amount of effort that has been spent by a rather sizable number of theoreticians° However_ the purely theoretical difficulties are not entirely to blame for lack of well established models_ since until now there has been a paucity of the kind of experimental data central to the problem; chat is_ a complete set of independently determined J values has never previously been known for such a complex structure° Thus the problem is usually bound up in a circle of ignorance_ attempts to deduce a set of J values from knowledge of the quantities TN _ 0 _ _ are based upon models of uncertain validity; the models are in turn uncertain because they have never been tested against a known set of J values of even, moderate accuracy° Furthermore_ with respect to study of the more complex struc- tures_ such as corundum._ %here are too many non-negligible J values to be determined from the three macroscopic parameters mentioned above° In the following discussion it will be shown how the essentially complete set of important J vaiues_ as determined by the experimental results of 109 - this study_ will allow independent evaluation of theories of exchange and theories of ordering_ at least for Cr203_ and possibly for the closely related compound _-_ _ as "_ - -'--2-_ _ ..... The calculation of exchange couplings in metal oxides necessarily becomes involved with the very complicated problem of super exchange: since many of the metal ion separations are too great to allow for direct overlap of their wavefunctions. Instead_ the exchange often must take place through one of the P orbitals of an intervening oxygen ion_ one lobe of the P orbital interacting with one of the metal ions7 the other, oppositely directed lobe interacting with the second metal ion. Thus_ a linear array (metal-oxygen-metal bond angle = 180 ° ) is expected to allow for the greatest super-exchange_ while a 90 ° bond angle is expected to give essentially zero exchange° Hence_ for bond angles in the neighbor- hood of 90°; if a coupling is nevertheless observed it is usually explained in terms of direct contact of the metal ion wavefunctions. The intermediate Case_ involving bond angles on the order of 135°_ is often considered to be ambiguous, since that range of bond angle usually suggests several distinc_ possibilities for the super-exchange mechanism_ as well as some- times allowing for a competing direct exchange° It has been suggested by J_ B, Goodenough 69 that the direct exchange is possible in oxides of the corundum family_ of which Cr203 and _-Fe205 are representatives, Goodenough classifies the exchange interactions according to the manner in which the two octahedra of oxygen ions sur- rounding the cations of a pair mate with each other; if they share a common face (first neighbors) or a common edge (second neighbors), direct cation-cation interactions are not only possible but made necessary as well by the fact that the bond angles in each instance are very nearly 90 ° . (The exact angles in Cr203 are 88 ° and 94o for the first and second neignbors_ respectively.) However_ if the two octahedra share only one corner (third and fourth neighbors); the bond angles are in the inter- mediate range (132 ° for third neighbors and i12 ° for fourth)_ and a super-exchange is indicated. Furthermore_ Goodenough predicts that all four exchanges will be antiferromagnetic_ with absolute values of J less than 350 cm-I for the direct exchanges_ and of the order of i00 cm -I for the two super-exchanges - II0 - The above prediction of direct exchange is well borne out by the -iJ value of -183.5 cm measuredfor the first neighbors. On the other hand_ the prediction about the J values for the two super-exchanges does not fare nearly as well_ since it runs counter to the fact that the exchange observed for the fourth neighbors is ferromagnetic. The contradic- tion most assuredly does exist_ since of all the J values given in Table 5.1 that listed for the fourth neighbors is the most certain. There- fore_ a better theory of the super-exchange must be found. W. P. 0smundhas suggested a rather ingenious schemeof super-exchange 7o relating to the third and fourth neighbor pairs of both Cr203 and _-Fe203;_ the chief justification of that scheme can now be the fact that it correctly predicts the signs of the third and fourth neighbor exchanges in Cr203 as given by the direct experimental observations of this work. The theory is particularly noteworthy for its ability to account for the observed difference in sign of the two exchanges in question_ in spite of the fact that the differences in atomic arrangement of the two corresponding pair types are rather subtle. The Osmund theory was originally suggested by_ and i_ i_Li_tely bound up with, the different antiferromagnetic spin arrangements of Cr203 and _-Fe203 as known from neutron diffraction studies (see Fig. 5.1). Before recounting the theory itself, we digress _omentarily in order to fill in the background relating to the problem of magnetic ordering. To facilitate reference to the various exchange integrals; the value for the .th m nearest neighbor pair will be refered to as J. in the following. 1 It will be noticed from Fig. 5.1 that the relative spin alignments of the second and fourth neighbors are different in the two compounds represented there. In addition to Osmund himself, a number of other writers have suggested that the sign of J4 may be important in establish- ing the difference in the two spin arrangements; among the others are Yo Y. Li 71, A. lida, 72 and E. F. Bertaut. 73-75 Thus a question first alluded to in Chapter I is raised once more: how many exchange inter- actions are required to determine the complete antiferromagnetic spin arrangement? - iii - J0 ][ Fez03 - _. TIT Cr203 FIG. 5.1--Antiferromagnetic spin ordering for -Fe203 and Cr203. The coupling labeled J0 should be called J6 to suit the conven- tion used here_ (_rom Bertaut74).• - 112 - Goodenough has suggested (on the basis of his assumption that all four J values are antiferromagnetic) that if J1 and J2 predominate, the arrangement found in Cr203 will occur; on the other hand, if J_w and 76 J4 predominate, the Fe203 arrangement must occur° Although correct as far as it goes, the above is misleading in that it does not consider all the possibilities. In fact, it can be shown that predominance of any two of the exchanges J1 ' J2 ' J3 and J4 will completely determine a spin arrangement, with one very important exception: when J1 and J3 predominate, a third exchange is required to completely determine the spin arrangement. Since this latter, exceptional case is just that indicated to occur in Cr203 by the J values listed in Table 5.1, it will be considered in greater detail. That the combination J1 ' J3 is insufficient to completely de- termine a magnetic ordering can be seen from the following: the Cr203 lattice can be viewed as a collection of bipyramids of the Cr203 molecule, each bipyramid containing one first neighbor Cr pair° Each bipyramid is surrounded by a first shell of six other bipyramids; connection betweeD the u_al bi_yr _ o_a +_ first shell is only by way of second and fourth neighbor types. As strange as it may seem (until one has looked at a three dimensional model of the Cr203 lattice)_ the third neighbors connect the central member to a second, outer shell. Thus_ there are two interpenetrating "lattices" of bipyramids, such that Jl and J3 by themselves can provide no link between the two, but only interconnect the various members within each lattice. Thus it would seem that the value (including sign) of J4 , or perhaps of some combination of J2 and J4 , is important to the de- termination of the complete spin arrangements of Cr203 and _-Fe203 In connection with this point it is interesting to note that Bertaut (reference 74) has hypothesized that the difference J2 -2J4 , which he considers to be positive in Fe203 and negative in Cr203, provides the determining factor. Indeed, the directly measured J values justify Bertaut' s assumption cozcerning the sign of J2 -2J4 for Cr203, Joe. -i from the values listed in Table 5.1_ it can be seen that J2 -2J4 = -25.5 cm One might go one step beyond Bertaut's contention to theorize that since - 113 - J2 probably is antiferromagnetic in _-Fe203 as well_ the difference in the two oxides will be brought about by a change in sign of J4 Wear_..... nn_. _-_. _ _±u__-- to appreciate 0smund_s theory_ which can be summarized as follows_ two of the three oxygen p orbitals will lie very nearly along the two almost perpendicular arms of the first neighbor metal-oxygen-metal bond _r_l_ +he +_-_.... . ........ normal to the plane of the bond angle. Each cation has available to it three t2g and two e orbitals.g Since the configurations of Cr 3+ and Fe 3+ are d3 and d5 _ respectively, both orbitals are half filled in Fe203_ but in Cr2Oq_ t2g is half filled and e is completely empty. Geometry of the corundum lattice and loca- g tion of the t2g and eg orbitals are such that the situations shown schematically in Fig° 5.2 are obtained for the third and fourth neighbor exchanges. For the third neighbors_ the p orbital whose axis is normal to the first neighbor metal-oxygen-metal bond angle connects t2g orbitals on both cations M a and M b of the pair; if the spin of Ma is called (+) _ then its already half filled t2g orbital can only accept a (-) spin from the oxygen_ making the opposite end of the p orbital (+) ; in accepting a (+) spin from the oxygen_ the t2g of Mb must orient its own spins in the (-) direction_ thereby making the exchange anti- ferromagnetic° For the fourth neighbors_ one of the other p orbitals is involved, and connects an eg orbital on Ma with a t2g orbital on M b This time the two oxides present two distinct situations: the half-full e g orbital of _-Fe203 leads to an antiferromagnetic exchange_ by the same mechanism listed for the third neighbors; but in Cr203 the empty e orbital of M accepts a spin of the same sign as that on M , g a a in keeping with Hund's rule_ and thereby leads to a ferromagnetic exchange. As for the geometry involved in making the oxygen and cation orbitals according to the above scheme_ the reader is referred to Fig. 2 of re- ference 70. In connection with this_ it is very helpful to have a three- dimensional model of the corundum lattice available for study. - i14 - Third neighbors 2P t2g (+)t t Ma Fourth neighbors 0 Ma I ! e g (+)t FIGo Mb 5_2--Schematic illustration of Osmund_s theory of super-exchange in the third and fourth neighbor pairs. The metal ion sites are labeled Ma and Mb. - im5 - Now to summarize_ Osmund_s theory of super-exchange in the third and fourth neighbor ion pair fits very well with all that is known or can be deduced from the experimental studies of this work. In tlhe first plac% it correctly predicts the signs observed in Cr203 for J3 and J4 Secondly_ from the Observed relative sizes of the four most important J values_, it is kmown +_tv.._Jl and J3 are the two predominating couplings Yet in that special case_ it can be shown that the sign of J4 must be positive (ferromagnetic) in Cr203 and negative in a-Fe203 in order to account for the difference in antiferron_gnetic spin arrangement found in the two oxides; Osmund_s theory predicts Just that change of sign for J4 The measured set of J values can be tested against a form_-la (I) for @ given by J. So Smart: 77 9 = - z.J. 0 (52! i I 3k i where S is the spin per atom (S = 3/2 for Cr 3+) and zi is the th number of i nearest neighbors per atom (in corundum_ zI = I z2 = z3 = 3 _ z4 = 6),, The test should be a good. one_ since expression (5_,2) is _hought to be exact° Inserting the J values of Table 5ol and -i 438OKusing the conversion factor i em = kT _ where T = I. _ we obtain @ = 565°K_ in excellent agreement with the measured value of -1 @ = 590 ± 20 cm _ as taken from the susceptibility measurements of Fo_x and Graff. 78 The data of reference 78 are reproduced in Fig° 5 3; note that the measurements extend to nearly 1300°K, a temperature nearly four times as large as T N _ thereby allowing a correct determination of @ (1)Expression (5.2) has been altered by a factor of two with the definition of J used here° to conform - 116 - o[, __.ooi __ oI 0 0 0 0 0 0 0 LO 0 0 0 0 0 0 0 Ol cO 0J o (D X I-4 o o .r_ >, ! - 117 - It should be noted here that an incorrect value of @ = 1070°K is often quoted in the literature; yet reference to the original measurements is hard to find. A review article bv Na_v_ o_ Yosida 79 finally led us to two papers which give graphs of I/X(T) in the range above T N : the Fo_x and Graff paper quoted above, and a more modern one by Troumson et al ....80 Although the authors themselves are nnt to _lo_,_, refer_mce 80 proved to be the source of error_ since the measurements of @ given there only cover the temperature range below ~2T N _ and yet the often quoted @ value had obviously been extracted from them. It should be realized that a correct value of @ cannot be deduced from such measure- ments, as can easily be seen from the way that the curve of Fig. 5.3 de- viates at temperatures below _2T N from the true straight line para- magnetic behavior seen at higher temperatures° To the best of our knowledge_ there is no adequate theory to relate TN to the measured set of J values; the only workable theories cor- respond to the much simpler situation in which one or at most two J values are important in the determination of even the long range ordering. (A good example of such a situation to which a successful theory has been 81 applied is the study by Harris, Lines and Owen of K21r CI 6 in solid solution with K2PtCI6. ) Despite the lack of a valid quantative theory for Cr203_ at least the following qualitative observation can be made: from the above discussion_ it has been seen that a relatively small ex- change coupling (J4) is required to complete the long range ordering; the Neel point may then tend to occur closer to a temperature at which this smaller coupling will be broken by thermal effects_ consistent with the fact that the observed ratio of TN/@ is much less than one. The problem in Cr203 may possibly be solved by a more complex version of the spin-wave theory applied in reference 81; the theory itself is described 82 in a paper by Lines. It is hoped that the set of J values determined by the experimental studies of this work will inspire the development of a valid theory of ordering in the corundum and other complicated struc- tures. Finally, it should be pointed out that the presence of anisotropic terms in the exchange hamiltonian_ as suggested by the observed splitting - 118 - Q of S levels in the first and perhaps other neighbors, may have an influence on the antiferromagnetic ordering. In the first place, such terms are usually required to establish the axis along which the spins are observed to align themselves; for Cr203 and _-Fe20 __ , this axis is Cx The anisotropic terms are also supposed to enter in to the determination of the exact value of TN ; for an example, see reference 81. 5.3. CORRESPONDENCE B_fWEEN THE MEASURED J VALUES IN RUBY AND THE ABSORPTION SPECTRUM OF Cr203 The optical_ absorption spectrum of Cr203 as measured and interpreted by Wiekersheim u3 is indicated by the energy level diagram of Fig. 5.4. Transitions originating on the ground state m s = - 3/2 level and terminating on the various exchange split components of the optically excited state were observed at 4.2°K and were found to be quite sharp. The one transition observed to originate from the m = - i/2 level S ..... i_ _ a_ted at temperatures above lO0°K and was very diffuse Figure 5.4 implies an interpretation based on the molecular fie±a model. That is, each transition is supposed to involve just one ion, where the energy levels of that ion have been split by an effective magnetic field created by all the neighbors exchange coupled to it. The concept of an effective magnetic field is the result of assuming that a time average of all the interacting spins can be taken; then the ex- change hamiltonian becomes )avH : z.J.. si s. (5.3)ex I iJ, 1 .th In (5.3) above, the levels of the j ion are split by the field created by all the spins s. If the axis of magnetic ordering is called z , 1 w then the average taken in (5.3) assumes that s. = _. = 0 , and that ix ly s. = ± s. at low temperatures. Iz 1 - 119 - 1.05±0.2cm -i _k I I I EEE (.) (.) c.) b- / I I I I p _| I I ' Ii / II I I VIBRONIC T +l/2,U_+II2,U+ I 183.1 cm -_ 139.4tcm-' [ 227.3cm -I , _ -1/2, u_ l ' iI --I/2, U+ f TT EE c..) _ _if') i_.1 _. D o +3/2 + I/2 -I/2 245 + 20 cm -I - 3/2 el Z/I t In 543 / unpol. o-_r Jl 21 7/'O" 6 unpol. ( '2) FIG. 5.4--Energy level diagram for Cr203_ - 120 - 83 (_rom Wickersheim ). Inserting the J values listed in the first column of Table _.l and the numbers zI = 1 , z2 = z3=3 , z4 = 6 • sI = s2 = s3 = - s4 = 3/2 into (_.3) yields a predicted splitting of _600 cm -1 between adjacent m levels_ The fact that the predicted splitting is more than twice s as great as that observed should not be at all surprising, since the approximations of the molecular field theory are gross, and no more than order of magnitude results are expected from them. In fact, the error here is very similar to that produced in a molecular field calcula- tion of TN ; in both instancest _the calculated values are on the order of two or three times too large. _lj Both R. L. White 84 and M. E. Lines 8_ have suggested to us that a better approximation may be provided by a model which views the first nearest neighbor pair as the entity subject to transition_ and calculates the effect of the molecular field created by all the other exchange coupled ions on the energy eigenvalues of that pair. The spin hamiltonian is then approximated by the form H = - Jl Sl " S2 + S--X(SIz - S2z) ' where ! and S is the magnitude of the average spin per site. Although it is not at all easy to calculate exact eigenvalues from expression (5.4), (_'l'Proponents of the molecular field theory may wonder if one of the less favored assignments for J3 listed at the end of Chapter IV would allow better agreement with the observed splitting in Cr203. The answer is no_ since the larger antiferromagnetic coupling would obviously give a worse result 3 and the ferromagnetic value of Jq = + 60 cm "I leads to a predicted splitting which is much too small (120 cm-l). Furthermore, as indicated previously_ we strongly reject the possibility of a ferromagnetic coupling for J3 , since such a coupling, when combined with the three more certain J values, leads to the wrong spin arrangement and would predict the wrong value for @ - 121 - an approximate treatment seemsto yield differences between the true ground state and the first excited state on the order of 3/2 Jl which would allow muchbetter agreementbetween the measuredexchanges and the spectrum of Cr203o Incidentally, the exact eigenvalues for a hamiltonian similar in form to (5.4) have been worked out for Cr203 by 86 Pratt and Bailey_ but '_fort'muately their calculation uses the J 6o values according to Statz, values which are inconsistent with those found by this study. Still better treatments of the problem are known, such as the Green function method recommended to us by Lines_ 85 but such a calculation is very difficult and is certainly beyond the scope of this work. However_ the calculation should be made_ and we hope that the unique opportunity to test the method against the empirical numbers of this work will inspire at least one ambitious theoretician. Of course, it is possible that the J values measured for isolated C_ ion pairs in ruby are not identical to those of pure Cr203; but until a truly adequate theoretical treatment has been applied to the problem of the ground state splitting in Cr203, and no reasonable agree- ment found between the observed splitting and that predicted from the J values given here_ we are free to continue to assume that measurements on the dilute solid solution are valid for pure Cr203. As evidence in favor of this very important assumption_ we can site two facts mentioned previously: first_ the fact that the J values measured in ruby fit all we know of the ordered spin arrangement in Cr2033 and second_ the close agreement between the predicted and measured values of the Curie-Weiss constant @ 5-4. THE PIEZOSPECTROSCOPIC SENSITIVITY OF THE RUBY PAIR LINES Next to the measured J values themselves, the piezospectrocopic sensitivities of the pair lines should be of great theoretical interest. A good theory of the piezospectroseopic effect is highly desired for several reasons: (i) The identification of optical transitions with given pair types should eventually be put on a better foundation than the - 122 - O f self-consistency argument used here, especially if it is intended to extend the method to the study OF other magnetic materials. (2) The piezo- spectroscopic effect will undoubtedly be important to the technology of tunable lasers some day; if the phenomenon is well understood, it may be possible to predict the existence of other tunable atomic fre- quencies useful as laser transitions. (3) The phenomenon should be understood for its own sak% and for the extra understanding of the optically excited state that would be created by a theoretical study of the pressure shifts. In order to facilitate discussion of the piezospectroscopic effect, it will be necessary to make a few definitions. First_ the piezospec- troscopic sensitivity itself will be defined as the frequency shift of the most intense line of a multiplet obtained for the standard pressure of i00 Kg/mm "_ and for the angle of applied stress in the mirror plane which yields the maximum shift. The exchange multiplet of a given pair type will refer to the quartet of lines originating on the lowest of the optically exc±_ea sL_L_ -_ud te__m_nt,_ng on the exchange split ground state. The center of gravity of an exchange multiplet will refer to the position of the single line that would be produced if the ground state exchange integral J were made zero, while the excited state exchange integral K retains its full strength. The term red shift will indicate the separation of the center of gravity of an exchange multiplet away from the R 1 line position. On the assumption that the inter-ion exchange represents the sole perturbation of the optical transitions of a pair away from the energy levels of an isolated ion, and also assuming that K ~ J , then the red shift for a given pair type should be approximately 5/4 J for antiferromagnetic coupling and 3/4 J for ferromagnetic coupling. How- ever_ the observed red shift for a given multiplet is usually about one order of magnitude greater than that amoun% and seems to be (roughly) in direct proportion to the piezospectroscopic sensitivity, as indicated in Table 5°2 below. - 123 - TABLE 5_2 Pair Type 4 2 3 I -i Red shift, cm 136 228 i_043 !_2!0 -i Piezo. sens._ cm -6 -ii -34 -23 -i J _ em +7 -ii -46 -183 The J values have been included in Table 5.2 since they also correlate with the piezospectroscopic sensitivities if the sign of J4 is ignored and if Jl is excluded. A good _heory of the piezospectroscopic effect will undoubtedly make use of_ and be able to explain, the correlations shown in Table 5o2_ in addition to accounting for the following equally important facts: (i) The observed pressure shifts are approximately proportional to the component of strain along the pair axis° (2) The various members of a given exchange multiplet exhibit a common piezospectroscopic sensitivity_ as discussed in Section 3°6 of Chapter III. (3) The piezospectroscopic sensitivity of the R lines is approximately -i - 3 cm _ and the shift is nearly independent of the direction of applied stress. The piezospectroscopic effect may involve modulation by the stress of one or more of the following: (i) A large exchange in the excited state (i.e., K >> J) o (2) A large crystal field directed along the pair axis. (3) Terms in the hamiltonian expressing the effects of covalent bonding. It should be understood that the distinction between (2) and (3) above is only in terms of the different mechanisms leading to a modified hamiltonian_ since the formal appearance of the terms introduced may be identical. - 124 - The first possibility mentioned above is probably not of major importance, since the partially known excited state splittings indicate that_ in general 3 K is of the same order of magnitude as J _ or at least not more than a few times J Possibility (2) seems to be ruled out by the lack of a good mechanism for the production of a strong axial field; for example_ the idea that lattice distortions brought about by the extra size of the Cr 3+ ion over that of the A13+ could cause a large odd field term seems rather implausible. The third suggestion can be supported in the following way: First, it is known that the Cr 3+ ion is about 20_ covalent bonded to the various near neighbor A13+ sites (first through fourth nearest neighbors) from the ENDOR studies of Laurance, McIrvine_ and J. Lambe.87 The covalent bonding of a Cr ion to one of its neighboring A1 ions may be drastically altered by substitution of a second Cr ion for the A1, on account of the different electronic configurations of the two ion types; thus the covalent bond between the two members of a Cr ion pair constitutes an odd member in the full set of bonds co all a_Ighbcring m_tal ion sites. From the above consideration one could then understand the relation of the pressure effect to the Cr ion pair axis, as well as the large red shifts. The covalency theory also suits the notion that covalency and super-exchange go hand in hand, as suggested by the fact that the values J2 ' J3 ' J4 are in proportion to the piezospectroscopic sensitivities of their associated pair types (although J2 may not represent a super-exchange). It should be understood that the above is offered only as a sugges- tion_ and that many other explanations are possible. At the time of this writing_ it must be admitted that the fundamental origin of the effect found so useful here still remains a mystery. 5°5° APPLICATION OF THE PIEZ0SPECTROSCOPIC EFFECT TO LASER TECHNOLOGY No matter what the ultimate explanation of the piezospectroscopic effect may be, the large frequency shifts and splittings produced and observed in the course of this study constitute a body of empirical fact that may be of great value to laser technology. Now it is true that - 125 - manyapplications merely require the production of large peak powers_ or the generation of a very pure and stable frequency_ where there is no requirement as to the exact frequency used (except perhaps that it lie somewherewithin a rather broad range to which a given detector type is sensitive). Yet manyother potential applications_ such as the use of lasers to rep!acc _±_re__7__ white light in spectroscopy_ require that the laser frequency be tunable_ usually over as large a range as possible. The need is particularly acute for a strong coherent source continuously tunable over the far infrared region of the spectrum_ where conventional sources are pitifully weak_ making impossible the sensitive_ high resolu- tion studies enjoyed in the visible and microwave regions° Although laser action has already been obtained at various spot fre- quencies in the far infrared_ it is hard to imagine any mechanismby which such transitions could be tuned over regions even several orders of magnitude smaller than say ± 50_ of the unperturbed frequency° How- ever_ if the far infrared radiation is produced by beating together two lasers operating in the optical region_ the laser frequencies need be tuned away from each other by just a few percent in order to cover an infrared region of several hundred wavenumbers. In the usual atomic transitions_ even that smaller degree of tuning is considered impossible of achievement. But the Cr ion pair transitions can be tuned over just such a range_ through the application of uniaxial stress° Furthermore, the pair lines are in general eminently suited for laser action by virtue of their sharpness and the fact that most terminate on levels that can be rather easily depopulated by cooling of the laser rod. The NI and N2 lines have already been madeto undergo laser action 50 and_ if only someway could be found to favor the installment of third neighbor pairs during the growth of a ruby crystal_ it should be possible to makethe highly pressure sensitive N line display0 laser action. Finally_ it should be pointed out that use of the pair lines would allow production of the two desired laser frequencies within one and the samelaser rod, thereby automatically guaranteeing the spatial coherence necessary to achieve efficient beating together of the two optical fre- quencies. - 126 - REFERENCES Q io . 3. 4. o 6. 7o 8. 9o i0. iio 12o 13. 14. 15. 16o 17. 18o 19o 20° 21. 23° 24. 25. 26. 27. 28. Edmond Becquerel, La Lumiere -- Ses Causes et Ses Effets, (Librairie de Firmin Didot Freres, Fils et Cie, Imprimeurs de L'Institut, Paris, 1867), Chapter IV. Lecoq de Boisbaudran, Comptes Rendues 103, ll07 (1886)o E. Becquerel, Comptes Rendues 103, 1224 (1886). Lecoq de Boisbaudran, Comptes Rendues _ 330 (1887). Ibi_., p. _54. Ibid., p. 824. Lecoq de Boisbaudran, Comptes Rendues 107• 468 (1888). Ibid., p° 490° G. C. Schmidt, Annalen der Physik 13, 622 (1904). Ho du Bois and G. J. Elias, Annalen der Physik 27, 233 (1908)o J. Becquerel, Phys. Zeit. 8, 932 (1907). A. Miethe• Deutch. Physik. Ges. Berichte 9_; 715 (1907). Jo Becquerel_ Comptes Rendues 151• 1344 (1910). J. Becquerel, Comptes Eendues 152____,183 (1911). H. du Bois and Go J. Elias, Annalen der Physik 35, 617 (1911). C. Eo Mendenhall and R. W. Wood, Phil. Mag. 30, 316 (1915)o Ho Bethe_ Annalen der Physik 3--,133 (1929). O o Deutschbein, Annalen der Physik 14, 712 (1932). Ibid.• p. 729 . O. Deutchbein• Annalen der Physik 20, 828 (1934). O. Deutchbein, Zeitschrift fiir Physik _ 489 (1934). O. Deutchbein, Physiko Zeitschrift 33• 874 (1932). Mo No Saha, Nature 125• 163 (1930). H. E. White, Phys. Rev. 33, 672 (1929). Y. Tanabe and So Sugano, J. Phys. Soc. Japan 9• 753 (1954)o Ibid., p o 766° Yo Tanabe and S. Sugano• J. Phys. SOCo Japan ii, 864 (1956). Yo Tanabe and H. Kamimura• J. Phys. Soco Japan 133 394 (1958). - 127 - 29. 30. 31. 32. 33. 34. 35° 36. 7. 38. 39. 40. 41. 44. 45. 46. 47. 50. 51. 52. S. Sugano and Y. Tanabe, J. Phys. Soc. Japan 13, 880 (1958). S. Sugano and I. Tsujikawa, J. Phys. Soc. Japan i__3,899 (1958). A° L. Schawlow_ J. Appl. Phys. __.33,Su_o._ _I, 398 x±_u_j.f_ D. S. McClure, Electronic Spectra of Molecules and Ions in Crystals, (Solid State Reprints, Academic Press, New York, 1964). M. Hammermesh, Group Theory and Its ApDlicat_on to P_@'sical Probiems, (Addison-Wesley, Reading_ Mass., 1962). S. Sugano, A. L. Schawlow, and F. Varsanyi, Phys. Rev. 120, 2045 (1960)o Ao Lo Schawlow, A. H. Piksis_ and S. Sugano, Phys. Rev. 122, 1469 (1961). G. F° Imbusch, Physics Department Dissertation and Microwave Labora- tory Report No. i190_ Stanford University (1964). I. Wieder, Rev. Sci. Instr. 30, 995 (1959). T. Maiman_ Phys. Rev. Letters _, 564 (1960). S. F. Jacobs, thesis, Johns Hopkins University_ Baltimore, 1957 (unpublished). A. L. Schawlow, D. L. Wood, and A. M. Clogston, Phys. Rev. Letters _, Y502 (1959). N. A. Tolstoi and A. P. Abramov, Opt. i Spektroskopiya 14_ 691 (1965) or (English translation) Opt. Spectry. 14, 365 (1963). L. R. Maxwell and T. R. McGuire, Rev. Mod. Phys. _ 279 (19:33). W. G. Wyckoff, Crystal Structures (Interscience Pub., New York, 1960), Chapter V, table p. 11. P. W. Anderson, "Exchange in Insulators/' in Ma_netism_ G° Rado and H. Suhl, eds. (Academic Press, New York, 1963), Vol. i_ p. 26. K.W.H. Stevens, _'Spin Hamiltonians," b_, p.20, ref. 44. J. 0wen, J. Appl. Phys., Suppl., 32, 2136 (1961). A. M. Clogston, "Emission Spectrum of Exchange Coupled Chromium lons in Ruby," (unpublished). A. L. Schawlow, private communication. A. L. Schawlow, "Infrared and Optical Masers," in Quantum Electronics (Columbia University Press, New York, 1960). A. L. Schawlow and G. E. Devlin, Phys. Rev. Letters _, 96 (1961). A. L. Schawlow, Sci. Am. 204, 6 (1961). R. T. Daly, Paper No. TBI6, Opt. Soc. of Am. meeting at Los Angeles, October 1961. - 128 - 53. 54. 55. 56. 57. 60. 63. 64. 65. 68. 69. 70- 71. 72 • 73. 74. P. Kisliuk, A. L. Schawlow, and M. D. Sturge in Advances in Quantum Electronics (Columbia University Press, New York, 1964), p. 725. P. Kisliuk and W. F. Krupke, "Exchange Interactions between Pairs of Cr Ions in Ruby," Internal Report, Aerospace Corp., E1 Segundo, Calif.(1964). We are indebted to H. Statz for transmitting this table to US o B. N. Brockhouse, J. Chem. Phys. _ 961 (1953). R. Nathan, T. Riste 3 G. Shirane3 and C. Go Shull, Bull. Am. Physo Soc. II, 2, paper FA4 (1957). M. Tachiki and Takeo Nagamiya, J. Phys. Soc. Japan 13, 452 (1958). L. Rimai, H. Statz, M. J. Weber, and G. A. deMars, Phys. Rev. Letters k, 125 (1960). H. Statz, L. Rimai, M. J. Weber, and G. A. deMars, J. Applo Phys. Suppl., 2186 (1961). J. Owen, J. Appl. Phys. 32, Suppl., 213S (1961). A. A. Kaplyanskii and A. K. Przhevuskii, Doklady Akademii Nauk SSSR 313 (1962) or (English translation) Soviet Physics-Doklady 7__ 37 (1962). H. D. Conway, Mechanics of Materials, (Prentice-Hall, New York, 1950)_ p. 30. E. A. Wood, Crystal Orientation _nual, (Columbia University Press, New York, 1963), p. 30. J. B. Wachtman, W. E. Tefft, D. G. Lsm_ and R. P. Stinchfield, J. Res. of National Bo Standards, 64A, 213 (1960). D. S. McClure, J. Chem. Phys. _ 2757 (1962). S. C. Moss and R. E. Newnham, Zeitschrift f[ir Kristallographie 359 (1964). J. o. Artmanand J. C. Murphy,Phys.Rev. A1622(1964). J. B. Goodenough, Phys. Rev. i17_ 1442 (1960). W. P. 0smund, Proc. Phys. Soc. of London _ 394 (1962). Y. Y. Li, Phys. Rev. _ 1015 (1956). A. lida, J. Phys. Soc. Japan ii, 1300 (1956). E. F. Bertaut, Comptes Rendues 252 , 252 (1961) and errata Ibid. 252, 2032 (1961). E. F. Bertaut, Phys. and Chem. Solids 21, 256 (1961). - 129 - 75. 76. 77. 78. 79. 80. 81. 82_ 83. 84. 85. 86. 87. E. F. Bertaut, "Spin Configurations of lonic Structures," in Ma_netism_ G. Rado and H. Suhl, eds., (Academic Press_ New York, 1963), Vol. III. J. B. Goodenough, "Magnetism and Crystal Structure in Nonmetals_ _ Ibid, p. 16_ ref. 75. J. S. Smart; '_valuation of Exchange Interactions from Experimental Data_ _'ibid, p. 66, ref. 75. G. Foex and M. Graff, Comptes Rendues 209, 161 (1939)• T. Nagamiya and K. Yosida, Advances in Phys. 4, 17 (1959). E. P. Trounson_ D. F. Bleil, R. K. Wangness, and L. R. Maxwell, Phys. Rev. 79, 543 (1950). E. A. Harris, M. E. Lines, and J. 0wen, "Paramagnetic Resonance from Three Exchange Coupled Lines," in Parama_netic Resonance_ W. Low, ed., (Academic Press, New York, 1963), Vol. II, p. 553. M. E. Lines, Proc. Royal Soc. London _ 105 (1963). K. A. Wickersheim, J. Appl. Phys. Suppl. 34, 1224 (1963). R. L. White; private communication. M. E. Lines, private communication. G. W. Pratt and P. T. Bailey, Phys. Rev. 131, 1923 (1963). N. Laurance_ E. C. Mclrvine, and J. Lambe, J. Phys. Chem. Solids 23; 515 (1962). f - 13o -

Exchange-coupled Chromium Ion Pairs in Ruby

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Exchange-coupled Chromium Ion Pairs in Ruby

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