ABSTRACT LOW-TEMPERATURE SPECIFIC HEATS OF HEXAGONAL CLOSE-PACKED ERBIUM—THULIUM ALLOYS By Akella V. S. Satya The specific heats of hexagonal close-packed erbium and thulium metals. and three of their isostruc- tural alloys were measured in the liquid—helium tempera- ture range between 1.3° and 4.2°K for examining the valid- ity of the localized 4f-band model. on which the current theories of the rare-earth metals are based. Barring possible uncertainties in the magnetic properties of the samples and their impurity contents. the coefficients of the Specific-heat component linear in temperature calcu- lated from the present data range in values approximately two to twenty times the constant electronic specific-heat coefficient predicted by the above model for all the hexagonal close-packed rare-earth lanthanides. Possible explanations for such discrepancies are discussed. An itinerant 4f—band model based on the one-electron-band l Akella V. S. Satya model suggested by Mott is proposed for the lanthanides as an alternative to the localized 4f-band model. LOW-TEMPERATURE SPECIFIC HEATS OF HEXAGONAL CLOSE-PACKED ERBIUM—THULIUM ALLOYS BY Akella V. S. Satya A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of ' DOCTOR OF PHILOSOPHY - Department of Metallurgy. Mechanics and Materials Science 1969 ACKNOWLEDGEMENTS The author can express only inadequately his gratitude to Professor Chuan-Tseng Wei for suggesting this problem. his constant guidance and encouragement. and his generosity in reading the manuscript. It is a pleasure to acknowledge his indebtedness to Professor Donald J. Montgomery. Chairman. Department of Metallurgy. Mechanics. and Materials Science. for several enlightening discussions. his constant interest. and the moral support received from him; and to Professor Austen J. Smith for many helpful discussions and valuable suggestions. The author also wishes to extend his gratitude to Profs. Frank J. Blatt. Harold Forstat. and Carl. L. Foiles. Department of Physics. for several stimulating discussions. Thanks are due to the National Science Foundation for making this work possible with its grant #GK-2224. It is a pleasure to acknowledge with thanks the prompt services received from the Division of Engineering ii Research: its staff. the machine shop. and the electronics shop. The close cooperation and help received in the past from Drs. Paul J. Tsang and Ram M. Srivastava are gratefully appreciated. Finally. the author would like to acknowledge the devotion of his wife. Mrs. A. Padma Satya. who has shared with a smile all the fruitless struggles when he was at- tending to his earlier research problem commuting from Port Huron. Michigan. where he served as the Chief Metal- lurgist in the Midwest Machine Company of Indiana. Inc.. for nearly two years. iii TABLE OF CONTENTS Page ACKNOWLEDGEMENTS. . . . . . . . . . . . . . . . . . ii LIST OF TABLES. . . . . . . . . . . . . . . . . . . v LIST OF FIGURES . . . . . . . . . . . . . . . . . . vi Chapter I. INTRODUCTION. . . . . . . . . . . . . . . . 1 II. EXPERIMENTAL TECHNIQUES . . . . . . . . . . 17 III. EXPERIMENTAL RESULTS. . . . . . . . . . . . 37 IV. DISCUSSION. . . . . . . . . . . . . . . . . 49 V. CONCLUSIONS . . . . . . . . . . . . . . . . 70 REFERENCES. . . . . . . . . . . . . . . . . . . . . 72 APPENDICES. . . . . . . . . . . . . . . . . . . . . 76 iv Table I—l II-l II-2 III-l III-2 III-3 III-4 III-5 IV-l LIST OF TABLES Electronic configurations and some physical properties of the lanthanides. Some low-temperature specific—heat data of the lanthanides. Spectrographic analyses of the pure metals. Analyses on the present melts Specific—heat data on Erbium. Specific-heat data on Er alloy. .75Tm.25 Specific-heat data on Er Tm alloy. 5 5 Specific-heat data on Er alloy. .25Tm.75 Specific—heat data on Thulium sample. Specific-heat contributions of the present samples in milli-cal/mole/°K. Page 10 18 24 41 42 43 44 45 55 Figure 10. ll. 12. 13. 14. LIST OF FIGURES Calculated energy—band for gadolinium. Fermi surface for holes in Tm from one- electron calculation and localized 4f-band model. Induction furnace. Experimental setup (schematic) Cryostat system. . . . Manometer system Circuit diagram. Calibration for Tm Specific-heat vs. temperature curve for Er and Er0.75TmO.25° Specific-heat vs. temperature curve for r . . . . . . . . . . . E 0.5Tmo.5 Specific—heat vs. temperature curve for Tm and ErO 25Tm0.75° . . . . . . Low-temperature magnetic structures of Er and Tm. . . . . . . . . . . . Curie temperature vs. composition for Ho-Er and Er-Tm systems. . . . . . . . . Cv/T vs. T2 curves and analyses for Er vi Page 21 27 29 33 35 39 46 47 48 51 51 56 LIST OF FIGURES (cont.) Figure 15. 16. l7. l8. 19. 20. 21. 22. Cv/T vs. T2 curves and analyses for Er0.75Tmo.25 Cv/T vs. T2 curves and analyses for ErO.5TmO.5 . . Cv/T vs. T2 curves and analyses for Er0.25Tmo.75 Cv/T vs. T2 curves and analyses for Tm Electronic specific—heat coefficient vs. %Tm Comparison of specific-heat data of different workers. . . . . . . Calculated and measured 3d—energy bands. Schematic itinerant 4f-band model. vii Page 57 58 59 6O 61 64 66 69 CHAPTER I INTRODUCTION The physical properties and the alloying behavior of a metal depend to a large extent on the structure of its electron energy bands. Significant progress has been made in recent years in both the theoretical calculation of the energy bands and the experimental investigation of the Fermi surfaces in various metals. The electronic structure of the rare-earth metals is. however. far from being completely understood. The lanthanides. from cerium to lutetium. and the actinides. from thorium to lawrencium. tOgether:flnm1the largest group in the periodic table. and are characterized by their partially filled f-shells in the atomic state. Even in the metallic state. the lanthanides have been traditionally viewed as consisting of trivalent atomic cores including the 4f—shell. with three conduction electrons per atom. This concept arose in an effort to ex- plain the similarities in the chemiCal and the physical properties of these rare-earth metals. Any dissimilarities are attributed to the difference in the number of electrons in the 4f-Shell. Nierenberg and co-workers [l] determined the ground-state electron configurations of all the lanthan- ides in the atomic state. by using an atomic-beam reson- ance method. The results are listed in Table I-l together with some of their physical properties. The physical properties of the lanthanides were reviewed early by Spedding et al. [2]. Kasuya [3] treated the magnetic properties and the electrical resistivities of the lanthanides based on an s-d exchange model assuming that the atomic picture is nearly applicable for the 4f— electrons in the rare earths even in the metallic state. Yosida and Watabe [4]. and Elliott and Wedgewood [5] at- tempted to explain some of the experimental data by assum- ing that only the 5d—6s electrons occupied the conduction bands. which were treated essentially as free-electron bands. perturbed perhaps by a small crystal potential. Several calculations [6-10] for the band structure of the rare earths have appeared in the literature since then showing that the Fermi surfaces of these metals are considerably modified from those of the nearly-free- electron model. Dimmock and Freeman [6] criticized the earlier nearly-free-electron calculations [4. 5] that they failed to explain the large electronic specific heat TABLE I-1.--E1ectronic configurations and some physical properties of the lanthanides =====: Atomic-state Crystal Curie Neel u gJ Atomic electron con- struc— temp. temp. Bogr Bohr No. Element figurations ture °K °K magne- magne- tons tons (Ref.1) (Ref.15) (Ref.15) 57 La 5dl6s2 hex A3' 1 1 2 58 Ce 4f 5d 65 , fcc A1 12.5 0.62 2.14 4f26s2 59 Pr 4f36sz hex A3' 2‘3" 3.2 60 Nd 4f46s2 hex A3' 7.5 2.3 3.27 61 Pm 4f56s2 hex A3' 6 2 62 Sm 4f 65 rhomb. 14.8 7 2 63 Eu 4f 65 bcc A2 90 5.9 7.0 7 1 2 64 Gd 4f 5d 65 hcp A3 293.2 7.55 7.0 65 Tb 4f96sz, hcp A3 221 229 9.34 9.0 4f85dl6s2 lO 2 66 Dy 4f 63 hcp A3 85 178.5 10.2 10.0 11 2 67 Ho 4f 65 hcp A3 20 132 10.34 10.0 12 2 68 Er 4f 63 hcp A3 19.6 85 8.0 9.0 13 2 a 69 Tm 4f 65 hcp A3 22 51- 3.4 7.0 60 70 Yb 4f14632 fcc A1 71 Lu 4fl45d16s2 hcp A3 aThulium-type anti-phase magnetic transformation temperature. of gadolinium [ll~l3]. The Specific-heat data. discussed in later sections. indicate a density of states at the Fermi surface of Gd about eight times that predicted by the free—electron model. Accurate measurements of the various properties of a screw—type spin ordering in the rare earths by Koehler et a1. [14. 15] prompted Kasuya to extend his s-d exchange model [3] to incorporate the s-f interaction [16]. Dimmock and Freeman [6] calculated the band struc- ture of gadolinium metal using the non-relativistic aug— mented—plane-wave (APW) approximation. Their results show that the 4f-band is only 0.05 eV wide. and is about 10.9 eV below the bottom of the conduction band. which is 5d—6s in nature. They stated that the seven localized 4f electrons per atom in gadolinium would account for the major part of the 7.5uB saturation magnetic moment ob- served in this metal [17]. However. their predicted value of the electronic specific-heat coefficient of Gd is only 40% of the value measured by Lounasmaa [13]. who also assumed that the three conduction electrons per atom alone occupy the hybridized 5d-6s bands. Dimmock and Freeman [6] attributed this disparity to an electron- phonon enhancement. Their representation of the density of states of the 5d-6s conduction band at the Fermi sur— face is shown in Fig. 1. from which they showed that the 5d band is about 6.8 eV wide and that it resembles the d-band in the transition metals. Based on their APW calculations Freeman. Dimmock and Watson [7] computed a Fermi Surface for the thulium metal as determined largely by the 5d band. Their Fermi- surface representation for the holes in the thulium metal is shown in Fig. 2. They agreed. however. that the posi- tion and the width of the 4f-band was strongly dependent on the periodic potential assumed. They further attempted to explain the anomalies in the temperature dependence of the resistivity [18-20] of the heavy rare earths (from Gd to Lu) in terms of the super—zone boundaries in the Fermi surface that appear as a consequence of the magnetic order- ing. and found but a qualitative agreement between their theory and the experimental data. Watson. Freeman. and Dimmock [8] then considered the perturbations of the 5d bands due to the ordered 4f—moments in the heavy rare earths. and suggested that these perturbations introduced gaps in the bands largely at or near the Fermi—surface sections along its basal planes. This. they claimed. :3- >~ E30. Gd é . 4 E10] 2 . . 2 f -0.4 Ene my -> (R ydbergs) Calculated energy band for gadolinium no.1 (3.9.6) Fermi surface for holes in Tm from one-electron calculation and localized 4feband model no. 2 (Ref. 7) destroyed or perturbed large sections of the Fermi sur- faces in these metals. Kim [9] treated the conduction electrons in the rare earths as being mediated by the exchange of Spin waves with the localized Spins. that might cause an en- hancement in the electronic specific heat similar to that of the electron—phonon type. However. he did not bring out any quantitative comparison between his analysis and the available experimental data. Andersen and Loucks [10] calculated the band structure of bcc eurOpium by using the relativistic APW method. and reported a density of states at the Fermi surface of 12 per atomic Rydberg. This value corresponds to an electronic Specific-heat coefficient of about 0.5 milli-calorie/mole/°K2. which is only one-third the value obtained experimentally by Lounasmaa [13]. Andersen and Loucks [10] attributed this difference to the electron—phonon enhancement. purity of the sample. and the steepness of the density—of-states curve. Herring [21]. on the other hand. believes that there is a quite narrow group of 4f-1ike bands in the rare earths appreciably hybridized with the s-p-d bands. The width of this 4f—complex is presumably small compared to the energy cost of placing other than the optimum number of 4f electrons on an atom. and only when the complex acci- dentally lies at the Fermi energy will there be any 4f-like portion of the Fermi surface. Herring objects the values of the widths of the 4f—1ike bands and their positions predicted by the one—electron calculations as being un- reliable. .Considering the available Hall coefficients and the low-temperature specific—heat data Gschneidner [22] attempted to interpret the band structure of the lanthanides from an experimental standpoint. He regarded the electronic specific-heat coefficients obtained from the low—temperature data as being unreliable. and hence used the room-temperature specific-heat analysis for his treatment. He concluded that the 4f electrons occupy either discrete energy levels. or very narrow one-electron bands as proposed by Mott [23]. According to Mott [23]. the overlap between incomplete 4f inner shells in the rare earths is so small that it is most unlikely that a 'metallic' type of wave function would be appropriate for the 4f electrons or that the 4f shells con- tribute to the Fermi surface. When the overlap between the atomic orbitals is small. an inner band would split into sub-bands of energy levels containing only one electron per atom. H D- (J I)‘ D | (I) .n’ l)! The low—temperature specific heats of all the lan- thanides. except Pm and Er. have been measured by Lounasmaa [24]. Similar measurements were also made by Dreyfus et a1. with Pr. Sm. Tb. Ho. Br. and Tm in the temperature range of 0.4 to 4.2°K [25]. Their results were reported in a sum- mary form only. The specific heats of Dy and Er were also reported by Parks [26]. Data of these groups are listed in Table I-2. In analyzing the low-temperature specific-heat data. Lounasmaa [24] and Parks [26] used a localized 4f—band model such as that discussed above and assumed a more or less constant electronic specific-heat coefficient for the lanthanides of the hcp structure. Such a coefficient has often been used for comparison with the values predicted by the theoretical work. The concept that the 4f shells in the lanthanides are partially filled and yet the 4f electrons contribute neither to the conduction band. nor to the low-temperature specific heat. is similar to the model proposed by Mott and Stevens [26] for the transition metals in which the d-electrons would be localized. Based primarily on the results of the low—temperature specific-heat work by Beck and co—workers [28]. this localized d-electron model was 10 TABLE I-2.--Some low-temperature specific-heat data of the lanthanides Element Lounasmaa et al.[24] Dreyfus et al.[25] and crystal Gamma(milli- Debye temp. Gamma(milli- Debye temp. structure cals/mole/°K) °K cals/mole/°K) °K (assumed) (assumed) Ce (fcc) 5.02 147 ---- --- Pr (hex) 5.83 152 4.54 85 Nd (hex) 5.38 157 ---- --- Pm (hex) ---- 162a ---- --- Sm (rhomb) 2.89 166 2.4 116 Eu (bcc) 1.37 --- —-—- --- Gd (hcp) 2.4 176 —--- -_- Tb (hcp) 2.2 181 ---- --- Dy (hcp) 2.3 186 2.2 207 Ho (hcp) 2.4 191 6.2 114 Er (hcp) --— 2.2 195a 3.1 134 Tm (hcp) 2.5 200 5.1 127 Yb (fcc) 0.7 118 --—- --- Lu (hep) 2.7 210 ---- --- a Interpolated values bBy Parks [26] 11 proved to be wrong and was corrected by Mott [29]. If the localized f-electron model in the rare earths is vindicated it would be a unique case in all metals. The present work. therefore. is aimed at a study of the low—temperature specific heat of some rare—earth metals and their alloys in the hope that the results will shed some light on the nature of their band structures. The only alloy system of the lanthanides which has been inves- tigated with the low-temperature specific-heat method is the Gd-Pr system by Dreyfus et a1. [30]. They did not try to establish the localized f-electron model. but used such a model for evaluating the hyper-fine coupling constants of their alloys. All the heavy rare-earth metals. except ytterbium. crystallize into the hexagonal close-packed structure. In the atomic state. the outer electron configurations of these lanthanides are 4fx6sz. except for Gd and Lu whose configur- ations are respectively 4f75dl6s2 and 4fl45dl6sz. If the 4f electrons are indeed localized. and hence do not con- tribute to the density of states at the Fermi surface. then isostructural hcp alloys of the erbium-thulium. thulium-ytterbium. and thulium-lutetium systems should have similar Fermi surfaces as assumed by Dimmock et al.[6]. 12 They should then reveal a constant electronic specific- heat coefficient. On the other hand. if the 4f electrons do form a band in the usual sense. and hence do contribute to the Fermi surface. then alloying thulium with erbium. which have complete solid solubility in each other. should gradually increase the number of electrons in the f band. The alloys should show variations in their electronic specific-heat coefficients. The specific heat of a metal can be expressed as = + = + + + + ... Cp CV Cd CE CL CM CN Cd (1) where Cp and CV are the specific heats at constant pressure and volume respectively separated by the dilatation term Cd. CE is the electronic specific heat. CL. the lattice specific heat. CM. the magnetic specific heat. and CN. the nuclear hyper-fine contribution to the specific heat. The dilatation term '2 cd—cp-cv—an'r ...(2) where 8 is the volume expansion coefficient. n. the bulk modulus. and V. the volume of the sample. is negligible for solids at liquid-helium temperatures. [31 two‘pn. f [201‘ 13 According to the free-electron theory Sommerfeld [31] expressed the electronic specific heat CE as 122 CE — yT - 311 k N(Ef)T ...(3) where y is the electronic specific-heat coefficient. N(Ef). the density of states at the Fermi surface. and k. the Boltzman constant. Thus N(Ef) is proportional to y. and N(Ef) = 1.785 7 ...(4) . . . . . o 2 when y 18 expressed in milli—calories/mole/ K . Stoner [32] derived a more general expression for the electronic specific heat considering an arbitrary— shaped band _ _122 2 CE — 7T — 3n k N(Ef)[1+6kT CN- —> water 4-argon FIG. 3 Induction furnace 22 Dennison et al. [42] studied the amounts of tung- sten and tantalum picked up by the lanthanides when the latter metals were held in either tungsten— or tantalum— crucibles at various temperatures above their melting points for an hour. They reported that about 0.1 wt% tungsten was picked up by the erbium and thulium metals when they were held molten at 50“C above their respective melting points for one hour. In View of the short time the present melts were retained in the molten state in the tungsten crucibles. it is expected that an insignifi- cant amount of tungsten was picked up by each of these samples. The samples were freed from the crucibles by chipping and grinding off the tungsten that might have adhered to the surface of the samples; any regions of tungsten to be removed at this stage were easily identi- fied by a quick etch with a 8%.hydrochloric acid solution. The samples were then sand-blasted to remove any surface contamination. wrapped in 0.01" thick tantalum foils. and were sealed in quartz tubes under helium at less than 10 torr. pressure. All the alloy samples were homogenized at 700°C for twenty—four hours after wrapping them in 0.01" 23 thick tantalum foils. and sealing them in quartz tubes under helium at less than 10 torr. pressure. Pure thulium and three erbium—thulium alloys were induction-melted under the above conditions. Table 11-2 shows their compositions as analyzed by Messrs. Atomergic Chemetals Inc. by mass-spectroqraphic technique after the heat-capacity measurements. To measure the specific heat of a sample. a known quantity of heat is supplied to the sample. and the cor- responding temperature rise is determined under adiabatic conditions. The heat is supplied to the sample in the form of electrical energy. The temperature of the sample is determined by measuring the resistance of a carbon re- sistor embedded in the specimen assembly. and the tempera- ture rise by the change in the resistance. The experimental setup was described by Wei [43] and by Tsang [33] except for minor modifications. As shown 1J1 Fig. 4. the apparatus consists of: a) a cryostat for cooling the calorimeter and maintain- ing it at liquid-helium temperatures. b) a low-vacuum system to lower the temperature of the liquid-helium bath from 4.2°K to 1.3°K. 24 TABLE II-2.—-Ana1yses on the present melts in ppm by weight Element Er Er0.75Tm0.25 Er0.5Tm0.5 Er0.25Tm0.75 Tm A1 75 160 32 110 18 Ca 18 550 93 260 170 Ce 37 34 40 37 68 Cu 100 6 1 13 18 Dy 120 190 160 170 410 Fe 400 19 30 43 31 Ho 97 56 50 31 2 K 0.23 22 2 7 130 La 8 9 15 9 22 Mg 0.3 8 7.5 11 22 Mn 12 18 36 39 62 Na 13 3800 10 29 510 Nd 110 390 580 940 2000 Ni 3 35 46 98 600 Pr 52 42 46 24 33 Si 32 32 13 37 17 Ta 29 86 3 3 14 w 140 700 500 620 750 y 1100 520 400 150 20 Zn 2.8 5 160 230 47 c 86 36 140 200 850 F 6300 8600 4500 3800 320 N 1300 54 21 7 1100 O 120 470 240 490 1400 25 c) a high-vacuum system to evacuate the calorimeter for isolating the cooled sample, d) a manometer system and a McLeod gage to measure the vapor pressure of the liquid—helium bath. and e) an electrical system to measure all the electrical quantities and the heating time. A pure copper disc. shown in Fig. 5(a). was Prepared to house the thermometer and the heating coil. An uncoated 1/10 watt carbon resistor was used as the thermometer. It had a nominal resistance of 56 ohms at room temperature. about 850 ohms at 4.2°K. and 50.000 Ohms at 1.3°K. This thermometer was lightly greased and snugly fitted into the hole in the copper disc to provide the thermal contact between the two. The disc was then non—inductively wound with a 40-gage formex-coated man— ganin wire to serve as the heater. Its resistance was 325 ohms at room temperature. Both the thermometer and the heater ends were soldered to cotton—insulated 40-gage copper wires. which in turn. to 1%" lengths of manganin leads. 26 Legend for Figure 4. HeC IP K3 LHe MG NC OM OS Pl. SG TT Cryostat System Low-vacuum system High-vacuum system Manometer system Electrical system Bellows Calorimeter Cold—cathode vacuum gage Cathetometer Counter Calorimeter valve Diffusion pump Frequency gage Fork-lift Helium gas cylinder Instrument panel K-3 universal potentiometer Liquid—helium container Liquid-nitrogen container McLeod gage Mercury manometer Nitrogen gas cylinder Oil manometer Oscillator P2. P3. P4 Pumps Recorder Stokes' gage Transfer tube 27 329.5303 9:: 32.0.5396 v.0: 28 Legend for Figure 5 10 11 12 13 14 SV IS Nine-pin connectors Flange to high-vacuum system Flange to manometer system Flange to low-vacuum system Protection tube Middle-jacket valve MJV Calorimeter conduit %"¢ Epibond-lOOA seal Calorimeter can Lower kovar-seal connector Sample Heater-thermometer assembly Liquid-helium cryostat Liquid-nitrogen cryostat Safety valve Inlet screw / //7 // \l m 29 ] S \AL \\\\\\\XX $5” \ \\\\‘ . \ \\\\\ \\\\ l\\\\ \\ . / V 6‘ 3 Q h V , if .I 1 ‘ 5 .1 v1.0 ' b - Q’s!) P . 7 l [1 L J] V - w ’\r 8 1+ 9 10 ell FIG. 5 Cryostat system 30 The sample under investigation was sliced into two cylindrical halves. and the parting surfaces were ground and polished flat to match the top and bottom sur— faces of the heater—thermometer assembly. The polished surfaces were lightly greased. and the heater-thermometer assembly was sandwiched between the two sample halves and tied together with 20-gage copper wires. The assembled specimen was suspended so as to be at the center of the calorimeter can by means of three single lengths of #50 nylon thread. The lead wires from the heater—thermometer assembly were soldered to the four pins of the lower kovar seal. The electrical connections from this kovar seal to the room-temperature region at the t0p of the calorimeter system were also made of manganin wires. Twisted pairs of the manganin wires were taken out into the liquid-helium bath through a 1/8" copper tube. which was sealed with Epibond-lOOA as described by Roach et al. [44]. The heater and the thermometer wires were separated at the top of the protection tube. and were passed through two bent stainless— steel tubes. at the end of which they were soldered to two separate connectors. 31 The manometer system was designed to provide an accurate measurement of the vapor pressure of the liquid- helium bath from 4.2°K down to 1.3°K. by incorporating a mercury and an oil manometer. and a McLeod gage. The mer— cury and the oil manometers could be operated together be- low 50 torru and the oil manometer and the McLeod gage. below 25 torr. The ratio of the density of mercury to that of the octoil used in the manometers was found to be 13.81. The details of this system can be seen in Fig. 6. The resistances of the heater and the thermometer and the currents passing through them were measured by means of a Leads and Northrup K-3 universal potentiometer. The circuit diagram of the instrument panel is shown in IFigu 7. and. the switch settings for the various functions are listed in Appendix A. A double-pole switch 81 was in— stalled to activate a CMC 800A electronic counter when the heater current was turned on. The counter was connected to a Hewlett Packard 200CD audio oscillator set at 1000 cps. whose frequency was continuously checked by a separate Hewlett Packard 3734A electronic counter with a :1 count accuracy. 32 Legend for Figure 6 To HeC MG MM MVl through MV5 MT P3. P4 To helium gas cylinder McLeod gage (Todd Scientific Co.) Mercury manometer Manometer valves Mercury-and-oil trap Mechanical fore-pumps To cryostat 92 MV3 12 l _(l)__ MG MM OM “G" b P4 “*— FIG. 6 Manometer system 34 Legend for Figure 7 BS1. C1 CCG CTR DCA FRG MCA MLA MSl ND OSC P1. R14. BSZ. P2. R16 Rh' Rt REC RSl. RS2 P3. BS3 P4 Battery substitutes (constant 3—volt supply) Capacitor Cold-cathode vacuum gage Electronic counter DC Amplifier Frequency gage Microammeter Milliammeter Main switch Null detector Oscillator Mechanical fore-pumps Decade resistors for the heater and the thermometer circuits respectively Heater resistance and thermometer resistance Recorder Standard resistances of 100.000 and 100 ohms respectively for standardizing the thermometer and the heater currents 81 through S19 Switches Eacuazu 2386 k .0: 36:32:20.. mix um H .u (a mim .0 x 35 <02 36 The basic standard for the emf measurements with the potentiometer was a low temperature coefficient cad- mium standard cell SC. which had an emf of 1.01925 volts i0.005% at 24°C and an internal resistance of less than 500 ohms at 20°C. For current measurements. two Leeds and Northrup standard resistances RS2 and R81 of 100 ohms 10.005% and 100.000 ohms i0.005% respectively were used as standards in the heating and the thermometer circuits. The accuracy of the present setup was discussed by Tsang [33]; and an over—all accuracy of 12% can be expected in the specific-heat measurement. The experimental procedure. as described in Ap— pendix B. essentially involved the preparation of the cryostat for the liquid—helium transfer. the calibration of the thermometer resistance between 4.2°K and the low- est temperature attained. and the measuring of the temp- erature response of the thermally isolated specimen when a known quantity of heat was supplied to it. CHAPTER III EXPERIMENTAL RESULTS The specific heats of the five samples described previously were successfully measured between 1.3“ and 4.2°K. The calculations and the analysis of the experi— mental data were performed on a CDC 3600 computer. The program is given in Appendix C. The input data for the calibration were the thermometer resistance and the liquid-helium vapor pressure. Above the X-point the vapor pressure data were corrected for the hydrostatic pressure of the liquid-helium head above the center of the calorimeter can. The temperature corresponding to the vapor-pressure values were calculated from the 1958 liquid-heliumIV temperature scale [45] by interpolation. The thermometer resistance and the temperature data were fitted to the Keesom-Pearlman relation [46] N (1ogR/T)s5 = 2 cn (log n)“ ...(11) to yield the coefficients Cn+1° The calibration data from all the experiments were found to satisfy (11) with N=l 37 38 with a scatterwof less than 10 milli-degrees. A typical calibration curve is shown in Fig. 8. The specific heat of the specimen assembly con— sists of the contribution from the sample and that from the heater-thermometer assembly including the copper wires tieing the specimen together .2 . = H + 1 Rht/J n CV A1 n h 1 2Cv AT ...(12) 1 2 where ih is the heating current. Rh. the resistance of the heater. t. the heating time. J. a conversion constant to yield the energy in calories. n1 and n2. the number of moles of the heater-thermometer assembly and the sample respectively. CVl and CV2' their respective specific heats. and.AT is the corresponding temperature rise. The Specific heat of the sample is then .2 CV — 1tht/n2JAT- nlCV /n2. ...(13) 2 l The heater-thermometer assembly weighed about four grams. of which the non—copper material was about 8% by weight. CV was assumed to be that of copper with neglig- 1 ible error. The specific heat of copper was determined by u—IN (IogR/T) 39 2'0 1'8 -.‘ b l 1 1'0 0'8 2-8 32 1 3'6 4'0 log R—-> FIG. 8 Calibration for Tm 44 4-8 40 several workers and the value reported by Isaacs and Mas- salski [47] was Cv = 0.0001668T + 0.00001159'1'3 ...(14) l in calories per mole per degree K. The five experiments were conducted with the speci- mens of the erbium-thulium system: two with the pure metals. and three with the alloys. The specific-heat data of the samples are listed in Tables III-1 through III-5. and the plots of the specific heat versus tempera- ture are shown in Figs. 9 through 11. 41 TABLE III-1.--Specific-heat data on Erbium Sample Date: 1—31-68 Weight of sample = 31.264 gm = 0.1869185 mole. Weight of Heater-Thermometer Assembly = 3.55 gm = 0.05587 mole. Heater resistance at 4.2‘K = 286.16, at 1.3°K = 285.95 ohms. Thermometer-calibration coefficients: C1 = -l.029l935 C2 = 0.6272095 Data Temperature Specific Heat Data Temperature Specific Heat No. °K m-cal/mole/°K N0. °K m—cal/mole/°K 2 1.4213 124.86 16 2.7872 193.07 3 1.4790 124.10 17 2.9272 207.52 5 1.6379 126.76 18 3.0666 222.22 6 1.7016 130.30 19 3.2069 237.60 7 1.7545 129.49 20 3.3774 255.18 8 1.8434 135.39 21 3.4645 270.06 9 1.9803 138.86 22 3.5779 288.01 10 2.1235 147.19 23 3.7090 315.25 11 2.2437 153.76 24 3.8172 326.12 12 2.3045 156.77 25 3.8963 335.80 13 2.4144 162.00 26 3.9836 364.35 14 2.5088 171.08 27 4.1136 383.94 15 2.6807 181.96 28 4.2005 404.36 42 TABLE III-2.--Specific-heat data on Er 7STm 25 alloy Date: 9-16-68 Weight of sample = 32.2917 gm = 0.192581 mole. Weight of Heater-thermometer Assembly = 4.7821 gm = 0.07526 mole. Heater resistance at 4.2°K = 285.755; at 1.3°K = 285.50 ohms. -1.0355476 ll Thermometer-calibration coefficients: Cl C 0.6301181 2 ._.1__ Data Temperature Specific Heat Data Temperature Specific Heat No. °K m-cal/mole/°K No. °K m—cal/mole/°K 1 1.3233 20.327 17 2.3500 28.052 2 1.3375 19.398 18 2.4585 30.392 3 1.3850 19.091 19 2.5889 33.637 4 1.4464 19.383 20 2.7165 37.177 5 1.4985 19.656 21 2.8758 42.432 6 1.5501 19.751 22 3.0915 50.732 7 1.5894 19.816 23 3.2297 57.325 8 1.6388 19.971 24 3.4097 65.249 9 1.6976 20.393 25 3.6295 77.270 10 1.7490 20.463 26 3.8014 87.999 11 1.8131 20.973 27 3.9177 96.319 12 1.8899 21.633 28 4.0061 102.417 13 1.9806 22.516 29 4.0755 106.604 14 2.0669 23.647 30 4.1400 111.803 15 2.1801 25.132 31 4.2023 116.161 16 2.2844 27.063 Table III-3.--Specific-heat data on Er 5 Tm alloy Date: 8-2-68 Weight of sample 28.516 gm = 0.1696401 mole. Weight of Heater-Thermometer Assembly = 3.7255 gm = 0.05863 mole. Heater resistance at 4.2°K = 285.955; at 1.3°K = 285.72 ohms. Thermometer-calibration coefficients: C1 C 2 -1.0336623 0.6298109 Data Temperature Specific Heat Data Temperature Specific Heat No. °K m—cal/mole/°K 1 1.3186 16.170 2 1.3633 17.443 4 1.4677 18.363 5 1.4779 19.008 6 1.5431 19.319 7 1.5745 20.380 8 1.6288 21.304 9 1.6554 21.782 10 1.6924 22.953 11 1.7379 23.922 12 1.7915 25.859 13 1.8395 26.694 14 1.8737 28.403 15 1.9239 30.123 16 1.9649 31.254 17 2.0177 32.817 18 2.0677 34.911 19 2.1331 38.165 20 2.2120 41.239 21 2.3112 45.627 No. °K m-cal/mole/°K 22 2.4275 51.617 23 2.5244 56.234 24 2.6431 62.883 25 2.7829 71.053 26 2.9257 79.876 27 3.0484 87.437 28 3.2234 99.597 29 3.3278 104.057 30 3.3917 110.607 31 3.5212 120.818 32 3.5992 124.213 33 3.6658 130.486 34 3.7299 138.970 35 3.8078 140.882 36 3.8674 151.890 38 3.9662 154.360 39 4.0227 158.044 40 4.0820 164.599 41 4.1261 161.679 42 4.1713 175.426 44 TABLE III-4.--Specific—heat data on Er 25Tm 75 alloy Date: 9-11-68 Weight of sample = 32.405 gm = 0.1922968 mole. Weight of Heater-Thermometer Assembly = 3.7715 gm = 0.05936 mole. Heater resistance at 4.2°K = 285.755; at 1.3°K = 285.545 ohms. -1.033711 0.630025 Thermometer-calibration coefficients: C1 C2 Data Temperature Specific Heat Data Temperature Specific Heat No. °K m-cal/mole/°K No. °K m-cal/mole/°K 1 1.3141 28.901 21 2.2736 64.034 2 1.3274 28.682 22 2.3518 66.011 3 1.3650 31.288 23 2.4375 69.581 5 1.4526 32.102 24 2.5623 75.066 6 1.5147 36.147 25 2.6535 75.232 7 1.5849 39.023 26 2.7662 79.628 8 1.6266 39.290 27 2.9045 83.591 9 1.6605 41.355 28 3.0392 85.571 10 1.6899 41.941 29 3.1746 88.834 11 1.7248 43.516 30 3.3366 90.022 12 1.7702 45.091 31 3.4848 96.076 13 1.7945 44.544 32 3.5951 100.006 14 1.8282 46.391 33 3.6627 97.744 15 1.8664 47.644 34 3.7635 101.249 16 1.9137 50.714 35 3.8573 104.191 17 1.9665 50.879 36 3.9396 104.422 18 2.0244 54.079 37 4.0140 103.868 19 2.1030 51.475 38 4.0979 106.948 20 2.1742 60.119 39 4.1631 107.394 45 TABLE III-5.--Specific-heat data on Thulium sample Date: 8-22-68 Weight of sample = 27.094 gm = 0.16038216 mole. Weight of Heater-Thermometer Assembly = 3.8885 gm = 0.061198 mole. Heater resistance at 4.2°K = 285.805; at 1.3°K = 285.585 ohms. -1.0341672 0.6297254 Thermometer-calibration coefficients: C1 C2 Data Temperature Specific Heat Data Temperature Specific Heat No. °K m—cal/mole/°K No. °K m-cal/mole/°K 1 1.3029 19.121 18 2.3697 38.042 2 1,3155 19.574 19 2.4944 40.446 3 1.3658 20.496 20 2.5822 42.779 4 1.4127 20.827 21 2.7223 45.484 5 1.4710 21.904 22 2.8829 49.550 6 1.5233 22.601 23 3.0107 52.650 7 1.5807 23.210 24 3.1418 56.157 8 1.6244 24.296 25 3.2995 60.480 9 1.6744 24.790 26 3.5023 66.401 10 1.7448 25.937 27 3.6824 72.241 11 1.8185 27.099 28 3.8105 76.917 12 1.8777 28.320 29 3.8845 79.329 13 1.9258 29.244 30 3.9542 82.633 14 1.9853 30.356 31 4.0284 83.291 15 2.0479 31.622 32 4.0896 84.370 16 2.1210 32.692 33 4.1547 86.619 17 2.2242 35.013 34 4.2175 91.545 Ni .10 .2802: \ 0.00 ‘I .3:Ev lull" .o-‘ U.‘—U-Qn 200 . 1 46 150- (milli—caIs/moh/OKl 100— Ehasn't-2:. - 3 e _ 1 .c U :0.- . 1 '5 e a an 50... .) 1 J «J c 1 l 1 J n L 1 l 0 l 2 3 4 Temperature —> (OK) FIG.9 Specific heat vs. temperature curve for Er and Er Tm 0-75 0°25 Specific heat ——-—* (milli—cals/mole/°K) 47 150 r § 1 8 - Temperature -——> PK) FIG. 10 Specific heat vs. temperature curve for E'O-STmOfi (milli-caIs/moIe/OK) Specific heat 48 200 150 § A l .l 0 I 2 ‘1- 1 7 Temperature ——> (°I() FIG. ll Specific heat vs. temperature curve for E'O-ZSTMOVS and Tm CHAPTER IV D ISCUSS ION The specific-heat data on erbium. thulium. and their alloys were given in the preceding chapter. A prOper analysis of these data for the separation of the various specific-heat contributions as discussed in Chapter I depends on a knowledge of the magnetic proper— ties of the specimens under investigation. Koehler and co-workers [14] studied the magnetic properties of the rare—earth elements by neutron-diffraction techniques. They found that erbium transformed from its high-temperature antiphase structure into a ferro—magnetic spiral configuration at its Curie temperature TC of 19.6°K with the basal components retaining their helical arrange— ments. The resultant magnetic moment lies on the surface of a cone generated around the c-axis. The half apex angle a of the cone was found to change with temperature. Koehler [15] also found that thulium adopted an antiphase domain-type structure below its Curie temperature of 40°K. in which four layers of north-pointing moments 49 50 were followed by three layers of south-pointing moments. The period of this magnetic structure was noted to be constant at seven layers over the temperature range. At 4.2°K each thulium atom had its maximum moment of 7.0 HB. parallel or antiparallel to the c-axis. Because of the incomplete cancellation. there was a net moment of 1.0 “B per atom parallel to the c-axis. These low—temperature magnetic structures of erbium and thulium are Shown in Fig. 12. and. their measured magnetic moments are com- pared with the ground state values gJ predicted from the Hund's rule in Table I-l. Bozorth and Gambino [48] studied the magnetic properties of the solid solutions of several heavy rare earths. They noticed a maximum in the Curie temperatures at 40%.Er in the erbium—holmium system. In the Er-Tm sys- tem they found that the Curie temperature decreased with increasing thulium content up to about 12% Tm as shown in Fig. 13. At this point there was a jump in the curve. and the new Curie temperature Tc followed a slowly de— 2 clining trend again with increasing thulium content. This Tc was suggested to be the thulium-type Curie temperature. 2 below which the saturation magnetization is very low due to the antiphase domain—type of magnetic ordering. K /_ 51 00 000 000000000 at ‘ Tm FIG. l2 Low-temperature magnetic structures of Er and Tm 40 — 32 T‘1 T‘z ‘L 1 ) I 20: ..F o I I I I I I L Ho 50 Er 50 Tm FIG. I3 Curie temp. vs. composition for Ho-Er and Er-Tm systems 52 Lounasmaa [49] studied the specific heat of several rare-earth metals also in the 3 to 25°K temperature range mainly to analyze the magnetic specific heat CM. He found that the magnetic specific heat of thulium was 1.5 T2'5 milli-cals/mole/°K in the 0.4 to 4°K temperature range. and 1.98 T2'3 milli-cals/mole/°K in the 4 tx> 20°K range 3/2 in contrast to the T dependence predicted by the spin- wave theory for the ferri— and ferromagnetic materials. and the T3 dependence for the antiferromagnets. Lounasmaa attributed the differences between his results and the theoretical predictions to the lack of validity of the existing theory to the antiphase magnetic structure such as that of thulium. One may. however. note the apparent closeness of the experimentally analyzed magnetic Specific heat values of thulium to that predicted by the spin-wave theory for the antiferromagnetic structure. In the case of holmium metal. which also has a ferromagnetic spiral type of structure similar to erbium below its Curie point of 20°K. Lounasmaa [49] obtained CM=0.36 T3'2 milli—cals/mole/°K in the 3 to 20°K tempera— ture range. and a T3 dependence for CM in the 0.4 to 4.2°K range. Kaplan [50] suggested that for a ferromagnetic spiral structure the dispersion relation between m(q) and 53 q'was linear for small values of the wave vector q even though the net spin was not zero. This dispersion rela- tion. Similar to that deduced for the antiferromagnetic case. was attributed to the normal modes corresponding approximately to oscillations of the components perpendic- ular to the magnetization. These components themselves formed an antiferromagnetic spiral. From this interpreta— tion. Kaplan [50] treated the ferromagnetic-spiral struc- ture as being similar to that of an antiferromagnet. the magnetic specific—heat contribution of which was propor- tional to T3 as noted in Chapter I. From the foregoing discussion. one may expect a T3 dependence of the magnetic specific heat for all the present samples. The specific-heat data on erbium. thulium. and their alloys. as presented in Tables III-1 through III-5. were therefore analyzed with CV/T = 7 + (01+II)T2 + vT-B. ...(15) Since the aim of the present work is to examine the validity of the localized 4f-band model. no assumption was made on the nature of the 4f band. 54 The main computer prOgram was written to analyze the present specific-heat data to obtain the electronic. lattice plus magnetic. and the nuclear specific-heat con- tributions according to equation (15). To analyze the magnetic term. the nuclear and the lattice specific-heat contributions were subtracted from the total specific heat. and the electronic and the magnetic contributions were easily separated from the remainder. The lattice specific heat was calculated from the Debye temperatures taken from Lounasmaa's work [24]. The analysis. as outlined above. is straightforward for the specific-heat data of the erbium. Er0.75Tm0.25. and the thulium samples. The Er and Er 0.5Tm0.5 0.25Tm0.75 samples. however. offer some difficulty. As shown in figures 16 and 17. the Cv/T versus T2 curves for these samples appear to be concave downwards in the middle similar to that for Gd in the work of Dreyfus et al. [29]. 0.23Pr0.77 None of the known theories seem to be able to account for such a feature. In order to obtain an upper and a lower limit for the electronic specific—heat coefficient. the Cv/T data of the Er sample was analyzed by first 0.5Tm0.5 fitting the values below 2.5°K to equation (15) to evaluate 55 the nuclear contribution. which was then subtracted from the Cv/T values for the entire temperature range. The re— mainder. consisting of only the linear and the T3 terms in CV. was analyzed by treating separately the data below and above 2.78iK for 7min. and Vmax. respectively. The true electronic Specific-heat coefficient would probably lie between these two values. For the Er sample. 0.25Tm0.75 7min. and vmax. were obtained by extrapolating the two branches of the Cv/T curve to 09K. Figures 14 through 18 Show the results of the analysis. The various Specific-heat coefficients are listed in Table IV—l. while the electronic specific-heat coefficients are plotted against the composition in fig. 19. TABLE IV—1.—-Specific-heat contributions of the present samples in milli-cal/mole/°K Electronic Lattice Magnetic Nuclear Sample -—————— -———————— T T3 T3 T‘2 . 3 3 -2 Erbium 4.1T 0.063T 1.23T 11.73T Er Tm 2.5T 0.0615T3 1.4T3 23.91"2 0.75 0.25 3 3 _2 BIG 5TmO 5 8.7:5.3T 0.060T 2.4:9.8T 9.29T O O 3 Er0.25Tm0.75 23.3:].87T 0.0591:3 ---- 3 - 2 Thulium 13.3T 0.058T 0.42T 1.76T’ cv/r (milIi-cals/molc/ 0K2) 56 Io Erbium _ 8 ._ 6 e 9 oo o o 0% 4 elec./ mag. nuc. 2 - lot. 0 ’—==: J - l l 0 4 8 12 lb °K2I T2—>( FIG. 14 Cv/T vs. T2 curves and analyses for Er 2O N U \“-C °E\ \\ ¥°\ .-mE\ .P\ ‘Iullllll Cv/T —> (milIi—cals/moII/oKz) 57 T I I I 25 - E'0-75T'“0-25 ~ mag. 20 ~ ‘ 15 - ‘ DUC. IO - " (poo 5 - <9 .. elec. \ [0}; OZ /__-_=_, i I I 4 0 4 8 12 lb 20 T2 -—> (0K2) FIG. 15 Cv/T vs. 1'2 curves and analyses for Ero,75Tmo,25 Cv/T -——> (milli-cals/mole/oKz) 58 50 4o) aor 20!- 10)- E'os'mo-s J . _ o o J at I elegmax.) nut. elec.(min.) \ '0'; #L-———— ; E l 4 8 I2 16 20 72 (°K2) FIG. 16 Cv/T vs.T2 curves and analyses for Er Tm 0’5 0'5 (miIIi—caIs/mole/° K2) Cv/T 59 I I I I 40— E"0.25““0-75 ‘ elec.(mox.) 30*- “J O o 0 total 00 O o 20- " elec.lmin.) IO- - o I I l I O 4 8 l2 I6 20 12 —> (°K2) 2 FIG. 17 Cv/T vs. T curves and analyses for E'O-25TmO-75 60 25_ Thulium _ o 00 or 20" 9 0° ‘ x o E a \5. total 3 IS— _ U I elec. E 1 ml . I— mag. > 5“ - nuc. '0’; o | 1' 1 _L 0 4 8 12 16 T2 (0K2) FIG. 18 ‘Cv/T vs. T2 curves and analyses for Tm 20 Electronic specific heat—s» (mC/male/oKz) N U! M C I ...e Ut O 61 present work I value predicted by Dimmock et al. [6-10] _‘ 00 25 50 75 I00 5' o . / ThulIum—> a FIG. 19 Electronic specific-heat coefficient vs. % Tm Tm 62 The results indicate that the 7 values of the samples range from 2.5 to 23 t 8 milli-cals/mole/°K2 Figure 19 also shows. for comparison. the constant value of 1 milli—cal/mole/“K2 predicted by the APW-calculations of Dimmock et a1. [14—18]. and the nearly constant value of about 2.5 milli—cals/mole/°K2 assumed by Lounasmaa [24]. both based on the localized 4f—band model. Although the differences between the theoretically predicted value and the experimentally obtained electronic specific-heat data [24. 25] for the rare earths are not entirely new. the present data indicate that the difference can be much greater for some of the alloys. The discrepancies have been so far attributed to the electron—electron enhance- ment [9]. the electron-phonon [10] and electron-magnon enhancements [l6]. and the impurity contents in the samples of the different workers [10]. Kasuya [16] estimated an electron-phonon enhancement of about 30%»and an electron- magnon enhancement of about 20% of the 7 value for gado- linium. These enhancements can account for at most a factor of two. The low—temperature specific heats of erbium and thulium reported by different workers are compared in 63 Fig. 20 with the present data. Lounasmaa [24] pointed out that discrepancies in the specific heats of such magni— tudes are not uncommon below 4°K for the rare earths. He attributed the discrepancies to the differences in the impurity contents in the samples. The effect of oxygen in gadolinium on its specific heat was investigated by Crane [51]. He reported an in- crease in the specific heat with increasing amounts of oxygen. This effect was explained in terms of the magnetic ordering of the Gd+ ions in the oxide. No specific-heat measurements on erbium and thulium oxides have been re- ported. If a quantitative analogy could be assumed be- tween the effects of oxygen on the specific heats of Gd and the Er-Tm alloys. about 0.1%.oxygen by weight might cause an enhancement in the specific heats by as much as a factor of two. With all the effects contributing to the uncertainties in the measured 7 values taken into consider- ation. a factor of not exceeding four can be included in the corrections. This is still insufficient to bring down the 7 values of the thulium. Er and 0.5Tm0.5' . . o 2 Ero.25Tmo.75 samples to the 1 milli-cal/male/ K range predicted by the localized 4f—band model. 64 80" PW: Present work PVV - L : Lounasmaa [24] Thufiunt D : Dreyfus et al. [25] 60' L 7 40' D NA M O \ .2 o q 20' \E U .5 B I I I I 1: ° 0 .g () Tenmud>-I 2 3 4. K '5 . . a Q . U, P P : Parks [26] 40“ E'bh"“ PW: Present work d Dreyfus et al. [25' V” I) 20- - 0 l l L l FIG. 20 Comparison of specific-heat data of different workers 65 As discussed in Chapter I. the current theories of the rare-earth metals are based on the localized 4f-band model. The 4f electrons are assumed to form partially filled bands and yet they do not contribute to the Fermi surfaces in the rare—earth metals. Only the three valence electrons form the conduction band. and hence a constant electronic Specific heat is predicted for all the hop lanthanides. The present results can not be explained by such a model. If the localized 4f-band model were valid. the 5d band of the rare earths should be the major contributor to the Fermi surfaces in these metals. and to their elec- tronic specific heats. According to the APW-calculations by Dimmock et a1. [14—18] using such a model. the 5d band has a width of 6.8 eV as shown in Fig. 1. One may compare this with the 3d band in the first long period transition metals. Belding [521 calculated the 3d band in bcc para— magnetic Cr by using the tight-binding approximation. and obtained a comparable band width of about 5 eV. A close resemblance can be noted between this calculated result and the experimentally determined 3d energy band of Beck and co—warkers [28] as shown in Fig; 21. .Based on this‘ 66 vocab xaeecoIpm peeonoufi pca po.o_:u_au pm .0: A>ev Ilo.|I xaeocu pl NI ml 3 _ l N \ Cu ..3: Juan pea ._e>> deep—U 13.13: 9:23 I V /Ae/quIa/) “solois- I° Aigsueg (ugds I O ev 1 mod ('0 ‘3' Z». ti< in IE! the m0) 110 I16 30 th SP In to 67 evidence Mott [29] corrected his earlier localized 3d-band model [27] for the transition metals. The calculated and the experimentally obtained 3d bands for the bcc transi- tion metals do not differ to any appreciable extent except in the Cr-Fe alloy system. On the other hand. all the ex- perimental evidence from the present work as well as the results of Lounasmaa [24] and Dreyfus et al. [25] indicate that the 7 values are much larger than what might be pre- dicted by the localized 4f-band model. All the heavy rare—earth metals. except Lu which has a filled 4f band. are known to have a net magnetic moment at low temperatures. A localized 4f-band model is not necessarily needed to explain the low-temperature mag- netic structures in these metals any more than a localized 3d-band model is required to explain the magnetic proper- ties of the transition metals. Instead. it is possible that a narrow 4f band could split into up—spin and down— spin half-bands. which may or may not overlap as a group. The relative positions of the two half-bands with respect to the Fermi surface may vary from one rare-earth metal to another depending upon such factors as the crystal struc- ture. the number of 4f electrons per atom and the exchange "I 68 interaction between the electrons. One may modify Mott's one-electron band model [23] so that each half-band. not necessarily localized. is built of seven overlapping one- electron bands. The density of states of each of these half-bands may contain peaks and valleys. When there is an integral number of 4f electrons in the metal. the Fermi surface is most likely to be near a valley. Such a model would explain the nearly uniform electronic specific-heat coefficients of the pure rare—earth metals as well as the large variations in the 7 values of the alloys as those that are observed in the present work. A schematic dia- gram of such a proposed itinerant 4f-band model is shown in Fig. 22. Additional work such as the room-temperature specific—heat measurements of alloys of the rare earths may help to confirm the proposed model. Should this itinerant 4f-band model be established. then it would not be necessary to resort to using the electron—phonon type enhancements in explaining the discrepancies between the theoretically predicted and the experimentally ob- tained electronic specific heats of the rare—earth lanthanides. 69 NEH 4* Efi N16) t FIG. 22 Schematic itinerant 4f-bcnd model CHAPTER V CONCLUSIONS The specific heats of hexagonal close-packed Er and Tm metals and three isostructural Er-Tm alloys were de- termined in the temperature range of 1.3 to 4.2°K. Barring possible complications due to the uncertainties in the magnetic properties of these samples and the effects of impurities. the electronic specific-heat coefficients obtained from the analyses of the specific- heat data range in values from 2.5 to 23.3 milli-cals/ mole/°K2. These values are two to some twenty times the constant value that might be predicted by a local— ized 4f-band model. Such discrepancies are too high to be accounted for by the electron-phonon and electron- magnon enhancements. It appears possible that the 4f electrons in the rare- earth lanthanides may form energy bands which do con— tribute to the Fermi surfaces and hence to the Specific 7O 71 heats in the usual sense. and that they are better de- scribed as itinerant rather than localized. Such an itinerant 4f-band model would explain the high electronic specific heats of the rare-earth metals and alloys. It may not introduce any more difficulties in explaining their complicated magnetic properties than the itinerant 3d-band model might have in the case of the transition metals. 11. l. 10. 11. CHAPTER VI REFERENCES R. Marrus. W. A. Nierenberg. and J. Winocur. U.C.R.L. 9207. May 1960; also quoted by B. B. Cunningham. Rare Earth Research. III (127). Macmillan Co.. N.Y.. 1961. F. H. Spedding. S. Legvold. A. H. Daane. and L. D. Jennings. Prog. Low Temp. Phys.. II. Ch. 12. Ed.: C. J. Gorter. 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Phy5.: S32 (48): 1961. Ibid.. . . . S34 (1335). 1963. W. C. Koehler. Phys. Rev.. 126 (1672). 1962. T. Kasuya. Megnetism. IIB (215). Ed.: Rado and Suhl. Acad. Press. N.Y.. 1966. H. Nigh. S. Legvold. and F. H. Spedding. Phys. Rev.. 132 (1092). 1963. P. M. Hall. S. Legvold. and F. H. Spedding. Phys. Rev.. 109 (971). 1958. R. V. Colvin. S. Legvold. and F. H. Spedding. Phys. Rev.. 120 (741). 1960. D. E. Hegland. S. Legvold. and F. H. Spedding. Phys. Rev.. 131 (158). 1963. C. Herring. Magnetism. IV. Acad. Press. N.Y.. 1966. K. A. Gschneidner Jr.. Rare Earth Research. III (153). Ed.: LeRoy Eyring. Gordon and Breach Inc.. N.Y.. 1965. N. F. MOtt: Phil. Mag.. E (306): 1961. 0. V. Lounasmaa. Phys. Rev.. 128:'#3 (1136): 1962 (3H0): Ibid. ... 129. #6 (2460). 1963 (:Gd.Yb; same as 13). Ibid. ... 133. #1A (211). 1964 (:Pr.Nd). Ibid. g3. #1A (219). 1964 (:Lu) Ibid. ... $32} #2A (502): 1964 (:CSIEU): Ibid. ... lag} #6A (1620): 1964 (Tm) Rare Earth Research. II (221). Ed.: Vorres. Gordon & Breach. IDC.: N.Y.: 1964 (C9:Pr:Nd:Eu:LU). 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 74 B. Dreyfus. B. B. Cunningham. A. Lacaze. and G. Trol- 1iet. Compte Rendu. Acad. des Sciences (1764). 1961. R. D. Parks. Rare Earth Research (225). Ed.g Nachman. and Lundin. Gordon & Breach. N.Y.. 1962. N. F. Mott and K. W. H. Stevens. Phil. Mag.. 2.(1364). 1957. C. H. Cheng. C. T. Wei. and P. A. Beck. Phys. Rev.. 120 (246). 1960. N. F. Mott. Adv. Phys.. 13 (325). 1964. B. Dreyfus. J. C. Michel. and A. Combiende. Proc. :X Intl. Conf. Low Temp. Phys. (1054). Plenum Press. X.Y.. 1965. A. Sommerfeld. Ann. d. Physiks. 28 (1). 1937. E. C. Stoner. Proc. Roy. Soc. (London). A154 (656). 1936. P. J. Tsang. Ph.D. Thesis. Mich. State Univ.. 1968. C. Kittel. Introduction to Solid State Physics. J. Wiley & SOUS: N.Y.: 1967. F. Bloch. z. physik..§2 (555). 1928. J. V. Kronendonk and J. H. van Vleck. Rev. Mod. Phys.. ‘30. #1 (1). 1958. D. T. Edmonds and R. G. Petersen. Phys. Rev. Letters. 2 (499). 1959. E. S. R. Gopal. Specific Heats at Low Temperatures. Plenum Press. N.Y.. 1966. W. Marshall. Phys. Rev.. 110 (1280). 1958. B. Bleaney. J. Appl. Phys.. 34, #2 (Part 2). (1024). 1963. 75 41. J. Mackowiak. Physical Chemistry for Metallurgists (160). Am. Elsevier Pub. Co.. N.Y.. 1966. 42. D. H. Dennison. M. J. Tschetter. and K. A. Gschneid— ner. J. Less-Common Metals.‘lg (108). 1966. Ibid. i ... ;1_(423). 1966. 43. C. T. Wei. Ph.D. Thesis. Univ. of Illinois. 1959. 44. W. R. Roach. J. C. Wheatley. and A. C. Mota de Vic- toria. Rev. Scientific Intr.. 35, #5 (634). 1964. 45. N. B. S. Monograph 10. The 1958 He4 Scale of Tempera- tures. H. van Dijk. M. Durieux. J. R. Clement. and J. x. Logan. 1960. 46. P. H. Keesom and N. Pearlman. Encyclopedia of Phys.. Ed.: S. Fluge. 1§_(297). 1956. 47. L. L. Isaacs and T. B. Massalski. Phys. Rev.. 38. #1A (134). 1965. 48. R. M. Bozorth and R. J. Gambino. Phys. Rev.. 147. #2 (487). 1966. 49. 0. V. Lounasmaa and L. J. Sundstrom. Phys. Rev.. 50. #2 (399). 1966. 50. T. A. Kaplan. Phys. Rev.. 124 (329). 1961. 51. L. T. Crane. J. Chem. Phys.. 29, #1 (10). 1962. 52. E. I. Balding. Phil. Mag.. &_(1145). 1959. 76 .umsnofi n how “GBOQ u a has n D “Hmucmo u o .msau was Ham 20 ms gouazm aw .snuocmum “on «so .mcaummn now 20.. mmo a mac zo flow v o n a o .. mocmumammm ummeOEHmnB mcflcuoowm mmo a who 20 I m a a D U I .owm 0cm HQAHAHQEG on no cofiumunflamo mac n zo 20 flow a o o o o I has umumsosumna maausmmmz mmo a zo I I I o o o zo mam “mucus mcausmmmz mmo a zo zo now I o a o I ucmuuso mcflusmmmz mo GOADMNwUHMUcmum mmo a zo I I I o o o zo unmuuso mcaummm mo cofluMNfloumocmum 20 D 20 I I I a a U I HmumEoHucwuom mo Godumuwoumncmum NH Ha as m n o m w m a GOHDUGSh goua3m may «0 mmcauumm . H 1H” mmUH>mG DZHMDm4m2 mm? m0 mZOHBUZDm mDOHm¢> mom wUZHBBmm mUBHBm fl XHDZMQQfl APPENDIX B EXPERIMENTAL PROCEDURE Considering the time at which liquid helium is transferred from the container to the cryostat as zero-hour. the experimental procedures are listed -48:00 Weigh the two sample halves. Prepare the men: Sandwich the sample-halves with the thermometer assembly. Weigh the specimen the below: speci- heater- and suspend it in the calorimeter. Solder the leads to the four kovar—seal connector-pins. Check the electrical system for open- and short-circuits by measuring the heater and the thermometer resise tances. Solder the colorimeter can in position. Load the calorimeter system into the inner dewar. Position the cathetometer and adjust its levels. Turn on the main-switch M81 (see figure 7) for the electrical system to stabilize. Evacuate the calorimeter and start the diffusion pump. 77 78 Appendix B (cont.) -42:OO -16:OO -4:00 Repair any leaks in the calorimeter system if necessary till a stable reading is obtained on the cold-cathode vacuum gage in the lower range of 10.5 torr. without the cold-trap. Turn off the diffusion pump. Prepare the dewar system for liquid-nitrogen transfer: Evacuate the middle jacket of the liquid-helium dewar. and flush it with dry nitrogen gas. Close valve 6 (figure 5) when the nitrogen inside the middle- jacket is at 200 microns. Evacuate the inner- dewar slowly by opening valves V1. V2. V3. and flush it three times with helium gas. Release the system to helium gas and maintain h—psig pressure. Flush and evacuate the calorimeter system with the helium gas three times and close the valve CV (figure 4). leaving the helium gas inside at 700 microns pressure. Transfer liquid nitrOgen into the outer dewar. Standardize the potentiometer. and adjust the decade-resistors R14. R16 to set the heater and 79 Appendix B (cont.) -1:00 0 .0 00 the thermometer currents at 0.1mA and 1.0uA re- spectively. Measure the thermometer and the heater resistances at the liquid-nitrogen temper— ature in order to check the electrical system. (See Appendix A.) Evacuate the calorimeter. and start the diffusion pump. (The end—vacuum should be in the lower range of 10-6 torr. with the cold- trap; if not. lower the outer—dewar to let the system warm up to room temperature. and go back to -42:00.) Refill the outer-dewar with liquid nitrOgen. Evacuate the liquid—helium transfer—tube jacket. Position the liquid-helium container on a fork- lift. flush the transfer-tube with helium gas and insert it in the container. Check the pressure of the helium gas in the inner-dewar for k-psig. Remove the inlet screw IS and insert the delivery end of the transfer-tube into the inner-dewar. Supply helium gas to the liquid-helium container at 2/3-psig to start the liquid-helium transfer. 80 Appendix B (cont.) (Stop the transfer immediately if the calorimeter vacuum deteriorates. or if the liquid-helium level begins to fall at any time. Go back to -42:00 in the former case. and start at -1.00 if a fresh liquid—helium container is available in the latter case.) Open valve MVl (figure 6). start pump P3. and open valve P3V slowly. Turn off the diffusion pump. Open valve CV to release helium gas into the calorimeter. and close valves CV and P2V to arrest about 400 microns of helium gas in the calorimeter. Collect the liquid helium up to 4” below the top of the inner—dewar. (The transfer usually takes about 10 mins.) Open the safety valve SV (figure 5). and retain it in that position. Remove the transfer tube. re-instate and tighten the inlet screw IS. 0:30 Standardize the potentiometer again. Set and note the decade-resistor R14 readings for 0.05. 0.07: 0.1: 0.2: 0.3: 0.4: 0.5: 0.7: and 1.0 milliamperes of current through the heater. 81 Appendix B (cont.) 1 00 Measure the heater resistance. Set the decade resistor R16 to get 1.0uA thermometer current. Measure the thermometer resistance. (Always maintain the sum of the last two readings cons- tant by compensating the R16 reading when meas- uring the thermometer resistance.) Check the recorder calibration with the potentiometer. Start calibration: Measure the level of the liquid—helium bath. Close valve MVl. Note the potentiometer reading. Take the mercury—manometer readings and the Stokes—gage reading. Start pump Pl. Drop safety-valve Sv. and simultaneously but slowly open the needle-valve V1 (figure 4) to pump the liquid-helium vapor at a slow rate. Balance most of the voltage-drop across the thermom— eter with the EMF from the potentiometer. As the unbalanced voltage-drop reading fed to the recorder slowly decreases and passes the zero of the re- corder scale at the SOuV-scale of the DC-amplifier close the valve MVl to the manometer. Note the liquid-helium head. the potentiometer. manometer. 82 Appendix B (cont.) 7:00 and the Stokes'—gage readings. Open MVl slowly. set the potentiometer reading at the next desired value. and open valves V1. V2. and V3 slowly as required. Compensate R14 for the thermometer- resistance. and take readings as described. Re— peat the procedure to collect about ten sets of data above the lambda-point at increasing gaps of potentiometer-readings. (Below the X—point of liquid helium the hydro-static correction need not be noted.) Switch over to the oil-manometer at below 40 torr.. and to the McLeod—gage at about 5-torr. liquid-helium vapor—pressure. Reset the heater current at 0.1mA. and measure the heater resistance at the lowest temperature attained. Turn off the heater and the thermometer currents. the high-sensitivity galvanometer key of the potentiometer. and turn on the shunt switch 510 (Fig. 7). Start pump P2. Open valve P2V. and turn on the diffusion pump to remove the exchange- gas in the calorimeter. Set the decade-resistor 83 R14 corresponding to the heater current of 0.0?mA. Turn off the shunt switch 810 and balance most of the voltage-drop across the thermometer wit} the potentiometer EMF so that the recorder pointer stays close to the zero on the ZOOuV scale. Start the recorder-chart. set the counter-reading to zero. and turn on the heater-current. Quickly compensate R16 for the anticipated change in the thermometer resistance. and turn off the heater- current when the recorder pointer approaches the extreme of the recorder-scale. 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LSQDEG.LP) DEMENSION YV(MNPC.NCG).XH(MNPC.NCG).NPC(4).PRM(4.3). LBL(15) COMMON/GRA/LSD DATA (NCURVE =0 LSD=LSQDEG INITIALIZATION CALL PLOT (o..o..o. 80.. 80.) CALL PLOT (o..-13.75.2) CALL PLOT (0..0..0) CALL PLOT (2..3..2) CALL PLOT (0..o..0) IUB=(NCRV/NCG)*20+20 CALL PLOT (IUB.X.3) GRID FN=NCH-l $ YLBY - FN*0.25+11.0 CALL CHAR (YLBT.0.0.LBL(1).8.0.0..15..1) CALL CHAR (YLBT.1.25.LBL(2).8.0.0..15..1) CALL CHAR (YLBT.2.50.LBL(3).8.0.0..15..1) CALL PLOT (0..0..2) FLN=10.0 IF (LN.NE.0) FLN=0.1 92 93 DO 2 I=2,10,2 A=I-l CALL PLOT (0.0,A,2) CALL PLOT (FLN,A,1) CALL PLOT (FLN,A+1.,2) CALL PLOT (0.,A+1.,1) 2 CONTINUE CALL PLOT (0.,0.,1) DO 3 I=2,10,2 A=I-l CALL PLOT (A,0.0.2) CALL PLOT (A,FLN,1) CALL PLOT (A+1.,FLN,2) CALL PLOT (A+1.,0.,1) 3 CONTINUE CALL PLOT (0.,0.,1) CALL PLOT (0.,0.,2) SCALE X CALL CHAR (-1.0,2.00,LBL(4),8,0.0.,15,.0) CALL CHAR ('1.0,3.25,LBL(5),8,0.0,.15,.1) CALL CHAR (-l.0,4.50,LBL(6),8.0.0,.15.,1) XS=0.0 XL=20.0 C=(XL-XS)/l0.0 ENCODE (6,4,IS)XS 4 FORMAT (F6.2) CALL CHAR (-0.25,-0.26,IS,6,0.0,1./8.,1./12.) B=xs DO 5 I=l,10 FF=I $ F=FF=0.26 $ G=B+C ENCODE (6.4,JS)G CALL CHAR (-0.25,F,JS,6,0.0,1./8.,1./12.) B=G 5 CONTINUE SCALE Y CALL CHAR (2.00,-l.5,LBL(7),8,90.,.15,.1) CALL CHAR (3.25,—1.5,LBL(8),8,90.,.15,.1) CALL CHAR (4.50,—1.5,LBL(9),8,90.,.15,.l) YL=20.0 YS=0.0 C=(YL-YS)/10.0 ENCODE (8,7,KS)YS 7 FORMAT (F8.3) CALL CHAR (0.0,—l.10,KS,8,0.0,1./8.,1./12.) B=YS DO 9 I=l,10 F-l $ G=B+C ENCODE (8,7,LS)G CALL CHAR (F,-l.10,LS,8,0.0,l./8.,1./12.) B=G 9 CONTINUE 94 PARAMETER LIST Do 20 I=1,3 YLB=FN*0.25+10.25 $ XLB=3*(I-1) $ LL=I+9 CALL CHAR ( YLB,XLB,LBL(LL),8,0.0,.15,.1) Do 20 NC=1, NCG FNl=NCG-NC $ YP=FN1*0.25+10.25 $ XP=XLB+1.25 IF (I.NE,3) GO TO 17 NCURVE=NCURVE+1 ENCODE (3,100.NCVE)NCURVE 100 FORMAT (I3) CALL CHAR (YP,9.870,NCVE,3,0.,1./8.,l./12.) IF (NCG.EQ.1) GO To 17 YSY=YP+0.1 $ XSY=9.0 CALL SYMBOL (NC,YSY,XSY,80.,80.) 17 NCMl=NC-l IF (NC.GT.1.AND.PRM(NC,I).EQ.PRM(NCM1,I)) GO To 20 ENCODE (8,15,JP1)PRM(NC,I) 15 FORMAT (F8.3) CALL CHAR ( YP, XP,JP1,8,0.,.15,.1) 20 CONTINUE CALL PLOT (0.,0.,2,80.,80.) CALL CURVE (YV,XH,NPC,MNPC,NCG,XL,XS,YL,YS,LP) CALL PLOT (O.,0.,0.80.,80.) CALL CHAR (12.5,0.,LBL(13),8,0.,0.5,1./3.) CALL PLOT (20.,o.,2) END SUBROUTINE CURVE (YV,XH,NPC,MNPC,NCG,XL,XS,YL,YS,LP) DIMENSION YV(MNPC,NCG),XH(MNPC,NCG),NPC(4) COMMON/CRV/Y(101,X(101,NCURVE DATA (NCURVE=0) SY=100./((YL-YS)/8.) SX=100./((XL-XS)/8.) CALL PLOT (YS,XS,0,SY,SX) Do 25 NC=1,NCG NCURVE=NCURVE+1 K=NPC(NC) DO 3 J=1,K Y(J)=YV(J,NC) 3 X(J)=XH(J,NC) IF (K.GE.101) GO To 9 D0 5 I=1,K CALL SYMBOL (NC,Y(I),X(I),SY,SX) 5 CONTINUE K=NPC(NC)-LP+1 CALL LSTSQ (K,XL,XS,LP) 9 DO 10 I=l,101 IF (Y(I).GT.YL-0.005) Y(I)=YL IF (Y(I).LT.YS+0.005 Y(I)=YS 10 15 17 19 20 25 100 105 110 95 CONTINUE CALL PLOT (Y(1),X(1),2,SY,SX) IF (Y(1).NE.YS.AND.Y(1).NE.YL) CALL SYMBOL (NC,Y(1), X(1),XY,SX) NP=K IF (K.LT.101) NP=101 Do 20 I=2,NP $ J1=1 Il=I+1 $ IMl=I-l YTl=(Y(I)=YS)*(YL—Y(I)) $ YT2=(YIMl)-YS)*(YL-Y(IM1)) IF (YT1.EQ.0.0.AND,TY2.EQ.0.0) J1=2 CALL PLOT (Y(I), X(I),X(I),Jl,SY,SX) IF (YT1.EQ.0.0.AND.I.NE.NP) GO TO 17 IF (YT1.NE.0.0.AND.I.EQ.NP) GO TO 10 $ GO To 20 IF (Y(IM1).EQ.YL.AND.Y(I1).LT.YL) GO TO 19 IF (Y(IM1).LT.YL.AND.Y(L1).EQ.YL) GO To 19 IF (Y(IM1).GT.YS.AND.Y(11).EQ.YS) GO TO 19 IF (Y(IM1).EQ.YS.AND.Y(II).GT.YS) GO To 10 $ GO To 20 CALL SYMBOL (NC,Y(1),X(I),SY,SX) CONTINUE CONTINUE CALL PLOT (YS,XS,2,SY,SX) SUBROUTINE LSTSQ (K,XL,XS,LP) DIMENSION W (101), R(2,101),C(10) COMMON/CRV/Y(101),X(101),NCURVE COMMON/GRA/N PRINT 100 FORMAT (l3HlLSTSQ OUTPUT,//) K=K+LP-l DO 5 I=1,K W(I)+l.0 KP=K-LP+1 CALL MCPALS (KP,N,0.0.W,X,Y,R,C,LP,IDEG) PRINT 105, NCURVE,(I,R(1,I),R(2,I),I=LP,K) FORMAT (4X,*CURVE NO.*,13,/,42X,5HERROR,15X,10HFRAC ERROR, // A (38X,I3,X,2(E12.4,8X))) IDEGl=IDEG+l PRINT 110, IDEG,(C(I),I=1,IDGE1) FORMAT (//,X,*IDEG=*,I2,/,X,*LSTSQ COEF*,/,(X,5 (E16.8,X))) DO 10 J=l,101 X(J)=(J-l)*0.01*(XL-XS)+XS $ Y(J)=0.0 IF (X(J).EQ.0.0) GO TO 9 DO 8 I=l,IDEGl Y(J)=Y(J)+C(I)*X(J)**(I-l) CONTINUE $ GO TO 10 Y(J)=C(l) CONTINUE END UlIbWNH 12 96 SUBROUTINE SYMBOL (NC,YI,XI,SY,SX) R=0.04 CALL PLOT (YI,XI,2,SY,SX) GO TO (l,2,3,4),NC CALL CALL CALL CALL CALL END CIRCLE (R,RI,XI) $ GO To 5 TRI (R,YI,XI) 5 GO To 5 SQU (R,YI,XI) $ GO TO 5 DIA (R,Y1,XI) PLOT (YI,XI,1,SY,SX) SUBROUTING CIRCLE (R,YI,XI) CALL PLOT (YI,XI,2,100.,100.) CALL PLOT (YI,XI+R,2) DO 12 I=10,360,10 A=I*3,l415926536/180. X=R*COSF(A)+XI CALL PLOT CONTINUE CALL PLOT (YI,XI,2) END $ Y=R*SINF(A)+YI (Y.X.1) SUBROUTINE TRI (R,YI,XI) CALL CALL CALL CALL CALL CALL END PLOT PLOT PLOT PLOT PLOT PLOT (YI,XI,2,100.,100.) (-(2./3.)*0.866*R+YI,XI+R,2) ((4./3.)*0.866*R+YI,XI,1) (-(2./3.)*0.866*R+YI,XI-R,1) (-(2./3.)*0.866*R+YI,XI+R,1) (YI,XI,2) SUBROUTINE SQU (R,YI,x1) CALL CALL CALL CALL CALL CALL CALL END PLOT PLOT PLOT PLOT PLOT PLOT PLOT (YI,XI,2,100.,100.) YI-R,XI+R,2) (YI+R,XI+R,1) (YI+R,XI-R,l) (YI-R,XI-R,1) (YI-R,XI+R,1) (YI,XI,2) SUBROUTINE DIA (R,YI,XI) CALL PLOT (YI,XI,2,100.,100.) CALL PLOT (YI,XI+R,2) 97 CALL PLOT (YI+R,XI,1) CALL PLOT (YI,XI-R,1) CALL PLOT (YI-R,XI,1) CALL PLOT (YI,XI+R,1) CALL PLOT (YI,XI,2) END SUBROUTINE MCPALS(M,N,EPS,w,x,Y,R,C,LP,IDEG) DIMENSION W(M),X(M),Y(M),A(10,10),SUMXSQ(19), C(10),R(2,M),B(10) SUMXSQ(1)=B(1)=0 NMX=N IF((M-N-1).LT.0) NMx=M-1 NMX1=NMX+1 MN=M+LP-1 Do 1 I=LP,MN R(2,I)=1.0 B(1)=B(1)+Y(I)+W(I) SUMXSQ(1)=SUMXSQ(1)+W(I) R(l.l)=B(l) NMN=1 IF(EPS.EQ.0) NMN=NMx Do 10 NN=NMN,NMx N2=2*NN Nl-NN+1 N21=N2~1 IF(EPS.EQ.0) N21=1 DO 2 J=N21,N2 J1=J+1 IF(J1.LE.NMX1) B(J1)=0 SUMXSQ(J1)=0 DO 2 I=LP,MN R(2,I)=R(2,I)*X(I) SUM=R(2,I)*W(I) IF(J1.LE.NMX1) R(1,J1)=B(J1)=B(Jl)+SUM*Y(I) SUMXSQ(J1)=SUMXSQ(J1)+SUM Do 3 I=1,N1 J1=I-1 Do 3 J=1,N1 A(I,J)=SUMXSQ(J1+J) CALL GAUSS (N1,A,B,C) DO 4 I=1,N1 B(I)+R(l,I) DO 8 I=LP,MN SUM=C(N1) Do 5 J=1,NN SUM=X(I)*SUM+C(Nl-J) SUM=Y(I)-SUM IF((ABSF(SUM).LT.EPS).OR.(NN.EQ.NMX))GO To 7 98 DO 6 J=l,NMXl 6 . R(1,J)=B(J) GO TO 10 7 R(l,I)=SUM 8 CONTINUE DO 9 I=LP,MN 9 R(2,I)=R(1,I)/Y(I) IDEG=NN RETURN 10 CONTINUE RETURN END SUBROUTING GAUSS(M,A,B,C) DIMENSION A(10,10),B(M),C(M) 101 FORMAT (//53X,30H***SINGULAR MATRIX IN GAUSS***//) Do 6 K=1,M C(1)=0 IMAX=K Do 1 I=K,M T=ABSF(A(1,K)) IF(C(1).GE.T) GO TO 1 C(1)=T 1 MAX=I 1 CONTINUE IF(C(1).NE.0) GO To 2 PRINT 101 RETURN 2 IF(K.EQ.IMAX) GO To 4 J=IMAX T=B(K) B(K)=B(J) B(J)=T DO 3 L=1,M T=A(K,L) A(K,L)=A(J,L) 3 A(J,L)=T 4 I=K+1 DO 5 J=I,M T=A(J.K)/A(K.K) B(J)+B(J)-B(K)*T DO 5 L=I,M 5 A(J,L)=A(J,L)-T*A(K,L) 6 CONTINUE J=M+1 Do 8 K=1,M I=J-K T=0 IMAX=I+1 DO 7 L=IMAx,M 99 T=T+A(I,L)*C(L) C(I)=(B(I)-T)/A(I.I) RETURN END