LI B R A R Y Micliigan State University mZ20Zéam J O/M% Major professor .’~ .tn' 9 J- ‘6... $uc‘ ‘l.~‘ ...e L retal D O ‘5“ ”I: \Q ..\ ~. A ~ ‘ b is us» . ‘ ~ .0 ABSTRACT THE ULTRALOW TEMPERATURE MAGNETIC SUSCEPTIBILITY OF COPPER TETRAPHENYLPORPHINE USING A 3He-4He DILUTION REFRIGERATOR BY Jeffrey L. Imes An investigation into the magnetic behavior of a class of dilute paramagnetic solids is presently in proqress. The ultralow temperature magnetic susceptibility of several meta110porphyrin and metallo-tetraphenylporphyrin compounds 3He-4He dilution refrigerator. has been measured using a This thesis reports the results obtained for the first of these compounds, copper a,8,y,6 - tetraphenylporphine (CuTPP), to be studied in both powder and single crystal form. The apparatus and techniques used are also presented in detail. The 3 He-4He dilution refrigerator is capable of reach- ing 8.4 mK in the continuous mode of Operation and 4.0 mK in the "single-shot", non-continuous mode of operation. Its mixing chamber was constructed in a unique dual-tail arrange- ment which allows a powdered sample and a powdered CMN thermometer to be in good thermal equilibrium with each other. Careful thermal calibrations of the refrigerator indicate the temperature of the two tails agree to within .5% at 4.0 mK. Besides the two conventional ac mutual-inductance magnetom- eters there is also a Superconducting Quantum Interference fl". ‘ to ‘S I p 9...... bib-L q a W‘?‘ it v I IV V “t u. D A‘w-a: 'todu ‘ R! "F u..,,"r‘ A Set“ “Me. L! ('1 .\ ‘_. .14?) q‘ ‘- (2‘0“ Device (SQUID) magnetometer mounted on the dilution Jeffrey L. Imes refrigerator. Measurements of the susceptibility paral- lel and perpendicular to the c axis of aligned 1 mm3 single crystals of CuTPP were made with this magnetometer. The future of SQUID magnetometers as a means of carrying single crystal measurements to ultralow temperatures seems very promising in light of our results. The parallel susceptibility of CuTPP was found to obey a Curie-Weiss law with all = -l.54 mK to about 20 mK. The Curie-weiss theta was determined from the intermolecular dipolar interaction by carrying out the appropriate lattice summations. The susceptibility perpendicular to the c axis is expected, on a theoretical basis, to be dominated by the hyperfine interaction between the c0pper electronic and nuclear spins. The experimentally measured perpendicular susceptibility indicates the presence of this type behavior. Both the parallel and perpendicular susceptibility crystal data were corrected for demagnetization effects. The demag- netizing factors were determined by utilizing a room tempera- ture technique involving a mock-up of the crystal made from mild steel. The results obtained for the powder suscepti- bility to 4.0 mK show no indication of a transition to the ordered state. Comparison of the single crystal and powder data indicates thermal equilibrium between the crystals and the dilute solution was maintained down to approximately 12 mK. THE ULTRALOW TEMPERATURE MAGNETIC SUSCEPTIBILITY OF COPPER TETRAPHENYLPORPHINE USING A 3He-4He DILUTION REFRIGERATOR BY an Jeffrey L1 Imes A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree DOCTOR OF PHILOSOPHY Department of Physics 1974 DEDICATION This thesis is dedicated to my wife, Dana, whose encouragement and assistance during its preparation were greatly appreciated. ii ACKNOWLEDGMENTS My sincere thanks go to Dr. William P. Pratt, Jr. who effectively guided me to the completion of this thesis. A very special thanks is extended to Gary L. Neiheisel and Paul R. Newman for the considerable time and effort they Spent helping me gather the necessary experimental data. I would also like to express my appreciation to Dr. Jerry A. Cowen, Dr. Thomas A. Kaplan, and Dr. Robert D. Spence for their advice and suggestions concerning various aspects of this thesis. I am also very grateful for the wonderful people in the Machine ShOp who were so helpful during the construction of the apparatus. iii TABLE OF CONTENTS LIST OF FIGURES LIST OF TABLES INTRODUCTION Chapter I. THE EXPERIMENTAL APPARATUS . . . . . . . . A. History of the Dilution Refrigerator: 3 Solutions of He in 4He . . . . . . . . Page vi xi B. Theory and Operation of a Dilution Refrigerator 11 C. Design and Operation of this Dilution Refrigerator . . . . . . . . . . . . . An Overview . . . . . . . . . . . . . . The Condensor, Heat Exchanger, and Still The Mixing Chamber . . . . . . . . . . . D. Measurements of Magnetic Susceptibility The Conventional Magnetic Susceptibility The SQUID Magnetometer . . . . . . . . . E. Thermometry and Thermal Equilibrium within the Refrigerator . . . . . . . . . . . . . Thermometry . . . . . . . . . . . . . . Thermal Equilibrium . . . . . . . . . . Background Susceptibility Measurements . iv Coils 21 21 24 28 40 4O 52 61 61 63 78 ”Labor .4" ~55 I v.. o.‘. Chapter Page II. THE MAGNETIC SUSCEPTIBILITY OF COPPER TETRAPHENYLPORPHINE . . . . . . . . . . . . . . 87 A. A General Survey of the Porphyrins . . . . . 87 B. Copper a,8,y,6 - Tetraphenylporphine . . . . 94 Introductory Remarks . . . . . . . . . . . . 94 Purification and Crystal Growth . . . . . . . 95 The Structure of CuTPP . . . . . . . . . . . 99 C. Theory of the Magnetic Susceptibility of Tetraphenylporphine . . . . . . . . . . . . 112 III. THE EXPERIMENTAL DATA . . . . . . . . . . . . . 124 A. Presentation of the Experimental Results for CuTPP . . . . .‘. . . . . . . . . . . . . . 124 B. Factors Affecting the Measurement of Magnetic Susceptibility at Ultralow Temperatures . . 133 Demagnetization Corrections . . . . . . . . . 133 Thermal Equilibrium of Single Crystals . . . 140 IV. AN ANALYSIS OF THE EXPERIMENTAL RESULTS . . . . 143 A. Comparison of Theory and Experiment. . . . . 143 B. Summary and Conclusions . . . . . . . . . . 154 LIST OF REFERENCES . . . . . . . . . . . . . . . . . . 157 APPENDIX A . . . . . . . . . . . . . . . . . . . . . . 162 APPENDIX B . . . . . . . . . . . . . . . . . . . . . . 169 .APPENDIX C . . . . . . . . . . . . . . . . . . . . . . 175 .APPENDIX D . . . . . . . . . . . . . . . . . . . . . . 184 LIST OF FIGURES Figure Page 1. The phase separation diagram at saturated vapor pressure for 3He-4He mixtures. The dotted line represents the phase separation of a 28% solution Of 3H8 in 4He 0 O O O O O O O O O O O O O O O O O O 2. A schematic diagram of a3He-4He dilution refrigerator indicating the flow of 3He through the various components . . . . . . . . . . . . . . 13 3. The 3He concentration of the still as a function of still temperature assuming the mixing chamber to be at a temperature of 0.0 K. . . . . . . . . l8 4. An overall view of the cryostat. The dilution refrigerator components are enclosed within a vacuum space and interconnected by capillary tubing which carries the flowing 3He. . . . . . . 23 5. A detailed drawing of a sintered copper heat exchanger showing the flow of liquid helium . . . 26 6. A schematic drawing of the dual tail mixing chamber. The arrows represent the flow of 3He through the mixing chamber. For clarity only the upper half of one of the magnetic suscepti- bility coils is shown . . . . . . . . . . . . . . 30 vi . v. .‘N re .“-i n 11 I §.. Figure 10. ll. 12. 13. 14. Page Results of a theoretical calculation of the expected thermal relaxation time of a dual-tail mixing chamber. Closed circles represent heat conduction through the coil foil and mixing chamber walls. Open circles represent heat conduction via the liquid helium. The total relaxation time is represented by a dashed line . 33 A detailed view of the sample chamber design. . . 38 A block diagram of the ac mutual inductance bridge circuit. All resistances are in ohms . . 42 A detailed drawing of the dilution refrigerator and susceptibility coils. A = c0pper flange with lead fusewire "O"—ring; B = stainless steel support tube sealed in epoxy; C = SQUID mounting port; D = copper inner vacuum can connected to the still; E = c0pper to epoxy joint; F = coil foil . . . . . . . . . . . . . . . . . . . . . . 45 A tOp View of the dilution refrigerator and susceptibility coils. The letters refer to parts which are similarly labeled in Figure 10 . . . . 47 A close-up view of the mutual inductance magnetic susceptibility coils detailing their construction 48 A block diagram of the SQUID magnetometer circuit. The SQUID sensor and sample coil are located inside the cryostat . . . . . . . . . . . . . . . 55 The SQUID sample chamber and its means of attach- ment to the mixing Chamber's dilute solution return line C O O O O O O O O O O O I O O O O O O 58 vii Figure Page 15. The results of the calibration of a germanium resistance thermometer against the absolute temperature in the range .3 K to 3.0 K . . . . . 65 16. Results of temperature calibrations of the refrigerator using 100% CMN thermometers. Open = .55570 gm CMN; M = .55555 gm circles: M 2 l CMN. Closed circles: M1 = .55555 gm CMN; M2 = .55570 gm CMN. The subscripts refer to tail #1 or tail #2 . . . . . . . . . . . . . . . 68 17. A comparison of the mixing chamber temperature as determined by 100% CMN and LMN-CMN thermom- eters. Closed circles: M1 = .75444 gm LMN-CMN; M2 = .55555 gm CMN. Open circles represent an indirect comparison of .90891 gm LMN-CMN and .55570 gm CMN I O O O I O O O O O O O O O O O O O O O O 71 18. Results of a temperature calibration of the refrigerator using LMN-CMN thermometers (M1 = .90837 gm LMN-CMN; M2 = .90891 gm LMN-CMN) . . . . . . . . . . . . . . . . . . . . 74 19. The inverse temperature at the SQUID magnetometer (l/T; ) as a function of the inverse temperature as defined by the LMN-CMN thermometer (l/T; ). Dashed lines represent the theoretically esti- mated effects based on siphon tube impedance measurements . . . . . . . . . . . . . . . . . . 77 20. The inherent background susceptibility of the mixing chamber. Open circles - tail #2; closed circles - tail #1. The coil constant used for coil #1 was .0833 emu/dial unit and for coil #2 was .0883 emu/dial unit . . . . . . 80 viii ime 26. ‘fi (1. 0“ Figure Page 21. The background susceptibility of the SQUID sample chamber at an applied field of 2.5 gauss 83 22. Structural representation of a porphine molecule. This structure forms the basis for all porphyrins 88 23. A comparison between the structure of a metallo- porphyrin and metallo-tetraphenylporphyrin molecule. The site labeled M represents a metal ion; Open circles are nitrogen atoms and closed circles are carbon atoms . . . . . . . . . 90 24. The structure of copper tetraphenylporphine. The relative positions of the numbered atoms are indicated in the text and in Table 6. . . . . . . 100 25. The manner in which the 3d orbital energy Splits in crystalline field environments of different smetry O O C I O I O O O O O O O O O O I O I 0 106 26. A perspective view of a CuTPP single crystal showing its relation to the crystalline axes . . 107 27. A unit cell of CuTPP. The small squares around each copper site represents the porphyrin ring plane 0 O O O O O O O O O O O O I O O O O O O O O 109 28. The projection of one unit cell of CuTPP on the (001) plane. Each line structure represents a different plane perpendicular to the c axis . . . 111 29. The manner in which the single crystals were mounted in the SQUID magnetometer for measure- ments of the parallel and perpendicular susceptibilities . . . . . . . . . . . . . . . . 126 ix LIE « w: I I a. a 3 1 11.4 Figure 30. 31. 32. The magnetic susceptibility data for CuTPP. The solid lines represent the theoretical susceptibilities without inclusion of the dipolar interaction and the dashed lines represent the theoretical susceptibilities including the effect of dipole coupling . The magnetic susceptibility data for CuTPP corrected for demagnetization effects. Open symbols represent data further corrected for thermal equilibrium effects. The +‘s and x's represent an "effective powder susceptibility" derived from the single crystal data. The high temperature single crystal and powder magnetic susceptibility data. The 9 values determined from this data are also shown. "effective powder susceptibility" was obtained from the crystal data after normalizing the The perpendicular susceptibility to 91.: 2.001 . . A "circuit diagram" representing thermal resistance to heat flow between the two tails Path A refers to heat flow through the liquid (Rd-RC-Rd) and path B to heat flow through the c011 fOil (RK-RCu-RK). of the mixing chamber. The response function of a SQUID magnetometer. The total flux (¢) through the superconducting 100p is plotted as a function of the external applied field . Page . 130 . 144 . 149 . 163 . 172 Table LIST OF TABLES The Entropy: Thermal Conductivity, and Viscosity 6.3% 3 of Concentrated He and a Dilute Solution of 3He in 4He at Low Temperatures . . . . . . . Coil #1 Background Correction Equations . . Coil #2 Background Correction Equations . . SQUID Background Equations (H = .25 gauss) Inverse SQUID Temperature as a Function of l/T The Nonplanarity of a CuTPP Molecule. The Deviations of Various Atoms from the (001) Plane Passing Through the Copper Ion are Listed. Relevant Bond Lengths are also Tabulated48 . Principle Values of the Interaction Tensors The Values in Parenthesis were Obtained During Our Investigations of CuTPP . . . . . . . . Results of the CR-lOO Germanium Resistor Calibration . . . . . . . . . . . . . . . . The Magnetic Susceptibility Data for CuTPP-- Single Crystal, Parallel Axis (Not Corrected for Demagnetization Effects) . . . . . . . . The Magnetic Susceptibility Data for CuTPP-- Powder and Single Crystal, Perpendicular Axis (Not Corrected for Demagnetization Effects) xi Relevant to CuTPP Along the Crystalline Axesss. Page 10 84 84 85 85 102 118 185 186 187 6. Page The Magnetic Susceptibility Data for CuTPP-- Single Crystal, Parallel Axis, (Corrected for Demagnetization Effects) . . . . . . . . . . . . 188 The Magnetic Susceptibility Data for CuTPP-- Powder and Single Crystal, Perpendicular Axis (Corrected for Demagnetization Effects) . . 189 The Effective Powder Susceptibility as Determined by the Demagnetization Corrected Single Crystal Data . . . . . . . . . . . . . . 190 xii Rece extended mo the fine very INTRODUCT ION Recent advances in refrigeration techniques have extended the range of temperatures which can be reached into the millikelvin region. The means of achieving the very lowest temperatures (~ 2 mK) is almost exclusively by the adiabatic demagnetization of a cerium magnesium nitrate (CMN) sample which is precooled by a 3He-4He dilution refrigerator. This important low temperature salt is also used to define the temperature scale down to about 6 mK by Curie law extrapolation of its magnetic susceptibility. However, thermometry below this tempera~ ture is rather uncertain.1 The prOperty of CMN which makes it so useful in ultralow temperature applications is its small paramagnetic ion density. Only for compounds of high magnetic dilution is the separation of the para- magnetic ions sufficient that their interaction is essentially due to weak classical dipolar coupling. These dipolar solids exhibit very low ordering temperatures; therefore a complete study of them must involve ultralow temperatures. Also, since the lowest temperatures attain- able by adiabatic demagnetization is limited by the ordering temperature of the salt, dilute paramagnetic systems with ordering temperatures lower than CMN will be 193855 cjo l in; ‘ - — necessary to attain lower temperatures.2’3 Interest in the properties of dilute paramagnetic solids has been further stimulated by the results of several ongoing investigations. For example, anomously good thermal contact between CMN and pure 3He at ultralow temperatures has been observed recently (see J. H. Bishop st 31.4 and the references therein). This unusual phenomena is probably due to magnetic coupling between the 3He atoms and cerium ions at the liquid-solid interface. For a complete understanding of this interaction surface studies must be undertaken between liquid 3He and other dipolar solids. Another very important discovery has been made in relation to the properties of pure 3He at ultralow tempera- tures. Liquid 3He was found to form, under pressure, two different phases, both of which are superfluid.5 However, 3He is very a complete study of the superfluid properties of difficult because it occurs at about 2 mK, the limit of temperatures presently attainable, and because of the lack of a good, practical thermometer which can be used in the low millikelvin temperature range.6 These problems indicate a need for new dipolar solids. Only a few such compounds are known at the present time.7'8'9 It is therefore essential to expand our knowledge of the nature of dipolar solids and to discover new compounds which may be useful in the production of lower temperatures, in the clarification of the absolute temperature scale at ultralow temperatures, and in the study of several important properties exhibited by pure 3He at ultralow temperatures. The magnetic systems presently under study are the meta110porphyrins; their zero—field magnetic susceptibility are being measured from about 4 mK to 4.2 K. Molecules of the porphyrins are characterized by a basic structure to which various ligands can be attached without changing the environment of the paramagnetic ion. The spin density of crystalline porphyrins can be varied from as low as approxi- 19 21 mately 10 spins/cm3 to as high as approximately 10 cm3 by attaching different ligands to the basic porphyrin spins/ structure. These spin densities imply that the porphyrins will be primarily dipolar solids. The manner in which the spin density of a porphyrin can be diluted is important. The normal procedure for diluting a paramagnetic system is to replace some ions at random with an isostructural diamag- netic ion. The resulting disordered system is difficult to analyse because of the random nature of the occupation of each paramagnetic site. However, the dilution of a porphyrin is not a random process because the ions are simply separated further by the addition of larger ligands and still maintain a well-defined crystal structure. This thesis presents the results of the magnetic susceptibility studies on the first porphyrin (Copper Tetraphenylporphine) for which we have completed both powder and crystalline measurements. «A? ‘9'!“ .Ju uL & OaPfierE ure-r AU"; bdbUL U I a.“ ‘1‘} vide‘Ad . 1 "I tun H «A 5..» '-.,. “I +155 ~ ~ r. “a ”Cs v .‘w,& The 3 He-4He dilution refrigerator has been designed and constructed for the initial purpose of carrying out ultralow temperature zero-field magnetic susceptibility studies on dilute magnetic systems. Its mixing chamber is a unique side-by-side arrangement designed to provide good thermal equilibrium between a thermometer in one tail and a sample in the other. Simultaneous measurements of the magnetic susceptibility of powdered samples located in each tail are made with two conventional ac mutual-inductance suscepti- bility coils. To obtain useful magnetic susceptibility information one must also make measurements along the magnetic axes of single crystals. A SQUID magnetometer is utilized to make susceptibility measurements on very small (~ 1 mm3), aligned single crystals. The difficulties encountered in the study of very small single crystals in the SQUID magne- tometer due to their physical size and effects such as demagnetization corrections are more than compensated for by the low temperatures to which the single crystal measure- ments can be carried. The large surface-to-volume ratio of these crystals means that they can be cooled to lower tem- peratures than is possible with larger crystals. These studies have shown that SQUID magnetometers are extremely useful devices in ultralow temperature studies of single crystal samples. _‘ Ln ... vol -. . A" 3 r“. Li I); (I? F?) CHAPTER I THE EXPERIMENTAL APPARATUS A. History of the Dilution Refrigerator: Solutions of 3He in 4He In the original theoretical work on the properties of 10 3 3 describe the He He-4He mixtures, Landau and Pomeranchuk particles as an impurity in the 4He with an energy contribu- tion given by: 3 “E30 I!) ll 2 * + p /2m3 In this manner the 3He atoms are treated as independent particles within the 4He sea, having a binding energy -E30 and a quasiparticle momentum p. The effective mass m; is a result of the interaction between a moving 3He atom and the 4He background. It was as a result of this theory that H. London11 3 4 later proposed that solutions of He and He be used to attain low temperatures. However, practical techniques for 3 cooling with He-4He mixtures had to await the discovery of the existence of the phase separation phenomena in certain 3He in 4He. This isotropic separation 12 concentrations of was first observed by Walters and Fairbank in 1956 and occurred at T = .83 K. Later specific heat measurements by 5 Hard 0 a, CA " .3153: ... '.‘ ' t... x. ,, 5:9.“ (.I‘ (D 1v- (D L" (11 _-" If, Edwards, SE 31.13 for various concentrations of 3He in 4He demonstrated the variation Of phase separation temperature with 3He concentration. The phase separation diagram has been determined experimentally by R. de Bruyn Ouboter, K. W. Taconis, 25.31.14 and is shown in Figure 1. These measurements were made at saturated vapor pressure. The 3 4He in the 4 finite solubility Of He in He-rich lower region at low temperatures, essential to the operation of a dilution refrigerator, is due to the binding energy Of an individual 3He atom being greater in pure 4He than in pure 3 3He and 4He atoms are different, He. Since the masses of the phase separation results in the lighter pure 3He phase rising above the 4He-rich phase in the gravitational field. These developments led London, Clarke and Mendoza15 to put forth the idea Of attaining low temperatures by "evaporation" of the upper, 3He-rich phase, into the lower, 4He region. Also, it was suggested that the 3 He in the upper phase could be replenished, thereby allowing for the possibility of continuous refrigeration. Since the Operation of a dilution refrigerator is intimately connected with the properties Of dilute solutions of 3He in 4 He and of pure 3He, much research was also being carried out to determine more about their properties. Some of the more important results, necessary for a better under- standing Of the Operation of a dilution refrigerator, are presented here and will be referred to later. The entropy, Figure l. The phase separation diagram at saturated vapor pressure for He-4He mixtures. The dotted line represents the phase separation of a 28% 3 4 solution of He in He. (K) —. I‘ESMI’F RA TURF ed vapor ted line (K) TEMPERATURE l.4 |.2 |.O NORMAL SUPERFLUID “El-WM HELIUM l I Y I I Y Z .4 6 .8 n X3 = 3 n3+n, |.O thermal conductivity, and viscosity (Table l) are useful in explaining the actual cooling process and thermal equilibrium within a refrigerator. These properties are quoted for temperature ranges over which the expected Fermi-Dirac statistics are most obvious. More detailed discussions Of these thermodynamic properties are presented by R. Radebaugh,l6 J. C. Wheatley,17 and W. E. Keller.18 The first dilution refrigerator was built by Hall, Ford, and Thompson.19 Their refrigerator cooled to 65 mK under continuous Operation. Within about a year the dilution refrigerator had been further perfected tO the Point where 20 had built a refrigerator which cooled Viches and Wheatley to 10 mK under continuous Operation and 4.5 mK in the noncontinuous mode of Operation. The author and the research group he is associated with have recently built a dilution refrigerator capable Of attaining temperatures of about 8.4 mK in the continuous Operation mode and 4.0 mK in the "one-shot", noncontinuous mode. 10 Table l. The entropy, thermal conductivity, and viscosity of concentrated 3He and a dilute solution Of 6.3% 3He in 4He at low temperatures. JTROPY per mole of He3 (joules/mole-K) scmzzr T£.|5K SdRfilOZT T lk .IOK THERMAL CONDUCTIVITY (erg/sec-cm) Kc R” I; T£.O4K T Kd 52’» 37g T£.O|5K VISCOSITY (dyne—sec/cmz) 2 xlo’6 T£.O7K 22 77c 77d :3 5on’ T£.O7K 11 B. Theory and Operation of a Dilution Refrigerator Dilution refrigerators have become important to low temperature research not only due to their ability to achieve very low temperatures, but also because they may be Operated continuously for extended periods of time. This continuous Operation mode requires the circulation Of 3He atoms within the refrigerator. The path of 3He flow through various parts Of the refrigerator is shown in Figure 2, a schematic diagram Of a dilution refrigerator. Gaseous 3He is returned to the refrigerator through a condensing capillary which is thermally tied to the 1° pot. The 1° pot, actually a small 4He evaporation refrigerator, is at a temperature of approximately l.2°K. A flow impedance located immediately following the capillary maintains the incoming gas at a high enough pressure to cause its liquifi- cation at the condenser. The liquified 3He is then allowed to pass through several stages of heat exchangers and then into the mixing chamber where the actual cooling occurs. The heat exchangers allow the incoming warm 3He to be cooled prior to its introduction into the mixing chamber. The cold 3He then returns via the heat exchangers to the still where it is evaporated from the dilute solution, compressed, and returned to the condenser, thereby completing the cycle. A 3He-4He solution in a dilution refrigerator is cooled initially by Operating the system as an evaporation refrigerator. The mixture cools by this method until it 12 Figure 2. A schematic diagram Of a 3He-4He dilution refrigerator indicating the flow of 3He through the various components. 13 Liquid He: fi I°Pot ' * Ab Condenser ‘u Helium-4 B I ‘ i-ie3 V°P9' Bifiiar Wound Flow Orifice Heater Impedance I L OFHC Copper Orifice a ---- ‘~--D ‘ A Still 'Q':~~- ~ I °/. Helium-3 1/, -43 Still / Heat Exchanger Capillary Heat Exchanger Mixing * Chamber 2?? Sintered Copper Heat Exchangers i IOO°/o Helium-3 w ——"_f 6.30/0 Helium-3 in Helium—4 14 reaches the appropriate phase separation temperature. At this temperature an upper, 3He-rich region will begin to form, ultimately reaching a concentration of 100% 3He at lower temperatures. Concurrent with this is the formation of a 4He-rich lower region which will eventually approach a concentration limit of 6.3% 3He in a "bath" Of 4He. The relative volumes Of the concentrated (3He-rich) and the dilute (4He-rich) sides Of the refrigerator determine the 3He concentration Of the mixture to be used for a given refrigerator. It is important for the maximum efficiency to place the phase separation level inside the mixing chamber and not in the heat exchangers or interconnecting capillaries. The cooling process which occurs within the mixing chamber can best be explained by referring to the properties Of the concentrated and dilute phases (Table 1). We can see that at a given temperature the migration Of a 3He atom across the phase boundary must result in an increase in entropy for that atom. This increase in entropy is Obtained by the absorption of heat from the dilute solution. The cooling process is made to be continuous by removing the 3He atoms from the dilute solution and returning them to the concentrated 3 He region via the condenser. The great advantage Of a dilution refrigerator over normal evaporation refrigerators, besides the possibility of continuous, long-term Operation, can best be appreciated by comparing 15 the temperature dependence Of the refrigeration capability Of each system. The lowest temperature evaporation refrigerator, a liquid 3He system, has a refrigeration capacity which is proportional to the 3He vapor pressure. -L /kT The vapor pressure is prOportional to e o , where L0 is the latent heat of the 3He liquid and k is Boltzmann's 3 constant. The ratio, Lo/k, is 2.5 K for He so for tempera- tures on the order of .25 K the refrigeration capacity is already tOO small to be Of much use because the vapor pressure is so low that the 3He cannot be pumped away efficiently. However, the refrigeration capacity Of a dilution refrigerator is approximately proportional to T2 at low temperatures due to the difference in the entropy per 3 4 atom of pure 3He and Of a dilute solution of He in 'He. Since this entropy is proportional to T for both the pure 3He and the dilute solution at low temperatures, their difference will also be proportional to T. This difference represents 3 the entropy change of the He as it migrates across the phase boundary. The entropy change can be related to the heat 3 absorbed by the He through the thermodynamic relation A0 = TAS. One immediately sees that the absorbed heat must be proportional to T2. Thus the dilution refrigerator can maintain a useful cooling capacity to much lower temperatures than is possible for a conventional evaporation refrigerator. 3 It is the purpose of the still to allow He to be pref- erentially removed from the dilute solution and circulated poo ' Opp. 'cly. 5v s,- _ I “.I ‘A ~.‘ ' ‘ 16 through the refrigerator. It generally Operates at a tem- perature Of about .65 K, which means the 3He concentration in the still will be less than 1%. Although this would seem to contradict the concentration as predicted by the phase separation diagram, in reality it does not. The phase diagram was determined at saturated vapor pressure with a uniform temperature throughout the mixture. Since there is a large temperature gradient between the mixing chamber and still, the phase separation diagram no longer applies in this portion Of the refrigerator. The proper- ties Of the dilute solution in a thermal gradient, assuming the mixing chamber is held at T = 0.0 K, are shown in Figure 3. In order for thermal and mechanical equilibrium to be simultaneously maintained in the system (i.e. the 4 chemical potential Of the He to remain constant) there must be a concentration gradient in a thermal gradient.21 4He in The condition of constant chemical potential of the the dilute solution return lines is required if there is to be no acceleration Of the superfluid in this region. This could result in convective heat flow from the still to the mixing chamber. In order to prevent any such heat flow, the dilute solution return lines are constructed from small diameter capillary tubing which effectively blocks the flow Of 4He normal fluid.22 3He be circulated through Since it is desired that only the system during its Operation, as will become apparent later, this low 3He concentration in the still would seem 17 Figure 3. The 3He concentration Of the still as a function of still temperature assuming the mixing chamber to be at a temperature of 0.0 K. function ; chamber (K) Tstiil .60 .40 .20 .IO .08 .06 .04 .02 .Ol 18 I I T I I I IT PHASE .. SEPARATION ... LINE\ L— _ Tmixing chamber = 0-0 K I L I l L l l L .0l .02 .04 .06 .08 .IO ...o‘-A‘ err“: quN a b J surf-n v D r I t.) eh fl kg‘h. .- b: V- ‘.*E 5‘ 3:1 19 to pose a problem. However it happens that due to the weak binding of the 3He quasiparticles to the 4He background in comparison tO the binding of a 4He atom to the 4He back- ground, the partial vapor pressure Of 3He above the liquid is much higher than the partial vapor pressure of 4He. Despite this fact there is another process which can work to decrease the high 3He partial vapor pressure. The 4He liquid in the dilute solution, being superfluid, can rise through the pumping-line orifice of the still by film flow and evaporate at a higher temperature inside the pumping line. This is the familiar thermo-mechanical or fountain effect Observed in superfluid helium. As the 3He concentration in the still is lowered by 3He the system will replenish the lost 3He evaporation of atoms in order to maintain the required equilibrium concen- tration at that temperature. An osmotic pressure gradient will be produced between the still and mixing chamber by the removal of 3He atoms at the still, thereby causing 3He atoms to migrate from the mixing chamber into the still. 3 The He atoms subsequently removed from the dilute solution region Of the mixing chamber can be replenished from the 3He region above the phase boundary to keep concentrated the dilute solution at its proper low temperature concentra- tion Of 6.3% 3He, and in the process will cool the mixture. This flow Of cold 3He from the mixing chamber to the still is used to cool the incoming warm liquid 3He through several stages of heat exchanging. The effectiveness of I...’ .t. .C 4M. r OAaH .4 uh I .v \ Q " tn ‘ “we. cgr .J‘e 20 ’the exchangers not only depends on their design, but also on the ability Of the refrigerator to prevent 4He from being circulated with the 3He. If 4He is present in the incoming warm liquid, the heat exchangers besides having to cool the liquid will also have to absorb the heat liberated as the-4He phase separates from the 3He. There- fore the 3He to 4He ratio in the circulated gas should be as large as possible. i.Ui d ~r. .' Ab and“ i .1“. .V'... i ‘v- N: 'n.‘ :1 FIG 21 C. Design and Operation of this Dilution Refrigerator An Overview The dilution refrigerator built by this research group was designed initially for studies of the magnetic susceptibility exhibited by dilute magnetic systems. The apparatus contains two conventional mutual-inductance magnetic-susceptibility coils, one being used to monitor the refrigerator's temperature and the other to study a sample. A Superconducting Quantum Interference Device (SQUID) magne- tometer was added for magnetic measurements on very small samples. A schematic view of the cryostat depicting the overall relationship of its major components is presented in Figure 4. The refrigerator is contained within two vacuum cans which separate it from the surrounding bath Of liquid 4He. The outer vacuum can also serves to support a magnetic shield arrangement necessary to isolate the susceptibility coils and SQUID magnetometer from external magnetic fields. This shield consists Of an inner layer of sheet lead covered by an outer layer of u-metal. The u-metal reduces the effect of the earth's magnetic field at the SQUID position to approximately .07 gauss, while the lead shield in its superconducting state prevents any changing external mag— netic field from being transmitted to the magnetometers. 22 Figure 4. An overall view of the cryostat. The dilution refrigerator components are enclosed within a vacuum space and interconnected by capillary tubing which carries the flowing 3He. 23 H93 Pumping Line 1 Liquid He4 I 1%”— LL I' —-'-‘Copper Screen rf Shield SQUID Sensor I Outer VacuunI Can \\Jylufion Refrigerator Cam onents / p f,4lnner Vacuum Can ‘ Ti" 4531‘ ....s '1 ans e 24 The refrigerator is also shielded from rf interference by a 16 mesh OOpper screen attached to the helium dewar. This is very important because rf induced eddy currents may cause internal heating within the metallic parts of the refrigerator, eSpecially those constructed of copper. Further and possibly more severe problems can develop due to rf heating Of the carbon resistance thermometers and interference with the magnetic susceptibility bridge electronics. During the Operation the entire apparatus floats on four columns of compressed nitrogen gas. This support system allows the resonance frequency of the apparatus as a whole to be lowered to about 1 hertz so it effectively 23 This becomes vibrationally isolated from the building. vibration isolation is necessary at the lowest temperatures where relative motion between refrigerator parts may be sufficient to cause internal heating. The Condensor, Heat Exchanger, and Still. The incoming 3He passes through a condensing impedance of 2 x lOlz/cm3 which produces a high enough 3He pressure (~ 40 mm Hg) to insure liquification of the at the 1° pot. This flow restriction requires a 3He circulation rate Of at least 3.0 x 10.5 moles/sec through the refrigerator. The 1° pot itself is patterned after the flash evaporation design of L. E. DeLong 23121.24. 25 The transfer Of heat between the incoming warm 3He and the outgoing dilute solution is accomplished through five stages Of heat exchanging. Initially this liquid is allowed to come into close thermal contact with the dilute solution in the still, which is Operating at a nominal temperature Of .65 K, by flowing through 100 cm Of .025 cm inside diameter (.0075 cm wall thickness) cupro-nickel capillary tubing wound inside the still. The precooled liquid is then allowed to pass into the capillary exchanger. This heat exchanger is composed of two series connected sections each containing two concentric thinwall cupro-nickel capillary tubes. In one section, which has an overall length of 150 cm, the inner tube has an inner diameter of .025 cm (wall thickness = .0075 cm) and the outer tube an inner diameter Of .119 cm. The other section is 100 cm long with the inner tube diameter being increased to .036 cm (wall thickness = .0075 cm). The incoming warm 3He liquid is made to flow through the inner capillary, and thus is completely surrounded by the cold dilute solution which flows through the annular region between the capillaries. The last three stages of heat exchange occur in identical exchangers made from sintered copper disks and interconnected by capillary tubing. The sizes Of these capillaries are similar to those reported for other dilution refrigerators.25 The basic design of these exchangers is shown in Figure 5. Both the incoming 3He and outgoing dilute solution pass through a volume filled with sintered copper powder.26 The 26 Carbon 2‘ Resistor Thermometer <——- i-—*- Scale « 0'7"- .. lf—J (Hf A“ Conc. He3 4’ l {emeym 1W 0 ' I . Sintered Copper 5; \ Dilute . ‘-—r Solution Region Soft ,//H .1? h o e KO _—‘fi_ Figure 5. A detailed drawing Of a sintered copper heat exchanger showing the flow of liquid helium. :eat exc *1 “In 27 heat exchanger volume must be relatively large (~ 1 cm3) in order to provide efficient exchange of heat between the 3He quasiparticles and the 4He phonons in the dilute solution. The large surface area of this sintered copper facilitates the transfer Of heat between the 4He phonons and the OOpper at low temperatures where Kapitza resistance becomes AT K increaSingly Significant. This reSistance (RK = ——~ 523756;) is due to acoustic mismatch between the COpper and 4He phonons at the interface of the dilute solution and copper metal.27 TO further enhance the heat flow between the sintered copper disks the body of each exchanger is constructed from high purity OFHC copper. The mechanism of heat transportation within the copper is via the conduction electrons. The disks were shrunk-fit into the bodies of the exchangers. At a mixing chamber temperature of 10 mK these exchangers Operate at about 70 mK, 45 mK, and 20 mK respec- tively. These components are mounted within the refrigerator on a framework mainly constructed from pressed graphite. This material has enough strength to easily support the refrigerator components. It also provides a means of ther- mally isolating them from each other because Of its very low thermal conductivity. The problem Of film flow into the still orifice and the Subsequent circulation of 4He seems to have been eliminated in our refrigerator. This process is countered by wrapping a heater around the pumping line orifice (see Figure 2). 28 This orifice heater drives the ever-present superfluid 4He film normal and effectively blocks its flow into the pumping line. By appropriate adjustment of the orifice 3He to 4He ratios Of 50:1 can easily be maintained in heat, the circulating gases. This ratio can be measured during the Operation of the refrigerator by a mass-spectrometer 28 allowing Optimum 3He-4He ratios to be leak detector established throughout the course Of an experiment. This removal of the 4He from the circulated gas allows Our refrigerator to operate with less heat exchangers than would normally be required. The Mixing Chamber The standard mixing chamber design has been modified in our apparatus to form a dual-tail arrangement which allows a cerium magnesium nitrate (CMN) thermometer (see page 61 for a discussion of thermometry) and a sample to be in close thermal contact with each other. The basic mixing chamber and susceptibility coil arrangement is shown in Figure 6. This sketch does not show the details of the mixing chamber, but is primarily for presenting the relative arrangement of components within the chamber. Also, the flow Of 3He within the system is indicated by arrows. During Operation there are actually two phase separation boundaries located approximately 1.0 cm above the samples in each tail. A Superconducting Quantum Interference Device (SQUID) Inagnetometer system has been installed above tail #2. The Figure 6. 29 A schematic drawing of the dual tail mixing chamber. The arrows represent the flow of 3He through the mixing chamber. For clarity only the upper half Of one Of the magnetic susceptibility coils is shown. SCiLE N PH 3% SQUID PORT ,’ 1 I I \\ // T I I I I I I coIL I I ran I I ReszANCE l Ifi’P” 0- I I THERMOMETERS I I I I I I I I I ixing I | | I l I | I I" °f . . ' ' I I . Ha- Ha I I aritY I I PHASEwNI I _I I“ I | tic BOUNDARY IF I I I II I I I r- | lI # II II I I. I I | I'— T I: I‘I l . q: I ”I I ////, I I ’//// I I SAMPLE I /CMN I . ////. I I ////. I TAIL TAIL L ‘I ‘2 L55 SECONDARY calL“2 LEADPLATED BRASS SHIELo‘““1————v PRIMARY con.*2“’”‘L——r‘ 31 dilute solution provides the means of maintaining thermal contact between the SQUID sample and the CMN thermometer. A complete discussion of the SQUID magnetometer and the question of thermal equilibrium between these parts will be presented later. The mixing chamber parts were molded and machined from Epibond lOO-A epoxy.29 COpper coil-foil was molded into the interior walls of the upper half of each tail to present a low resistance thermal path at higher temperatures. The coil foil consists Of strands of copper wire bonded parallel to each other so as to form flat sheets. This design allows heat to flow along the wires and avoids the possible formation of eddy currents within the copper due to time-varying external magnetic fields. The upper ends of this coil foil are intimately attached to a copper block containing a germanium resistance thermometer. In order to take experimental data, the mixing chamber temperature must be stabilized and thermal equilibrium must be established between the tails. Stabilization of the temperature is accomplished by heating the 3He as it flows through a capillary tube into the mixing chamber. The tem- perature will remain constant when the applied heat is adjusted so that it is exactly balanced by the cooling rate at the phase boundaries. The results Of an estimation of the thermal relaxation times within this system (see Appendix A) as a function Of temperature is shown in Figure 7. As the mixing chamber Figure 7. 32 Results of a theoretical calculation of the expected thermal relaxation time of a dual-tail mixing chamber. Closed circles represent heat conduction through the coil foil and mixing chamber walls. Open circles represent heat conduction via the liquid helium. The total relaxation time is repre- sented by a dashed line. Relaxation Time (sec) Thermal 33 I04. I ' T ' l ' I 1 - a .I l _ ........... _,. _ __ __ .jes.flqsiflng.ti_m§_: I- . a r q - 0.9.? ° ° ° ° ° ’ ° ° .9? )L . '\. \. .. \. .I \. \ \ I \\ o 3.. .— lO - f" \.\ .. I \ I I \ ”I \\c '- r' \\ -I I \\ "I \3 r I I Tmax.~ l hour IOZJ' .— 1 _ I. q I- -I I! IO 1 1 l 1 l I l 1 O 50 I00 I50 200 250 'I" (mK) 34 is being cooled from one temperature to another, a thermal gradient may be produced between the two tails. Any such temperature difference must be allowed to become small before a susceptibility measurement can be made. Assuming the temperature difference will relax exponentially, the relaxation time is defined as the time necessary for the initial temperature difference to drOp to l/e of its -t/T where T is the thermal original value (i.e. AT = ATo e relaxation time). This estimation was based on a calcula- tion of the relaxation time for heat flow from one tail to another assuming parallel paths of heat flow through the coil-foil and liquid. This calculation does not take into account the flow of 3He through the mixing chamber. The results show that the longest relaxation times occur at approximately 30 mK with values on the order Of one hour. At high temperatures the copper coil-foil path provides the least resistance to heat flow between the tails, while at low temperatures, where Kapitza resistance between the copper wires and the 3He becomes appreciable, the dilute solution offers the path of least thermal resistance. An alternative means Of estimating the relaxation time between the tails can be made on the baSlS the 3He flow rate. The time necessary for the 3He liquid to divide,flow through each tail, and recombine in the siphon tubes is also a measure of the refrigerator's relaxation time. Calcu- lations show that at temperatures above 10 mK and a flow rate 35 of 5 x 10.5 moles/sec, this process requires about 6 x 103 sec. Of course in reality the relaxation time is 3He flow through the a result of a combination of both mixing chamber and thermal conduction. Observations of the refrigerator in Operation indicate that the relaxation times are on the order of two hours. In actual practice we find the major thermal relaxation problems are not caused by the relaxation times within the refrigerator, but are usually due to the relaxation times associated with Kapitza resistance at the surface of powders and crystals being studied. In order to circulate the cold dilute solution around the samples and to maintain the positions Of the phase separation levels near the samples, siphon-tube return lines were placed immediately above the sample sites (Figure 6). These tubes were machined from Epibond lOO-A epoxy and con- structed so as to minimize the total flow impedance presented to the mixture. The tubes have an inner diameter of .272 cm and an outer diameter of .406 cm. They are positioned con- centrically inside the tails, which have a .635 cm inside diameter. The flow impedance in each annular region was measured by allowing nitrogen gas to flow through them at room temperature and utilizing the following expression for impedance: z = 9% (fig-I (cgs units) 36 where n is the viscosity Of the gas, and Av/At is the volume flow rate for a pressure difference Of Ap , which is assumed small compared to the total absolute pressure. It was found 3 and 3.56 x 104/cm3. that the impedances were 2.92 x 104/cm Since the viscosity Of both pure 3He and dilute solutions of 3He increases as l/T2 at low temperatures, these impedances become important in considering the question of viscous heating of the flowing solutions in the tails at low tempera- tures. An estimate of the thermal gradient which may be produced can be made by utilizing the following expression which relates the thermal gradient in the mixture (or pure 3He) to the impedance presented to the flow Of liquid: 2 n3/Ao 20 mK 4 AT “§“§) I -5 I I T ’ ‘f 10 cm 10 <<]_ moles/sec In this equation T is the temperature in mK and K is a con- stant which is .07 for pure 3He flow amd .54 for the flow of dilute solution. With this expression we calculate that the temperature difference between the two tails with the phase separation levels at approximately 1.0 cm above each sample is at most only .1% at 10 mK as a result Of impedance effects. The dividing Of the mixing chamber into two tails and the resulting formation of two phase separation lines therefore would seem to present no serious thermal equilibrium problems between the tails. More will be said later about the effect 0f the impedance Of the siphon tubes on the temperature of the SQUID. The chamber: first tI leak ti would I] grevioz Ill. Struct; epoxy j holder still 37 The samples being studied are contained in sample chambers which were designed with two main criteria in mind; first to minimize the difficulty of changing samples in a leak tight system and second to make sure the mixing chamber would not be contaminated by the sample which had been previously studied. Both problems have been solved quite well. Figure 8 shows the details Of the sample chamber con- struction. The samples are packed into independent, removable epoxy holders. They may either be packed directly into these holders, or, if a smaller mass of sample is desired while still retaining it in the shape of a right circular cylinder, packed into smaller units designed to slide inside the usual holders. The samples are covered with and placed on filter paper to trap all loose particles. To further avoid problems, no unit is used twice, a new one being made for each different sample. The problems associated with installing and removing samples from the refrigerator and still maintaining the necessary leak tight joints have been solved by using a remarkable glue made from glycerine and Ivory soapflakes.3o Both tails Of the mixing chamber are threaded at the sample chamber to allow easy access when replacing samples. To assure a leak-tight fit, the female parts Of the threads were tapped, then the male parts were cut with a lathe until a snug fit was Obtained. The seal is made by warming the glue until it is a liquid, then coating the male thread with FHter Paper Scale in Cm. Figure 8. 38 W Epoxy Insert for Small Samples Detachable Powder Sample Holder Mixing Chamber Tail A detailed view of the sample chamber design. 39 the glue and screwing the entire unit, with the sample in place, into the refrigerator. After the glue sets it is possible to evacuate the refrigerator, but we usually wait until the apparatus is cooled to 0°C before evacuation. This type of seal has proven to be extremely reliable in this apparatus. It should be mentioned also that equally reliable seals, although of the permanent type, were made during the con- struction of the mixing chamber by using Epibond 121 liquid epoxy.29 This epoxy is the standard glue used for joining pieces of Epibond lOO-A epoxy. The convenience and ver- satility of these epoxy resins and the glycerine and Ivory soapflakes glue should not be underestimated in the building of low temperature refrigerators. Modular mixing chambers designed for various types of experiments are a realistic possibility through utilization of these materials. 40 D. Measurements of Magnetic Susceptibility The Conventional Magnetic Susceptibility Coils The conventional methods of measuring the magnetic susceptibility of a substance are based on techniques where the sample's magnetization changes the mutual inductance between two coils. One coil, the primary, is used to pro— duce a small ac magnetic field at the sample. The other coil, the secondary, is wound coaxial with the primary in two sections which are connected in series and are in Opposition. Thus to first order the total primary-secondary mutual inductance is zero. The primary-secondary mutual inductance is influenced by the temperature dependent magnetization of the sample which is located within one of the secondary sections. In order that the epoxy mixing chamber tails do not couple magnetically to the secondary coil they extend below the samples so that equal amounts of epoxy are interior to each secondary coil.31 A schematic diagram of a susceptibility coil and the associated mutual inductance bridge circuit necessary for these measurements is shown in Figure 9. A fraction of the reference mutual inductance signal is tapped by the ratio transformer and used to null the change in the primary- secondary mutual inductance produced by the magnetization U of the sample. Since the complex susceptibility (x = x + ix") has a resistive loss component as well as an 41 Figure 9. A block diagram of the ac mutual inductance bridge circuit. All resistances are in ohms. 42 isolation transformer f' ----- ‘1 : I attenuator RI! I I— ” --------- 1 I l s K I ' I I ' <—-’\N\NV\N\— ' f I l I I I ____' I I 5" * I _____ : I 200 I 4,. , I L+—-——--—--~-’ I on ‘ W‘— I 1" lock-in oscillato; I voltmeter at I? z I I secondary I primary I I I rat‘ I0 I transformer I 3.....- I I l I I I reference I <—-I-——- I I r ' coil L—---' foil 43 inductive component, one must ensure that the reference mutual inductance signal is only being used to balance the signal produced in the secondary due to the inductive coupling between the paramagnetic sample and secondary coil windings. For this reason the primary and secondary circuits are coupled via an isolation transformer by which the resistive component can be nulled. The phase adjustment R¢ is used to ensure that the isolation transformer signal is indeed the "90° phase" resistive component. The susceptibility coils, their placement relative to the refrigerator being shown in Figures 10 and 11, were patterned after those reported by A. C. Anderson, R. E. Peterson, and J. E. Robichaux.32 One major addition which was made, because of the proximity of the two coils to each other, was to surround both coils with superconducting lead-plated brass shields. This is necessary to avoid interaction between the coils due to the primary of one coil coupling into the secondary of the other. A more detailed view of the coils, Figure 12, shows their construction much more clearly. The coils were carefully made to be as nearly identical as possible. The primaries were wound on forms of Epibond lOO-A machined to 3.175 cm outside diameter. The overall length of each primary is 10 cm, being composed of 2480 turns of #32 copper wire wound in six layers. Each layer is separated by .25 mil. mylar with 1.0 mil. mylar being used under the Figure 10. 44 A detailed drawing of the dilution refrigerator and susceptibility coils. A = copper flange with lead fusewire "O"-ring; B = stainless steel support tube sealed in epoxy; C = SQUID mounting port; D = copper inner vacuum can connected to the still; E = copper to epoxy joint; F = coil foil. 46 Figure 11. A top view of the dilution refrigerator and susceptibility coils. The letters refer to parts which are similarly labeled in Figure 10. 47 .‘I‘ \ \ //2o2m «worm 3.2303 448 3‘ III 0.. I1 I l °‘ I | | I \ PRIIARY COIL ASTRTIC SECONDARY /... REFERENCE X COIL LIADPLA‘I'ED IRA. 8 OIIIILO Figure 12. A close-up view of the mutual inductance magnetic susceptibility coils detailing their construction. 49 first and over the last layer. Mylar is used to prevent electrical shorts from forming between layers, which would occur if the copper insulation weakens under thermal stress and to allow for some flexibility as the coils contract during cooling. In order to ensure interwinding stability and to further protect against shorts, each layer is heavily covered with a "glue" composed of 50% GE 7031 varnish and 50% toluene. Wrapped around the outside of the primary is a Faraday shield constructed from a layer of copper coil foil situated so that its wires run parallel to the axis of the coil. The two backwound sections of the secondary coil each contain 3800 turns of #36 COpper wire. The primary-secondary mutual inductance with this design is approximately 82 mhenry for each section in the absence of the lead shield. Each section is 3.0 cm in length and positioned such that their center-to-center separation is 4.25 cm. The toluene and GE 7031 varnish glue was again liberally applied to ensure rigidity with respect to thermal stressing, but mylar was only used under the first and over the last layer of each section. The windings were supported by teflon split rings during their construction until the glue had set firmly. To mini- mize uncontrolled shifts in the balance point of the suscepti- bility bridges due to primary-secondary capacitive coupling, the outer layer of the primary and the inner layer of one secondary section are interconnected and grounded to the cryostat. 50 A reference mutual inductance was made by winding 100 turns of #36 copper wire on the secondary section which does not contain the sample. The reference mutual inductance is .55 mh with the superconducting shield around the coils. The entire coil system is mounted inside a vacuum space and is thermally attached to the still. The superconducting shields which surround each coil consist of a thinwall .079 cm brass tube with an inner diameter of 4.76 cm. The inner surface of the tube was electroplated with a .003 cm layer of pure lead. The length of the tube, 14.29 cm, allows it to extend beyond the ends of the coil and better shield against nearby magnetic objects and the magnetic field produced by the neighboring coil. In order to test the effectiveness of this supershield arrangement, a .794 cm steel ball hearing was lowered to the shield's edge while the coil was at liquid helium temperatures. The result was an introduction of only a .04 uhenry imbalance of the coil. Thereafter we were confident that any effect on the balance condition of the coil due to a nearby sample would be negligible. The presence of the superconducting shield tends to reduce the magnetic field produced by the primary coil at the sample. The effective field inside the primary now becomes approximately: ~ Beff ~ Bp(l - Ap/As) 51 where Bp is the field produced without the superconductor being present and Ap/AS is the ratio of the cross-sectional area of the primary coil and shield. For our coils this means that Beff = .513 Bp. Detailed computer calculations of the effect of superconductors on magnetic fields has 33 His results recently been carried out by Todd I. Smith. show that a further effect of the superconducting shield is to reduce the variation of the magnetic field over the volume of the sample. These coils have behaved well despite the rigors of many cooling and warming cycles. Their balance point at 4.2 K changes very little from experiment to experiment, an indica- tion of good interwinding stability. Sensitivity is also quite good. Using the coils in conjunction with a mutual inductance bridge operating at 17 Hz and an Ithaco Dynatrac 391 Lockin Voltmeter, we are able to achieve a sensitivity of -4 10 uhenry, which corresponds to a susceptibility of about 10.8 emu. Calibration of the coils was first done by averaging the data obtained on powered CMN samples during the investigations of thermal equilibrium within the mixing chamber. These calibrations determine constants for the coils which relate unitless dial changes on the susceptibility bridge ratio transformer to mutual inductance changes within the coil due to the presence of samples. These measurements gave coil constants for coil #1 and #2 respectively as .0919 i .0013 emu/dial unit and .0883 i .0010 emu/dial unit. 52 It was later found necessary to determine the coil constant more accurately for the coil which was being used to take measurements on various samples. This was accomplished by utilizing a .87724 gm CMN single crystal, shaped into a cylinder, mounted so that the known susceptibility (perpen- dicular to its c axis) was being measured. The calibration obtained for coil #1 with this crystal was .0833 t .0002 emu/dial unit. The SQUID Magnetometer The circumstances which eventually led to the deve10pment of the Superconducting Quantum Interference Device (SQUID) were initiated by B. D. Josephson34 in his paper on the theory of tunneling currents between two superconducting materials separated by a thin insulating region. Josephson's theory predicted that even with zero voltage across the junction, a dc supercurrent could exist across the junction. This current could have any value from zero to a certain maximum value. If a nonzero voltage were applied to the junction, the dc supercurrent would still be present, but an ac supercurrent with an amplitude equal to the maximum possible dc supercurrent and a frequency 2 eV/M was superimposed on it. For practical purposes, a more convenient weak link connection, having properties similar to the thin insulation region, can be constructed by forming a point-contact between two super- Conductors. 53 A SQUID magnetometer is a device which utilizes the properties of a point-contact junction in a superconducting loop to measure very small changes in applied flux. A I block diagram (Figure 13) shows schematically how this is achieved. A change in the magnetization of a sample will induce a current in the superconducting sample coil (comparable to the secondary windings of the conventional coil mentioned previously). Since the sample coil and signal coil comprise a superconducting circuit, this current will couple, via the signal coil, a flux into the SQUID sensor which contains the superconducting loop and point contact. Also coupled to the SQUID sensor is a rf tank circuit. The voltage response of this tank circuit is periodic with changes in the total flux coupled into the SQUID sensor with the period being o0, the unit of flux quanta (¢o = 2 x 10.7 gauss-cmz). The SQUID is normally operated in a "locked-on" mode. This is characterized by the use of a feedback circuit to Stabilize the SQUID's operation at a particular dc flux level. The audio oscillator (1000 Hz) feeds a current into the rf ¢ COil and is adjusted so the external flux varies by i"% abolltthe static dc flux. If the dc flux now changes due to a safluile's magnetization being coupled to the SQUID sensor Via the flux transformer, the feedback servo of the lock-in amplifier will put out a current proportional to this change. St -. . ab11lzation of the dc flux level is accomplished by feeding 54 Figure 13. A block diagram of the SQUID magnetometer circuit. The SQUID sensor and sample coil are located inside the cryostat. 55 rf oscfllator l9MHz S<2UIE> 3 Sensor I'""""""""'I ~|F ~ . I . ' : SIgnal, ' I‘III’I Co” I ' I I I L..-_.-_._..- -_..I- .. _..I Flux Transformer @ Sample Con II o I Audio oscillator I000 Hz Pre — amp Detector 56 the current into the rf coil in order to counter the change in flux. The feedback current is monitored by a digital voltmeter. In this manner changes in the magnetization of a sample, and therefore its magnetic susceptibility, can be measured once the SQUID has been calibrated with a sample of known magnetization. A more rigorous treatment of the theory of point-contact SQUID magnetometers is presented in (Appendix B. Most of the components of this SQUID system were manufactured by SHE Corporation.35 The actual construction of the SQUID flux transformer and sample chamber was not done by the author; a more com- plete discussion of the SQUID system will be presented else- where.36 However, it is essential to know the details of the SQUID sample chamber construction and its position relative to the CMN thermometer in the dilution refrigerator. This chamber was also designed for easy access to the samples With nunimal disruption of the leak-tight seals necessary in the refrigerator. As such, it was built utilizing the glycerine-soapflakes glue described in a previous section. The SQUID magnetometer coil is wound on a form machined from Epibond 100-A epoxy, having a .254 cm inside diameter sample Chalnber region. The coil consists of two sections, each constructed of 21 turns of .0097 cm diameter niobium wire and ‘“"“N3 in series Opposition. The resulting sensitivity of the 8 SQUII) sensor is approximately 10-2 flux quanta (~ 10- emu) £01” a 2.5 gauss field. The sample coil is coupled to a 117 57 turn signal coil located in the SQUID sensor by a pair of tightly-twisted niobium leads which are shielded by encasing them in high purity superconducting indium foil. Figure 14 shows the sample chamber and its means of attachment to the refrigerator. The position of the SQUID chamber port in the refrigerator was shown in Figure 10. The samples are inserted into the SQUID coil on epoxy fingers with replacable tips designed specifically for a given sample or different orientations of the same sample. The coil and sample are entirely surrounded by a niobium superconducting shield used to trap the desired dc magnetic field at the sample site. This is accomplished when the cryostat is below 4.2 K by applying the appropriate field to the niobium cylinder with a magnetic-field solenoid wound on the inner vacuum can. The field is applied parallel to the cylinder's axis. The current producing the field is monitored by a digital voltmeter during the field application. When the field has reached the desired value, the niobium cylinder is heated to drive it into the normal state. The onset of the nornual state can be seen by observing the resistance of a niObium sensor wire placed in close thermal contact with the niobiumcylinder. After the niobium has become a normal cominctor, the heat is removed allowing the niobium to again become superconducting, and in the process, trapping the applied field within the niobium cylinder. The external field 18 then turned off. 58 3% (I SCALE IN ASTATIC NIOBIUM COILS SAMPLE «I NIOBIUM CM. SHIELD /\/\/\/\/ \/\/\/\/‘ g, j\ SQUID PORT F igure 14. The SQUID sample chamber and its means of attachment to the mixing Chamber's dilute solution return line. 59 There are several criteria which must be met when determining the magnitude of the magnetic field to be trapped in the cylinder. One must be certain that the field is not so large as to produce non-linear field effects in the susceptibility measurements. An estima— tion of these effects can be made by expanding the magne- ti zation of a simple spin 35 system as given by the Brillouin function to the first non-linear term in the field (order H3) . The applied field must be small enough so that at the lowest temperatures attainable this term will be insignificant. Such an expansion will result in a susceptibility of the following form: x=§%Ti%i (l-%(9T:%)2) From this expansion it can be seen that non-linear effects may occur due to the sample's having too large a suscepti- bility, or from the application of too large a field, or both. Therefore, care must be taken to be sure that the low temperature susceptibility being measured in the SQUID is indeed the zero-field susceptibility by an approPriate a 8 '- O L' °o : I' o " I- .- - d I- o d D . d I- . 'I _ 8 _ Q! . 3 I- o - I- o .. -I I 0 oo-I . 'I I. q l l L l 1 l L 0. o. o N (mus shour) KIIIIqIIdeosns punomxoog IO .IO T," (KI .OI .OOI 81 was not machined away from the Epibond lOO-A epoxy after it was molded. As with the thermal calibration graphs, the abscissa is plotted logarithmically to allow the entire temperature range to be easily scanned on a single graph. The background measurements have also been converted to correction equations which apply in specific temperature ranges. These equations, useful for computer analysis of raw data, are listed in Tables 2 and 3. The corrections for tail #1 are added to susceptibility measurements (x1 corr. = X1 + Axl) while those for tall #2 are sub- tracted from the susceptibility measurements (X2 corr. = X2 — sz). These corrections were applied to our previously presented temperature calibrations. One further calibration experiment was made to determine the background susceptibility of the SQUID sample chamber. The results of this experiment for a 2.5 gauss trapped field are shown in Figure 21, and as with the mixing chamber back- ground, indicates the presence of small number of paramagnetic impurities in the system. Table 4 lists the appropriate background equations and their range of validity. The corrected magnetic susceptibility is found from xs corr. = ;x8 + Axs * H/2.5 where the background is assumed to be ILinear in the magnetic field, H. The background suscepti- bimlity was also measured at 25 gauss to ensure that this assumption was reasonable. The maximum correction term, aSsuming an arbitrary zero correction at l/T; = 200, is 3.7 82 Figure 21. The background susceptibility of the SQUID sample chamber at an applied field of 2.5 gauss. 83 O T I I r r I I I I ‘— 1 L- .. r. -| r- 0 1 t 3 .. 0 ID a: O - o a - i— O o d .- o .1 u— o m -J r- 0 . - O N r— O _. 8 + I- " -1 I L o - X _ _<_> ” " ' c r- O - N h an F i— 0 - - -1 O O I q, : . - 4 .. O .. L— .1 O - <45 v- d L l 1 l 1 1 1 _L 4 5 to <1- r0. N —. 00 ° I (nma 9_o| x) Mmqudaosns punomaooa 84 Table 2: Coil #1 Background Correction Equations Background Correction (emu) Temperature Range (K- Axl=+l.08x10-6(1/T§)+l.07x10-5 0 < 1/15 < 1.00 Axl=+2.08x10'6(1/T5)+1.00x10‘5 1.00 < 1/T3 < 1.23 Ax1=+l.21x10-5(l/T§)-2.42x10-6 1.23 < 1/T5 < 1.55 Axl=-1.75x10’7(1/15)+1.70x10'5 1.55 < 1/15 < 6.40 Ax1=-1.98x10'7(1/T§)+1.78x10'5 6.40 < 1/15 < 50.0 Ax1=-9.17x10'8(1/T§)+1.23x10‘5 50.0 < 1/T5 < 108. Axl=-2.67x10'8(1/'1'§)+4.83x10’S 108. < 1/T3 < 195. Axl=0.0 195. < 1/T5 Table 3. Coil #2 Background Correction Equations Background Correction (emu) Temperature Range (K- Ax2=+1.38x10’6(1/T§)+1.27x10'5 0.0 g_1/T§ : 1.62 Ax2=-6.62x10’5(1/T§)+2.56x10’5 1.62 < 1/T5 < 1.70 Ax2=-9.53x10’3(1/T3)+1.46x10’5 1.70 5 l/T§ :_7.00 Ax2=-1.43x10'7(1/T§)+1.49x10'5 7.00 < 1/T3 : 70.0 Ax2=-4.33x10'8(1/T3)+3.27x10'5 70.0 < 1/15 1 200. Ax2=0.0 200. < * l/T2 85 Table 4: SQUID Background Equations (H=.25 gauss) Background Correction (emu) Temperature Range (K-l) Axs=-3.73x10"6(1/'.r§)-3.71x10'6 0.0 3 1/15 3 .285 Axs=+1.07xlo"7(1/205)-3.44xlo"6 .285 < 1/15 3 .780 Axs=-3.62x10-7(1/T§)-3.08x10-6 .780 < 1/T3 : 1.02 Axs=+1.19x10-6(l/T§)-4.66x10-6 1.02 < 1/T3 : 1.19 Axs=+5.07x10-8(1/T§)-3.31x10-6 1.19 < 1/T5 : 2.48 Axs=+3.42x10—8(1/T§)-3.29xlO-6 2.48 < 1/15 5 38.0 Axs=+2.61x10-8(1/T5)-2.98xlO-6 38.0 < 1/T3 g 62.5 AXS=+1.59x10’8(1/T5)—2.25x10'6 62.5 < 1/T5 3 115. Axs=+3.48x10’9(1/T§)-8.28x10’7 115. < 1/T5 Table 5: # Inverse SQUID Temperature as a function of l/TE 1) Inverse SQUID Temperature (K- g l/T; = * l/Ts l/T; 1/T5 +.94444(1/T§) + 6.11116 +.73476(1/T§) + 45.41778 Temperature Range (K- 0.0 < 1/‘1‘1'2b i 110. 189 * < 1/T2 110. < 1/T§_<_ 189 l ) 86 6 emu at 2.5 flux quanta, which correSponds to 3.4 x 10- gauss. The high temperature data obtained during the thermal gradient calibration experiment is useful in calibrating the SQUID coils. A calibration constant of 9.1289 x 10"7 emu/flux quanta was found for these coils for a field of .25 gauss. CHAPTER II THE MAGNETIC SUSCEPTIBILITY OF COPPER TETRAPHENYLPORPHINE A. A General Survey of the Porphyrins The structural base on which all porphyrins are built is the configuration called porphine. This basic molecular foundation consists of four nitrogen pyrrole rings inter- connected by four methane-bridge carbon atoms, referred to as the a, B, Y , and 6 meso-positions. The geometry of the molecule is planar, with hydrogen atoms attached to the four meso-positions and to the eight pyrrole sites as shown in Figure 22. The porphyrins are formed by substitution of various ligands at the hydrogen atom sites. Although no naturally occurring porphyrins are found with substitutions at the a - 6 meso-positions, they invariably have substitu- tions at all eight pyrrole hydrogen atom sites. Only three of the many possible free porphyrins, that is, those which are not metal complexes, are found to occur in nature, these being present in leguminous plants and in urine under certain pathological conditions. 44, the metallOporphyrins Metal complexed porphyrins (Fig. 23), are found to be more prevalent in nature and in fact are very important biologically especially as molecules involved in energy transfer processes. There are two means by which free porphyrins can take up a metal ion. One Imethod involves the dissociation of the two central hydrogen 87 Figure 22. 88 V‘ 4! \ I H ,’ ’ I I I B I I H I I I, H\ ’1’ 9 5 \ H 7 7 6 H H H Structural representation of a porphine molecule. This structure forms the basis for all porphyrins. Figure 23. 89 A comparison between the structure of a metallo- porphyrin and metallo-tetraphenylporphyrin molecule. The site labeled M represents a metal ion; Open circles are nitrogen atoms and closed circles are carbon atoms. KPyrrole Ring METALLOPORPHYRI N \ ' /e\ 2/0 ‘ METALLO- TETRAPHENYLPORPHYRIN X‘ X 91 nuclei from the porphyrin in solution followed by a metal ion - free porphyrin reaction. In the other method a complex is formed by the coordination of a metal ion with the two nitrogen atoms. This structure then stabilizes itself by the total absorption of the metal ion and subse- quent exclusion of the hydroqen nuclei. One of the most important classes of porphyrins which can be produced is the iron complexes, which biologically are the haems. The usefulness of the haem structure becomes apparent when the proper protein side-chains are added to form haemoglobin, myoglobin, the cytochromes, etc. Another biologically significant molecule results from replacing the hydrogen nuclei with a magnesium atom. This is not a true porphyrin, but a hydrOporphyrin, since one of the pyrrole rings bonds is altered slightly. This molecule is either chlorOphyll-a or chlorOphyll-b, depending on which substitution is made at the third coordination site. As is well known, chlorophyll is the pigment in leaves which converts sunlight into a useful form of energy for plants. Another group of porphyrins can be formed which do not occur in nature and are therefore synthetic porphyrins. This is done by introducing ligands at the a - 6 meso-positions (Fig. 23). The first successful synthesis of a, B , y, 6 - Tetraphenylporphine (TPP), the relevant ligands being phenyl 45 rings, was reported by P. Rothemund and A. R. Menotti in connection with their studies of chlorophyll and 92 photosynthesis. Further work on the porphyrins led to a later paper46 on the preparation of metal complexed salts. Three methods of preparing tetraphenylporphyrin metal complexes were presented, involving refluxing of the free base with the apprOpriate alkaline medium. The spin density of a porphyrin crystal can be varied by attaching different ligands to the central porphine base, so that the volume occupied by an individual molecule is changed. This method of separating the paramagnetic ions is superior to the usual method of diluting paramagnets by replacing paramagnetic sites at random with an isostructural molecule containing a diamagnetic ion. By increasing the ligand size one can change the spin density in a uniform, controlled manner while retaining a regular crystalline structure. Such a process could be very useful in the study of magnetic interactions. For example a paramagnetic ion could be studied in a variety of crystalline structures. This would be possible, because the introduction of different ligands at the a - 6 meso-positions will generally not seriously affect the intra-molecular environment of the ion. Alternatively, different electronic spin states of the same paramagnetic ion could be studied in very similar crystal structures. This possibility arises because the net electronic moment of some paramagnetic ions in a porphyrin structure can be changed by the introduction of various ligands at the 5th and 6th coordination sites without 93 increasing the molecule's size to any appreciable degree. Even in the more concentrated metalIOporphyrins (i.e. those with no attached ligands), the spin densities are low enough that the intermolecular interactions will be mainly due to weak, long range dipolar forces, with exchange coupling usually being small in comparison to the dipolar coupling. This means that most porphyrins will likely have very low ordering temperatures and must be studied at ultralow temperatures if their magnetic behavior is to be determined completely. Certain of these compounds may aid in future investiga- tions into the nature of the anomalous thermal contact observed between CMN and liquid 3He at ultralow temperaturesa. This phenomena is believed to be due to magnetic coupling between the liquid 3He and CMN at the liquid-solid inter- face. Other surface studies would also seem feasible because many porphyrins can be sublimated at about 300°C without destroying their molecular structure. The con- trolled sublimation of the surface layer of a crystalline porphyrin would result in a very clean surface for the investigations of interface phenomena. It should also be possible to construct thin films of porphyrin molecules for monolayer experiments by sublimation techniques. 56‘; C011 its IE‘. ex} wi‘ la 1131 C1‘: 94 B. Copper 0, B, y, 6 - Tetraphenylporphine Introductorpremarks COpper Tetraphenylporphine (CuTPP) is one of several metallo-tetraphenylporphyrin and meta110porphyrin compounds whose magnetic susceptibility has been measured with our apparatus. Magnetic susceptibility measurements along crystalline axes are invariably necessary if one is to obtain useful results. Since CuTPP was the first porphyrin which was successfully grown as a single crystal, its analysis was completed first and the results are reported here. The highly anisotrOpic hyperfine coupling exhibited by the COpper ion makes CuTPP a useful porphyrin with which to initiate a series of investigations into the ultralow temperature magnetic properties of the porphyrins. This anisotrOpic coupling dominates the magnetic suscepti- bility along the a and b crystalline axes at low tempera- tures, while the intermolecular dipolar coupling dominates the susceptibility along the c axis. Thus these effects on the susceptibility can be observed approximately indepen- dently of each other. Since the influence of the hyperfine coupling on the perpendicular susceptibility can be calcu- lated, the low temperature behavior of CuTPP will yield useful information such as the thermal equilibrium of single crystals. General instructions for the purification of CuTPP Samples prior to the growth of single crystals will be given, 95 followed by the procedures used to grow the single crystals. Similar procedures apply to the preparation of any of the porphyrin samples for crystallization, the main difference being the choice of solvents for a particular compound. Although the preparation of CuTPP is not a difficult process in itself, the subsequent isolation of the pure complex from TPP may present problems that make the synthesis of these compounds undesirable to many researchers. We are grateful to Dr. Richard Yalman and Mr. Gordon Comstock of Antioch College for supplying us with our first porphyrin samples. Many of the porphyrins are now available from Strem Chemicals Inc.47. However, the method by which the copper complex of TPP is prepared is presented here. The copper complex of TPP is easily prepared from TPP by first forming a solution of 500 mg of TPP in 50 m1 of chloroform, then adding to this a hot solution of 200 mg of COpper acetate in 50 m1 of glacial acetic acid. The resulting mixture is heated in a Soxhlet refluxing apparatus for two hours to insure complete metal ion replacement. The solution is then concentrated to 50 ml and cooled to allow microcrystals to precipitate. Purification and Crystal Growth Before any attempts at growing crystals is made, it is advisable to purify the tetraphenylporphyrin complex in order to remove any impurities which may be present in 96 the form of tars. The presence of these tars is undesirable, not only because of their possible effects on magnetic suscep- tibility measurements, but also because they may present problems during crystal growth by providing too many nuclea- tion sites. One of the major problems encountered while trying to grow large single crystals of CuTPP is that many microcrystals invariably grow instead of a few good single crystals. The removal of these impurities was found to be important, although their removal did not guarantee good crystal growth. The manner in which the CuTPP powder was purified in preparation for recrystallization is as follows. The sample was first put into solution with benzene. As with most porphyrins, the solubility of CuTPP is rather low in many solvents. This means that simply mixing a solvent with CuTPP may not be sufficient to dissolve enough material to be of use. In order to increase the concentration of the solution, it is refluxed for several hours in a Soxhlet extraction apparatus. A rough rule of thumb for the porphyrin complexes is to use no more than 1 gram of the salt per liter of solvent. Generally, it is convenient to work with quantities on the order of 250 ml. After getting as much of the salt into solution as possible, the solution is filtered while still warm to remove the impurities. Filtering is done through activated alumina contained in a sintered glass disk filtration funnel. The alumina is 80-200 mesh, chrom- atographic grade and should be heated for several hours at 97 200°C to drive off any water which may be present. Filter paper should be used to prevent the alumina from lodging in the glass sintered disk. After purification the solution must be further concen- trated in preparation for crystal growing. There are two basic methods by which the crystals may be grown. In the first method, the solution is returned to the Soxhlet flask, but now the refluxing system is replaced by a condensing tube arrangement to trap the evolved benzene vapors. Assuming 250 ml is the original volume of the solution, it should now be concentrated to about 60-80 ml. One way to determine if the solution is concentrated enough is to look for the presence of small crystals just above the surface of the liquid on the walls of the flask. When this crystalliza- tion occurs, the solution should be removed, a small amount of benzene added, and allowed to cool. The solution can then be transferred to small beakers for recrystallization. These beakers should be covered with a layer of parafilm with holes punched into it to adjust the evaporation rate. A close watch should be kept on beakers containing benzene as the solvent, since benzene vapors tend to decompose the parafilm cover within a few days. The beakers usually need to be cleaned of microcrystals every few days, and the best seed crystals transferred to a new beaker into which the solution has been filtered. It is often necessary to use a microsc0pe to identify the good seed crystals. 98 The second method involves the partial replacement of the first solvent by a second solvent, in which the por- phyrin has a much lower solubility. The effect is to put more material into solution in the second solvent than would normally be possible. Generally this solvent also has a much slower evaporation rate than the original solvent, which allows the crystal growth to be more easily controlled. As before, the original solution is returned to the Soxhlet flask for the concentration process. However, as the benzene is removed, it is slowly replaced by xylene until roughly an 8:1 ratio of xylene to benzene is obtained. The amount of replacement does not seem to be critical. The resulting mixture is then cooled and placed in parafilm covered beakers for crystal growth. It is much easier to regulate the rate of evaporation with this method and in some cases the evaporation rate is so slow as to not require a cover, although one is usually present to prevent dust from entering the beakers. CuTPP crystals were found to grow relatively easily from a concentrated solution containing only benzene, with the major difficulty occurring while the technique of manipulating micro-seed crystals was being mastered. The crystals were large enough for our purposes after about two to three weeks. microcrystals had to be removed from the beakers every two days to prevent them from twinning with the good crystals. 99 The Structure of CuTPP Molecules of CuTPP do not appear to have the exact planar structure exhibited by the basic porphine skeleton. The non-planarity exhibited by the porphyrin ring system containing a copper atom at the central position can best be seen by considering the deviations of various atoms with respect to a plane passing through the four nitrogen sites. This plane is found to be parallel to the (001) crystal plane. The copper atom is located at -.05 A below the plane, while the pyrrole carbon atoms labeled 2 and 3 in Figure 24 are located at +.20 A and -.13 A respectively. Therefore the pyrrole rings are slightly twisted out of the porphyrin plane. Fleisher 22.21348 have concluded from their work that the presence of the copper metal atom in the porphyrin structure is not the main reason for its non-planarity. Instead, they believe it is mostly a result of the desirability of a closely packed crystal structure, which is interfered with by the presence of the phenyl ligands. The close proximity of these ligands to the porphyrin plane apparently causes it to distort considerably. Further information about the relative positions of the porphyrin and phenyl rings can be obtained by referring to Figure 24. The phenyl rings, attached at the a - 6 meso-positions, are tilted such that their plane is almost perpendicular to the (001) plane. The angle 100 Z IO 9 Figure 24. The structure of COpper tetraphenylporphine. The relative positions of the numbered atoms are indicated in the text and in Table 6. 101 between the CS-C6 carbon-carbon bond and its projection onto the (001) plane is 13°. Similarly, the projection of a line through the C8 and C10 carbon atoms makes an angle of 72° with this plane. Table 6 presents a more complete listing of the deviations of the atoms from a planar structure and some relevant bond lengths. Fleischer49 has 0 also stated that the C -C bond length of 1.51 A is evidence 5 6 for electronic isolation of the porphyrin ring and phenyl groups. However, this is apparently disputed by the ESR 50’51. These studies of CuTPP and its p-chloro derivative measurements tend to indicate that the phenyl rings are electronically coupled with the porphyrin ring resonance structure. 9 transition metal system (effective The Cu2+ ion is a 3d electronic spin S=l/2, nuclear spin I=3/2) and is contained in a square-planer tetracoorinate environment within the porphyrin ring. The nature of the bonds between the Cu2+ ion and the pyrrole ring ligands can be quantitatively determined from molecular orbital theory using linear combinations of atomic orbitalssz. If two atomic orbitals overlap, two molecular orbitals are formed, one having a lower energy than the lowest energy atomic orbital and the other having a higher energy than both atomic orbitals. These molecular orbitals correspond to new electronic wave functions constructed from the addition and subtraction of the atomic wave functions. The 102 Table 6. The nonplanarity of a CuTPP molecule. The deviations of various atoms from the (001) plane passing through the copper ion are listed. Some relevant bond lengths are also tabulated48. Atom Deviation from Bond Bond the Cu ion length (K) (K) Cu 0.00 Cu-N I.957 t .Ol3 N -.04 N-C. L396 1 .Ol3 C. .26 N-C, L396 2 .Ol3 C; .23 C,-C2 L427 1' .Ol6 C3 -.09 02-03 L335 3 .023 C, -,24 03-0, L427 1' .Ol6 05 -.42 q,- c5 L398 1‘ .013 c6 -.75 C7 —|.42 103 molecular orbitals derived from the addition of the atomic orbitals, called bonding orbitals, are mostly localized between the ion and ligand and are therefore mostly covalent in character. The molecular orbitals obtained from the subtraction of the atomic orbitals are antibonding orbitals because they are localized on the COpper ion, and in fact resemble the free ion orbitals. The five 3d orbitals that one would associate with the free COpper ion, the dxy' dxz' dyz’ de-yz' and d22 orbitals are situated so the four lobes of the de—yz orbital are pointing towards the four nitrogen atoms along the x and y axes. This allows the de-yz orbital to form a very strong covalent bonding orbital with the sp2 hybrid orbitals of the nitrogen pyrolle ligand. The unpaired electron of the CuTPP resides in the high energy antibonding o orbital associated with this same combination. There seems to be no in-plane n bonding to the nitrogen atoms with the dxy'dxz' or dyz orbitals. The effect of the filled d22 orbital, which points perpendicular to the plane of the nitrOgen atoms, is to have a repulsive influence toward any ligands which may try to bond at the fifth or sixth coordination sites. This is consistent with the crystal field theory approach to the energy of the 3d COpper ion orbitals in the porphyrin structure. In this picture the 3d free ion orbitals are assumed to be located in a crystalline electric 104 field due to negative point charges whose symmetry is determined by the symmetry of the ligands surrounding the ion. The manner in which the energy of the five-fold degenerate 3d orbitals splits in crystalline fields of different symmetries is shown in Figure 25. The dx2_y2 orbital has the highest energy in the square-planar environment because its four lobes point directly toward the point charges. Therefore, the unpaired electron would again be expected to reside in this orbital. These energy levels of the crystal field theory approach correspond to the antibonding molecular orbitals found from molecular field theory. The COpper a, B, y, 6 - tetraphenylporphine crystals grown from benzene were found to be tetragonal. The blue-violet crystals had highly degenerate faces at (0,i1,il) and (11,0,31) giving the crystals the shape of a tetragonal bipyramid. A perspective drawing of the crystals illustrating the direction of the crystalline axes is shown in Figure 26. The crystals we grew were on the order of 1 mm3 in size with the base of the pyramids being .l-.2 mm thick. The space group is 142d (No. 122 International Tables) and unit cell dimensions are a=15.03i.01A; c=13.99:.OlA. There are four molecules per unit cell (z=4) for this structure, resulting in an 8.4; nearest neighbor distance within a crystal. Figure 27 indicates the positions of the COpper sites in a unit cell and also 105 Figure 25. The manner in which the 3d orbital energy splits in crystalline field environments of different symmetry. 106 EASE 335;“. otosum otoaum _otumcoaoo 1.: I], NxaNx // III 2.3.: I; is: «N // \\ ’// / \ / / \\ / / \ / /\\ >x // \>/ \\ um \ «N / \ \\ // \ x // \ 4 / \ 0 xx \ . A 3 \\ u>lux «N \ \ \ \ \ «also. \ \ \ \ 107 J. AXIS H AXIS \‘ Figure 26. A perspective view of a CuTPP single crystal showing its relation to the crystalline axes. 108 Figure 27. A unit cell of CuTPP. The small squares around each copper site represents the porphyrin ring plane. 110 represents the planes of the porphyrin rings by small squares drawn around the copper atoms. The planes of all these porphyrin rings are situated perpendicular to the c axis, which is the four-fold axis of the crystal. This configuration determines a spin density in the magnetically concentrated single crystal of only 1.27 x 1021 spins/cm3. The four-fold axis of a crystal is relatively easy to identify because it is perpendicular to the base of the bipyramid structure. The positioning of the porphyrin rings within a unit cell and the manner in which the nearest neighbor distance is increased due to the presence of the attached phenyl rings is shown by Figure 28, which is a projection of one unit cell on the (001) plane. Molecules which lie at different positions along the c axis are represented by different line struc- tures in this drawing. The phenyl rings of any one molecule are tilted by 72° to the (001) plane and situated so as to lie over or under a neighboring molecule. Thus the effect of the phenyl ligands is to limit the distance of closest approach of the paramagnetic ions within the porphyrin rings and therefore reduce the intermolecular dipole-dipole and exchange coupling. The phenyl rings are believed by J. Assour51 to diamagnetically shield one molecule from another, resulting in much less dipolar interaction broadening in ESR spectra than would ordi- narily be expected for crystals of this magnetic concentra- tion. 111 Figure 28. The projection of one unit cell of CuTPP on the (001) plane. Each line structure represent5°a different plane perpendicular to the c axis. 112 C. Theory of the Magnetic Susceptibility of COpper Tetraphenylporphine The application of a magnetic field to a material can result in a magnetization being induced in the material. The magnetic susceptibility is in general a tensor quantity which relates the induced moment per unit volume, or the magnetization, to the applied field: In many materials, as in CuTPP, the induced moment is parallel to the applied field along at least one set of axes so that the magnetic susceptibility for that material becomes a scaler: 3M0 Xena.) = '33; where 0 represents a component of the field. The magneti- zation of a quantum mechanical system can be expressed as: lo: (Ma) = (in Z) 1;. 8v 0’ Ha where B = l/kT, V is the volume of the material, and z is the partition function, which is prOportional to the probability that a given energy state is pOpulated. The partition function may be determined once the energy levels are known and is given by: Z(T) = tr e’BH f—h 113 where H is the total spin-hamiltonian for the system. We will first consider the general effective spin hamiltonian which can be used to describe the energy of the Cu2+ ion in its molecular and crystalline environment and then decide which terms will make important contribu- tions to the susceptibility. This hamiltonian has the form: 4 2 Hi = §°§°§ + s-g-g +n£1 §'§n'§n + D{sz - 1/3S(s+1)} 2 . + Q{Iz - 1/3I(I+l)} + I gN g + g Hij from which the total spin hamiltonian for the crystal can be obtained by a sum over all copper ion sites (H = 2 Hi). Although this general expression is quite complicated, it can be simplfied to some extent for CuTPP. It should be mentioned that the spin hamiltonian is inherently an approximation to the actual hamiltonian of an ion in a crystalline electric field potential. The manner in which this approximation is made will be briefly outlined below. The individual terms will then be explained and discussed in the following paragraphs as they relate to CuTPP. A more detailed discussion of the effective spin hamiltonian 53 and W. Low54. can be found by referring to G. E. Pake In the absence of spin-orbit interactions the crystal field interaction will quench the orbital angular momentum 114 (i.e. = 0) and therefore quench the magnetic moment of the copper ion. The effect of the spin orbit inter- action, which is much weaker than the crystal field inter- action in this case, is to reintroduce a small amount of orbital angular momentum into the system. The energy of the electron in a magnetic field is not changed to first order by the spin-orbit coupling, but a second order calcu- lation does yield an energy shift due to the perturbative admixing of excited orbital sites. It is found that this effect can be included in a Spin hamiltonian formalism by assuming the electron to have its free value electronic spin (S=8) and to couple with an external magnetic field via an anisotropic tensor (§°g-H). This is the electronic Zeeman interaction term of the spin hamiltonian. Because of the axial symmetry associated with a square-planar environment, the g tensor is also axially symmetric allowing the Zeeman interaction to be expressed in terms of gll' the principal value of the g tensor perpendicular to the porphyrin ring (parallel to the c axis) and g1, the iso- trOpic value of the g tensor in the plane of the porphyrin ring (perpendicular to the c axis). The Zeeman term now becomes: HZee = gllu Ssz + giu(SxHx + S H) yy where u is the Bohr magneton. 115 Another term which arises from this perturbation analysis of spin-orbit coupling is the crystal field term (DfSi - 1/38(S+l)}). This term represents the effect of a splitting of the electrons orbital ground state due to the spin orbit coupling in the crystalline field environ- ment. The form in which it appears here is valid only if the crystalline field environment possesses axial symmetry, which is the case for the square-planar coordinated copper ion in CuTPP. Since the crystal field interaction does not split the energy levels for a spin 8 system, it does not have to be considered in a discussion of CuTPP. Two of the terms in the general equation can be eliminated because their interaction energies are always small compared to the temperature at which we are experi- mentally able to investigate the magnetic susceptibility of this material. The nuclear Zeeman interaction (S-gN-I) expresses the coupling of the copper nuclear moment to the applied external field. However, the nuclear magnetic moment, being on the order of 2000 times smaller than the electronic moment guarantees this term will always be small. The quadrupole interaction (Q{I§ - l/31(I+l)}) is a result of the coupling between the nuclear electronic quadrupole moment and the gradient of the crystalline electric field at the nucleus. An upper bound on the strength of this interaction, obtained experimentally by 1c t} 51 SE pl (f- [(1') 116 P. T. Manoharan and M. T. Rogersss, is Q 5'4 x 10.4 cm-l. This corresponds to a temperature T :_.58 mK. At the very lowest temperatures which we can obtain, the magnitude of the interaction which this term represents could be sufficient to make it observable. An estimation of its effect, assuming the to-be-mentioned hyperfine coefficient, B, equal to zero, has shown it to produce no contribution to the parallel susceptibility. It is also found that in the same approximation the quadrupole interaction increases the perpendicular susceptibility by .4% at 20 mK and 1.5% at 10 mK. It should be emphasized that this is the maximum effect possible because Q was assumed to equal the upper bound value 4 x 10-4 cm-1 for this calculation. The result to this point is that for CuTPP the hamiltonian is of the form: 8 The magnitude of these interactions is sufficient to allow the possibility that any or all of them may be significant at 10 mK. Of the remaining terms that have not yet been mentioned, §~A°I represents the COpper electronic spin-nuclear spin 56’57i.e.the hyperfine coupling. In the same interaction, Imanner as was seen for the Zeeman term, the hyperfine inter- action tensor may be expressed in terms of the principal 117 values of a hyperfine interaction tensor perpendicular and parallel to the porphyrin ring allowing the hyperfine term to become: thp = ASZIz + B(SxIx + Sny) where A and B are the hyperfine coefficients parallel to and perpendicular to the c axis respectively. The magnitude of the coefficients in this term and the other interactions relevant to CuTPP are listed in Table 7. Of the remaining two terms, one represents the super- hyperfine coupling between the copper electron and the nuclear Spin of the attached pyrrole rings' nitrogen atoms (2 S-An-In). The magnitude of the interaction energy, on n=l ~ ” ~ the order of 2.5 mK, is significant at low temperatures. An exact calculation of the effect of this term is extremely difficult, but an estimation was made by utilizing the Laplace transform computational method presented by P.H.E. Meijer58 (assuming the hyperfine coefficient, B, was zero). The results of this calculation59 show that it reduces the zero-field parallel susceptibility with the reduction being of order XII Ai/Tz. The magnitude of this contribution at 15 mK, the approximate limit of the validity of our theo- retical results, is about .5% and at 10 mK is about .8%. The contribution to the perpendicular susceptibility assuming B-O has also been estimated. It is found that this 118 31¢. H CLO. M “to.“ I. I I Aon)cv TmONV Nootwv nwfimv 0.3. .6. 6.4. on «ma nnod an; no. am. lam. In: J: 4m. 1.1m A_qu.fb_xv 222323 .760 .o. .3 “7:3 110:: 223325 3:32:30 3:22:03 3227.9 322.2 33.32.35 2:22;: sternum. .mmeU .HO ncowummwumo>sfl H50 onwuso omswmuno mums mammnpsonmm ca mosam> 0:8 mm psm>oaon mHOmcou :ofiuomumusw on“ no mosam> mamfiosfium mmxm ostHmumMHo on» mson mmasu on .h OHQMB 119 susceptibility will be reduced by approximately .7% at 20 mK and 2.0% at 10 mK. It should be mentioned that if this term were large, one would not expect the crystal field approximation, and therefore this spin hamiltonian, to be valid for this system. This would be true because a basic assumption of the crystal field theory is that the ion under consideration is entirely separate from the source of the crystalline electric field, and can be treated as a free ion in the crystalline electric potential. The only term which has not been discussed is Z'Hij' In general this term expresses the intermolecular coapling of the copper ions. It is composed of both the classical dipole-dipole coupling and exchange coupling. Even the large nearest neighbor distance (8.43) in crystalline CuTPP does not prevent the effects of the dipolar coupling from becoming significant at low temperatures. However, it does mean that the effects of exchange coupling between COpper atoms, which approximately drops off exponentially, will probably be very small in this material barring any unusual super-exchange effects. Such effects could possibly occur if two COpper electrons could interact through the phenyl rings. However, there is no experimental evidence that such an interaction occurs in CuTPP. Assuming the molecules only interact by classical dipole-dipole coupling we can write 120 the interaction at the ith ion due to all the other copper ions as: H = Z.(yi-yj) - 3(gi-r-.)(u.-r--) i j r 3 ij where j is summed over the crystal lattice and ”i is a ~ vector whose components are gk S (k = x,y, and z). The k. manner in which this summation islcarried out in actual calculations is presented in Appendix C. The presence of the dipolar coupling term disallows a solution of the magnetic susceptibility for this system in closed form. Since the Zeeman and hyperfine terms are by far the larger of the interactions we treat dipole-dipole interactions as a perturbation in order to estimate its effects. The hamiltonian that we must work with in the absence of such perturbations is: Ho = gllu Ssz + 91-11(stx + SyHy) + A8212 + B(SxIx + Sny) The zero field magnetic susceptibility can be easily calculated by working in the basis in which the hyperfine interaction is diagonal. The result of this calculation for the susceptibility parallel to the c axis is: 2 . . 2 2 [e-BA + §_2 {BB SinhBa + coshBu} + Slnh B] gIIu 20 A5 Ba BB (X°)II = 4kT -BA [e + 2 coshBa + cosh BB] 121 This susceptibility exhibits a Curie law behavior in both the high and low temperature limit, with the effective 2 2 9 u Curie constant decreasing by about 7% from II in the 92 “2 4k high temperature limit to II ( 12 2) in the low 4k l+33 /A temperature limit. The zero field susceptibility perpendicular to the c axis was determined to be: 2 2 9L“ [%‘i (e‘BA + 2 coshBa) X - Oi 4a(Bz-a2)(e BA + 2 coshBa + coshBB) (éigfifii) sinhBa - Aa coshBB + 23a sinh BB] which has a Curie law behavior in the high temperature region, but becomes essentially temperature independent at about 10 mK. The details of the manner in which these calculations were carried out is presented in Appendix C. The modification of these results due to the effects of the classical dipole-dipole interaction in a crystalline solid can be estimated by utilizing Van Vleck's moment expansion technique60. This method involves treating the dipole-dipole coupling as a perturbation on the Zeeman and hyperfine interactions, and expanding the partition function in powers of l/T. In this manner the effect of dipolar coupling can be determined for temperatures such that the 122 coupling energy is small compared with kT. Van Vleck's result showed that if a system had a Curie law behavior (x = c/T) at high temperatures then the effect of a dipolar term would be to modify the magnetic suscepti- bility according to 29 T X = C/T (1 + + ...) where A is an appropriate summation of the dipole inter- actions over the crystal lattice61. If this expansion is carried to first order, the result looks like a low tempera- ture expansion of the Curie-Weiss law (x = Egg) to first order. In practice the dipole lattice summation is often used to define an effective Curie-Weiss theta (0 = cA). The results of applying this expansion technique to evaluate the dipolar contribution to the parallel and perpendicular magnetic susceptibilities of CuTPP to first order are: xll = no)“ {1+ no)” All + ...} Xi.= (x°)l.{1 + (X°)i_AI. + ...} 2 _ r2 x2 _ r2 where A - X ( 13 1:3) and A = Z ( 13 11) are the ij ij appropriate lattice summations for a crystal of CuTPP. In these summations rij is the distance between the ith and 123 3th atoms and has components xij' yij and zij' These summations define effective Curie-Weiss thetas of -l.54 mK and +0.62 mK respectively. The method by which this lattice summation and the susceptibility calculations was carried out is presented in Appendix C. CHAPTER III THE EXPERIMENTAL DATA A. Presentation of the Experimental Results for_CuTPP Once the direction along which magnetic susceptibility measurements are to be made is decided and related to the external morphology of the crystal, it must be prOperly mounted within the SQUID. The crystal shape and relative orientation of the porphyrin rings with respect to the crystalline axes determine the difficulty encountered with the alignment of the crystal. There is also the additional problem of the crystals' small size. Due to the symmetry of the CuTPP crystals and the fact that the porphyrin rings are perpendicular to the c axis, this task was not too difficult for this compound. Drawings of the SQUID sample holders and mounted crystals for measurements of magnetic susceptibility paral- lel and perpendicular to the planes of the porphyrin rings are shown in Figure 29. The sample holders were designed to make the alignment process as simple as possible. Crystal alignment within the holders was accomplished by adjusting the crystal relative to the sample holder axis with a small probe while observing the results through a microsc0pe. The accuracy of the alignment was then checked by placing the sample and holder in an optical goniometer 124 125 Figure 29. The manner in which the single crystals were mounted in the SQUID magnetometer for measure- ments of the parallel and perpendicular susceptibilities. 126 4} Parollei Axis -+ i"-- -- ----q /\ Perpendicular Axis -—> 127 and determining the position of the crystal with respect to the axis of the sample holder. It was found to be relatively easy to attain an accuracy of i4°. Calcula- tions of the maximum effect of a misalignment of this magnitude results in an uncertainty in the magnetic susceptibility of about .2% at 10 mK. The crystal was secured to the holder by applying a small drop of glycerine and soap flakes glue. Before magnetic susceptibility measurements were made on the CuTPP crystals, they were checked by ESR techniques to determine for certain that the c axis was indeed obvious from the external crystal morphology. The hyperfine splitting was found to be maximum along the direction we had chosen as the four-fold axis, thereby indicating this direction was perpendicular to the plane of the porphyrin rings and was the c axis of the crystal. These ESR measure- ments yielded electronic g-values of gII = 2.175 t .017 and 91.: 2.052 t .017. The uncertainty in these numbers does not include the effects of crystal misalignment within the ESR apparatus, which could not be determined. The hyperfine constants were also determined to be A = (205.1 1 14.2) x 10'4 cm“1 and B ~ 30 x 10’4 cm'l. Since the hyperfine splitting parallel to the porphyrin planes could not be resolved in the magnetically concentrated crystal, the hyperfine coefficient in this plane, B, was found by extrapolation of the out-of-plane data. These numbers are 128 comparable with the values obtained by P. T. Manoharan and Max T. Rogersss. They measured gII = 2.179 and gL|= 2.033 for the electronic g-values and A = 212.2 x 10- cm.1 and B = 30 x 10"4 cm.1 for the hyperfine con- stants. Having confidence in our crystals we then aligned then in the dilution refrigerator. Two crystals were used for parallel susceptibility measurements, one having a mass of approximately .6 mg and the other having a mass of 1.25 mg. The .6 mg crystal was originally a larger crystal which had fractured so that it no longer had the tetragonal bipyramid shape, but instead was very roughly pyramidal. The 1.25 mg crystal had the normal tetragonal bipyramid shape and was also used for susceptibility measurements perpendicular to the c axis. This crystal was estimated to be about 1mm3 in volume and to have a surface area of 5.2 x 10_2 cm2. Both the parallel and perpendicular susceptibilities were measured with a .25 gauss dc magnetic field trapped in the niobium cylinder. The systematic errors in the zero-field magnetic susceptibility due to the presence of this small finite field is negligible even at the lowest temperatures attained during the experiment. A sample of powdered CuTPP was placed in the conven- tional magnetic susceptibility coils of the refrigerator. The sample was formed by pressing .25009 gm of CuTPP into a .793 cm right circular cylinder with diameter equal to 129 height. The resulting filling factor for this sample was 44.5%. The CuTPP was not actually powdered by us but was used exactly as it had been formed by the preparation process. The grain size obtained during a rapid precipi- tation of the CuTPP is very important if the precipitate is to be used directly at ultralow temperatures. If the individual grains are small enough and internal relaxation processes are not too long, powdered samples are the only means of properly thermally tying a material to the cold dilute solution at extremely low temperatures. A comparison of single crystal and powder data then becomes useful as an indication of where the single crystals might be going out of thermal equilibrium with the refrigerator and there- fore of the range of validity for these single crystal measurements. The average grain size for CuTPP was estimated using a high power, oil drop microsc0pe to be on the order of 1 micron. During these experiments we found no indication of a thermal equilibrium problem with regard to the powdered CuTPP sample. The powder susceptibility measurements were made in a 1.6 gauss ac magnetic field in the temperature range from 4.2°K to .3°K, where careful calibration of the 90% LMN - 10% CMN thermometer is necessary, and in a .4 gauss field at lower temperatures. The experimental zero field magnetic susceptibility data obtained for CuTPP is shown graphically in Figure 30, which 130 50 T T i f O O O A A 40" O a: O O O 30" ‘ a o _I o z \ D 2 E >‘20I o o q x etr-‘*:;;;;;:»u._f_ o *--—;:;;___ (X); IO” 6 mg Single Crystal 4 0 I25 mg Single Crysiol 0 Powder 0 A J L l O 50 I00 ISO 200 mvenss T' (K") Figure 30. The magnetic susceptibility data for CuTPP. The solid lines represent the theoretical susceptibilities without inclusion of the dipolar interaction and the dashed lines represent the theoretical susceptibilities including the effect of dipole coupling. .250 131 is a plot of susceptibility in emu/mole against the inverse magnetic temperature. The data are also presented in tabulated form in Appendix D. The magnetic susceptibility perpendicular to the c axis (i.e. in the plane of the porphyrin rings) became approxi- mately temperature independent at about 10 mK after attaining a maximum susceptibility of 19.3 emu/mole. In the tempera- ture range from .7°K to .25°K, where deviations from Curie-law behavior are not yet apparent, this data yields gi.= 1.963 t .008. The most likely reason for this g-value being dif- ferent from the ESR results is that the .25 gauss magnetic field was not trapped properly in the SQUID magnetometer. An error in the trapped field of only .02 gauss could easily account for this discrepancy. This effect is possible because the u-metal shield, which must be moved after each experiment, may be positioned slightly differently from one experiment to another. However, this presents no serious difficulty with the data since the g-factor can be renor- malized if necessary. The susceptibility parallel to the c axis (i.e. perpen- dicular to the porphyrin planes) was found to continue rising, at least initially, in a Curie-like manner down to the lowest temperatures. The value obtained for gII by using the high temperature data measured on the 1.25 gm crystal was 2.177 i .012. This compares favorably with the previously mentioned values obtained by ESR methods. Since 132 the .6 mg crystal could not be massed prOperly, the parallel susceptibility data acquired from this crystal was nor- malized such that it agreed with the other crystal data at high temperatures. The two sets of data deviated slightly from each other at the lowest temperature; however, one might expect this as demagnetization corrections for each sample (see page 133) are different. The powder susceptibility continues rising down to at least 4.2 mK with no indication of the onset of a transition to the ordered state being apparent. We believe the powder data to be very reliable even at temperatures below 10 mK, when the refrigerator is operating in the "single shot" mode, due to the expected good thermal contact between the powder grains and dilute solution. This seems to be verified by the fact that during the experiment the relaxation time of the CuTPP powder sample was observed to be at least as rapid as the relaxation time associated with the LMN-CMN thermometer. 133 B. Factors Affecting the Measurement of Magnetic Susceptibility at Ultralow Temperatures Demagnetization Corrections An important consideration that must be taken into account when studying single crystals at very low temperatures is the presence of demagnetizing fields within the crystalssz. The magnetic susceptibility one measures experimentally is determined with respect to the external magnetic field (Ho) that is applied to the sample as a whole (x = M/Ho). However, a paramagnetic ion ext within the crystal does not respond to the influence of the external field, but instead to some resultant field interior to the crystal which depends on the crystal shape (i.e. boundary conditions at the crystal surface) and the relative positions of the paramagnetic ions (i.e. the crystal lattice structure). Therefore the experimentally measured magnetic susceptibility must be corrected so that it reflects the influence of the local field (H100) at the paramagnetic ion. These corrections can be estimated in the following manner. Consider an ion at the center of the crystal. The magnetic field at the site occupied by this ion is less than the applied external field due to the effect of induced "magnetic poles" at the crystal surface. This opposing field is prOportional to the intensity of magnetization so 134 that one can express the internal magnetic field in the crystal as: where N is the demagnetization factor. Now we must con- sider the effect of all the other paramagnetic ions on this site. This can be determined by dividing the influence of these ions into several contributions. First we imagine removing all the ions in a spherical region, which is microsc0pically large but macroscopically small centered about this site. The contributions to the local field are now due to the ions outside the sphere, on the surface of the spherical region, and interior to the spherical volume. The ions outside the sphere are assumed far enough from the site at which we are determining the local field so that they can be considered as a continuum of magnetic dipoles. An integration over this volume shows it produces no net field at the center of the sphere. The surface of the sphere on the other hand contributes a field of gl-M at its interior. The effect of ions interior to the sphere is determined by adding the contributions from each ion. This is related to the classical dipolar lattice summation of page 122. The local field at the ion can now be expressed as:60 _ _ I Hloc - Ho + (4fl/3 N + A )M whe Si] til di do bi Ci r l 135 where MA' is the field determined from the lattice summation. Since the effective field derived from this lattice summa- tion has been accounted for previously by the expansion of dipole-dipole interaction term of the spin hamiltonian, we do not need to consider this term here. The actual suscepti- bility can now be calculated: = xext x 1°C 1 + (4n/3 - N)x ext where we define Xloc = M/Hloc' Our problem now is to determine the demagnetizing factor for the particular crystals we study. Demagnetizing factors can only be calculated for samples which are ellipsoids of revolution or limiting cases of this shape (i.e. a flat disk or long needle). The complications arising from such a complex surface as that exhibited by CuTPP crystals make a calculation of N essentially impossible. In order to evaluate the effect of this correction an alternate method must be found other than a direct calculation. Since N is only a function of the shape of the sample we can estimate it by a room temperature experiment as follows. A mockup (scaled to 1 mm = 1 inch) of the CuTPP crystals was made from mild steel. The magnetic induction internal to the steel "crystal" upon the application of an external magnetic field is: ...}1 whe app in Now mucl inti 1/u; The] app] app: Stet app: 900d iSOt 136 B. = H. + 4nM i (1 + 41rxi)Hi where the internal field Hi is related to the external applied field Ho through the demagnetizing coefficient (Hi = Ho - NM). We can also write the magnetic induction in terms of the external field: U1 II HO + (4n-N)M (1 + (4n-N)xext)no Now for steel the internal permeability (pi = l + 4nxi) is much greater than 1 so combining the two expressions for the internal magnetic induction and applying the condition l/ui Z 0 will yield the result: N~_1__=.‘i<1 xext M Therefore the shape dependent factor N can be determined for a steel mockup of the crystal and this demagnetization factor applied to the CuTPP crystal data. Of course there are some approximations inherent in this calculation. For the mild steel we used the permeability was determined to be approximately 250 so that the approximation l/ui = 0 is good to about .5%. It is also assumed that the steel is an isotrOpic medium. by 56: in ma WI] Or. 00 mu 84 di re 137 The demagnetizing factor was determined experimentally by the following method. The steel mockup of the CuTPP samples and a similar size steel sphere were alternately inserted into a pickup coil in the presence of an applied magnetic field. Their magnetization was determined by the flux changes they induced in the pickup coil as they were placed into and removed from it according to the following relation: A M = 4nan where A6 is the flux change, V is the volume of the sample, n is the number of turns per unit length in the pickup coil and f is a factor which depends on the coil geometry: L/2 f = (82 + to _oim>to ransom 29.5 as 3.. o_oEm" o:.@. O (31owxnw3) "‘A x 0.. N._ 150 13-14 mK for the 1.25 mg crystal and at about 11 mK for the .6 mg crystal. This indicates the .6 mg crystal was probably in better thermal equilibrium than the 1.25 mg crystal which is, of course, reasonable since the .6 mg crystal has the smaller heat capacity and a larger surface-to-volume ratio. Below these temperatures it is apparent that thermal equilibrium can no longer be main- tained between the respective crystals and the surrounding liquid helium. Now at the temperatures at which the lack of thermal equilibrium becomes noticeable, the perpendicular suscepti- bility has become approximately temperature independent. Therefore, one would expect the experimentally determined perpendicular susceptibility to deviate very little from the actual value to even lower temperatures than is possible for measurements of XI . This allows us to manipu- late the data in the following manner in order to estimate the parallel susceptibility to lower temperatures. One can shift the effective powder data to higher temperatures (at a constant susceptibility value) until it coincides with the experimental powder susceptibility. Assuming the shift to be due solely to a lack of thermal equilibrium during the measurements (i.e. that the crystals were actually at the temperature defined by the shifted value), we can apply this temperature correction to XII and xi. Since the different crystals came out of thermal equilibrium at 151 different temperatures, these corrections will be different for each crystal. The effect of applying the thermal equilibrium corrections is shown in Figure 31. The Open symbols on this graph represent thermally corrected data. Notice that now the .6 mg and 1.25 mg data coincide to temperatures on the order of 8.5 mK. Although it may not be valid in this temperature range, the theoretically determined parallel susceptibility has been extended into this region for comparison with the corrected experimental data. This extension of the theoretical XII continues to follow the data down to about 8.7 mK. The obvious next step in the analysis of these results is to attempt a calcu- lation of the effect of the superhyperfine term when B # 0. However, such a calculation is extremely difficult and has not been carried out as yet. Also, it will most likely be very important to consider the l/T3 dipolar interaction term. Up to this point we have largely been concerned with uncertainties in the experimental data. However, there are also uncertainties associated with the coefficients used to determine the theoretical results. The theories therefore may be somewhat modified by the possible variations in these coefficients. The theoretical perpendicular susceptibility was found to rise above the experimental data below 30 mK. Since this theoretical X1 is mainly controlled by the magnitude of the hyperfine coefficient A, one might ask how it would be 152 affected by a change in the value of A. Adjustments of A, within its uncertainty, were made and its effect on the X1 theory was observed. In order to preserve the low temperature behavior of xi as indicated by the experimental data, and also not change XII significantly, it is necessary to constrain the ratio B/A. If A is increased by 4% over the value used to obtain the theory plotted in Figure 31, then B must also be increased by 4%. This 4% increase is the limit allowed by the uncertainty in our EPR determina- tion of A. Since P.T. Manoharan and M.T. Rogers55 do not indicate the uncertainties in their measurements of A and B for magnetically concentrated single crystals, it is not known if these increased values would lie within the range of their uncertainties. The effect of such an increase is to reduce the X1 theory so that it roughly follows the previous calculation of (xo)i. However, it still would not fit the experimental data to within the error bars, but it is an improvement. One can therefore not say that an adjustment of these coefficients within the range of their uncertainties would be sufficient to explain the experi- mental data, but only that it is a possible factor in the X1 theory not agreeing with the experimental data at the lowest temperatures. It was estimated that the theoretical parallel suscepti- bility was on the average about 1.3% lower than several experimental XII data points in the temperature range around 153 30 mK. If the experimental data is not corrected for demagnetization effects the theory and experiment would agree much better in this intermediate temperature region. (The same statement applies to xi.) However, one finds that the effective powder susceptibility as determined by the crystal data in the absence of demagnetization corrections does not agree with the experimental powder data nearly as well over this temperature range as does the demagnetization corrected data. It therefore seems reason- able that the application of the demagnetization corrections is essentially correct. If the uncertainty in the experi- mental data is taken into account, the discrepancy between the experiment and theory is as small as .9%. One should also recoqnize that the maximum uncertainties in our tem- perature measurements (11%) occur over the approximate temperature range 30-100 mK (see Figure 18). The tempera- ture uncertainties will be reflected in the measurements of x. In light of the foregoing discussion of this data it seems that further analysis requires a calculation of the effect of the superhyperfine coupling and nuclear quadrupole interaction with B # 0. Only after this calculation is complete can one make more definite statements about the behavior of the parallel and perpendicular crystalline sus- ceptibilities from the intermediate to the low temperature regions. We believe this data in general to be fit rather 154 well by the theoretical calculations. The basic behavior of the parallel susceptibility appears to be explained by the dipolar interaction, while the perpendicular suscepti- bility is primarily a result of the highly anisotropic hyperfine coupling. B. Summary and Conclusions The magnetic susceptibility of CuTPP parallel to the crystalline c axis has been observed to follow the first order expansion of a Curie-Weiss law behavior down to a temperature of approximately 15 mK. The susceptibility perpendicular to the crystalline c axis became approxi- mately temperature independent at about 10 mK. Theoretical calculations show this behavior for XL is primarily a result of the hyperfine interaction. An extension of the theoretical description of the parallel and perpendicular susceptibilities to lower temperatures will require a calcu- lation of the superhyperfine interaction effect when B # 0, and a calculation of the l/T3 term in the expansion of the dipolar interaction. The powder data shows no signs of the onset of a transition to the ordered state to 4.0 mK. The powder data is expected to be valid to this temperature because of the good thermal contact between the powder grains and the dilute solution. A comparison of the experimental powder data with an effective powder susceptibility deter- mined from the single crystal data indicates the single 155 crystals were in thermal equilibrium to approximately 12 mK. The results of these single crystal experiments have proven the SQUID to be a very valuable tool for investigating the magnetic behavior of single crystal samples at ultralow temperatures. This thesis has presented the initial results of a comprehensive study of metallOporphyrin and metallo- tetraphenylporphyrin compounds which is presently under way. The information obtained and techniques develOped ..“ during the investigation of CuTPP is proving useful for the further study of these compounds. However, the study of CuTPP itself is undoubtedly not completed with the publishing of these results. The magnetic susceptibility along the crystalline c axis of CuTPP has been observed to continue rising at least down to 7 mK. It would cer- tainly be very interesting to carry these measurements to even lower temperatures. A possible method by which this could be accomplished is the adiabatic demagnetization of a single crystal using the SQUID magnetometer. Such experiments as this may be forthcoming in the future. It should be noted that adiabatic demagnetization of a powdered sample of CuTPP would not be useful as a means of producing lower temperatures, because the magnetic entrOpy vs. temperature relationship perpendicular to the c axis is such that adiabatic demagnetization would produce warming in all microcrystals for which the external magnetic 156 field is oriented in this direction. The relatively good thermal equilibrium observed between the single crystals and dilute solution at low temperatures indicates good internal heat transfer mechanisms within the crystals. Therefore it seems reasonable that this material could be of use in investi- gating the anomalously good thermal contact shown to exist between some salts and pure 3 He 9 One may find that CuTPP single crystals could be studied at lower temperatures by utilizing 3He to provide better thermal contact at ultralow temperatures. It should be emphasized that the feasibility of a complete study of the metal complexed porphyrins at ultra- low temperatures was critically dependent on the outcome of the single crystal measurements attempted on CuTPP. Since large single crystals of these compounds cannot be grown, they cannot be studied in conventional magnetometers. 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A complete description of the SQUID system will be presented in the Ph.D. thesis of Gary L. Neiheisel. The basic design and Operational characteristics of a SQUID system is explained in a paper by R. P. Giffard, R. A. Webb, and J. C. Wheatley, J. Low Temp. Phys. 6, 533 (1972) and in references therein. A. C. Anderson, J. Appl. Phys. 32, 5878 (1968). W. R. Abel and J. C. Wheatley, Phys. Rev. Letters 21, 597 (1968). R. A. Webb gt al., Phys. Letters, 215, l (1972). CR-100 3He gas-filled germanium resistor. CryoCal Inc., P.O. Box 10176, 1371 Avenue "E", Riviera Beach, Florida, 33404. MKS Baratron pressure meter-type 77, MKS Instruments Inc., Burlington, Mass. R. H. Sherman 22 21,, J. Res. Natl. Bur. Std., 685, 579 (1964). Abraham £5 al., Phys. Rev. 181, 273 (1969). J. E. Falk, Porphyrins and Metalloporphyrins (Elsevier Publishing Company, 1964) 160 45. P. Rothemund and Amel R. Menotti, JACS 62, 267 (1941). 46. P. Rothemund gt al., JACS lg, 1808 (1948). 47. Strem Chemicals Inc., 150 Andover Street, Danvers, Mass., 01923. 48. E. B. Fleischer et al., JACS 86, 2342 (1964). 49. E. B. Fleischer, JACS 85, 1353 (1963). 50. D. J. E. Ingram et al., JACS 18, 3545 (1956). 51. J. M. Assour, J. Chem. Phys. 43, 2477 (1965). 52. R. McWeeny and B. T. Sutcliffe, Methods of Molecular ‘ Quantum Mechanics (Academic Press, 1969); also F. Basolo and R. Johnson, Coordination Chemistry (W. A. Benjamin, Ind., 1964). 53. G. E. Pake, Paramagnetic Resonance (W. A. Benjamin, Inc., New York, 1962). 54. W. Low, Paramagnetic Resonance in Solids, Seitz and Turnbull, Editors (Academic Press, 1960). 55. P. T. Manoharan and M. T. ROgers, in Electron Spin Resonances of Metal Complexes (Plenum Press, 1969). 56. A. Abragam and M. H. L. Pryce, Proc. Roy. Soc. (London) 5205, 135 (1951). 57. B. Bleaney, Phil. Mag. 43, 441 (1951). 58. P. H. E. Meijer, Phys. Rev. B, 6, 214 (1971). 59. W. P. Pratt, Jr., Michigan State University, private communication. 60. J. H. Van Vleck, J. Chem. Phys. 5, 320 (1937). 61. J. M. Daniels, Proc. Phys. Soc. (London) A64, 673 (1953). 62. 63. 64. 65. 66. 67. 68. 161 D. De Klerk, Handbuch der Physik, Vol. 15 (1956), p. 117. To be published: A 4.47 mg single crystal of the compound CaCu(OAc)4-6H20 has been studied in crystal— line form and found to remain in thermal equilibrium below 10 mK. R. De Bruyn Ouboter and A. Th. A. M. De Waele, Progress in Low Temperature Physics V01. VI (North-Holland Publishing Co., Amsterdam, 1970), p. 243. A. H. Silver and J. E. Zimmerman, Phys. Rev. 151, 317 (1967). J. E. Zimmerman, P. Theine, and J. T. Harding, J. Appl. Phys. 21, 1572 (1970). R. M. White, Quantum Theory of Magnetism, (Mc Graw-Hill 1970). p. 44. J. R. Peverley, J. Computational Physics 2, 83 (1971). APPENDIX A APPENDIX A THERMAL RELAXATION WITHIN THE MIXING CHAMBER A calculation of the thermal relaxation times for heat flow between the CMN thermometer and sample positions in each tail was carried out while designing the refrigerator's mixing chamber. The feasibility of using the dual tail arrangement, which allows for a compact system, depends critically on the rate at which the refrigerator can come into thermal equilibrium at low temperatures. It was assumed for purposes of this calcu— 1ation that heat was applied to one tail and then allowed to flow so as to establish thermal equilibrium with the other tail. The conduction of heat between the two tails can be approximately assumed to occur in two parallel paths, one through the dilute solution and concentrated 3He, and the other through the copper coil foil which surrounds each sample chamber and forms a continuous path between them via the resistance thermometer mounting parts (see Figure 6). A "circuit" diagram is presented in Figure A.l which repre- sents each impedance to heat flow as a thermal resistance. The thermal relaxation time for heat flow through a material can be expressed as T = RC where R is the thermal resistance of the path and C is the heat capacity of the material. 162 163 V U Rd Rd V NJ _M RK RK TAIL”: TAIL *2 Figure A.1. A "circuit diagram" representing thermal resistance to heat flow between the two tails of the mixing chamber. Path A refers to heat flow through the liquid (Rd-RC-Rd) and path B to heat flow through the coil foil (RK-Rcu-RK). 164 The calculation is by no means exact but is a rough overestimation of the thermal equilibrium problems one might expect from this mixing chamber design. The flow of 3 He through the mixing chamber has been ignored in this calculation. The time necessary for the 3He atom within the mixing chamber to be replenished by the incoming flow of liquid 3He during cooling is also a measure of the thermal equilibrium times of this refrigerator. The thermal relaxation process acting to bring the two tails into equilibrium occurs simultaneously with the cooling at the phase boundaries as 3He is circulated. Thus in reality the total thermal equilibrium time is a function of both these processes. We will first consider the flow of heat through the liquid helium. From the heat flow diagram we see that there are effectively three paths in series, Rd-Rc-Rd, where the subscripts represent dilute solution and concentrated 3He, respectively. The worst possible case in the low temperature region, where the major heat flow is through the liquid, is obtained by assuming the entire region is filled with dilute solution. This is true because the dilute solution's thermal conductivity is smaller than the thermal conductivity of the con- centrated 3He at very low temperatures. It is also assumed that the heat capacity of the entire volume of dilute solution within the mixing chamber will contribute 165 to the thermal relaxation time. Therefore all the dilute solution is treated as if it were located at the sample positions. This is obviously an overestimate of the true situation. The relaxation time for path A can then be expressed as: TA = Rdcd lenl £2n2 Cd = ( A + A ) (X_) 1 2 d where Kd is the thermal conductivity of the dilute solution and Cd is its heat capacity per mole of solution. The ratios Ill/A1 and 2.2/1;2 are the length to cross-sectional area ratios of the different regions connecting the two tails. The number of moles of dilute solution present in each region (n1,n2) must be estimated. At low temperatures n the 3He concentration (x3 =-———;—) in the dilute solution n3+n4 is small (~ .064) so we can approximate the number of moles of dilute solution by its maximum valuezs: n : 31.. 1 V4,o and identically n :4:£_ 2 v4,o where v4 0 = 27.6 cm3/mole is the limit of the molar volume I of 4He at T = 0.0K and V1 and V2 are the volumes of regions 166 1 and 2, respectively. Combining these results gives us an expression for the relaxation time for heat flow through the liquid: T = (-Q) (cgs units) The values of the molar Specific heat and thermal conduc- tivity have been experimentally measured in the range 3 mK to 250 mK, and were obtained from graphical presentations of this datal7. The second path of heat flow must include the effects of Kapitza resistance between the dilute solution and c0pper coil foil (RK) and thermal resistance along the coil foil path between the tails. The total series resistance is: R=2RK+RCU An estimation of the thermal time constant associated with heat flow along this path B can be expressed as: TB = (ZRK + RCu)cd where Cd is the heat capacity of the dilute solution in the mixing chamber region. This heat capacity, being much larger than the coil foil's heat capapity, effectively deter- mines the rate of heat flow along this path. The Kapitza resistance between copper and dilute solution has been measured and can be formulated as:25 167 RK = 1.5 x lo-S/AT3 where A is the contact area. The resistance to heat flow within the copper is: R = R/A'K Cu Cu where R/A' is the length to area ratio of the coil foil and KCu is its thermal conductivity. For temperature less than 1 K we can assume KCu I 1.4 x 107T (erg/sec-cm-K). Using the same procedure as before to evaluate the number of moles of dilute solution within the sample chamber leads to an expression for the thermal relaxation time associated with this path in terms of the dilute solution's heat capacity: The results of these calculations can now be combined into a single expression for the total relaxation time for heat flow between the tails. By using the apprOpriate dimensions for our mixing chamber we find: TA = 3.8 Cd/Kd and -5 T = ( 3 + 5.6x10 ) C T T d 168 Since the two paths are in parallel, the total relaxation time for heat flow between the two tails can be found from r-ilH %-+ A B HIH The result was presented in Figure 7. As has been mentioned, the presence of two phase lines, one in each tail, means that each tail is cooling simultaneously and we therefore have overestimated the problem of thermal resistance. In practice one must distinguish between thermal relaxation and the rate at which cold 3He passes through the mixing chamber as it is cooled. Typically we find that at tempera- tures near 10 mK it may take roughly two hours for the mixing chamber to cool from one temperature to another. However, the maximum relaxation time for this mixing chamber seems to be only on the order of one hour. This calculation indicates, and indeed it has been observed, that this design presents no thermal equilibrium problems. APPENDIX B APPENDIX B THEORY OF POINT CONTACT SQUIDS The phenomena of superconductivity can be explained as a condensation of electron pairs (Cooper pairs) into a single quantum state64. A superconducting region can then be described by a complex order parameter, 0(r,t) = wo(r,t) eie, where Iwo(r,t)l2 is the supercurrent charge density and 6 is the quantum mechanical phase. If one is dealing with a superconducting loop, the condition of single- valuedness of the order parameter requires that the integral of its phase around the loop must be Znn, where n is an integer. This implies a quantization of the total flux (¢) trapped within the loop, and indeed this flux is found to be quantized in units of ¢o = h/2e. The usefulness of a circuit element consisting of a superconducting 100p closed by a point contact junction (weak link) in magnetometry lies in its ability to pass a supercurrent (is) which is a function of the phase difference (A8) across the junction. Since any such junction will also have an associated resistance and capacitance, there is in general also a normal current (in) and a displacement current (id) through the junction. The SQUID built for this refrig- erator is an rf-biased single-junction device which uses a pointed niobium screw contact to provide the weak link in a superconducting 100p. The low inductance junction produced 169 170 by this contact is useful in low frequency applications. The low frequency mode of Operation is characterized by the supercurrent being the dominant current in the junction. For this condition the total current becomes 1 = iC sin 0 where iC is the critical supercurrent for the junction. It can be shown that the total flux change through the lOOp is related to the external flux applied perpendicular to the lOOp by the expression65: . . 2n _ _ LiC Sln Eg-(¢+k¢o) - ¢ ¢ext where k¢o = 0, i ¢o’ i 2¢o -°° . This response function follows from the quantization of the integral of the canonical momentum of a Cooper pair around the loOp (again the condition for single-valuedness of the wave function). Under the normal operating conditions of an rf-biased circuit, which is characterized by the requirement that ZnLiC/q>O be greater than one, the response is as shown in Figure B.l. This type of response is obtained by adjusting the niobium screw contact tension until the critical current meets the above criteria. The dotted lines indicate transi- tions correSponding to single flux quantum shifts (Ak = :1) which occur when the critical flux, or critical current, is reached in the loOp. In order for this device to be useful, one must be able to measure the flux changes introduced into the supercon- ducting lOOp from the magnetized sample via the signal coil. 171 Figure 8.1. The response function of a SQUID magnetometer. The total flux (O) through the superconducting loop is plotted as a function of the external applied field. 172 3 1* 03/21 magnetomta 9’2. uperconduiz; the em" 173 This is accomplished by inductively coupling the loop to a tank circuit which is resonant at the rf signal frequency66’36. As the rf current amplitude in the inductor is increased, the superconducting loop will set up an Opposing field by the creation of a supercurrent within the loOp. Suppose the dc flux level in the 100p is at point A in Figure‘B.l. As the rf flux amplitude about point A is increased the rf voltage across the tank circuit increases linearly until the rf amplitude reaches A-3 at which the rf voltage will be VA’ At this point an irreversible transi- tion occurs. The energy required for this transition is drawn from the tank circuit with a resulting decrease in the rf oscillation level. This process repeats itself as long as the rf drive is insufficient to overcome the loss in energy per rf cycle due to the traversing of one hysteresis 100p. In this region of Operation the rf voltage is no longer a linear function of rf current ampli- tude, but is approximately constant at VA. Now assume the dc flux level is at B. The same process as before occurs except that now the maximum critical flux level will be reached when the rf flux amplitude equals B-3 and an irreversible transition about two hysterises lOOps occurs. The maximum voltage across the tank circuit at this point will be V > V . The distance B-A represents one-half of B A a flux quantum (¢o/2). Since the points A and A', B'and B", 174 etc. are equivalent points in the response function, it is apparent that the maximum voltage across the rf tank circuit is periodic in the dc flux level with period ¢o' One can therefore utilize this pr0perty to measure the dc flux change through the loop. APPENDIX C - {I Afit {v APPENDIX C THEORETICAL CALCULATION OF THE MAGNETIC SUSCEPTIBILITY OF CuTPP We have seen that the spin hamiltonian which describes the energy of the paramagnetic copper ion at the ith site [ in a crystal of CuTPP is: H + S H ) + AS I 1 . x y. y z z. i 1 i 1 + B(sX I + s I ) + f Hij 1 xi yi yi j (u -u.) - 3(u.-r..)(u,.r..) where H.. = *1 ”3 “1 ~13 ~J ~11 13 r. lj The total spin hamiltonian for the crystal is simply H = 2 Hi' It will be useful to write the single ion spin 1 hamiltonian in the form: where Hz refers to the Zeeman term, ”0 is the hyperfine i 1 interaction and Hd represents the intermolecular classical i dipole-dipole coupling. The zero-field magnetic suscepti- bility along the ath axis can be obtained from the partition 175 176 function for this system. The exact partition function is: and the zero-field susceptibility is obtained from the well-known expression: . 1 32(2n2) xa=11m WT— Hd+o aHd where a = x, y, or z axis. In practice, one finds that it is impossible to find a ba31s which is composed of eigenstates of the total hamiltonian. Therefore the evaluation of the trace involves more than a simple sum of diagonal elements. In fact the dipolar coupling term guarantees that no such basis can be found. Since the susceptibility cannot be calculated in closed form with this hamiltonian, one must assume the dipole coupling is a perturbation and expand the partition function. A particularly good method for many applications is to expand the partition function in powers of the applied magnetic field and retain only terms to first order in the dipolar interaction. Of course only the terms to order H: should contribute to the zero-field susceptibility because of the second derivative which is involved in its derivation. 177 Another method which is closely related to the previous one involves expanding the expectation value of the magnetization in an identical manner. The advantage of this method lies in the fact that it is only necessary to expand the magnetization to first order in the applied field to obtain the susceptibility. The magnetization of the crystal can be expressed as: M=-gp a 0: i101 where u is the Bohr magneton and 9a is the apprOpriate com- ponent of the g-tensor. The zero-field magnetic suscepti- bility is then obtained from the first derivative of this quantity: BMa H +o 3H d a In order to see how this expansion may be carried out, 8(A+B) where consider a general operator of the form e- A and B are two operators which may not commute. Also assume that B is small compared to A. We can expand this exponential Operator in powers of B by the following pro- cedure. We can express this Operator as: e-B(A+B) = e-BA¢ (B) 178 where ¢(B) is a function of B, which can be written in a manner that initially seems to complicate the issue, but will prove useful: ¢(B) = eBA e-B(A+B) The derivative of ¢(B) with respect to B can be used to derive a recursion relation for the expansion of ¢(B) in powers of B: dgés) = _eBA B e-B(A+B) -B(B) 45(8) where B(8) = eBA B e-BA. Integrating this equation and recognizing that ¢(0) = 1 will give the result: a ¢ = 1- [o B(s) ¢(s> as which is the recursion relation for an expansion of ¢(B) in powers of B. This expansion can now be applied to the expectation value of the spin operator where the terms H21 and Hdi are considered to be small. The expansion and evaluation of the appropriate matrix elements will be carried out in the basis defined by the eigenstates of Hoi. The small term approxima- tion is valid for the zero field limit because Hz. is a 1 179 linear function of the applied magnetic field and "d i involves the dipolar coupling interaction energy which is much smaller than the hyperfine interaction energy. Assuming the applied field to be along the c axis (i.e. Hz = gap Ha 8a ) the expansion can be shown to yield: 1 i -Bi(Hoi + HZi + Hdi) tr[e 2 Sa ] <2 S > = l 1 i “i -BZ(H + H + H ) . o. 2. d. tr e 1 1 i l ‘82 Ho. 8 - (1,) tr e 1 l { - X 9 pH I ds S (s) S Z .. a a o d. a. o 13 j i z I8 I8 + g uH ds ds' (S (s) H (s') S ijk a a o o aj dk Oi + Hd. (5) Sa (5') $0.) + ... j k i s; Hoi -sZ Ho. -B§ H where Sa (5) = e Sa e 1 1 , etc. and 20 = tr e i i Only terms to first order in the field and to first order in the dipole interaction which contribute to the susceptibility have been kept. Due to the requirement that there be no net spontaneous moment, all terms which are odd in Spin Operators will be zero. The first term in this expansion represents the susceptibility with only the hyper- fine and Zeeman interactions present, and the other terms represent the first order perturbative effect of the dipolar coupling. I?“ 180 The simplification that can be made by utilizing this expansion over an expansion involving the partition function is apparent from the fact that lower order terms are suf- ficient for the calculation of the susceptibility (i.e. O(Ha) instead of O(H:)). Once this expansion has been obtained, the remainder of the calculation involves evaluat- ing the trace over the eigenstates of HO . The eigenstates i of ”O can be constructed from linear combinations of the i well-known eigenstates of S2 (I i ms, 1 mI>, where ms = i 1%. mI = i3/2, :8). First the hyperfine interaction is written in terms of raising and lowering Operations and its matrix elements in this basis are determined: _ B HO — A 52' I2 + 5 (5+. I_ + s_. I+ ) 1 l J. l .1 1 l and S: Ims,mI> = [s(S+1)-ms(msil)]l5 |m511,mI> % I: Ims,mI> = [I(I+1)-mI(mIil)] Ims,mIi1> From an appropriate arrangement of this basis one can form three 2 x 2 subspaces and two 1 x 1 subspaces along the diagonals of the matrix. The entire matrix can no he diagonalized by individually diagonalizing each subspace. The eigenstates of this newly formed matrix comprise the 181 the desired basis which diagonalizes the hyperfine inter- action. One should obtain: <¢iHO.|¢> = where: |1/2,3/2> 9- H II 2 ¢2 = 1 { §%—.;-1/2,3/2> + (a+A/2) |1/2.1/2>} /2a(a+A/2) 2 ¢3 = 1 { (a+A/2) |-1/2,3/2>-3%— |1/2,1/2>} Ga (a-I-A/Z) —l— { |-1/2,1/2> + |1/2,-1/2>} /§— 6- .6. ll 182 ¢5 = ,zi { l-1/2.1/2> - I1/2.-1/2>} 2 ¢ = l { (a+A/2) l-l/Z —1/2> + 332 |1/2 -3/2>} 6 /20(d+A/2) .1f- 1 332 ¢7 = { ‘Z""1/2'“1/2> - (a+A/2) |1/2,-3/2>} /2d(c+A72) ¢8 - I'l/21-3/2> and y = A/4 a = 8 (25.22.3132)15 One can now utilize this basis by expanding the formal trace Operation into a sum over the eigenstates of Ho . 1 Although this expansion can rapidly become very complicated, especially when evaluating the term involving the dipolar coupling, the calculation can be carried through to obtain the susceptibility results presented on page 122. It is useful for the purposes of this calculation to express the dipolar coupling interaction in terms of raising and lowering Operations67. For each calculation, the perturba— tive term will separate into a product the original unper- turbed susceptibility (xo ) and (x0 ) , and a lattice summation of the dipole iiLeraction over the crystal (All and AL). ‘5‘" - . ‘_ [- . l l' 183 The lattice summations were carried out over the CuTPP crystal lattice within a spherical region centered about the ith site using the method proposed by J. R. Pevere1y68. His technique involves using a convergence factor to enhance the convergence of the lattice summation without having to carry the calculation over an extremely large number of unit cells. The summation was carried out over several different radii to ensure that covergence had been obtained. The results indicated the lattice summations had converged correctly. The effect of the superhyperfine coupling has not yet been determined in detail. This calculation rapidly becomes extremely complicated and one must resort to the more powerful Laplace transform computation technique presented by P.H.E. Meijersa. Despite using this technique we have only been able to estimate the superhyperfine coupling effect in the special case of B = 0 along the parallel and perpendicular axes. APPENDIX D APPENDIX D TABULATION OF PERTINENT DATA The measurement of temperature within the dilution refrigerator is intimately connected with the CR-100 germanium resistor. The results of a careful calibration of this resistor over the temperature range .3K to 3.0K is tabulated in this appendix. Also, the experimental magnetic susceptibility data obtained for COpper tetraphenylporphine is presented here. Both the raw experimental data and the single crystal data after application of the demagnetization corrections has been tabulated. The effective powder susceptibility as calculated from the demagnetization corrected data is also listed. 184 185 Table D.l. Results of the CR-100 Germanium Resistor Calibration. Resistance Inverse T* Resistance Inverse (ohms) (K‘l) (ohms) (K'l) 207.1 .312 2054.2 1.439 231.1 .348 2323.1 1.512 256.2 .385 2626.2 1.584 282.8 .421 2966.4 1.657 310.5 .457 3351.2 1.730 339.3 .494 3785.5 1.802 370.1 .530 4276.5 1.875 402.3 .566 4832.7 1.948 436.3 .603 5465.0 2.020 471.6 .639 6180.0 2.093 510.2 .675 6989.7 2.166 549.8 .712 7906.7 2.239 592.0 .748 8946.3 2.311 636.3 .785 10,123.0 2.384 683.2 .821 ll,464.0 2.457 732.1 .857 12,977.0 2.529 783.5 .894 l4,706.0 2.602 839.6 .930 16,661.0 2.675 898.0 .966 18,882.0 2.747 959.6 1.003 21,383.0 2.820 1094.0 1.075 24,260.0 2.893 1244.3 1.148 27,505.0 2.966 1413.5 1.221 31,120.0 3.038 1603.1 1.293 35,210.0 3.111 1816.5 1.366 186 Table D.2. The Magnetic Susceptibility Data for CuTPP-~ Single Crystal, Parallel Axis (Not Corrected for Demagnetization Effects) .6 mg Sing1e Crystal 1.25 mg Single Crystal * Inverse T XII XII Inverse T (emu/mole) (K‘l) (emu/mole) (8‘1) .725 1.634 .390 .874 .738 1.662 .471 1.067 .864 1.964 .546 1.243 1.000 2.256 .707 1.583 1.098 2.483 .839 1.881 1.242 2.798 1.007 2.258 1.343 3.042 1.192 2.693 1.532 3.468 1.186 2.682 2.183 4.989 1.226 2.769 4.613 10.526 1.316 2.966 5.974 13.737 1.510 3.401 7.663 17.746 2.271 5.118 9.610 22.571 3.965 8.978 12.745 30.718 6.165 14.044 16.898 42.122 11.169 26.111 20.023 51.964 15.953 38.793 22.303 59.388 23.890 62.946 24.942 68.829 31.544 93.333 26.324 74.370 31.905 96.159 27.623 79.384 34.977 109.550 28.706 83.291 40.004 137.910 30.406 90.394 40.465 140.290 33.343 102.980 42.882 164.130 35.262 113.050 41.621 151.670 43.122 170.720 43.744 177.300 44.339 183.960 1' 187 Table D.3. The Magnetic Susceptibility Data for CuTPP-- Powder and Single Crystal, Perpendicular Axis (Not Corrected for Demagnetization Effects) 1.25 mg Single Crystal Powder Xi Inverse T* Xpow Inverse T* (emu/mole) (K-l) (emu/mole) (K-1) .602 1.656 .233 .584 .644 1.793 .290 .732 .740 2.042 .354 .892 .785 2.186 .665 1.670 .897 2.490 .769 1.928 .970 2.695 .883 2.218 1.034 2.862 1.003 2.521 1.123 3.105 1.045 2.629 1.118 3.093 1.132 2.849 1.357 3.816 1.990 5.021 1.294 3.590 2.147 5.436 1.821 4.931 3.971 10.214 3.129 8.924 7.730 20.276 4.961 14.020 7.910 20.728 6.855 19.387 11.332 30.999 9.082 26.400 14.452 41.357 11.999 36.984 16.853 51.506 14.966 51.080 19.817 66.094 16.612 62.346 22.555 83.358 17.987 75.443 24.313 96.711 18.969 93.526 25.793 110.310 19.299 110.800 28.657 140.580 19.287 135.380 31.291 170.880 19.242 146.130 33.013 191.510 19.104 167.430 34.955 224.600 18.740 186.470 35.497 233.840 ..vrfi» m". t 188 Table D.4. The Magnetic Susceptibility Data for CuTPP-- Single Crystal, Parallel Axis, (Corrected for Demagnetization Effects) .6 mg Single Crystal 1.25 mg Single Crystal * xl' Inverse T XII Inverse T (emu/mole) (K-l) (emu/mole) (K-1) .726 1.634 .390 .874 .739 1.662 .471 1.067 .866 1.964 .546 1.243 1.002 2.256 .708 1.583 1.101 2.483 .840 1.881 1.245 2.798 1.008 2.258 1.347 3.042 1.194 2.693 1.537 3.468 1.188 2.682 2.194 4.989 1.228 2.769 4.661 10.526 1.318 2.966 6.054 13.737 1.513 3.401 7.796 17.746 2.277 5.118 9.819 22.571 3.985 8.978 13.116 30.718 6.213 14.044 17.554 42.122 11.327 26.111 20.954 51.964 16.278 38.793 23.465 59.388 24.625 62.946 26.404 68.829 32.839 93.333 27.958 74.370 33.230 96.159 29.428 79.384 36.576 109.550 30.660 83.291 42.110 137.910 32.607 90.394 42.621 140.290 36.008 102.980 45.311 164.130 38.257 113.050 45.858 151.670 47.687 170.720 48.449 177.300 49.180 183.960 * Table D.5. X1 (emu/mole) .601 .643 .739 .784 .895 .968 1.031 1.120 1.145 1.352 1.290 1.812 3.103 4.897 6.734 8.870 11.632 14.399 15.916 17.175 18.068 18.367 18.356 18.315 18.190 17.859 189 The Magnetic Susceptibility Data for CuTPP-- Powder and Single Crystal, Perpendicular Axis (Corrected for Demagnetization Effects) 1.25 mg Single Crystal Inverse T (K'l) 1.656 1.793 2.042 2.186 2.490 2.695 2.862 3.105 3.093 3.816 3.590 4.931 8.924 14.020 19.387 26.400 36.984 51.080 62.346 75.443 93.526 110.800 135.380 146.130 167.430 186.470 * xpow .233 .290 .354 .665 .769 .883 1.003 1.045 1.132 1.990 2.147 3.971 7.730 7.910 11.332 14.452 16.853 19.817 22.555 24.313 25.793 28.657 31.291 33.013 34.955 35.497 Powder (emu/mole) * Inverse T (K'l) .584 .732 .892 1.670 1.928 2.218 2.521 2.629 2.849 5.021 5.436 10.214 20.276 20.728 30.999 41.357 51.506 66.094 83.358 96.711 110.310 140.580 170.880 191.510 224.600 233.840 190 Table D.6. The Effective Powder Susceptibility as Determined by the Demagnetization Corrected Single Crystal Data .6 mg Sing1e Crystal 1.25 mg Single Crystal X365. Inverse T* xggi' Inverse T* (emu/mole) (K-l) (emu/mole) (K-1) .650 1.634 .352 .874 .662 1.662 .427 1.067 .780 1.964 .494 1.243 .898 2-255 .634 1.583 .988 2.483 .751 1.881 1.114 2.798 .900 2.258 1.209 3.042 1.068 2.693 1.379 3.468 1.066 2.682 1.978 4.989 1.100 2.769 4.042 10.526 1.181 2.966 5.298 13.737 1.354 3.401 6.833 17.746 2.037 5.118 8.627 22.571 3.477 8.978 11.374 30.718 5.439 14.044 14.685 42.122 9.884 26.111 17.025 51.964 13.762 38.793 18.601 59.388 19.266 62.946 20.277 68-829 23.390 93.333 21.116 74.370 23.595 96.159 21.851 79.384 24.767 109.550 22.400 83.291 26.743 137.910 23.250 90.394 26.900 140.290 24.627 102.980 27.691 164.130 25.458 113.050 27.923 151.670 28.414 170.720 28.606 177.300 28.780 183.960 1.9.5,;