NUCLEAR MAGNETIC RESONANCE STUDY OF ANTlFERRDMAGNETIC Rb 2MnCB=4 ‘ 2H2 Q Thesis for the Degree :1? Ph. D. MICHfiGAi‘é STATE UNWERSHY fiOHN ADDiS CASEY 1967 rHESIS This is to certify that the thesis entitled NUCLEAR MAGNETIC RESONANCE SI'UDY OF AN'I‘IFEILROMAGNEI'IC szl'lnC14' 21120 presented by John Addis Casey has been accepted towards fulfillment of the requirements for _P_hD___ degree in Jhxsfiss. WE Major prof sor Date May 15, 1967 0-169 t ’roton. —.'.t.,- .m u . . (M) have been .1 a. 32320 which become 1 .. . . . 3.21%. At 1.1:1' :hn tr . . . dad [8.30 “H: and the!» ‘4‘. . _ . . I. . ‘0‘ field {mes th 1....11. 3.. .' .-- ‘ ¢. l . "files were fOUHd 3n "Moo, ._ e» .L ‘ 0.63 an, id.” wz, o:- . u.-. '3» ‘mu. t lots of three liner- ruv- ,--' o u 7‘ . fl - ’ fl aha-felllv tummy“... . . ’ 0. "A ' no second most a .4» wt... , . -d a“ .' 9.: :1: " 1,” the lines sauce. :; .’ . ,. .9. M L; a €. , , €- to the pun W 1W x . III: were observed a that «yew-oi. t ‘67.}. .m' M! lines at 2.38 an, an?! fl. 1' 13"? as ""1: the ordered state u Li't um 5%..‘lfng’n7tfih ' '- ABSTRACT NUCLEAR MAGNETIC RESONANCE STUDY OF ANTIFERROMAGNETIC szMnCL4-2HZO by John Addis Casey Proton, chlorine and rubidium nuclear magnetic res- onance (NMR)have beenstudied in triclinic crystals of 1 'szMnCL4-2H20 which become antiferromagnetically ordered “at TN = 2.24°K. At 1.1°K two proton lines were found at 118.96 MHz and 18.30 MHz and their angular dependence in a "Small applied field fixes the magnetic space group as Psi. MSix C835 lines were found at frequencies 6.25 MHz, 7.72 MHz, 8.56 MHz, 9.63 MHz, 10.48 MHz, and 10.72 MHz. These consti- tute two sets of three lines, each set corresponding to one ‘ -of the two chemically inequivalent chlorine sites in the f lattice. The second moment of the Hamiltonian was used to '~ properly group the lines subject to the condition that it .textrapolate to the pure quadrupole frequencies of 3.85 MHz - ~Lend 5.34 MHz in the paramagnetic state. The rubidium res- 87 r ..onance spectrum contained lines due to both Rb with I = g g' and Rb85 with I = %. Two quadrupole lines at 2.6743 MHz 'Land 3.2229 MHz were observed in the paramagnetic state for lbe85 and four lines at 2.38 MHz, 2.78 MHz, 3.20 MHz, and 7 ..79 MHz in the ordered state at 1.1°K. The Rb8 quadrupole John Addis Casey resonance was found at 3.1161 MHz and six lines were ob— ' served at 1.22 MHz, 3.89 MHz, 4.18 MHz, 5.12 MHz, 5.42 MHz, sand 9.26 MHz at 1.1°K. The angular variation of the resonance lines in the presence of a small eXternal field was analyzed using the second moment of the Hamiltonian to determine the direction f of the internal fields at the chlorine and rubidium sites. The chlorine internal fields are nearly parallel and with this taken as the direction of magnetization, com- puter programs were used to calculate the magnitude and I l ! I Q NUCLEAR MAGNETIC RESONANCE STUDY OF ANTIFERROMAGNETIC RbZMnC£4°2H20 By John Addis Casey A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Physics 1967 ch authu: H1snc. _ ‘R. D. Spcncu _,; ."8fl this wcrh, :9 fitting the collectte/ '1'. .l. A. Coven. v- ... p '5 H. CIIISGU for g. 'm. ’4'in:- (n a '5', '- m '3 ACKNOWLEDGMENTS The author wishes to express his gratitude to »*-:fessor R. D. Spence for his continuing guidance and ygypfidit in this work, to Dr. V. Nagarajan for his diligent "{erts during the collection and reduction of the data “u~1to Dr. J. A. Cowen, Dr. H. Forstat, Dr. P. M. Parker, hwétfir. E. H. Carlson for their assistance in preparing s‘: ; - ii LIST 0F . LIST OF \ o I Chapter I I .II 0 TABLE OF CONTENTS ACKNOWLEDGMENTS . . . . . . . . . . . . TABLES . . . . . . . . . . . . . . . . FIGURES . . . . . . . . . . . . . . . . INTRODUCTION . . . . . . . . . . . . THEORY . . . . . . . . . . . . . . . . A. The Hamiltonian . . . . . . . . B. Method of Moments . . . . . . . C. Magnetic Space Group. . . . D. Dipolar Anisotropy Energy . . . AP PAMTUS O I O I I I O O I O O O O O O SAMPLE PREPARATION . . . . . . . . . . A. Crystal Growth and Chemical Analysis B. Crystal Structure . . . . . . . . . RESULTS AND DISCUSSION . . . . . . . . A. Proton Resonance . . B. Chlorine Resonance . . C. Rubidium Resonance . D. Internal Fields . . . E. Magnetic Space Group . F. Discussion of Errors . G. Anisotropy Energy . . O I I o o I v I e I o I o e o .I o I I I I I II o I I I I o I a I I I I I I I I l I I H; 'REFBRENCE S I O I O C C C O I O O I I O O O I O LIST OF TABLES Coefficients for Calculating the Pm . Results of Chemical Analysis . . . Assignment of C835 Lines . . . . . . 5 and Rb87 . . .-. . 85 and Rb87 Comparison ongb8 Pure Quadrupole Lines of Rb Orientation and Magnitude of Internal Fields I I O C I I O Q l O I C I I I Calculated Internal Fields . . . . . Fields 0 ' e e e e I e e I e I e e I I ‘Differences Between Calculated and Observed Page 15 25 26 27 35_ 44 44 Figure 10. 11. 12. 13. 14. 15. LIST OF FIGURES Page Structure of RbZMnC14'ZHZO . . . . . . . . . . 16 Temperature Dependence of the Proton Resonance Frequencies . . . . . . . . . . . . . . . . . 19 Proton Resonance Diagram in an Applied Field of 82 Oersteds . . . . . . . . . . . 20 Temperature Dependence of the C8 35 Resonance Frequencies . . . . . . . . . . . . . . . . . 23 Temperature Dependence of the Rb87 Resonance Frequencies . . . . . . . . . . . . . . . . . 29 Relations between Transitions and Energy Levels for Rb87 . . . . . . . . . . . . . 31 Temperature Dependence of the Rubidium Internal Field . . . . . . . . . . . . . . 32 Variation of the Rb87 Energy Levels with the Internal Field . . . . . . . . . . . . . . . 33 Temperature Dependence of the Rb85 Resonance Frequencies . . . . . . . . . . . . . . . . . 34 Magnetic Bravais Lattice PS . . . . . . . . . 38 Chemical Cell with an Antitranslation along One Edge . . . . . . . . . . . . . . . . . . . 38 Chemical Cell with an Antitranslation along a Face Diagonal . . . . . . . . . . 39 Chemical Cell with an Antitranslation along the Body Diagonal . . . . . . . . . . 39 Stereogram Showing the Observed Internal Fields and the Calculated Rubidium Fields . . 42 Spin_ Arrangement for the Magnetic Space Group Pa bl . . . . . . . . . . 43 '- I. INTRODUCTION This section discusses in general terms some of the information about antiferromagnetic materials obtainable with nuclear magnetic resonance. szMnC84-2H20 is among those hydrated transition metal salts which are of interest partly because proton magnetic resonance is easy to inter- pret when applied to the investigation of ordered states. Protons are ideal probes of the magnetic fields in the cry- stal because they are not effected by a quadrupole inter- action and so respond only to the local field. The transition temperature can be measured by ob- serving the abrupt change in the resonance lines as the temperature is changed through the transition. Pure quad- rupole resonance is ideal for this purpose, because at the transition the quadrupole energy levels are split by the appearance of an internal field and the resonance will sud- denly disappear to reappear at other frequencies. Ordered state resonances are usually unsatisfactory for this purpose because the lines weaken and disappear at a temperature somewhat below TN. Proton magnetic resonance in the para- magnetic state can be used for this measurement, but the applied field may shift the transition temperature. Perhaps the most precise measurement that can be made with NMR is the temperature dependence of the sublattice l magnetization. Since resonance can be done without an ex- ternal field, the nuclei respond to the local internal field which is a direct measure of the magnetization. For most nuclei the effects of the quadrupole interaction have to be considered in deducing the magnetization from the resonance spectrum, but the proton frequencies are directly proportional to the magnetization. The direction of the internal field at nuclei lo- cated within the distribution of magnetization about the magnetic ions is a good approximation to the direction of magnetization. Even a weak transferred hyperfine interac- tion produces fields at these nuclei much larger than the dipole fields so that the internal field at such a nucleus is nearly parallel to the spin of its neighboring magnetic ion. The distribution of internal fields and the magnetic spin arrangement are determined by the crystal‘s magnetic space group. The number of possible magnetic space groups is restricted by the symmetry of the magnitudes and direc- tions of the internal fields. If the atomic coordinates are known, calculation of the local fields predicted by each possibility may select the correct space group. Combination of the crystal structure and NMR data may suggest possible exchange paths in the crystal. For example an atom situated between two magnetic atoms and sub- ject to a particularly large hyperfine field is almost certainly a link in a super-exchange path. Finally NMR can be used to map a crystal's magnetic fifihase diagram by observing the antiferromagnetic to spin- Lilop and spin-flop to paramagnetic transitions as functions .Ief the temperature and externally applied field. At the - .gpin-flop transition the magnetic spins align themselves ‘ffibrmal to the applied field; the size of the field required .gtto produce this effect is a measure of the anisotropy ener- :gf‘holding the spins along the magnetization direction. II. THEORY A. The Hamiltonian The nuclear magnetic resonance spectrum for the nuclei studied in szMnC84-2HZO is determined by a Hamil- tonian composed of Zeeman and electric quadrupole inter— actions. H=-E-fi+Q:vE (1) where E is the nuclear magnetic moment, E the magnetic field at the nuclear site, Q the nuclear quadrupole moment tensor, and VE is the electric field gradient tensor. In general two very different effects contribute to E, the di- polar field due to the magnetic ions throughout the lattice, and the transferred hyperfine interaction. The magnitude and direction of the dipolar field depends on the position of the nucleus with respect to the magnetic ions in a purely classical way while the transferred hyperfine interaction arises from the exclusion principle. Insofar as their configuration and spin spaces over— lap the free ion wave functions of the electrons of the C8- and Mn++ ion are non-orthogonal. To achieve the orthogonality required by the exclusion principle the wave functions of C8” electrons whose spins are parallel to the d electrons L..- A (f the Mn++ must contract towards the chlorine nucleus. The wave functions of C£_ electrons whose spins are anti- parallel to the d electrons of Mn++ mix with those of the d electrons to form covalent bonds with the result that their density at the chlorine nucleus is reduced. The en- hancement of parallel spin density and the reduction of anti-parallel spin density in the neighborhood of the chlorine gives rise to the hyperfine interaction. B. The Method of Moments The energy levels of the Hamiltonian depend in a complicated way on various parameters. However analysis of the resonance spectrum can be greatly simplified by re- course to the moments of the Hamiltonian. The mth moment is defined: rm = trace (Hm) (2) Brown and Parker1 express the Fm in terms of mth degree polynomials in the resonance frequencies, but they can also be expressed in terms of mth degree scalars involv- ing H and VE as follows: r2 = £2 fi-fi + g2 VEsz (3) r3 = {3 fifizvfi + g3(VE-VE):VE (4) Since the Hamiltonian is traceless, F1 = 0. The coefficients involve only functions of the nuclear spin and the coupling cODStants a = 7% and b = %% and are evaluated by comparison with Brown and Parker as shown in Table I. F“ Table I. Coefficients* for Calculating the Fm Coefficient I = E I = E 2 2 35 2 f2 5a —7a 2 2 14 2 32 3" rib £3 6a2b 84a2b 4 3 g3 0 73b “Values are in frequency units If v1, v ........ ,VZI are the frequencies of 2, transitions between adjacent energy levels, then: 1 21 21 2I r2 = W X g ; Ci(n)Cj (n)\)i\)j (5) n=0 1-1 J-l 21 2I 21 21 1 r3=(21—+1)3 Z Z Z Z Ci(n)Cj(n)Ck(n)\)i\)jvk (6) n=0 1=1 J=1 k=1 where Ck(n)=k ;kin Ck(n) = k - (ZI - 1) ; k > n In the paramagnetic state H = 0 and, representing the pure quadrupole frequencies for I = g as v1 and VII and for = .3. ' I 2 as VQ‘ 7 2 2 p = 2 _ e n =§ r2 vQ ‘%% (1+‘3 ’ I 2 (7) T p - 4 v 2+v v +v 2) = 21{E%% 2 1+3: I=§ (8) 2 ' 3 I I II II 23\ 3 ’ 2 p = =1 r3 0 .IZC9) r P = 2 2v +3v v 2- 3v 2v —2v 3 = 4 Eg% 3(1- 2)- 1=5 (10) 3 s I II I II I II 23 n ' 2 where 3E _ 2 q — 32 and 3E 8E ) __£ ‘ __Z 8x 82 ” 8B __5 32 In the antiferromagnetic state for I = % equations (3) and (5) combine to yield: a 3 2 2 3 2 1 r2 IVl + v2 + Iv3 + v1"2 + 2V1V3 + v2V3 5(2—1 + er The magnitude of the internal field is found from 2 n 1 + —3) (11) : equations (7) and (11) 2 . 2n a p . _ 3 = s In order to find the direction of the internal field, a small external field is applied 6H< —-> 11i —®'uo (19) The elements of(E§ are determined by the magnetic space group. The dipolar internal field is: 3¥i?. I + 14(‘51 ' 7)Q- (20) l r. r. 1 1 For a nucleus such as chlorine the internal field is due predominantly to the transferred hyperfine interaction and so is nearly parallel to the direction of magnetization. Thus the direction of go is approximately the direction of the chlorine internal field. In some crystals of sufficiently high symmetry it may be obvious that the magnetization direc- tion must lie along a particular symmetry axis; however this is not the case in triclinic szMnC14-2H20. D. Dipolar Anisotropy Energy The direction of magnetization is determined by the crystalline field of the coordination group about the mag- netic ion and by the dipole-dipole interactions among the magnetic ions. In cases where the crystalline field effect 11 is small then the spins align themselves in such a way as to minimize the energy of the lattice of dipoles. If a magnetic moment 30 is placed at the origin, then the field at the origin due to all the dipoles £1 at positions ;1 is given by equation (20). If the anisotropy energy is deter- mined by such dipole—dipole interactions, then Ho will align itself parallel to H, in the following way, +-> + _ 3riri I + Auo = Z: 5 I __3 <:i°uo (21) 1 r. r. 1 1 For a triclinic crystal 1 o o g + C2,: 0 l 0 ; for ui parallel to “o o o 1 -1 0 0 + g C).= 0-1 0 ; for u. antiparallel to u 0 0-1 1 ° It is clear from equation (21) that the direction of mag- netization is one of the principle axes of the tensor T: i 3¥i?i I T = X:—;—§— - ;—3 'G2_ (22) i i III. APPARATUS The NMR data was taken on a Pound-type marginal oscillator spectrometer. Signals of sufficient strength could be displayed directly on the oscilloscope, and weaker signals could be recorded with phase sensitive detection while the oscillator frequency was motor driven. Frequen- cies were measured by means of a short length of wire placed near the grid circuit in the oscillator whose pickup was amplified and put into an electronic frequency counter. The oscillator r.f. coil was wound directly on the crystal, the number of turns varying with the working fre- quency range desired. The tuning capacitor in the oscilla- tor normally provided a frequency range of about 4 MHz. For a given coil, the frequency could be extended downward considerably by the addition of extra capacitance in parallel With the tuning condenser. The crystal and coil were mounted at the end of a transmission line consisting of a german silver tube with Center conductor. The transmission line was run down into a dewar so that the crystal could be immersed directly in liquid He4. By pumping on the He4 bath the temperature of the Crystal could be varied from 4.2 to 1.1 degrees Kelvin. 12 13 For the experiments involving angular variations the crystal and coil were mounted on a pulley wheel at the end of the transmission line. A string, attached at one end to a spring inside the top of the dewar, was wrapped around the pulley and connected to a brass rod running out the dewar head through an O-ring seal. By moving the rod up and down the crystal could be rotated. This allowed measurements to be made in two perpendicular planes without removing the crystal from the dewar to remount it in a new orientation. Zeeman modulation was used in all the experiments described here. The modulation was provided by a pair of Helmholtz coils placed on the outside of the dewar and free to rotate about the axis of the dewar. In the experiments requiring a small external d.c. field, one coil of the pair carried the a.c. modulation current, while the other carried direct current. v"‘——————-— IV. SAMPLE PREPARATION A. Crystal Growth and Chemical Analysis Solutions were prepared by adding RbC£ to a warm saturated solution of MnC£2 in sufficient amount to satu- rate it. The solution was allowed to evaporate at room temperature in an open beaker. In twelve to twenty-four hours, seed crystals had formed and were allowed to de- velop until some of the larger ones could be tied and hung in the solution. Sizable crystals could be grown in a comparatively short time; however well shaped single cry- stals were difficult to obtain perhaps because the crystals tended to grow too fast. The chemical composition of the crystals obtained in this way was checked by sending a sample for commercial analysis. Table II compares the calculated percentages with the analyst's measurements. The crystals grew as elongated four sided prisms with no well developed end faces. Saunders2 reported on the morphology but his results are very incomplete because his crystals had insufficient faces and had a tendency to twin. 14 15 Table II. Results of Chemical Analysis Element Calculated Percentage Measured Percentage Rubidium 42.3 42.0 Manganese 13.6 13.9 Chlorine 35.1 34.6 Water 8.9 8.8 B. Crystal Structure Jensen3 studied the structure of this compound and he found it to be triclinic with space group PT, the only symmetry elements being inversion centers at each corner of the cell, at the face centers, and at the body center of the cell. He reports the chemical cell dimensions as a = 5.66 X, b = 6.48 X, c = 7.01 X, a = 66.7°, B = 87.7°, and y = 84.8°; he also gives the atomic coordinates of all the atoms except hydrogen and lists his calculated and ob- served structure factors. The oxygen and chlorine atoms lie near the corners of a square with manganese at the cen- ter and two oxygens lie above and below the plane of the square. Figure 1 shows the crystal structure projected into a plane normal to the a-axis. The oxygen atoms lie the proper distance from chlorine atoms on different octahedra to form hydrogen bonds. The hydrogen atoms were assumed to lie along these bonds at a ‘ lb Figure l.--Structure of RbZMnCK4 -2H20 l7 distance of 0.987 A from the oxygen as suggested by El Saffar.4 In order to determine the morphology, crystals were mounted on an X-ray diffractometer. Reflections were iden- tified by comparing the relative intensity of various orders with Jensen's data and comparing the 26 values with calculated interplanar spacings. In the ten crystals of various sizes studied in this way it was found that in every case the larger face was (100) and the smaller (011) with the elongated direction being the [Oil] zone axis. It was also found that every crystal was twinned about the [Oil] axis. The crystal which was used for angular measurements was far too large to insert directly in the X-ray beam; it was mounted so that x-rays could be diffracted from one cor- ner. Five corners of the crystal were examined in this way F and its orientation was determined from the results. Re- flections from four of the corners resulted in identical assignment of axes. These corners all showed very much f 'Weaker twin reflections. On the fifth corner, chosen because of its irregular appearance, the reflections from the two kinds of twin material were of nearly equal intensity, making assignment of axes impossible. Apparently the smaller pro- portion of twinned material became reoriented at some tem- perature, because at liquid helium temperatures, the angular Variations of the resonance lines showed no evidence of twinning. I V. RESULTS AND DISCUSSION A. Proton Resonance At l.l°K two proton resonances were seen in zero applied field. The frequencies of these lines were 18.96 MHz and 18.30 MHz and since H = ng, the magnitudes of the of the internal fields at the proton sites are 4,453 oer- sted and 4,298 oersted. Figure 2 shows the temperature dependence of the frequency of the two lines and the curve is extrapolated to TN = 2.24°K. Figure 3 is the resonance diagram obtained by apply- ing an external field of 82 oersteds in the plane normal to [011]. The diagram shows that the magnetic Ordering is antiferromagnetic as each resonance line is split into two lines by the external field. This splitting is a maximum when the applied field is parallel to the projection of the internal field into the rotation plane. Thus the projection of the larger internal field lies at ¢ = 110° and the smaller at 6 = 82°. Rotation of the field in another plane is then sufficient to determine the direction of the internal fields at the proton sites. The magnitudes and directions 0f the proton, chlorine, and rubidium fields are summarized in Table VI, which appears in section V-D. 18 . 19 f 20 l8 - l6 r- l4 - I2 _' \\\ _ \‘:\\ \\ N I'O _ ‘\“‘\ 2 __ ‘3‘ n. ‘3. CT ‘u 22 8 ' \\ LL ‘3. - ‘3. ‘.‘~ 6 — “ii ‘. 4 — : i _ I i 2 ‘ : E l I 0 J 1 L 1 l I L 1 1 1 l ' I-l l2 l3 l-4 l-5 l-6 L7 L8 I.9 2.0 2.: 2.2 2.3 T (°K) Figure 2.--Temperature Dependence of the Proton Resonance Frequencies h |94.. I713- ITIB- ISO |60 I40 |20 IOO 80 60 4O 20 in an Applied Field of 82 Oersteds 21 The small splitting of the resonance lines in Figure 3 is due to the dipole-dipole interaction between protons on the same water molecule. That is each proton in a water molecule is effected by the dipolar field of the other pro- ton in the molecule in addition to the much larger fields arising from the magnetic ions. The Hamiltonian for the two protons is: + + “ = ‘“1'H1 ' “2' 2 ' s + —“3‘ (27) where the subscripts identify the magnetic moments-and in- ternal fields at the two proton sites and f is the proton- proton vector. The small splitting in Figure 3 arises from the third and fourth terms in the Hamiltonian which were evaluated neglecting very small terms. 2 2 Av = _g_%_ (3 cos 61 cos 62 - cos 012) (28) r where Si is the angle between Hi and f and 012 is the angle between H1 and H2. The observed splitting is 40.1 kHz for the higher frequency line and 40.4 kHz for the lower while eValuation of Av yields 36 kHz. This indicates that the Pclsitions assigned the protons are not precisely correct, bllt this calculation gives no good indication of how they can be corrected . Figure 3 also shows that the magnetic space group CcIntains only elements which change a given field into an 22 Oppositely directed field. Furthermore it is clear that an anti-inversion center cannot support a non-zero axial vec- tor, since the vector would be transformed by the center into its negative and cancel itself. Thus the sites of the magnetic ions in the lattice cannot be anti-inversion cen- ters.5 These two facts lead to the choice of P51 from among the triclinic groups6 as the only possible magnetic space group, but the direction of antitranslation in this group remains to be determined. B. Chlorine Resonance In the paramagnetic state two weak C235 pure quad- rupole resonances were observed at 3.853 MHz and 5.335 MHz. In the antiferromagnetic state a total of twelve chlorine resonances were observed. Six of these resonances were strong and six rather weak, arising from C835 and C£37 re- 37 has somewhat smaller magnetic and quad- spectively. CK rupole moments than CZSS and is about one-third as abundant; both isotopes have nuclear spin I = g. Thus for every strong C£35 resonance there is a weaker C237 resonance at 35/037 W 1.2. The C837 reso- a lower frequency such that v nance is of little interest beyond identifying the resonant nuclei as chlorine. At 1.1°K the six C235 resonances occur at 10.72, 10.48, 9.63, 8.56, 7.72 and 6.25 MHZ. Figure 4 shows their temperature dependence and the two pure quadrupole lines q -MH% I +- ///// Rm l l 1 LI |.2 L3 L4 l.5 LS LT l.8 l.9 2.0 2.: 2.2 2.3 T (°K) Elgg:g_i.--Temperature Dependence of the C£DS Resonance Frequencies 24 above TN. The ordered state lines gradually disappear as the temperature approaches TN so that the resonances cannot be observed above l.SS°K. Of the six C235 resonances, three must be associated with each of the inequivalent chlorine sites in the chemical unit cell. The proper selections were made based on the fact that, when extrapolated into the paramagnetic state, the second moment equals the square of the pure quadrupole frequency. The internal field at the proton and chlorine sites may be written: 212+ u Ep (29) and ch = EC£ (30) Where is the expectation value of the spin of the Mn++ ion. The proton field is simply the sum of the dipolar fields of the magnetic ions throughout the lattice while the chlorine field is due principally to the transferred hy- Ikirfine interaction with a small contribution from dipolar fifislds. However C and ECK are vector functions of the lat- tlile parameters and atomic coordinates and are assumed in- dependent of temperature. Thus 2 _ 2 HCK — K Hp (31) 25 and from equation (11) for a temperature T: 2 2 YCK v 2(T) + vQ2(l +.fl§) (32) F (T) = 5K(———- 2 Yp P The selection is made by a trial and error choice of three lines from the chlorine spectrum from which F2 is calculated at various temperatures and plotted against the value of vp2(T) at the same temperatures. If a prOper v choice of v and us has been made, the graph will be 1’ 2’ a straight line and can be extrapolated to vpz = 0 where the intercept will be the square of one of the pure quadrupole frequencies. The results of this process of selection are shown in Table III. Table III. Assignment of C£35 Lines Site Transition* v at 1.1-MHz Calc. vQ-MHz Obs. vQ-MHz A (-:;’—— - %) 10.72 A (é- + %—) 6.25 3.85 3.85 .A. (+% — + g 8.56 B (% _ - 31g.) 10 48 B (é _ + %) 9.63 5 s3 5 34 B 0%.- + :3.) 7.72 *In.the high field labeling 26 The fact that the lines in each set are not approxi- mately equally spaced indicates that the angle between the internal field and the z-axis of the electric field gradient tensor is N54° at which angle equal splitting occurs only at very high fields. The magnitude of the fields at the two chlorine sites is found from equation (12), and the direction of the fields is found from equation (16) as described in Section II-B. The magnitude and direction of the chlorine fields are listed in Table VI. C. Rubidium Resonance Resonance lines were observed for two isotopes of rubidium whose properties are summarized in Table IV. In 85 addition to the fact that Rb has nuclear spin I = g and Rb87 has I = g, the relative quadrupole contribution to the tOtal Hamiltionian is much larger for Rb85 than Rb87. Table IV. Comparison of Rb85 and Rb87 IScitope NMR Frequency Abundance Spin Quadrupole Moment MHz/lOkG % in 7% exlO-24 cm2 _-‘ Rb85 4.111 72.8 g 0.28 Rb87 13.932 27.2 g— 0.14 \ 27 In the paramagnetic state three quadpolar resonances were observed at 3.2229, 3.1161 and 2.6743 MHz. They cor- respond to the 1% — 1% and 1% — 1% transitions for Rb8S the 1% — 1% transition for Rb87. and The ratio of the quadru- pole moments of the two isotopes has been reported by Meyer-Berkhout7 from beam measurements. 85 RQ = §§7 = 2.0669 1 .0005 (36) RQ can be calculated from equations (7) and (8) in terms of the quadrupole resonance frequencies. Only the assignment of frequencies shown in Table V yields the proper ratio, R = 2.07 1 .06. Q . 85 87 Table V. Pure Quadrupole Lines of Rb and Rb IsotOpe Spin Transition Frequency 85 5 3 1 Rb 7 VI (17 — 17) 2.6743 1 .0018 85 5 5 3 Rb 7 011 (:7._ 17) 3.2229 1:.0060 11 87 3 3 _ l _ b 2 0Q (17 17) 3.1161 '*.0016 \ An alternate method of selection is based on Cohen's8 tables of energy levels as a function of asymmetry parameter, I), for I = g and I = %. Two frequencies can be arbitrarily aSSigned to the two Rb85 transitions and the tables yield a 28 value for n. Using the ratios of the quadrupole moments and 87 this value of n, the frequency of the Rb transition can be found from the I = % table. The assignment shown in Table V is the only possibility with this internal consistency. The value of the asymmetry parameter obtained is n - 0.78. Equations (8) and (10) can also be combined to yield an expression for the asymmetry parameter. r n2)3 p 3 1 + 3 _ 400 (12 ) . _ 5 2 - p 2 , I - 7 (37) (1 _ n2 9,261 (r3 ) L With sz and P3P calculated from the frequencies assigned to Rb85 this yields n = 0.78. In the antiferromagnetic state the Rb87 resonance Spectrum consists of six lines. The temperature dependence of these lines is shown in Figure 5. Some of the lines Could be seen very near the transition temperature so the TN was measured by watching for the disappearance of the quadrupole lines and the appearance of the ordered state ilines as the temperature was slowly lowered. Accurate de- t'eir‘mination was hampered by the nearness of TN to the super- 4 flliid transition of the He bath, making the pumping rate difficult to control in this region. A value of TN - 2.185°K COlnpares with 2.24°K obtained by Forstat, Love, and McElearney9 fI‘om the specific heat. IO 9 #— aI— ? I— \ \ \ \ h N6 '- \\ 3C \ 2 ‘I I \ 35 I- ‘I h- | U- ‘. I I 4 L. '. I I I \ I :\ :‘L_. 3 _ ‘§\ 3 \\\ I \\ '3 ‘ I \ I \\ I, 2 — ‘I I n u I | \ I I — i | I I I '. O 1 I I L I I L I I I 1 L '. IJ L2 L3 L4» L5 L6 lf7 L8 L9 2‘) 2J 2L2 2:3 T (°K) ‘Figure S.-- Temperature Dependence of the Rb Frequencies 87 Resonance 30 The six lines in the spectrum represent all possible transitions among the four energy levels for I = g and, if they are de51gnated v1 > 02 > 03 > 04 > us > 96, they are interrelated. (38) (39) The transitions and their relations are illustrated in Figure 6. The magnitude of the internal field was calculated from equation (12) and this field is plotted as a function of temperature in Figure 7. The energy levels were calcu- lated from the frequencies at various temperatures and combined with the internal field at those temperatures to yield Figure 8 which shows the variation of the energy levels in frequency units with the internal field. The direction of the internal field was calculated from equation (16) as described in Section II-B. The results are sum- marized in Table VI. The resonance Spectrum of Rb85 in the ordered state consists of four lines, and their temperature dependence is shown in Figure 9. The frequencies of these lines vary little with the temperature because the Hamiltonian is dominated by the temperature independent quadrupole inter- . 85 . action. The Rb lines were a good deal weaker than the 87 Rb lines because, while Rb8S is nearly three times more 5| j \ \ \ \ \ \ \ \ \ \\ V4 \ V2 \ ,L “ 5° / / / / / / Ea / I 14 V6 ”0 E 3 r I \ \ \ \ \ \ \\ \g - EO I; 1/ 3 I I Us I I / I I I / [I E4 I / Figure 6.--Relations between Transitions and Energy levels 8 7 for Rb uI Ix) 'o I HinI- koe. o CO I 0.3r- OHZF- 0.0 J l 1 I #1 I I I I I I I I I I I I I I I I I I I I I I I e L J I I l l I.I I.2 L3 L4 I.5 I.6 T (°K) I.7 L8 L9 243 ZJ 21! 23 I igurx: 7.-- fennn3rat11ro [kniontknice (1f tIu: RIH1idiLun Intxsrnal Field -3~—- EI «IEO I I..- 52 £3 '50 / I I I I *1 I I I I I [.8 L6 |.4 I2 I0 08 0.604 0.2 0.0 .. . . - \‘7 . . I-_1gure_8_.—-V:II‘IutIon ol the Rb" Iznex'gy Levels Is'ItII the Internal 7 _a— I:IL‘I\I .5 moflucoisop; 03:35.an nnmE one 00 933......Hmubz 9533.33.97...mlwmI~w.rflM OE .r mm mm _.N ON a. m.— h._ 0.. 0. ¢.. n._ N._ _._ 4 . d d _ d . q 1 _ . 2 _ l1 \ x a I TOIL. . . a O o O O O o I z o a I 0 O. o 0.. O .l._ O O O \ no \ o. I x o o on oo o . o o o. oo o 00 o o o I vIIoll . s L I I I I I D DI IIIO III. N.~ N. 0. 0. w. '0 '0 N N 'sz —'baI_.3 V’ '0 o.n 0.0. 35 abundant, the ratio of the transition probabilities is pro- portional to the square of the ratio of the magnetic moment. W85 Y85 2 W a —"8_7' = 0.087 (40) Y D. Internal Fields Table VI summarizes the magnitudes and directions of the local fields found at the proton, rubidium, and chlorine sites. The magnitudes given in the table were measured at l.l°K and the directions are given in terms of the angles x, W, and w measured from the 3, b, and Z axes respectively. The directionscxfthe measured fields are shown in the stereogram in Figure 14. Table VI. Orientation and Magnitude of Internal Fields Magnitude at X I w Site l.l°K (koe) with a with b with c proton I 4.439 60 30 55 proton II 4.296 35 65 54 rubidium 1.879 128 123 70 chlorine A 19.49 62 39 42 chlorine B 21.63 57 42 44 36 E. Magnetic Space Group The magnetic space group has been fixed as PSI by the proton applied field resonance patterns, but the direc- tion of antitranslation represented by "5" remains to be determined. According to Opechowski10 there is only one nontrivial magnetic Bravais lattice in the triclinic system. It is generated by three primitive translations one of which is primed. Thus the magnetic lattice may be described by two normal translations and one antitranslation. The set of primitive translations which characterize u on o o - + + a triclinic lattice 15 not unique. If the set a1, a2, and + . a3 generate a lattice, a number of other sets can be made from these vectors which will also generate the same lat- t. h + ++ + —> + ++ ++ -> + ice, suc as (al 32)' 32, as, or (31 a2 a3), al, a3. In general the new sets of primitive translations can include the edges, the face centers, and the body diagonal of the original primitive cell. The set of primitive translations which describe the chemical lattice of RbZMnCK4-2HZO was chosen to make the lengths of the translations as short as possible. These ,,, , -+ + + primitive translatlons, namely a, b, and c are not neces- sarily identical to the translations which generate the magnetic lattice. The magnetic lattice is subject to two conditions. One of its primitive translations must be primed and secondly these translations must be the edges, 37 face diagonals, or body diagonal of the chemical unit cell. There exists a lattice in which the magnetic symmetry can be described with an antitranslation in only one direction, but the same symmetry described in the particular lattice formed by 3, b, and 6 may require more than one direction of antitranslation. By considering the primed direction of the magnetic lattice to be successively an edge, a face diagonal, and a body diagonal of the chemical cell, it can be shown that the magnetic symmetry when represented in the chemical unit cell may require one, two,or three directions of antitranslation. Figure 10 shows the triclinic magnetic lattice P5' The open and solid circles are used to distinguish primed from unprimed translations. Translations between two cir- cles of different kinds are antitranslations while those between circles of the same kind are normal translations. Thus the magnetic lattice is generated by a primed transla- tion i1, and two unprimed translations i2 and is. The chemical cell is shown in Figure 11 with the antitranslation chosen as an edge. The vectors represent the primitive translations of the magnetic lattice. The Same arrangement of open and closed circles can be repre- Sented by the chemical cell with one direction of antitrans- lation. The antitranslation has been chosen as a face diagonal of the chemical cell in Figure 12. This symmetry Figure IU.—-Mugnetic Bravais Lattice Pg Iibgu}}L4LI.—-Iflunnicul Ikrll witdi an thtitInnIslutirn1 uILnIg One Edge 3 E) Figure 12.—-(IheIIIical Cell with an .-'\ntitI‘anslatien along a Face Diagonal Figure 13.-—(?IIeIIIical (Iell with an Antitranslation along the Body Diagonal 40 can be described by one direction of antitranslation using the set of primitive translations shown, but in the chemi- cal cell it requires two directions of antitranslation. In Figure 13 the body diagonal of the chemical cell has been chosen as the antitranslation. It requires three primed translations to represent the same symmetry in the chemical lattice. I The preceding figures have shown that in the tri- clinic system the magnetic symmetry, while expressible in a proper magnetic lattice, may require one, two, or three antitranslations for its description in the chemical lat- tice. Thus the direction of antitranslation in the PSI can be any of seven possibilities: 3, b, 3, 3 enul b, 3 and E, b and 6, and 3, b and E. The dipolar local fields at the proton and rubidium sites were calculated on an electronic computer for each of the possible antitranslations. The chemical cell dimensions, the positions of the manganese, hydrogen, and rubidium atoms, and the direction of magnetization were read into the com- puter which formed the sum shown in equation (20) for all Inagnetic ions within a sphere of fixed radius. The direction <>f magnetization was taken parallel to the chlorine internal Ifiields and the radius of the sphere was 40 A which included I'O‘ughly one thousand magnetic ions. The magnetic moment at 1- 1°K was calculated by extrapolating the temperature de- pefndence of the proton frequencies to 0°K, and calculating 41 the fractional change between 0° and l.l°K. Then 0 (l.l°) 0(1.1°) = {$7567qu (41) P where the saturation magnetic moment was taken as five Bohr magnetons. The proton local field is largely determined by the orientation of the nearest magnetic spin and is not sensi- tive to the various antitranslations. Thus reasonable proton fields were obtained for each of the seven possibilities, indicating that the direction of magnetization was chosen properly, but not selecting the prOper directions of anti- translation. The directions of the calculated rubidium local fields, however, were strongly dependent on the direction of antitranslation. Figure 14 shows in a stereographic projec- tion the measured rubidium field and the fields calculated for each of the possible antitranslations. Only that field corresponding to antitranslations in both the 3 and b direc- tions lies close to the measured field. Table VII shows the magnitudes and directions of the fields calculated for this set of antitranslations. The magnetic spin arrangement is Shown in Figure 15. Figure 14.--Stereogram Showing the Observed Internal Fields —‘_._.§—_ .—————-——————-" ‘ o a . - u and the Lalculated Rubidium Fields Figure 15.-~8pin Arrangement for the Magnetic Space (iroup I’ l (I , I1 44 Table VII. Calculated Internal Fields Magnitude x W m Site Oersteds with a with b with c proton I 4,177 54 38 51 proton II 4,223 35 75 53 rubidium 1,768 117 102 44 F. Discussion of Errors The agreement between the calculated internal fields and the observed fields is indicated in Table VIII which lists the percentage differences in the magnitudes and the angular differences measured along a great circle. The agreement is quite good for the protons particularly since their positions are only based on a hydrogen bonding scheme. This indicates that the direction of the chlorine internal fields is a good approximation to the magnetization. Table VIII. Difference between Observed and Calculated Local Fields Angle along a Site Magnitude % Great Circle proton I 5.9 7.5 proton II 1.7 10.0 rubidium 5.9 24.5 45 The direction of the rubidium field is not in such good agreement. This is not simple to explain, because the direction of the internal field was determined more accur- ately than the protons and presumably the X-ray determina- tion of the rubidium atomic coordinates is quite accurate. The method of moments can be applied to any three lines in the spectrum of a nucleus with spin I = 3/2. The direction of maximum splitting was found for five Rb87 lines and dif- ferent sets of these lines were used in three calculations of the internal field direction. The field directions ob- tained from these calculations were all within ten degrees of each other. The rubidium internal field was calculated for a few directions of antitranslation with the magneti- zation direction taken along the manganese to oxygen bond. The magnitudes and directions differed slightly from those calculations with the magnetization along the chlorine fields but not enough to explain the discrepancy in terms of inac- curacy in the magnetization direction. One possible explanation that remains is that the rubidium field is not due entirely to magnetic dipolar ef- fects. The fact that the observed and calculated field di- rections and the magnetization direction are nearly coplanar, 5Suggests that the observed internal field is the result of dipolar fields and a nonvanishing magnetization at the rubi- dillm site. Even a small value of the magnetization could be Snifficient to combine with the dipole field to yield a net fieald in the observed direction. 46 G. Anisotropy Energy The exchange field He can be estimated from the transition temperature UH = kT (42) where u is the manganese magnetic moment and k is Boltz- mann's constant. This yields a value for He = 6.66 koe. The spin flop critical field, Hc’ is related to the exchange field and the anisotropy field Ha.11 HC ” ZHeHa (43) This relation is based on the molecular field approximation and is most nearly exact at absolute zero. Since a search for the spin flOp transition at l.l°K indicates that the critical field is not below 8.3 koe then the anisotropy field must be greater than 5.2 koe. Computer calculation of the principle axes of the tensor in equation(22)indicates that the dipole-dipole ef- fect does not make a significant contribution to the aniso- tropy energy. None of the axes lies very close to the direction of magnetization and the field produced at one manganese site by the other manganese ions in the lattice is only about 750 oersteds. The axis of local symmetry at the manganese site is the manganese to oxygen bond. The effect of the crystalline field is to align the spins in this direction and it is 47 significant that this bond direction is close to the chlorine internal fields as shown in Figure 14. Thus the direction of magnetization is determined almost entirely by the crystal- line field. In conclusion, continued investigation in this area seems worthwhile. The search for the Spin flop transition should be extended to higher fields,in order to map the phase diagram; and the effects of the relative magnitudes of the exchange and anisotrOpy fields may bear further study. 10. 11. (DOW REFERENCES C. Brown and P. M. Parker, Phys. Rev. 100, 1764 (1955). E. Saunders, Am. Chem. Jour. 14, 127 (1892). . J. Jensen, Acta Chem. Scand. 18, 2085 (1964). . M. El Saffar, J. Chem. Phys. 45, 4643 (1966). Donnary, L. M. Corliss, J. P. H. Donnay, N. Elliot, and J. M. Hastings, Phys. Rev. 112, 1917 (1958). . A. KOpsik, Shubnikov Groups (Moscow University, Mos- cow, 1966). . Meyer-Berkhout, Z. Physik 141, 185 (1955). . H. Cohen, Phys. Rev. 96, 1278 (1954). Forstat, N. D. Love, and J. N. McElearney, private communication. . Opechowski and R. Guccione, Ma netism (Academic IIA Chap. Press, New York, 1965) Vol 3. Kanamori, Magnetism (Academic Press, New York, 1963) Vol. I Chap. 4. 48