Nuclear spin-lattice relaxation time in copper below 10−20 K

NUCLEAR SPIN-LATTICE RELAXATION TIME IN COPPER BELOW 10- °K A. DUPRE, G. P I T S I d A. V A N ITTERBEEK, and H. VREYS~

T H E nuclear spin-lattice relaxation in metals is governed primarily by the interaction of the nuclear spins with the conduction electrons. The corresponding relaxation time has been calculated by Korringa, 1 who gave an expression for T which is inversely proportional to the conduction electron temperature and to the square of the Knight shift. The theory has been modified subsequently by Pines,2Redfield, a and Hebel and Slichter. 4 Experimental ~.ork on copper at low temperatures has been done by Anderson and Redfield 5 in the liquid helium temperature range and by Kurti 6 and others in the adiabatic demagnetization temperature range down to 1.2 x 10-20 K. The first authors used resonance techniques for the determination of the relaxation time, whereas the method of the Oxford group was based on a nuclear cooling of the copper spin system. Also in the investigations reported here a nuclear cooling method was used. The nuclear spin system was precooled by rotational demagnetization of CMN (cerium magnesium nitrate) which is highly anisotropic and which has only small interactions, so that temperatures down to a few millidegrees can easily be attained. The nuclear stage consisted of a bundle of copper wires in thermal contact with the salt. Copper was used as a nuclear sample because of its large nuclear susceptibility. Both salt and copper were located in the pole gap of an electromagnet, which could be rotated for precooling the salt and removed after the nuclear demagnetization. The spin-lattice relaxation time was determined by recording the decay of the nuclear susceptibility of the copper after the final demagnetization. The susceptibility was measured by means of an electronic mutual inductance bridge, the output of which was detected and recorded. A ballistic method was used for the determination of the susceptibility of the salt. The corresponding temperature was calibrated in the usual way by comparison with the vapour pressure of the liquid helium between 1° K and 4"2° K. The nuclear spin temperature was not calibrated; it was calculated using the theoretical value of the nuclear spin susceptibility and the sensitivity of the mutual inductance measurement. The salt temperat Aspirant bij het belgisch Nationaal Fonds voor Wetenschappelijk Onderzoek. :~Stagiair bij het belgisch Interuniversitair Instituut voor Kernwetenschappen en bij het Nationaal Fonds voor Wetenschappelijk Onderzoek. Received 22 May 1967. 336

ture was measured for some time after the recording of the nuclear susceptibility decay and was extrapolated to an earlier instant. The internal field was determined by means of a method used by the Oxford group2 A series of nuclear demagnetizations was performed from identical initial conditions to different non-zero fields. This way of measuring the internal field has the advantage that the nuclear temperature needs no calibration.

Review of Theoretical Results The spin-lattice relaxation in metals is possible mainly by interaction between the nuclear spins and the conduction electrons. Korringa I has calculated an expression for the relaxation time z corresponding to this relaxation mechanism as a function of the Knight shift A H / H [AH~ 2

r(---~)

h

~,~

= 4nk-----T'7~

. . . (1)

z is seen to be inversely proportional to the conduction electron temperature T and is field independent. Pines 2 has modified Korringa's theory by taking into account electron correlation effects. In deriving expression (1) the assumption is made that the magnetic field is large compared with the internal field, which describes the dipole-dipole interaction. Redfield3 and Hebel and Slichter 4 have extended the relaxation theory to include low field relaxation. The field dependence of the relaxation time can in a first approximation be written as: z(H)

=

H2+ C ZKH2+ 2C

so that the relaxation time in high fields (rK) is to be compared with twice the relaxation time in low fields.

Experimental Details The electromagnet was capable of producing a field of 21 kG in a 10 cm gap. The gap diameter was 25 cm, which was sufficient to have both the salt and the copper between the pole pieces, even when the two stages were far enough apart for the temperature measurement to be C R Y O G E N I C S • DECEMBER 1967

/

) Figure 1. Schematic representation of the sample consisting of 6 monacrystals of CMN and copper wires

carried out without a serious mutual influence. The magnet could be rotated about a vertical axis to perform the precooling, and translated over a few metres so that one could get rid of the remnant magnetization when recording the nuclear susceptibility decay. The ordinary glass dewars for nitrogen and for helium at 4"2° K surrounded the actual metal cryostat which contained helium pumped to about 1"15° K. Many precautions were taken to avoid a large stray heat leak to the sample. The top of the dewar carried a large weight and was suspended from the ceiling by means of springs to avoid vibrations in the sample. A radiation trap was built in the pumping line. The sample was surrounded by a perforated brass shield which was cooled by some ferric ammonium alum and acted as a trap for the residual gas. All these arrangements made it possible to bring the stray leak down to about 2 ergs/min. The CMN sample consisted of 6 monocrystals. Naturally grown crystals of CMN have a disc-like shape, with the main surfaces perpendicular to the trigonal axis. The growing of the crystals was stopped when they were about 4 mm thick and 30 mm in diameter; they were ground and stuck together with their crystalline axes C R Y O G E N I C S • D E C E M B E R 1967

parallel to each other so as to build up a sphere. A 2 mm hole was drilled through the crystals and a perspex rod of the same diameter, which was the only support of the sample, was inserted. Although the length of rod between the CMN and the iron ammonium alum cooled shield was only about two millimeters, the small heat leak mentioned above could be maintained. Copper wires of 0" 1 mm were stuck in between the four largest crystals and bent at their lower end as shown in Figure 1. Since the magnet could obviously be removed for only one direction of the magnetic field, the sample had to be mounted with its crystalline axis parallel to that direction (Figure 2), because the value of the g factor of CMN is a minimum along its axis. The ratio of the g values in the perpendicular and the parallel direction (with respect to the trigonal axis) is known to be about 60 for an ideal crystal. The apparent ratio of the g values for our sphere consisting of 6 monocrystals was between 20 and 40, this smaller value being due to imperfections in the crystals and to errors in their mutual alignment and in their overall direction with respect to the laboratory reference system. A perfect crystal (gper))/gpar = 60), mounted with an error of one degree, has indeed an apparent g ratio of only 40. A good tool for orienting the crystals would be a measurement of the susceptibility as a function of the angle since the susceptibility is proportional to g2. However, the crystal is mounted at room temperature, where the susceptibility of this salt is rather small. In the experiments reported here, the crystal was oriented by feeding the electromagnet with a.c. (9 c/s) and looking for a minimum in the pick-up signal of a coil consisting of one turn of copper wire pressed between two of the monocrystals. With this procedure one supposes that the naturally grown surfaces are exactly perpendicular to the crystal axis. The temperature of the salt as well as that of the copper was determined by means of a mutual inductance measurement. The coils were wound on the outside of the metal cryostat so that they were immersed in the liquid helium at 4.2 ° K. Each secondary coil consisted of two parts wound in opposite directions in order to eliminate the signal originating from inductance in the empty coils. The upper mutual inductance--the salt thermometer--was measured by means of a ballistic galvanometer. For the calculation of the salt temperature, the magnetic-thermodynamic temperature relation of Frankel, Shirley, and Stone 8 was used. The lower mutual inductance--the copper thermometer--was measured by means of a commercial mutual inductance bridge operating at 17 c/s. The output was connected with a phase sensitive

Finalfield direction 7-

f----f

/~~

Trigona[ axis of CMN nitiai field direction

/ /

Magnetrails

Figure 2. Schematic drawing illustrating the mutual directions of the crystals, the initial and final field, and of the horizontal movement of the magnet 337

detector and this d.c. signal was recorded. The current through the primary coil produced a magnetic field of 0.1 G, which is small compared with the internal field in copper.

Experimental Method The experimental procedure for measuring the relaxation time at the lowest temperature was the following: a few minutes after a first demagnetization experiment, when the nuclear spin system was warmed up to the lattice temperature, the mutual inductance bridge was balanced. The magnet was then brought in position again and rotated over 90 °. The magnet was switched on, producing a magnetic field perpendicular to the crystal axis. When the heat of magnetization was removed, the sample was isolated and the magnetic field lowered to a quarter of its original value in order to make the paramagnetic salt of the shield work. After waiting for five minutes the magnet was rotated over 90 ° in about ten minutes. After another interval of about fifteen minutes the magnet was switched off and removed. The electronic devices were started and the decay of the nuclear susceptibility was observed. The reduction of the field to a quarter of its initial value and the intervals between the different operations turned out to be necessary for the copper to get into thermal equilibrium with the salt. Not following this procedure resulted in a final nuclear temperature which was much higher, whereas a reduction of the field to less than a quarter or an interval of more than fifteen minutes gave no further improvement. For measurements at higher lattice temperatures, the starting position of the magnet was such that the field made a smaller angle with the crystal axis. This results in both a higher salt and a higher nuclear temperature. The high temperature measurements are limited by the small specific heat of the CMN. It is important to note that during the nuclear demagnetization the salt, too, cools down and the measured relaxation time is to be related to this final temperature. In our experimental procedure the initial temperature for the nuclear cooling (that means the temperature after the rotation, if there is thermal equilibrium between the salt and the copper) is not measured because of the difficulties in operating the salt thermometer when the mutual inductance coil is in the pole gap. This initial temperature can be calculated by the following procedure. If :/'1 is the helium temperature, T2 the initial temperature (after rotation, i.e. about 10-20 K), and T3 the final temperature of the salt (a few millidegrees), the apparent g ratio can be calculated by equalizing the entropies in situation (1) and (3). With this value of the g ratio, T2 can be calculated by equalizing the entropies in the situation (1) and (2) or (2) and (3). The results of both calculations differed by only a few per cent. The internal field of copper nucleus experiences can easily be deduced from the value of the final nuclear temperature obtained by an adiabatic demagnetization to zero field from a known initial temperature and field. Since the demagnetization is an isentropic process //l

h = T '7, in which h is the internal field, and the subscripts i a n d f 338

refer to the initial and final condition. This way of evaluating h requires the exact knowledge of the final temperature which cannot be calibrated accurately. In a method proposed by Froidevaux and Keyston7 only a quantity proportional to Tt must be known. If the frequency of an a.c. method for measuring the nuclear temperature is large compared with 1/r (the inverse of the spin-lattice relaxation time), the susceptibility actually measured is an adiabatic susceptibility, which is related to the isothermal susceptibility through the formula h2 Xa~

=

Xisoth h2 + H2

For the nuclear spin system of copper in the present experiment, Curie's law is a very good approximation, so that hVisotla= h/T, ~ being the Curie constant. We can apply the expression for Zao to the situation after a demagnetization to a non-zero final external field HI, the corresponding temperature being Tt. Rewriting 2 h2 Zaa = Tr h z + H i

and using the relation between H I and TI for an isotropic demagnetization to a non-zero external field Ht = (h +

x T, I H ,

we get + h 2)

=

If (Z&a)-2/3 is plotted against H i one gets a straight line which intersects with the H i axis at - h 2. Values of a quantity proportional to (Zaa)-2/3 yield a straight line with a different slope, but which still intersects with the H,? axis at - h 2. In our experiment the magnet was removed and replaced by a set of Helmholtz coils to produce the final field//i. The recorder deviation ~t, calibrated in mutual inductance uni(s was extrapolated to the time the magnet was switched off and ¢t-2/3 was plotted against H i.

%

"O..~ 5

10

t~min

15

20

Figure 3. Log co(co is the r e c o r d e r deflection, w h i c h is proportional to t h e nuclear s u s c e p t i b i l i t y ) as a f u n c t i o n of t i m e f o r a d e m a g n e t i z a t i o n w i t h a 21 ° rotation CRYOGENICS

• DECEMBER

1967

Results A typical relaxation curve for the nuclear spin system is shown in Figure 3. For a pure exponential decay of the nuclear susceptibility the (log ~,t) curve should be a straight line, the slope being - r - L All the experimental decay graphs show a curvature similar to that of the 21 ° demagnetization of Figure 3 : the slope is decreasing at the beginning and becomes constant after about ten minutes. TABLE

7". (10 -60 K) (seconds) Te (10 -8o K)

r

rTe (seconds degrees)

1

90 °

45 °

30 °

21 °

10 °

4.4 765 3"52

3"9 750 4"62

5.2 720 6.58

5"5 600 8.47

7.2 545 22"0

2.69

3.46

4.74

5.08

12.0

Table 1 gives a survey of the results of relaxation time measurement. The angle tp over which the magnet has been rotated is indicated on top of each column. The first row gives the final nuclear temperatures Tn, extrapolated to t = 0, the time at which the magnet was removed. These figures should be considered as approximate values since the nuclear temperature was not calibrated. The second row gives the value of the relaxation time 3, calculated from the slope of the straight part in the (log ~,t) graph. In the third row, the salt temperature Te at t = 15 min (straight part of the curve) is indicated. These values are obtained by extrapolation (because of the heat leak which was about two ergs per minute) of the values of the salt temperature which are measured when the nuclear susceptibility decay is over. The fourth row gives the product Ter. For some unknown reason, the 90 ° demagnetization resulted in a final nuclear temperature which is too high when compared with the values of Tn in the other experiments. Also the salt temperature was higher than the expected value. The most probable reason is that all the heat of magnetization had not been removed when the sample was isolated. The values of Tr must be compared with a value of approximately 0.4 measured by Kurti et al. in zero field

40

I

Hf2(geal~2}

Figure 4. ~-=/3 as a function of

- DECEMBER

40

1967

I

60

I=

H; (HI is the

field) CRYOGENICS

I

20

20

final magnetic

above 10-2o K and with a value of 0.47 found by Anderson and Redfield s in low fields in the helium temperature range. In a separate series of demagnetizations the internal field was measured by demagnetization from identical initial conditions ( H = 15 kG and T = 1.15 ° K) to magnetic fields of 1, 3, 4-5, and 8 G. Figure 4 shows the results in a graph of ~-2/3 against H i (final field). The points are within the limits of accuracy on a straight line which yields h = 6.1 G. The value of the internal field measured by nuclear cooling experiments above 10-2o K in Oxford gave a value of 3.1 G.

Discussion Relaxation time. The main experimental results are the larger value of the product T. z and the fact that it is temperature dependent. In a search for experimental errors, we shall start with an examination of the procedure in determining T. Since we have no direct means to measure the conduction electron temperature of the copper, we can only determine the temperature of the salt and select these values of the relaxation time which are measured at a time at which we have reason to believe that the conduction electrons are in thermal equilibrium with the salt. Let us suppose that the conduction electrons are at a lower temperature than the salt, which would result in a smaller value of the product T . r . If the thermal contact between the salt and the copper is bad, the conduction electrons will indeed be cooled by the nuclear spin system. The warming up of the copper is now governed by the thermal resistance between copper and salt. Kurti 6 has calculated the warming up rate of the copper in this case : it shows a pronounced curvature in a graph of log ~ against t, and thus cannot be a description of our experimental points which lie on a straight line, at least after some minutes. The curvature in our experimental plot of log ct against t during the first minutes has the opposite sense and can be accounted for by a cooling of the conduction electrons immediately after the nuclear demagnetization, due to the cooling of the CMN to its final temperature. The stray heat leak into the copper must not be considered, because accounting for this effect would result in an even larger value of the product T . r. Finally, we can suppose that the recorded signal is not due to the nuclear susceptibility, but to the presence of some superconducting solder inside the mutual inductance coils. Eddy currents, produced by the final demagnetization, can decay in a time comparable with ~ and give rise to an exponential signal on the recorder. However, this possibility was eliminated by performing an experiment in which the thermal contact gas was left in the apparatus all the time; the nuclear demagnetization produced no notable signal in this case. Since the nuclear susceptibility decay is exponential after some minutes, the conduction electron temperature is constant by that time. If the conduction electrons are at a constant but lower temperature than the salt, this constant temperature must be the result of a simultaneous and equal heat flux from the salt to the conduction electrons and from the conduction electrons to the nuclear spin system. Since the temperature difference between electrons and salt is constant, the flux through this resistance will be constant, whereas the temperature difference between 339

TABLE 2 ~p A (arbitrary units) B (arbitary units)

A/B

• 1 (seconds) r~ (seconds) • lTo (seconds

degrees)

90°

45°

30°

21 °

10°

316 404 0.78 143 765

384 504 0.76 143 750

183 412 0.44 153 720

143 425 0.34 125 600

262 158 1.66 93 545

0.50

0.64

0.99

1.00

1.78

2.69

3.46

4.74

5.08

r=T~5 (seconds

degrees)

12.0

electrons and nuclear spins is decreasing exponentially. The result of both processes cannot give rise to a constant electron temperature. Thus we may safely conclude that there is thermal equilibrium between salt and copper when the susceptibility decay is exponential. Consequently, it is the slope of this straight line in the plot of log ~ against t plot which determines the relaxation time corresponding to the salt temperature. In this reasoning we have neglected the influence of the enamel of the copper wires on the recorder signal. This possibility needs serious investigation, since the enamel of the copper wires contains an amount of hydrogen, the total nuclear susceptibility of which is about one third of the copper susceptibility. The proton spin system will certainly have a much longer relaxation time r2 than the copper spin system (r]). The recorder deflection ~ is then a sum of two exponentals = A exp ( - t/r]) + B exp ( - t/rz) and log 0t is no longer a linear function of t. This would provide an alternative explanation of the curvature in the experimental (log ~,t) graphs (we supposed before that it was due to a changing conduction electron temperature). The experimental decay graphs can be described very accurately with this expression for ~, which is to be expected in view of the large number of constants. The values of A,B, rl, and z2, giving the best fit, are shown in Table 2. r2T~5 is the product of the relaxation time, ascribed to the proton spin system, multiplied by the salt temperature at t = 15 rain (see also Table 1). rlTo is the product of the relaxation time, ascribed to the copper nuclear spin system, multiplied by the salt temperature at t = 0. Its values for large ~0are seen to be closer to Kurti's value of 0.4 and Anderson and Redfield's value of 0-47. There is no agreement however for small tp. If the initial field and temperature are the same for both spin systems, the ratio A/B depends on the relative amount C of copper and hydrogen atoms, on their Curie constants 2, and on the final temperatures of both spin systems, i.e. on their internal field h.

A/B = C1).lh2/Cz22h1 in which Ct2dCz22 is known to be 3. The ratio A/B should be constant for demagnetizations with different tp. The experimental values of A/B, however, are seen to vary strongly with the angle of rotation. Moreover, the internal field ratio hi/h2, calculated from the experimental A/B values, varies from about 2 to about 10, whereas a rough estimate of h2 from nuclear resonance line widths in enamel at liquid helium temperatures gives a value of ]12 which is of the order of h~. 340

In order to explain the tp dependence of A/B, we can suppose that the proton spin system has a higher initial temperature than the copper spin system, the difference being larger with lower temperature, due to th e long relaxation time of the protons. This would result in an even smaller value of A/B for perfect temperature equilibrium, which can be explained only by assuming an even smaller value of h2. For all these reasons we think that the susceptibility of the enamel does not play an important role in these nuclear demagnetization experiments.

Internalfield. The value of the internal field is larger (6-1 G) than the value measured by the Oxford group (3.1 G). The magnetic field produced by the primary coil is 0.1 G, which is small even when compared with the internal field value of 3.1 G and therefore cannot cause an important error. The extrapolation of the nuclear susceptibility to the time the magnet was switched off, can give rise to an inaccuracy in the determination of the lowest nuclear temperature, since the log ~ against time graph is not linear in this range. However, the corresponding error in h is certainly much smaller than a factor 2. The only possibility for an inaccurate determination may be the larger stray heat leak during the experiments for determining h than in those for measuring z. Yet we think that the difference with the 3-1 G value is significant. It should be noted that the internal field measurements were carried out at the lowest conduction electron temperature attained (rotation over 90°), which is about 3-5 mdeg. Conclusion Because of the larger value of T. z and of the internal field h that we found at these low temperatures, it might be worth while to investigate other properties of copper at the same temperature and which depend on the same fundamental physical quantities as the relaxation time or the internal field. An anomalous behaviour of the conduction electrons at a few millidegrees is one of the first possibilities. Knight shift measurements below 10 -2° K would probably throw more light on the present results. The authors wish to thank the Belgian 'Interuniversitair Instituut voor Kernwetenschappen' for financial support of the present work and for a fellowship to one of us (H.V.) Two of the authors (G. P. and H.V.) thank the Belgian 'Nationaal Fonds voor Wetenschappelijk Onderzoek' for a fellowship. REFERENCES 1. KORRINGA,J. Physica 16, 601 (1950) 2. PINES, D. Solid State Physics (Eds. Seitz and Turnbull), Vol. 1 (Academic Press, New York) 3. REDF1ELD,A. G. IBM Journal, 19 (1957) 4. HEBEL, L. C., and SLICHTER, C. P. Phys. Rev. 107, 901 (1957); 113, 1504 (1959) 5. ANDERSON, A. G., and REDFIELD, A. G. Phys. Rev. 116, 583 (1959) 6. KURTI, N. Cryq~enics 1, 2 (1960) 7. FROIDEVAUX, C., and KEYSTON, J. R. Proc. VII int. Conf. Low Temperature Physics, p. 93 (Toronto University Press,

1960) 8. FRAN~EL,R. B., SHIRLEY,D. A., and STONE,N. J. Phys. Rev. 140, A1020 (1965) C R Y O G E N I C S • DECEMBER 1967