3).. u: Q: \L v Em .. a. a . bflw s.- L. a L. a. I. ., . x $35. , o L ¢lr . . a. :4 {1:1 em 3w? , . am. «3%. . ._.s.....t}nL3¥ ; .. . .amawunnnfi. .mufig . 1 1:53... 2 39.3.1 . . . r lfitn’fll 2%. I15 .223. , 2.... 3 "55.4.1! . I -. Ami}... n “W. hm. «.4: 4.5.? f x: l\ u. . _ a: .H 4,1». .mdmmfi. was! Yak? “Wu“ .‘ ’4 u: , v3.0.3.1» 155.! :i it: 1"; A 9'. . ‘5‘ al'..| . a afiwhvll 1; ., . a , ‘ ,Lrhd ‘. . . .3hl....,c.. .‘ .uI-WK :jsvas.lm W. ...1hfinw..n1.=¢.~nnl.f . , . .A . . .25! 1| . ‘ v :I. I!1..In.. E..~..vr I I < 2. .5 < I I Z §_I LA..."" I L--_::LI l \ I .---- SQUID L. .......... o- in: PORT Figure 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- tization of a simple spin 8 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=bi§723§3 (1-%(1%)2I From this expansion it can be seen that non-linear effects may occur due to the sample's havingtxx>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 adjustment of the applied magnetic field. Of course the possibility of doing finite-field magnetization studies also exists for this system. When the single crystals are being studied, further problems may arise, due to their size and shape. In general 60 the crystals should be as small as possible with respect to the diameter of the coils in order to minimize violations of the assumption that each sample can be considered to be a small dipole source. Also, the crystal shape may make demagnetization corrections necessary. These corrections are virtually impossible to calculate for most crystals shapes and are usually roughly estimated by assuming the crystal is an ellipsoid of revolution with axial ratios similar to those of the crystal. However, other methods can be used to estimate demagnetization effects, as will be shown later. Even with these problems the SQUID proves to be an extremely valuable tool in the study of dilute magnetic systems, of samples which may be very expensive in quantity, and of small single crystals which may be difficult to grow to a size necessary for their use in more conventional magnetometers. In most instances it may not only be neces- sary, but desirable to use very small crystals for low temperature magnetic studies. Kapitza resistance between a crystal and the dilute solution, and poor internal heat transfer mechanisms may make it impossible to use a large crystal at low temperatures. Small crystals have a larger surface area per unit volume of material. Since Kapitza resistance varies inversely as the surface area, and the heat capacity is proportional to the volume, a small crystal will come into equilibrium with the surrounding dilute solution much sooner than a large crystal. 61 E. Thermometry and Thermal Equilibrium within the Refrigerator Thermometry Low temperature thermometry in our system is by Curie law extrapolation of the susceptibility of powdered cerium magnesium nitrate (CMN). At high temperatures the measured CMN susceptibility is plotted as a function of the inverse absolute temperature. A least squares fit of this data yields a Curie constant for the thermometer. This linear behavior with inverse temperature (x = C/T*) is then assumed to hold at low temperatures, allowing a magnetic temperature (T*) to be measured. Recent investigations of the CMN temperature scale indicate this Curie law extrapola- tion, and therefore the condition T* = T, to be valid to about 6 mK.l Problems such as powder demagnetization effects on non-spherical thermometers are usually also present at these temperatures. Experimental studies have been made by A. C. Anderson37 and by W. R. Abel and J. C. Wheatley38 on the magnitude of the demagnetization corrections to cylin- drical, powdered CMN thermometers. Estimations based on their measurements indicate that the magnetic temperature may be less than the absolute temperature by as much as .2 mK for a right circular cylinder thermometer with diameter equal 39 to height. R. A. Webb 32 al. have recently investigated these demagnetization effects further by comparing a similar 62 CMN thermometer with the temperature as measured by a Johnson device noise thermometer. They estimate the demag- netization correction to be 0 i .12 mK in the range 8 mK to 20 mK. This means that at temperatures on the order of 10 mK the absolute temperature is most likely not known to better than 2% when powder thermometers are used. However, powder thermometers must be used in ultralow temperature experiments in order to minimize the effects of Kapitza resistance between the thermometer and dilute solution. The means by which the CMN thermometer is tied to the absolute temperature scale at high temperatures, which must be done during each experiment, is to measure the CMN powder susceptibility against a 3He gas-filled germanium resistor4o calibrated in the temperature range .3 K to 3.0 K. This important resistor was calibrated in the following manner. 3 3 The germanium resistor, a 2 cm He vapor pressure bulb, and a 2.9146 gm single crystal of CMN were placed in the refrig- erator so as to be in good thermal contact with each other. 3 . The He vapor pressure, measured With a mercury manometer 41 and an MKS Baratron pressure meter with a 0-30 mm Hg pressure head, was used to accurately determine the CMN's Curie constant over the temperature range .7 K to 3.0 K. This tied the absolute temperature as defined by the T62 3He vapor pressure scale42 to the magnetic temperature as measured by the CMN. The CMN was then used to calibrate the germanium resistor over the temperature range .3 K to 3.0 K. The 63 calibration is presented graphically in Figure 15, and is also tabulated in Appendix D. Once this resistor was calibrated it could be used in later experiments to cali- brate CMN thermometers. Thermal Equilibrium A thermal analysis of the refrigerator was under- taken to investigate the possibility of a thermal gradient between the two sample positions in tail #1 and tail #2. Low temperature thermometers were constructed from CMN which had been put through a 45 micron sieve (sieve #325) to ensure a small grain size. In order to make the thermometers as nearly identical as possible, the two pills were simul- taneously pressed into the shape of a .794 cm of a right circular cylinder with diameter equal to height. Their masses were determined to be .55555 gm and .55570 gm, result- ing in filling factors of approximately 68%. The temperature calibrations were carried out by cooling the refrigerator with a CMN pill in each tail and plotting the ratio of the temperatures determined by each thermometer against the temperature of one of them. As has been mentioned before, all these temperatures were determined from a least-squares fit of Curie's law in the temperature range .3 K to 3.0 K. Then the CMN thermometers were interchanged and the entire process repeated. Of course if the tails were in perfect thermal equilibrium and the thermometers identical, 64 Figure 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 lo” Iqa.a. q j .4..4..J . _7_41_d_ a O . to. . 3 O 0 O 0 O O 1 0 0 O O O O . 10. o 2 O o 0 o O 0 l O O O O O . O 0 III. 0 O O o D O o O O O 0 O. o L O O O O O p_.L__L » p..p_—Lh r ...._... . O 4 3 2 O O 0 3:23 3:223". 336063.; 5320830 I K") 66 one would expect Ti/Tfi = l at all temperatures. Figure 16 shows the results of the calibration. This analysis indi- cated that some problem existed with CMN thermometers causing the temperature difference between the tails to reverse upon interchange of the thermometers. This implied that one or both CMN thermometers at low temperatures were giving incorrect temperatures for some unknown reason. However, the symmetry of these calibration experiments about the Tl/T2 = 1 line is evidence for thermal equili- brium between the tails. The results of this temperature calibration and frus- tation over the long thermal relaxation times between the CMN thermometers and the dilute solution at low tempera- tures, which limited the lowest temperatures that we could reach usefully, led us later to change to a thermometer of powdered 90% lanthanum and 10% cerium magnesium nitrate. This powder was prepared by dissolving the pure lanthanum and cerium salts in water at a molar volume ratio of 9:1 respectively. The solution was allowed to slowly evaporate and deposit large crystals of nominal 90% LMN - 10% CMN. One well-formed crystal was then ground in a mortar and put through a 45 micron sieve in preparation for being pressed into pills. These 90% LMN - 10% CMN samples were made into .953 cm right circular cylinders with diameter equal to height and had masses of .90891 gm and .90837 gm. It is known that by lowering the density of cerium ions the 67 Figure 16. Results of temperature calibrations Of the refrigerator using 100% CMN thermometers. Open circles: M1 = .55570 gm CMN; M2 = .55555 gm CMN. Closed circles: M = .55555 gm CMN; M = .55570 gm CMN. l 2 The subscripts refer to tail #1 or tail #2. 68 C: 0.. A... -qJ+. - l l l I l l l I ' I l l I I. _oo. .3. .3. N L. l .03 JH * #5.. 1N0. 69 heat capacity of the powder can be reduced to the point where more rapid thermal equilibrium between the thermometer and dilute solution could be attained at low temperatures.43 Initially a LMN-CMN pill was compared with the .55555 gm CMN thermometer. Careful measurements indicated that the CMN thermometer agreed with the LMN-CMN thermometer to within approximately .5% at a temperature of 10 mK and to within 1.5% at 7.0 mK (Figure 17). It was also very apparent that the relaxation time of the 100% CMN ther- mometer was considerably larger than the LMN-CMN thermometer. Later indirect measurements indicated to us that the .55570 gm CMN sample agreed with the LMN-CMN thermometer to 1.0% at 7.0 mK. This comparison was made by using two sets of susceptibility data obtained for a powder sample we were studying. For one set of data the .55555 gm CMN sample was used for the thermometer and the other data was obtained when the LMN-CMN was being used as a thermometer. It is not known for certain why the temperature as determined by the two CMN thermometers were in disagreement with each other, but showed little disagreement when com- pared individually with LMN-CMN thermometers. However, the combined results of these experiments indicated to us that the temperature differences we had originally measured between the two tails were possibility due to the large thermal relaxation times of the 100% CMN powder pills. This explanation is consistent with the fact that Figure 17. 70 A comparison of the mixing chamber temperature as determined by 100% CMN and LMN-CMN ther- mometers. 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. 4 '- 44~'-—1—va-v—‘fi _“V‘->T-Y."V_”" V 71 ‘ 0. 0.. 0.. .0. .00. .qqqu . a..qulJ. - d.-..- _.-JJ—__ .I 1mm. 0 0 . l O 00 Law. I. o .7 .l 0 mi 00 U o o . o T O 0.. 1 m o as. o o .+ u o o o .l. . lullllllllllllllllllololllnlIIIQIIIIO lllll 0 ...I100./ o oo o to I. r o m U I e 1.0.. o .- LN0.. ...p- P L.._»IF— . phpppp PbL-pbb - 72 * * LMN-CMN/TCMN Since it was also apparent that the LMN-CMN pills had T < 1 over the low temperature region. a relatively short relaxation time even at the lowest temperatures attainable by the refrigerator we were now confident that better thermometers could be made with this diluted CMN salt. The results of a subsequent recalibration of the refrigerator to check for possible thermal gradients between the tails using two LMN-CMN samples as thermometers is presented in Figure 18. These results, using much more reliable thermometers, conclusively show that no significant thermal problems exist due to the dual tail arrangement in spite of the presence of two phase separation levels in the refrigerator. It should be noted that most of the scatter in this high temperature data, as with the other calibra- tions, occurs above .8 K. This is very likely a result of 3He-4He mixture which are temperature fluctuations in the apparent before and during the onset of phase separation. In each case the thermal calibrations indicate better thermal equilibrium immediately after phase separation occurred than existed before. The calibration shows that to a temperature below 5 mK we can still expect the tempera- tures in the two tails to agree within 1%. In a typical experiment the conventional magnetic susceptibility coils will be operated at 1.6 gauss (20 mA amplitude current) in the region where calibration of the LMN-CMN thermometer is being done. This ensures the 73 Figure 18. Results of a temperature calibration of the refrigerator using LMN-CMN thermometers (M1 = .90837 gm LMN-CMN; M2 = .90891 gm LMN-CMN.) 74 C: 0.. 0.. .0. .00. .1..-u . - fl.dq.-. _ —uqdu~.d - -....- - J l law. 0. . O I .r 1m¢ f . . L O O. C. C ‘ it? *NL; u lllllllll ml lllllllll [or I. IL... I. l.oo._ II C C , l.. . . 0 *.rr . . .x. r O .0 .0... L O C O. . . < 50.. T I 1N0.— . p n b . - PL- p b P p P —r_ p - F P b ppp p b - p . 75 necessary sensitivity to calibrate the dilute magnetic system against the germanium resistor. The field is then reduced to a .4 gauss field (5 mA amplitude current) before continuing to lower temperatures. Thermometers made from dilute mixtures of CMN in LMN seem to be very promising as ultralow temperature ther- mometers. By increasing the nearest neighbor distance between cerium atoms in this manner one can lower the heat capacity of a thermometer so it can be more rapidly cooled at low temperatures. Also, this dilution process depresses the magnetic ordering temperature.2 It is expected that the LMN-CMN thermometer's susceptibility will obey Curie's law to a lower temperature than 100% CMN although no direct experimental evidence is available to support this assump- tion. This could possibly extend to lower temperatures the range over which such thermometers are useful (CMN is believed to begin deviating from Curie's law by about 6 mK). At the very least the comparison of the 100% CMN and 90% LMN - 10% CMN thermometers indicate that the LMN-CMN ther- mometers are no worse than 100% CMN in the temperature range over which we trust the 100% CMN thermometers. Further investigations of this material as an ultralow temperature thermometer will be carried out in the future. Referring to the SQUID position in the refrigerator as shown in Figure 10, it should be noticed that the SQUID sample chamber is connected to the siphon tube return lines 76 at a distance of about 6.35 cm from the thermometer. As might be expected, this placement results in a temperature gradient from the thermometer to the SQUID sample at low temperatures due to viscous heating of the dilute solution as it flows from the thermometer to the SQUID port. Despite this fact, the SQUID must be placed at this point due to lack of space between the vacuum-can walls and the mixing chamber. Although this thermal gradient is undesirable, it can be dealt with easily. To further understand the nature of this thermal gradient, the temperature of the SQUID sample chamber was measured against the main LMN-CMN ther- mometer by using a small sample of powdered LMN-CMN (4.44 mg) in the SQUID. Originally a sample of 100% CMN was used in the squid for this calibration, but the thermal relaxation problems which became apparent with the pure CMN thermometer at ultralow temperatures in the mixing chamber convinced us to redo this calibration with the dilute CMN powder. The results of this calibration is shown graphically in Figure 19 where the inverse temperature as determined by the mixing chamber thermometer is plotted against the inverse tempera- ture determined by the SQUID thermometer. Also, for com- parison purposes, the theoretically derived estimation of the SQUID temperature based on siphon tube impedance measure- ments (2 ~ 6.4 x 105/cm3) is shown. The calculation has been carried out for two possible flow rates. At temperatures 3 above 10 mK the He flow rate through the SQUID siphon tube 77 I r r 1 l 200- 1 C o / I/Ts‘ = I/T2 / Iso- \ / _ / o /’ / // . -6 A n3=8.5x|0 moles/sec / / / : I60” /’ /’ “ x. /’ " ./ / / / fi :3 // : I40” // .. ./ // // \Theoretical effect of 4}], siphon tube impedance 120- // (ha: 2 x no” moles/sec) « // / / x’ / 00 L l J l 1 ' IOO l20 I40 l60 I80 200 I/T’; (K") Figure 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/Tg). Dashed lines represent the theoretically esti- mated effects based on siphon tube impedance measurements. 78 is 2 x 10"5 moles/sec, but in the single-shot mode of operation below 10 mK the 3 He flow rate while the refrig- erator is cooling is reduced to about 8.5 x 10.6 moles/sec. However, even this flow rate will indicate a larger tem- perature gradient than is observed because at temperatures below 10 mK the refrigerator is brought into equilibrium to take a data point by further reducing the 3He circula- tion rate. Thus the theoretically calculated thermal gradient would overestimate this effect at the lowest temperatures. This calibration now allows one to define a SQUID temperature (T;) as a function of the magnetic tem- perature (T5) in the thermometer tail of the mixing chamber. These temperature correction equations for the SQUID are shown in Table 5. Background Susceptibility Measurements Further testing of the refrigerator in the form of background susceptibility measurements was required before any serious data analysis could be undertaken. The informa- tion obtained from the two background experiments is pre- sented in Figure 20. These background experiments showed the presence of small paramagnetic impurities in each tail. The maximum correction due to these impurities is no larger than 1.67 x 10"5 emu. Also one of the tails, labeled tail #1, exhibited a small diamagnetic shift at approximately T = 0.7 K. This is believed to be caused by the supercon- ducting transition of a minute piece of aluminum foil which Figure 20. 79 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 O r I l T I T __ r- r j .1 r - F o i O .- I— 8 C '1 o 3 o _o : 8 o 1 - O "‘ I ' ° .- . 3» y ’ 8 . O .. O .- O O Inn- 8 -' C o 32 : °o :-' v n. : .N . d E— h . q i- 0 CI . a P I— . 4 O O .. 8%, o :5 g o , - . o 0.+ r- 000? 1 l l 1 l L l 1 5 O O. o. o- N (nuns Sp”) KTngqudaosns punomxong 81 ‘Jas not machined away from the Epibond 100-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. Tuie background measurements have also been converted to cc>rrection equations which apply in specific temperature raznges. These equations, useful for computer analysis of rarw data, are listed in Tables 2 and 3. The corrections fc>r tail #1 are added to susceptibility measurements ()<1.corr. = X1 + Axl) while those for tail #2 are sub- txracted from the susceptibility measurements (X2 corr. = ‘XZ -Ax2). 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. = xs + Axs * H/2.5 where the background is assumed to be linear in the magnetic field, H. The background suscepti- bility 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 I I V T r T r I 7 -— .- -1 *- «II! P o g t 8 _ O (D a) O ... o :3 “—1 r- 0 -I b- 0 g 4 F- o d r- o ' _ O N o I i— a— A X v _O r; x'- O o - N I ~ " L— o -I O O ._ o __O_ >- (b : - " '1 .- O - L— d O - 6 L l 1 l L l L _l L 6 n q- rO N —' oO (nwa g_o| x) Mmqudaosns punmfixooa 84 Table 2: Coil #1 Background Correction Equations Background Correction (emu) Temperature Range (K- Ax1=+1.08x10'6(1/T3)+1.07x10"5 0 < 1/T5 < 1.00 Axl=+2.08x10’6(1/T§)+1.00x10'5 1.00 < 1/T5 < 1.23 Axl=+1.21x10'5(1/'r§)-2.42x10'6 1.23 < 1/T5 < 1.55 Ax1=-1.75x10’7(1/T3)+1.70x10'5 1.55 < 1/T5 < 6.40 Ax1=-1.98x10'7(1/T5)+1.78x10’5 6.40 < 1/T5 < 50.0 Ax1=-9.17x1o’8(1/T§)+1.23x10‘5 50.0 < 1/T5 < 108. Axl=-2.67x10’8(1/T§)+4.83x10’5 108. < 1/T5 < 195. Ax1=0.0 195. < 1/T5 Table 3. Coil #2 Background Correction Equations Background Correction (emu) Temperature Range (K- Ax2=+1.38x1o'6(1/T5)+1.27x10’5 0.0 :_1/T5 3 1.62 Ax2=-6.62x10'6(1/T§)+2.56x10’5 1.62 < 1/T5 < 1.70 Ax2=-9.53x10’8(1/T5)+1.46x10'5 1.70 :_1/T§ 3 7.00 Ax2=-l.43x10-7(l/T§)+1.49x10-5 7.00 < 1/T5 i 70.0 Ax2=-4.33x10'8(1/T5)+8.27x10‘5 70.0 < 1/T5 : 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/T3)-3.71x10-6 0.0 3 1/T5 3 .285 A)($=+1.07x10’7(1/T§)-3.44x10"6 .285 < 1/T5 3 .780 Axs=-3.62x10‘7(1/T§)-3.08x10'6 .780 < 1/T3 3 1.02 AXS=+1.19x10—6(1/T§)-4.66x10-6 1.02 < 1/T3 3 1.19 AXS=+5.07x10"8(1/'r§)-3.31x1o’6 1.19 < 1/T3 3 2.48 AXS=+3.42x10—8(1/T5)-3.29x10_6 2.48 < 1/T5 3 38.0 Axs=+2.61xlO-8(1/T§)-2.98x10-6 38.0 < 1/T5 3 62.5 Axs=+1.59x10—8(1/T§)-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 1/T5 Inverse SQUID Temperature (K- 1 ) Temperature Range (K- 1 ) * 1/TS * 1/TS l/T; l/Tg +.94 +.73476(1/T§) + 45.41778 444(1/T5) + 6.11116 -._l * 0.0 < l/T2 189 110. < l/Tg * < l/T2 3 110. 3 189 86 flux quanta, which correSponds to 3.4 x 10.6 emu at 2.5 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. ‘I 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. Metal complexed porphyrins44, the metalloporphyrins (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 method involves the dissociation of the two central hydrogen 87 F igure 22 . 88 x‘ 4! \ I H ,’ ’ I I B I l “ I I I, H\ 4’ V‘ \ \ a ” 7 7 B 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. 90 K Pyrrole Ring METALLOPORPHYRI N \ A . ./ /. ‘ METALLO- TETRAPHENYLPORPHYRIN 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 hydrogen 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, m§’oglobin, the cytochromes, etc. Another biologically S ignificant 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 03-: 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 chlor0phy11 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 1. s changed. This method of separating the paramagnetic ions 1'. s superior to the usual method of diluting paramagnets by 1‘ e placing paramagnetic sites at random with an isostructural mo Ilecule containing a diamagnetic ion. By increasing the 135— gand size one can change the spin density in a uniform, controlled manner while retaining a regular crystalline S"Caz-ucture. Such a process could be very useful in the study of magnetic interactions. For example a paramagnetic ion ‘:<:Vlnld be studied in a variety of crystalline structures. Tl"tis would be possible, because the introduction of different l-jLsgands at the a - 6 meso-positions will generally not $5fiar'iously affect the intra-molecular environment of the ion. Pilternatively, 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 meta110porphyrins (i.e. those with no attached ligands), the spin densities Eire 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 .1. ikely have very low ordering temperatures and must be studied at ultralow temperatures if their magnetic behavior 1 s to be determined completely. Certain of these compounds may aid in future investiga- 1i-‘—:.i.ons into the nature of the anomalous thermal contact 03:3 served between CMN and liquid 3He at ultralow temperatures . Th is phenomena is believed to be due to magnetic coupling be tween the liquid 3He and CMN at the liquid-solid inter- face. Other surface studies would also seem feasible be cause many porphyrins can be sublimated at about 300°C without destroying their molecular structure. The con- t~J':'c>11ed sublimation of the surface layer of a crystalline IP<3rphyrin 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. 94 £3. COpper a, B, y, 6 - Tetraphenylporphine IntroductogyiRemarks Copper Tetraphenylporphine (CuTPP) is one of several metallo-tetraphenylporphyrin and metallOporphyrin c3<3mpounds whose magnetic susceptibility has been measured turzith our apparatus. Magnetic susceptibility measurements along crystalline axes are invariably necessary if one is to obtain useful results. Since CuTPP was the first E><=>rphyrin which was successfully grown as a single crystal, i—it:s analysis was completed first and the results are be ported here. The highly anisotropic hyperfine coupling E-‘-3=~=:hibited by the c0pper ion makes CuTPP a useful porphyrin v"'~‘:iL.1:h which to initiate a series of investigations into the “l:l~‘tralow temperature magnetic properties of the porphyrins. urlhlnis anisotropic coupling dominates the magnetic suscepti- lDijLLlity along the a and b crystalline axes at low tempera- tures, while the intermolecular dipolar coupling dominates ‘tfikle susceptibility along the c axis. Thus these effects on ‘Zlne susceptibility can be observed approximately indepen- €1ent1y 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. £3imilar procedures apply to the preparation of any of the ‘Eyorphyrin 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 c>u15 these compounds undesirable to many researchers. We are grateful to Dr. Richard Yalman and Mr. Gordon Comstock of Arxtioch College for supplying us with our first porphyrin Many of the porphyrins are now available from S amples. However, the method by which the S":rem Chemicals Inc.47. chper 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 chper acetate in 50 ml of glacial acetic acid. The J:‘esulting 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 izhe form of tars. The presence of these tars is undesirable, trot only because of their possible effects on magnetic suscep— t;ibility measurements, but also because they may present problems during crystal growth by providing too many nuclea- t;:ion sites. One of the major problems encountered while ‘tzzcying to grow large single crystals of CuTPP is that many ITLjLCIOCIYStaIS invariably grow instead of a few good single czczrystals. The removal of these impurities was found to be i_111portant, although their removal did not guarantee good c=Jcfiysta1 growth. The manner in which the CuTPP powder was E>1;l:rified in preparation for recrystallization is as follows. The sample was first put into solution with benzene. As vVii—1m most porphyrins, the solubility of CuTPP is rather low iilnl many solvents. This means that simply mixing a solvent w:i~‘l:.h CuTPP may not be sufficient to dissolve enough material t3<3> be of use. In order to increase the concentration of the SQ Jution, it is refluxed for several hours in a Soxhlet e3-“’=:‘t:.ra.ction apparatus. A rough rule of thumb for the porphyrin <=<3mnplexes is to use no more than 1 gram of the salt per liter C>if solvent. Generally, it is convenient to work with cI‘uantities 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. .ZXsssuming 250 ml is the original volume of the solution, it Should now be concentrated to about 60-80 m1. One way to determine if the solution is concentrated enough is to look if<:>r the presence of small crystals just above the surface of title liquid on the walls of the flask. When this crystalliza- t:j_on occurs, the solution should be removed, a small amount of benzene added, and allowed to cool. The solution can tflaen 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, ‘NhiCh 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 c0pper 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 gg‘gl.48 have concluded from their work that the presence of the c0pper 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 IO 9 Figure 24. The structure of c0pper 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 C and C10 carbon atoms makes an angle 8 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 studies of CuTPP and its p-chloro derivativeso'SI. These measurements tend to indicate that the phenyl rings are electronically coupled with the porphyrin ring resonance structure. The Cu2+ ion is a 3d9 transition metal system (effective 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 Bend the Cu ion length (R) (K) Cu 0.00 Cu-N LS5? 2: .0l3 N -.04 N-C, L396 : .0I3 C. .26 N-C‘. 1.396 1' .0l3 02 .23 C.- C2 L427 1' .Ol6 03 -.09 C.‘,--C3 |.335 t .023 C4 -.24 03-04 L427 1' .Ol6 C5 -.42 C‘— C5 L398 2': .013 Cs —.75 C7 -I.42 103 Inolecular 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’ dx2_y2, and dZZ orbitals are situated so the four lobes of the dx2_y2 orbital are pointing towards the four nitrogen atoms along the x and y axes. This allows the ax2_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 0 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 dzz 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,il,il) and (il,0,il) 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=15.03:.01;; c=l3.99:.01§. 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 35:. 335;“. otezuw otoauw .otnmco.oo 513. III 1.5. II III 1.3:? I; .2: «N // \\ .// / \\ / / \ / / \ / /\\ >x // \/ \ \ / N / \ om \ u \ \\ // \ \ // \\ xx \ / .oo. \ \\ 6.7.»wa \ \ \ \ \ «alux \ \ 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 c0pper 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 21 spins/cm3. The four-fold axis of a crystal 1.27 x 10 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 51 to diamagnetically shield one are believed by J. Assour 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 represents-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: 3M 0 x0“) = T; where 0 represents a component of the field. The magneti- zation of a quantum mechanical system can be expressed as: _ .i__ (Ma) — 3H0 (in Z) 03H <| where 8 = 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 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 + s-g-g +n£1 s-gn-gn + D{sz - 1/3s(s+1)} 2 I + Q{IZ - l/31(I+l)} + 3 EN H + 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 c0pper 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=k) and to couple with an external magnetic field via an anisotr0pic tensor (S-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 gi, 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: H =g||11 SZHZ+giu(SxHX+S Zee Hy) Y where u is the Bohr magneton. 115 Another term which arises from this perturbation analysis of Spin-orbit coupling is the crystal field term (D{S: - l/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/3I(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 ch :5 RD 0 3r 116 P. T. Manoharan and M. T. Rogersss, is Q 3_4 x 10.4 cm-l. This corresponds to a temperature T 3 .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: W=§98+§84+ £9455 +§'W 1 13 ~ 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, S-A-I represents the c0pper electronic spin-nuclear spin 56'57i.e.t.he hyperfine coupling. In the same interaction, manner 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 (Z S-An-In). The magnitude of the interaction energy, on n=1 7 7 ~ 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). 59 show that it reduces the The results of this calculation zero-field parallel susceptibility with the reduction being of order XII Afi/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 380 has also been estimated. It is found that this ..U-n.v>l.-—p:v.- as .hflJcmd-naHeHd nun-u Cuba-Ninevc-hfi MUP~J~ LE...» .L-b-dfi aw) c~tqflNfiviin~i ,h-h |\. ut‘fi‘u..- 118 36. H CLO. H “50...... I. I I. 50m)(. Tmomv Nmotm. mkfiN. 0.3 _.o_ 9! on Nd.“ 308 me; In elm as .n. 4 4a: 3m ._ch.r0_xv 23.0.2»: .750 .o. x. . 780 To. .3 223525 2:32:03 2:22:03 3227.0 3232 65232.3; 05.32.: 23:36 .mmBDU MO mcoflummwuno>cfl Moo oceuso omsHmuoo ouoz memonusoumm cw mosam> 0:9 mm uno>0Hou whomcmu cofiuomnouse may no mosao> mameooflum moxm onwaamumhuo mop macaw mmasu on .5 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 c0pper ions. It is composed of both the classical dipole-dipole coupling and exchange coupling. Even the large nearest neighbor distance (8.4;) 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 c0pper 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: ~ where j is summed over the crystal lattice and "i is a vector whose components are gk Ski (k = x,y, and z). The manner in which this summation is carried 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: H = gllu Ssz + giu(SxHx + SyHy) + A8212 + B(SxIx + S o Iy) Y 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 . . gfluz [e-BA + §_2 {3g SinhBa + coshBo} + Slnh B] _ 20 A 80 BB (x°)|| 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 g u Curie constant decreasing by about 7% from ll 2 2 4k 4k l+382/A in the high temperature limit to 2) in the low temperature limit. The zero field susceptibility perpendicular to the c axis was determined to be: 2 2 91? [%2. e.BA + 2 coshBo) X - °i_ 40(B2-02)(e BA + 2 coshBa + coshBB) (Aigégi) sinhBa - A0 coshBB + 2B0 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 cA X=C/T (1+? +000) 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— _£L. T-0 order. In practice the dipole lattice summation is often ture expansion of the Curie-Weiss law (x = ) to first 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: XII = (x0)ll {l + (Xo)'| All + ...} X1: (x0)i {1 + (xo)L Al- + ...} Z2 - r2 X2 - r2 _ ' ij ij _ ' ij 11 where All — 2 ( ) and AL — Z ( ) are the j 5 j 5 r.. r.. 13 13 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 Parallel Axis —-> l I I ’ I .o/ /\ 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 14°. 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 9" = 2.175 1 .017 and gi,= 2.052 3 .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 f 4 1 and B ~ 30 x 10"4 cm'l. Since the 14.2) x 10' cm” 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 9" = 2.179 and gl = 2.033 for the electronic g-values and A = 212.2 x 10.4 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 . r. T ' A A O A A 40r o - O O O 30" J I: o .1 o 2 \ a 2 E ‘ .. i 20 r A _______ 9.- 3 - 3 3 o 0 K“ .0.. 6 mg Single Crystal .4 o l25 mg Single Crystal 0 Powder 0 L J l A 0 50 I00 I50 200 INVERSE 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 91.: 1.963 1 .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 gll by using the high temperature data measured on the 1.25 gm crystal was 2.177 t .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 crystals62. 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 (Hloc) 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 macrosc0pically 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 éE-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 60 as: _ _ I H10c — Ho + (4n/3 N + A )M 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 + (4w/3 - N)xext 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: 136 w II H. + 4nM 1 (1 + 47rxi)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: tn ll Ho + (4n-N)M (1 + (477-N)xext)Ho Now for steel the internal permeability (pi = 1 + 4nxi) is much greater than 1 so combining the two expressions for the internal magnetic induction and applying the condition l/ui I 0 will yield the result: H .9. M N~_l_.—_ Xext 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. 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: M=__Al. 4nan where A0 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/Z f = (82 + (L/2)2)35 where L is the length of the coil and R is its radius. Two pickup coils were used during the measurements. One coil was a mockup of a section of the SQUID sample coil to the same scale as the crystals. The other was a much larger coil (R = 19.9 cm) in which the diameter of the sample was much smaller than the coil diameter. This con- dition is assumed to hold for the above equation which relates magnetization to the flux change produced by a sample of volume V. For a spherical sample, if these conditions are valid, one would expect N = 4n/3. Therefore a measurement of the demagnetizing factor for the steel 138 Sphere provides a means of calibrating the pickup coils. In order that our susceptibility correction equation be valid for the susceptibility of CuTPP as measured in emu/mole we may define the following relationship: 8 = (40/3 - NID/m where m/p is the molar volume of CuTPP (p = 1.43 gm/cm3; m = 676.8 cm3/mole). The susceptibility of CuTPP corrected for crystal shape effects is then: Xexp Xloc (emu/mole) 1 + e xexp For the large pickup coil it was found that for the sphere Es = -.0003 while for the small coil, for which the assumed conditions are not valid, but more closely approxi- mate the experimental arrangement for measurements on CuTPP, we found Es = -.0013. These values could now be used to correct all other measurements to satisfy the condition Es = 0. The corrected values of 8 measured for the parallel axis direction of the 1.25 mg crystal was all = -.0013 t .0003 for both coil measurements and the perpendicular axis measurements gave CL = +.0027 for the large coil and + .0023 for the small coil, for an average value of 61.: +.0025 i .0005. All measurements were first with the external field in one direction, then again with the field reversed. This was necessary to ensure there would be no problems due 139 to residual fields in the steel samples. The differences in these measurements were not significant. The magnetic susceptibility of CuTPP corrected for crystal demagnetization effects can now be expressed as: exp x z Xll ll 1 -.0013 xlleXP and Xiexp X = i 1 +.0024 xi°Xp where the susceptibilities are in emu/mole. The correction was also estimated for the fractured .6 mg CuTPP crystal by machining a steel mockup crystal which approximated the shape of this CuTPP crystal as closely as possible. For this shape ell was found to be —.0022 i .0003. The uncertainty in this result is small because it does not reflect the difficulty of correctly reproducing the irregular shape of the fractured CuTPP crystal. However, upon application of this correction to the experimental data one finds that the .6 mg data even more closely follows the 1.25 mg data to low temperatures. This is evidence that the deviations of the original data obtained from each crystal differed because of the effects of demagnetization of the different crystal 140 shapes. A plot of the magnetic susceptibility of CuTPP corrected for demagnetization effects is shown in Figure 31 page L44,and is tabulated in Appendix D. Also shown on this graph is the theoretically calculated susceptibility including the contribution from the dipolar coupling term. The powder susceptibility measurements have not been corrected for demagnetization effects. These corrections are very difficult to estimate for powder samples. Attempts have been made to measure demagnetization effects in powdered right circular cylinders of CMN with diameter equal to height37-39. These measurements were mentioned with respect to thermometry. One might expect the corrections for a similar shaped CuTPP powder sample to be even smaller because of the more isotropic nature of a CuTPP powder grain as compared to CMN. It will definitely be a much smaller correction than is necessary for the single crystal measurements . Thermal Equilibrium of Single Crystals Another serious problem which arises when one attempts to study single crystals at ultralow temperatures is the difficulty of maintaining thermal equilibrium with the dilute solution. The very small crystals employed in the single crystal SQUID measurements allow thermal equilib- rium to be maintained to much lower temperatures than is Ipossible for the large crystals which would be required in 141 the conventional magnetometers. However, due to the rapid increase in Kapitza resistance for heat flow across the crystal surface to the dilute solution, even these crystals eventually reach a temperature below which thermal equilib- rium is no longer possible. One can estimate the relaxation time necessary for thermal equilibrium to be achieved between the crystal and dilute solution. Assuming the Kapitza resistance to be on the order of 10-5/AT3, approximately valid for many dilute solution-solid interfaces, the relaxation time will be: where C is the heat capacity of the crystal. The heat capacity of the 1.25 mg crystal was estimated assuming the entire contribution is due to the hyperfine coupling (i.e. ignoring the dipolar interaction). In the temperature range around 10 mK it was found to be about 5 x 107 ergs/mole—K. An estimation of the 1.25 mg crystal's surface area based on the crystal's dimensions as measured by a travelling micro- scope indicates that A ~ 5.2 x 10-2 cm2. Thus at 14 mK one would expect a relaxation time for this crystal on the order of two hours, but at 10 mK it would increase to 6 hours. This is consistent with our experimental results. At 14 mK the crystal's thermal relaxation time is on the order of the relaxation time of the refrigerator (see Figure 7 and Appendix A), but at 10 mK it has become much larger. Our 142 experimental data indicates the powder data do begin to deviate from the effective powder susceptibility as deter- XII 2XI = _._ + ) at 3 mined by the single crystal data (xp 3 about 14 mK (Figure 31). The data obtained with the .6 mg ow crystal, which has a smaller heat capacity than the 1.25 mg crystal, indicates thermal equilibrium with this sample was maintained to approximately 11 mK. This is verified by observations of a chart recorder which continuously monitors the SQUID sample response and LMN-CMN thermometer response to changes in temperature. No indication of unusually long relaxation times were noticed at temperatures above 14 mK. It should be mentioned that measurements on other crystals63 have shown it is possible to carry some single crystal measurements to below 10 mK. The value of the SQUID magne- tometer as a tool for studying crystals at ultralow tempera- tures is very apparent from these results. CHAPTER IV AN ANALYSIS OF THE EXPERIMENTAL RESULTS A. Comparison of Theory and Experiment The experimental magnetic susceptibility data corrected for demagnetization effects is presented in Figure 31 which is a plot of x in emu/mole vs. the inverse magnetic tempera- ture (l/T*). The uncertainty in the parallel and perpen- dicular susceptibility data is represented by error bars. Some of this uncertainty arises from the posSibility of crystal misalignment within the SQUID sample holder (i4°). Other sources of error from which these error bars are derived include uncertainties in the measurements which are necessary for the determination of the demagnetizing factors, and in the approximation 1/11i = 0. A probable major contribution to the uncertainty in this data, not included in the error bars, concerns the reliability of these demagnetization corrections in their application to the CuTPP crystal. It was assumed that measurements on a nearly isotropic medium, steel, can be directly applied to an anisotropic medium such as CuTPP. Error bars on the parallel susceptibility data below 14 mK on the 1.25 mg crystal data and below 11 mK for the .6 mg crystal data are no longer useful indicators of the accuracy of this data due to the onset of systematic errors arising from the lack of thermal equilibrium at these temperatures. 143 144 50 ' r a t A A A A A 40* - 30" «A a .J O 2 \ D 2 E >EZO’ d X IO" .1 O L l .L A 0 50 :00 :50 200 250 INVERSE T‘ (K") Figure 31. 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. 145 The susceptibility perpendicular to the crystalline c axis becomes approximately temperature independent at 10 mK. The uncertainty in this data due to the previously mentioned sources is .3% at about 20 mK and increases to .6% at 10 mK. The basic behavior of this susceptibility can be understood from the hyperfine interaction between the copper electronic and nuclear spin. The theoretical perpendicular susceptibility including the contribution from the dipolar-interaction lattice summation has been plotted in Figure 31 for comparison with the experimental data. The theory rises above this data below about 30 mK, following the data but remaining above it to the lowest temperature. Estimations of the effect of the nuclear quadrupole interaction in the approximation B = 0 indicate it will increase the XL theory by approximately .4% at 20 mK and as much as 1.5% at 10 mK. The contribu- tion from the superhyperfine interaction in the same approximation will reduce this theory by about .7% at 20 mK and 2% at 10 mK. Remembering that the estimation of the quadrupole interaction effect was carried out assuming the maximum value possible for Q, one can say that at the least the overall effect of these terms will be to decrease the X1 theory by about .3% at 20 mK and .5% at 10 mK. The susceptibility parallel to the crystalline c axis continues to increase in a Curie-like manner to the lowest temperatures we were able to attain. Estimations of the 146 expected uncertainty in this data indicate they are uncertain to within 1% at 20 mK, increasing to approxi- mately 1.5% at 12 mK. The theoretical parallel axis magnetic susceptibility is probably valid to approximately 15 mK. Below these temperatures it is possible that contributions from higher order terms in l/T will become important and should be considered before this theory can be extended to lower temperatures. Calculations of the effects of the superhyperfine and nuclear quadrupole interactions indicate that this theoretical XII should be reduced by about .4% at 20 mK and as much as .6% at about 14 mK in the approximation B = 0. The entire contribution is a result of the superhyperfine coupling, the quadrupole interaction giving zero contribution in this approximation. The possible correction of this theory due to the presence of the superhyperfine interaction is indicated by arrows in Figure 31. Of course in the actual case, where B # 0, one would expect this effect to be somewhat modified from our calculation. However, we do not believe this modifica- tion would be large. The manner in which B affects XII was shown in Figure 30. The effective Curie constant was changed by -7% from the high to the low temperature regimes. The superhyperfine coupling is an even higher order effect at these temperatures. The maximum deviation of the theory and experimental data for the parallel susceptibility, not including the estimated effect of the superhyperfine 147 coupling, is about 1.3% down to 16 mK, the theory lying slightly below the experimental data. This maximum deviation occurs at about 30 mK. The importance of the temperature independent behavior of the perpendicular susceptibility at low temperatures in estimating the behavior of the parallel susceptibility to even lower temperatures will now be made apparent. As has been mentioned previously, the powder sample is expected to be in good thermal equilibrium down to 4 mK. If we form an appropriate summation of the parallel and perpendicular axis crystal susceptibility (x = XII/3 + 2 xi/3), we can pow compare the experimental powder data with an "effective powder susceptibility" as determined by the single crystals. For this comparison it was necessary to normalize the XL data so that the experimental and effective powder data were equal over the temperature range of 3 K to .3 K. This is tantamount to requiring the condition ggow = gfI/3 + Zgi/B to hold at high temperatures. The perpendicular 9 value was adjusted to 2.001. Figure 32 shows the susceptibility data over this temperature range. The results of this summation is shown in both Figure 31 and Figure 32 where +'s indicate the 1.25 mg parallel axis crystal data were used and x's indicate the .6 mg parallel axis crystal data were used. In both instances the perpendicular axis data were obtained from the 1.25 mg crystal. The effective powder data begins to deviate from the measured powder susceptibility at about Figure 32. 148 The high temperature single crystal and powder magnetic susceptibility data. The 9 values determined from this data are also shown. The "effective powder susceptibility" was obtained from the crystal data after normalizing the perpendicular susceptibility to 91,: 2.001. 149 7-x. ah mmmm>z_ 0.m 0.N . 0.. 0N. d . . H . 1 q . H a - .1339... 0333;: IV. .. X A w 16 a W n l 1 W 0 1. .3 Susan. 0 1m ( .0.»...0 0.9.5 OE mm. o L .3230 6.9.5 as w. d 10.. IN... 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 XII. 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 i 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 1/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 XL theory was observed. In order to preserve the low temperature behavior of X1 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 x data points in the tem erature ran e around II P 9 153 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 recognize that the maximum uncertainties in our tem— perature measurements (i1%) 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 i 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 X1.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 i 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 3He 3 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. Even if this were possible, the lack of thermal equilibrium at relatively high temperatures would make an attempt at studying large crystals useless. The success of the single crystal studies of CuTPP using the SQUID magnetometer has shown it is possible to study very small single crystals at ultralow temperatures. LIST OF REFERENCES 10. 11. 12. 13. 14. LIST OF REFERENCES . A. Fisher, E. W. Hornung, G. E. Brodale and . F. Giauque, J. Chem. Phys. 33, 5584 (1973). R W M. Kolac et al., J. Low Temp. Phys. 3;, 297 (1973). B. M. Abraham et al., J. Low Temp. Phys. 31, 387 (1974). J. H. BishOp et al., J. Low Temp. Phys. 39, 379 (1973). H. Kojima, D. N. 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Wheatley, Progress in Low Temp. Physics Vol. VI, C. J. Gorter, Editor (North-Holland Publishing Co., Amsterdam, 1970) p. 77. W. E. Keller, Helium 3 and Helium 4 (Plenum Press, New York, 1969). H. E. Hall, P. J. Ford, K. Thompson, Cryoqenics g, 80 (1966). O. E. Vilches and J. C. Wheatley, Phys. Letters 225, 440 (1967); 233, 344 (1967). J. C. Wheatley, Am. J. Phys. 3g, 181 (1968). J. C. Wheatley et al., J. Low Temp. Phys. 1, l (1971). Lansing Research Corporation, 705 Willow Avenue, Ithaca, New York, 14850. L. E. DeLong et al., Rev. Sci. Instr. 5;, 147 (1970). J. C. Wheatley, O. E. Vilches, and W. R. Abel, Physics 3, l (1968). H. Sheinberg and W. A. Steyert, Los Alamos Scientific Laboratory Report LA-4259-MS. G. L. Pollack, Rev. of Mod. Phys. 1;, 48 (1969). Norton Vacuum Equipment Division, Newton, Mass. Furane Plastics, Inc., Los Angeles, Calif., 90039. A. C. Mota, Rev. of Sci. Instr. 33, 1541 (1971). R. J. Commander and C. B. P. Finn, J. Physics E: Sci. Instr. 3, 78 (1970). 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 159 A. C. Anderson, R. E. Peterson and J. E. Robichaux, Rev. of Sci. Instr. 2;, 4 (1970). T. I. Smith, J. Appl. Phys. 11, 852 (1973). B. D. Josephson, Physics Letters 3, 251 (1962). SHE Corporation, 3422 Tripp Court, Suite B, San Diego, California, 92121. 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. g, 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 3;, 597 (1968). R. A. Webb 22 gl., Phys. Letters, 215! l (1972). CR-lOO 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 §£_§l,, J. Res. Natl. Bur. Std., £35, 579 (1964). Abraham ggflgl., Phys. Rev. 321, 273 (1969). J. E. Falk, Porphyrins and Metalloporphypins (Elsevier Publishing Company, 1964) 160 45. P. Rothemund and Amel R. Menotti, JACS Q}, 267 (1941). 46. P. Rothemund EE.El-' JACS lg, 1808 (1948). 47. Strem Chemicals Inc., 150 Andover Street, Danvers, Mass., 01923. 48. E. B. Fleischer et al., JACS fig, 2342 (1964). 49. E. B. Fleischer, JACS 33, 1353 (1963). 50. D. J. E. Ingram et al., JACS 8, 3545 (1956). 51. J. M. Assour, J. Chem. Phys. 43, 2477 (1965). 52. R. McWeeny and B. T. Sutcliffe, Methods of Molecular Qpantum 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) gagg, 135 (1951). 57. B. Bleaney, Phil. Mag. 3;, 441 (1951). 58. P. H. E. Meijer, Phys. Rev. B, Q, 214 (1971). 59. W. P. Pratt, Jr., Michigan State University, private communication. 60. J. H. Van Vleck, J. Chem. Phys. 3, 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 Vol. VI(North-Holland Publishing Co., Amsterdam, 1970), p. 243. A. H. Silver and J. E. Zimmerman, Phys. Rev. 331, 317 (1967). J. E. Zimmerman, P. Theine, and J. T. Harding, J. Appl. Phys. 3;, 1572 (1970). R. M. White, Qpantum 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- lation 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 c0pper 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 4)) l2 > R0 /( ‘VVVL RC0 II 5 Rd Rd V AwJ LAA RK RK TAIL a"I 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 3 of He through the mixing chamber has been ignored in this 3He atom within calculation. The time necessary for the 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 22.1n1 £2n2 Cd =(A +A)(f<—) 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 Rz/A2 are the length to cross-sectional area ratios of the different regions connecting the two The number of moles tails. of dilute solution present in each region (n1,n2) must be estimated. n the 3He concentration (x3 = fi-:%—' 3 4 At low temperatures ) in the dilute solution is small (~ .064) so we can approximate the number of moles of dilute solution by its maximum valuezs: n : XL. 1 v4,o and identically n :vi 2 3 V4,o where v4 = 27.6 cm3/mole is I of 4He at T = 0.0K and V and 1 the limit of the molar volume V2 are the volumes of regions 166 l and 2, respectively. Combining these results gives us an expression for the relaxation time for heat flow through the liquid: T = (-—) (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 data17. 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: T = (2RK + R B Cu)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 RR = 1.5 x 10‘5/AT3 where A is the contact area. The resistance to heat flow within the copper is: R = i/A'K Cu Cu where z/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 T = (2.11:10‘5 5.6x1o’5) C B 3 d T T 168 Since the two paths are in parallel, the total relaxation time for heat flow between the two tails can be found from H|H HIH %+ A B 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, w(r,t) = wo(r,t) eie, where IwO(r,t)I2 is the supercurrent charge density and 0 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 100p must be 2nn, where n is an integer. This implies a quantization of the total flux (¢) trapped within the 100p, and indeed this flux is found to be quantized in units of 00 = 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 (A0) 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 i = ic sin 0 where 1C is the critical supercurrent for the junction. It can be shown that the total flux change through the 100p is related to the external flux applied perpendicular to the loop by the expression65: Lic sin %3 (¢+k¢o> = ¢-¢ ext where koo = 0, i 00, i 200 °°° . This reSponse function follows from the quantization of the integral of the canonical momentum of a C00per pair around the 100p (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 ZflLiC/¢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 100p. In order for this device to be useful, one must be able to measure the flux changes introduced into the supercon- ducting 100p from the magnetized sample via the signal coil. 171 Figure 8.1. The response function of a SQUID magnetometer. The total flux (0) through the superconducting 100p is plotted as a function of the external applied field. 172 95.. 173 This is accomplished by inductively coupling the 100p 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 100p will set up an Opposing field by the creation of a supercurrent within the 100p. 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 loop. 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 100ps 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 00. One can therefore utilize this prOperty to measure the dc flux change through the 100p. APPENDIX C 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 . x y. y z 2. H. =g||USz Hz+gJ-IJ(SX . l 1 1 l l . l + B(SX I + S I ) + 2 H1 1 J. yi yl J J (U-‘U ) - 3(u.-r..)(p,.;..) where H., = ~1 ~J ~1 ~13 -3 ~13 1] 3 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, H0 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: Z = tr e-BH 2 (Ho + H + H = tr e-B i i i i and the zero-field susceptibility is obtained from the well-known expression: 2 xa = lim %V a ginZ) H +0 3H d. a where a = x, y, or z axis. In practice, one finds that it is impossible to find a ba51s 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 CI 0. ila where u is the Bohr magneton and ga 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: 0) la 9 Q In order to see how this expansion may be carried out, -B(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) 3 e'BAMB) 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: - + ¢ = eBA e B‘A 3’ 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: dgéB) _ _eBA B e-B(A+B) -B(B) ¢(B) where 8(6) = eBA B e-BA. Integrating this equation and recOgnizing that ¢(O) = 1 will give the result: 8 ¢(B) = 1- IO B(S) ¢(S) ds 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 Hz and Hd are i 1 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 Ho . The small term approxima- i tion is valid for the zero field limit because ”2 is a i 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. ”Z = gau Ha Sa ) the expansion can be shown to yield: 1 i tr[e-B§(Hoi + Hzi + Hdi) Z S ] . (1. (ZS >= 1 l 1 “1 -BZ(Ho + H2 + Hd ) tr e i i i i -82 Ho 8 = (i ) tr e i i { - 2 pH I ds S (s) S Z go a O a a 0 ij j i 3 s + E g uH f ds I ds' (S (S) H (S') S ijk “ “ ° ° 0‘3' dk a1 + Hd. (3) Sc (3') 50.) + ... j k i where Sai (s) = e Sui e , etc. and 20 = tr e 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. 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. 0(Ha) instead of 0(H:)). Once this expansion has been Obtained, the remainder of the calculation involves evaluat- ing the trace over the eigenstates of H . The eigenstates o. 1 of “O can be constructed from linear combinations of the i well-known eigenstates of S2 (I i ms, t mI>, where ms = 1 it, mI = 13/2, i%). First the hyperfine interaction is written in terms of raising and lowering operations and its matrix elements in this basis are determined: and [5(S+1)-ms(msn)]l5 Im i1,mI> Si Ims,m > s I 8 I: lms,mI> = [I(I+1)-mI(mI:1)] lms,mI:1> From an appropriate arrangement of this basis one can form three 2 x 2 subspaces and two 1 x l 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: <¢|Ho.l¢> = where: |1/2,3/2> ~9- ...- I 2 { 3B ¢2 = 1 —Z— ;-l/2,3/2> + (a+A/2) |l/2,l/2>} /2a(a+A/2) 2 ¢3 = 1 { (a+A/2) |—1/2,3/2>-3%—- |1/2,1/2>} Ga(d+A/2) ¢4 =.7%: { |—1/2,1/2> + |1/2,-l/2>} 2 182 ¢5 - /:l { |-1/2,1/2> - |1/2,—1/2>} 2 1 332 ¢6 = { (a+A/2) l-1/2,-1/2> + T |1/2,-3/2>} /2d?a+A7§3 1 3132 ¢7 = { T |-1/2,-1/2> - (oH-A/Z) |1/2,-3/2>} /2dTO+A72) ¢8 = I‘l/2,-3/2> and y = A/4 a = A; (A2+3B2);5 One can now utilize this basis by expanding the formal trace Operation into a sum over the eigenstates of H01. 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 Operation567. For each calculation, the perturba- tive term will separate into a product the original unper- turbed susceptibility (Xoll) and (x0 ) , and a lattice summation of the dipole interaction over the crystal (AI' and AL). 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. Peverely68. 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-lOO 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 OOpper 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.1. Results of the CR-lOO Germanium Resistor Calibration. * Inverse T Resistance 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 11,464.0 2.457 732.1 .857 12,977.0 2.529 783.5 .894 14,706.0 2.602 839.6 .930 l6,661.0 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 Single Crystal 1.25gmg Single Crystal * XII Inverse T XII Inverse T (emu/mole) (K—l) (emu/mole) (K-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 * 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 35.497 233.840 188 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 * i: Inverse T XII Inverse T XII (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 (Rd) .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 Single Crystal 1.25 mg Single Crystal 353‘ Inverse T* :33’ 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 2.256 .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 v 0"? ‘JtHVE-RS‘T‘ (IL MICHw -‘ 3’ DE" ARSWIL‘“ EAST LAN " I 1”” J) 2' M|CH|GAN LL __. "7111111111111“)11))“