Nuclear magnetism in copper at nanokelvin temperatures and in low external magnetic fields

Physica 126B (1984) 51-61 North.Holland, Amsterdam

51

NUCLEAR MAGNETISM IN COPPER AT NANOKELVIN TEMPERATURES AND IN LOW EXTERNAL MAGNETIC FIELDS M.T. Huiku Low Temperature Laboratory, SF-02150 Espoo 15, Finland

Helsinki University of Technology,

Investigations of nuclear magnetism in metallic copper at nanokelvin temperatures are reviewed. Particularly, the recent measurements in the antiferromagnetically ordered phases are discussed. The survey includes the static susceptibility experiments in low magnetic fields in the single crystal and in the polycrystalline specimens. The magnetic field vs. entropy phase diagram is presented. The entropy and temperature measurements are discussed and results are compared to theoretical predictions.

I. INTRODUCTION

1.2

Recent progress in refrigeration and in superconducting magnet technology has made possible investigations of ordering phenomena in nuclear spins. The system of nuclei can be very different from the electron spins. We shall first discuss these differences.

The fundamental difference between electronic and nuclear systems is the thermodynamics of the spins. First, in order to investigate nuclear spins thermally isolated from the lattice and conduction electrons, the spin lattice relaxation time Xl must be considerably longer than the spin-spin relaxation time T2; the latter can be assumed equal to the decay time of all nondiagonal operators in the density matrix. This leads to the concept of spin temperature.4 In other words, the probability of finding a spin in an energy level E is given by the 1 Boltzmann distribution exp(-Ei/kT) , which defines the spin temperature T. In metals, coupling to conduction electrons is reduced by Fermi statistics and there ~I is inversely proportional to electron temperature Te: ~ = /Te, where ~ is the Korringa constant. 5 ~or copper, with Te = 50 ~K, T I can be as long as 2 hours 6 in zero field, whereas ~2 is roughly 80 usec. 7 There is thus a well defined nuclear temperature of copper, which is important because the concept of spin temperature provides the link between nuclear magnetism and thermodynamics.

1.1

Energies of Nuclear Interactions

Perhaps the most obvious difference between nuclear and electronic spins is that the nuclear magnetic moments are about 1000 times smaller than the electronic ones. This makes the system difficult to study because the transition temperature T to the ordered state corresponds c to the interaction energy between two neighboring spins. The latter is proportional to the square of the magnetic moment and, therefore, T can be a million times lower in c the nuclear spins. As an example of nuclear ordering in a metallic dipolar system, copper nuclei undergo a magnetic transition at the exceedingly low temperature of 60 nK I and in small external magnetic fields below 0.27 mT. 2 It is thus clear that an effective and sophisticated cooling apparatus is needed for investigating nuclear ordering in copper. Presently micro- and nanokelvin temperatures can be reached only by nuclear adiabatic demagnetization techniques ~see the review article by Andres and LounasmaaJ).

0378-4363/84/$03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

Thermodynamics

Second, a system of isolated nuclei is best described by a different set of thermodynamic variables than is used for electrons. While electrons are in good thermal contact with a reservoir, nuclei are thermally isolated from the external world over timescales much shorter than ~I" Then the entropy S is a better variable

52

M. T. Ituiku

than

the spin

temperature

well

defined

function

magnetic their

field

B,

Nuclear magnetism in copper

T; S, of course,

of

T

which

is a

arid the external

couples

the spins

to

surroundings.

1.3

Interactions

The

most

determines

the

spin a r r a n g e m e n t

state

~nd

the

transition

q

has

been

copper, methods

in

in the order~d

temperature

7' . w~,,

found by two independent 12 Otaniemi, [n addition to th(~

entirely different experiments ai.1-3 All *~h~A~e~' m< ~ s u r e m e I ~ t s

of Andrew ~;t ar'~ in good

agreement. spins

significant

different

classical cannot even

from e l e c t r o n i c

dipolar be

anisotropic

neglected.

In force

the

makes

ones

interaction

dipole-dipole

dominates

mediated

fact which

is that

between many

the

nuclei

cases

the

is c o m p a r a b l e

indirect

by the s u r r o u n d i n g

nuclear

or

interaction,

electrons.

The nuclear

magnets

categories: 2) magnets

I) purely c l a s s i c a l dipolar systems, in which both dipolar and e x c h a n g e

can be c l a s s i f i e d

into

three

forces are of importance, dominated systems. Copper

and 3) exchange and other simple

metals belong to the diamagnetic insulators 8 to

second category, the first, and van

Vleek p a r a m a g n e t s 9 and solid category.

3He ]0 to the third

In

simple

metals,

conduction

the e x c h a n g e force, interaction (RK),I] hyperfine conduction and can

called which

electrons

mediate

the R u d e r m a n - K i t t e l is p r o d u c e d via the

coupling of the nucleus and th~ electrons. By assuming s-electrons

a s p h e r i c a l Fermi surface of radius k F one derive the following form for the (xchange

The

first

the f a c t isotope

method IJ

seen by t h e s e c o n d ( 6 % *h.,. . . o f a b o u t ! 5 mT t h e d l f f ( r ~ m t ('163/2~ give nlse Fig.1 .

ti~e ii~tensity r a t Jo el t h ~ ,:~ . ~ H ~ ) / ( ~ . . M ~ . / = i).~,I, b~ b~ Oj Oj where H. is the m a g n e t i z a t i o n of the isotope i i (H i is p r o p o r t i o n a l to the naturdl <:~bundinc~). However, in high polariT.atiens il was obscrv<.d that the Cu-65 line was enhanced with respect to Cu-63. 12 The effect ~an be understood within .,i~ internal field model. 7h~ e x c i t a t i o n field at Without interaction lines is given by

isotope B 1 and exchang~ isotope

i, Bi,

consists

o

(L-D)

S

interaction of

between

quadrupole

the dipole-dipole interactions. The

effects

nuclei

--

4

.~

2

thus

consists

has excellent possibilities of testing t h e o r e t i c a l phase t r a n s i t i o n models.

The

:ice

vers~):

value of q fixes

the

of

the

IN COPPER

Hamiltonian

B7~ : D 1

L and [ are for the Lor{ntz factors, respectively. [~}, I

1

I

U

=0.8

,J [

0.4

and the approximate RK expression for the dipolar

OF THE RK I N T E R A C T I O N

ac fi{ ]d

(Ju~ tx~ the I:;K irlteraction with

in the

interaction is known with certainty. Often (as in copper) deviations from a spherical Fermi surface are small and this simple model d e s c r i b e s the system s u f f i c i e n t l y well. One then

2. S T R E N G T H

(and

I

o absence

:)f the ~external

interna[ fields (R) and dipoi :~r

the

+~oH (R+L-D)._ Here J and demagnetization

1{

full

s ,~n fJr'st ~'[~ '~{J

in an ~xternal f i ~ ,I ~ ' s o f t h e isotep~<%

63Cu

~RK = q ' ~ [..------~(P{)2f ( 2 k F r u ) I i "]j (1) iS r . . iJ where f(x) = eosx - sinx/x, r.. is a lattice vector, n is a m e a s u r e of the s rength of t m ~ exchange, and ~ is the nuclear g y r o m a g n e t ic ratio. The RK i n t e r a c t i o n is of long range (]/r 3) and isotropic in this approximation.

r~ii

11.3 H H z / T a n d ~65,,'!r: = li .1 H H z / T ) to separate NHR p e a k s a s is, s h o w n i n

Hamiltonian

The

Lised in Otani(mi

that in ~{ t w o s tep<, syste[T, th~ (63Cu) influesc{~s tLe cffc~tiv(~

and

_

t7 Fig.

I.

The

16 15 B (ml) susceptibi}ity

14

x''(S

arbitrary units measured by t~chnique with field swe~p polarization of t~i,* sample has and 0.1. The sca]e of' ~''(B) has

}f cepp~r

i;

the SQUID NHR 12 method. Th~ been 0.~, 0.-', b
by a factor of' 2.5 for t.%e p=O.] Tile ratio ~;' the intensities ~f t,h~ 63Cu ":rod 65(,u i ine:~ d : thus

high

polarization

gives

h

0.7"].

M.T. Huiku

/ Nuclear magnetism in copper

53

w r i t i n g PoM.= x.B* we see that the effective .... 3 .j i excltatlon actlng on isotope i is m o d i f i e d by a

relaxation the oxide

factor [I . (R + L - D)X~] -I. Far from the r e s o n a n c e ~ the s u s c e p t i b i l i l y X: ~ (m*. - m) -I is small and, we may approximate J (for Jspherical samples) Bi= B.(I+Rx ). Thus for R < 0 the I J excitation at the hlgher resonance frequency (65Cu) is larger than at the lower frequency (63Cu). The intensity ratio of the lines at high

crucial for observing the o r d e r e d state.

polarization

gives R = -0.43 c o r r e s p o n d i n g to

times x I in zero field than without treatment. This turned out to be

Our p o l y c r y s t a l l i n e specimen consisted of eight Marz grade copper foils, 17 0.125 mm thick and 5 mm wide. The coils for s u s c e p t i b i l i t y measurements were wound on a silver support so that the e x c i t a t i o n and pick-up fields were aligned with the 5 mm side. The astatic pick-up was connected superconductively

= 0.73.

to the signal coil of a SQUID.

The second method 12 used in Otaniemi gives R = -0.41 or rl = 0.71 in excellent agreement with the above e x p e r i m e n t s in higher external fields. Therefore,

the s t r e n g t h of the RK interaction in

the paramgnetic problem. The

state

does

not

present

a

heavier slmple metals as platinum (fcc) and

B:B~

t h a l l i u m (hcp) have larger RK strengths, qTI = 370 and ~ Pt = 28, respectively, e s t i m a t e d from the spin lattice r e l a x a t i o n rate. In the lighter elements such as sodium (bcc) ~ ~s typically less than in copper (qNa = 0.4). 14 Parameter n describes the relative magnitudes of the RK and dipolar forces. We see that copper is in the region in which these interactions are competing. It experimentally

_

is thus instructive to study the nuclear ordering in this

~

regime.

3.

EXPERIMENTAL SETUP

Our m e a s u r e m e n t s were carried out in a two-stage nuclear d e m a g n e t i z a t i o n refrigerator, described in detail elsewhere. 15 Our copper sample was employed as the second nuclear stage connected to the first stage by welding, i.e without any heat switch. By means of slow d e m a g n e t i z a t i o n of the first stage from 8 T to 0.1 T, the c o n d u c t i o n e l e c t r o n temperature could be reduced to 50 pK. Rapid d e m a g n e t i z a t i o n of the copper sample was then performed in three steps, separated by short waiting periods at 10 mT and at I mT. The purpose of these steps was to reduce noise in the detection electronics. The final demagnetitization from I mT started with an entropy S = 0.10 Rin4. An ordered state was reached below B = 0.25 mT. Two different natural copper specimens have been used in our static s u s c e p t i b i l i t y measurements. Both were selectively oxidized. 16 By this technique we obtained about 30 times longer

_

z

-

e

x

c

.

, ~ (q) pick-up /

-exc.

Fig. 2. The sample geometry and the astatic longitudinal (z) and transverse (x-y) pickup coils. The crystal axis of the single crystal specimen are about 10 degrees off the space axis. The external field is in the z-direction. The ac excitation fields are roughly along the x-, y-, and z-axis.

M.T. Huiku / Nuclear magnetism iH copper

54

The coils and the sample geometry in the next series of e x p e r i m e n t s 17 are shown in Fig.2. A single crystal of dimensions 0.5 x 5 x 20 mm 3 along the x-, y-, and z-axis, respectively, was employed. By X-ray d i f f r a c t i o n it was found that the cubic [100] crystal axis is 4 ° above the (x,y) plane and 13 ° off the x-axis, and that the [001] axis is about 8 ° off the z-axis. Separate transverse and longitudinal SQUID systems were connected to two a s t a t i e a l l y wound signal coils, (x,y) and z, respectively. Three different excitation coils, along the x-, y-, and z-axis, were used; the alignment of the excitation field was s e l e c t e d by switches and variable resistances. The transverse and longitudinal signals could be measured during a single run. The static s u s c e p t i b i l i t y X was measured using low frequency (10 Hz) a c - t e c h n i q u e s and a SQUID in the flux locked mode. I

I

the schematic behavior of x(t) just after demagnetization to zero field. During the warm up, entropy S and temperature T are increasing with time. At the beginning the s u s c e p t i b i l i t y increases and after 5-8 min reaches its maximum. This is c h a r a c t e r i s t i c for antiferromagnetic states in the electronic systems. Therefore, we also concluded that the nuclear system in copper is antiferromagnetic. 4.1.

Experimental entropy vs. t e m p e r a t u r e curve

The

first

step

to

proceed

in the e x p e r i m e n t s

were measurements of entropy S and temperature T in zero field. The starting point was the second law of thermodynamics: T : 6Q/6S, where oQ is the heat pulse given to the nuclei and 6S the corresponding change in the entropy. The measuring technique depended on whether we were in the paramagnetic or in the a n t i f e r r o m a g n e t i c region in zero field. In the paramagnetic state a thermodynamic cycle was made in the B-S plane. It consisted of two adiabatic magnetic field sweeps between zero and ! mT and of a perioC in zero field for' heating the spins, in ] mT we m e a s u r e d the entropy change 6S. The entropy susceptibility

m I CD

can be determined from the X, Iaeasured in a high enough

field (at I mT), p a r a m a g n e t i e state: -I

X

X

using

~oPMsat] -i (

(L

Bef f

I

5 10 t (min) S

-

T

-

,

R)

,,

;'

V -

eoth(x)]

,

: ~

I

15

where x = 7~Beff/2kBT, Ms0 t is the s a t u r a t i o n magnetization, Bef f = B + ,L + B)~oPMsa t is the effective static field in She z - d i r e c t i o n (D z =0); L = I/3, and D = 0.15 in the direction of .

Y

.

the transverse e x c i t a t i o n Fig. 3. The schematic behavior of the static susceptibility measured at 10 Hz as a function of time after d e m a g n e t i z a t i o n to zero field. The entropy and temperature are increasing with time. The increase in X(10 Hz) at the beginning of the experiment indicates that the spins are a n t i f e r r o m a g n e t i c a l l y ordered.

4.

-["

"

P = ~[4eoth(4x)

I

the equations of the

l

STATIC S U S C E P T I B I L I T Y RESULTS

In all our e x p e r i m e n t s we m e a s u r e d the static susceptibility X as a function of time during the w a r m - u p after demagnetization. Fig. 3 shows

We

were

thus

field.

able to ea2culate an

entropy S I

from the susceptibility X at 1 mT: first, the polarization was iterated from Eq.(2) and (3) and then the I-I c o r r e s p o n d e n c e of the entropy S and the polarization p was used to find 51 . Second, the field was swept to zero, where a heat pulse 6Q was given d i r e c t l y to the spins by NMR absorption. After this, the field was again swept adiabatically to I mT to find an entropy S 2 and the entropy change 6S = S 2- S I due to the heat pulse. The heat pulse 6Q was calibrated against the high temperature expansion of the entropy 18 at temperatures between ] and 5 uK. Due

to

the

eddy

current

shielding

a

low

M.T. Huiku / Nuclear magnetism in copper

frequency

(50 Hz) had to be used for heating the

4.2

Static susceptibility of the single crystal specimen

spins. In the ordered state the temperature, now written T : (6Q/6x)/(6S/6×), was measured in two steps because the field sweep from I mT to zero was found to be nonadiabatic. First, 6Q/6x was found by giving heat pulses during the warm-up at B = 0; no field sweep was needed to measure the relation of the heat pulse and the ensuing change in the susceptibility. Second, the entropy as a function of X was measured by sweeping the field adiabatically to I mT at different values of X- The resulting S vs. X curve gave 6S/6×, and finally the temperature in the ordered state.

6

The static susceptibility xi(t) (i = x, y, and z) was now measured in the three cartesian directions. 2 A characteristic increase of ×i(t) with time was again observed at the lowest entropies. In Fig. 5 the time dependence of ×~(t), corrected for internal fields b~ means o~ [ x ~ ( t ) ] -1 = [ x i ( t ) ] -1 - (g - D i + R ) , 1~ i s shown in the x-, y- and z-directions in three different external fields, 0, 0.15 mT, and 0.20 mT. The salient features of the data can also be described by three other quantities: omax the maximum value of the susceptibil×i

Our experimental data are shown in Fig. 4, where we have plotted S vs. T in zero field both in the low temperature Daramagnetic region and in the ordered state. 1'6 A characteristic rapid change in the entropy was observed at 60 nK, which is the critical temperature for the first order phase change. The entropy during the transition changes from Scl : 0.49 Rin4 to Sc2 = 0.65 R~n4; this corresponds to a latent heat L = 0.11 uJ/mole. Because of the large scatter in the measured points and the small entropy range ((0.25 - 0.42) R£n4) covered by the experiments, it was not possible to find the temperature dependence of S in the antiferromagnetic phase.

55

AFI

0 I

2 '

I

1.0

'

~.,

•, ~

,~

/"

0.90

/

~;~-

'

I

d- /

\ \

\\

AF2

2

4

6

8

t (min)

o~o ~d

1.0/Ix'F

I E

-~05

I .

l~/, I , i , ~ , E 0

06

'

~

B=0.15 mT

,/

080

o

I

8

"

O8

07

4 6 t (min)

J

'

'

'

J

'

'/

/ / / /

07.

;

o

I~

~B=CL2OmT

/

;~ 20

~

\/ AF3

, , , . , .

O,o

03

1

0

Tc

[ , l , l Z,0 60 80

, , lJ 100

120

T (nK)

Fig. 4. Provisional entropy diagram of the nuclear spin system of metallic copper below 120 nK. 6 The antiferromagnetic transition occurs at T 60 nK. Dashed lines are drawn only to c emphasize the first order phase transition.

2

~ 5 t (rain)

8

Fig. 5. The three characteristic static susceptibilities x?(t) (i=x,y, and z) in arbitrary units in the three external fields B = 0, B = 0.15 mT, and B = 0.20 mT. The plots correspond the different ordered regions AFI, AF2, and AF3 in the single crystal specimen. The suggested spin arrangements for the ordered phases are schematically shown on the right.

56

M. 7< ttuiku

IAFll,

,o

,/Nuclear

I AF21 IAF31 P

AFII,

0.12 I-T

~°°6i- i ,,~ '

io

i

//

~

oo t/

0

OO.q)._~O 0

i i •

×

° £

P

I I

\-\ o

(

\

o. 0.3

0.2 8[.. - '

[

!

'

I

'

E

,fb-o ! -

L

"h~ 0.2

i

0.3

e(mT) ,

!

,

!

,

.

~0.96

~

i

L

>,0.94

ci x"

× 0.92

ON

0

I

'

0

0.1 0.2 0.3 B(mT)

0

0.1 0.2 0.3 B(mT)

o/t\ (i=x,y, Fig. 6. The static s u s c e p t i b i l i t y ×it., and z) of copper nuclear spins as a function (of the external magnetic field B. In the t r a n s v e r s e geometry, shown in the three figures on the left, open circles are for the y - d i r e c t i o n and filled circles for the x-direction. Data in the longitudinal (z) direction are shown on the right. Akio denotes omaxthe d i f f e r e n c e of the m a x i m u m s u s c e p t i b i l i t y ×i , o b t a i n e d at time At. after i the end of the demagnetization, and the initial susceptibility x?(t=0). In the x-direction At i shows the

length of the plateau.

The dashed

lines in the At. vs. B plots indicate the 1 expected behavior e x t r a p o l a t e d from the region above 0.I mT. For all data S. = 0.I0 RZn4. 1

- x~(t : 0), the net rise ity, hXio Xio m a x the s u s c e p t i b i l i t y after demagnetization, a n d omax At, the warm-up time needed to reach Xi In Fig. 6 we show these quantities as a function of the magnetic field B for the transverse (x,y) and longitudinal (z) directions. in

, "-~ 1.00 Kgzq.

o.98

Q96

'o\

i 0.1

0

I I

0.98

O

.

o.o2,,VL6

B(mT)

E

O I

o ,o

""O

•

1.00(

I I

I

°x" 0.06-

&,

o!

i

0.08 -

<1

i 0.1

I

0.10-

m

i/O..q~,

IA%201 I AIF3 I P

0.12

C3 II

81o 0.10~-?x. ! ~ 008~ j" \

:oo41-T

magnetism in cot)/wr

× 0.92

0

'

0.1 0 2 0.3 B(mT)

0

0.1

0.2 0 3

B(mT)

Three c h a r a c t e r i s t i c <~urve~. c:~

Xir,tl it-! E:-F:;.

correspond to different o r d e r e d regions. ~n ~,~ fields below 0.04 mT, a <:]ear inc.'rease of ~i after demagnetization wa:.; obtained only in ihe y-direction; the growth of ~°(t) is about IC ~. Y This suggest a spin arrangemant, AF!, in wh] 'h the staggered m a g n e t i z a t i o n is along the y-uxi:. This conclusion is based on the behavior o~ ~ i.q the antiferromagneti( two sublattiee system: x-perpendicular to sublsttice magnetization stays constant below T .2 whil<, x-para[],:i approaches zero as T ~ 0. A second a n t i f e r r o m a g n e t i c phase, AF2, was found between 0.II and 0.!6 roT. We S
M.T. Huiku / Nuclear magnetism in copper

the time after demagnetization. A small increase can be seen in ×.(t) in contrast to the paramagnetic like behavior in ×z(t). Therefore, the spins are leaning towards B = Bz. AF3 has a net magnetic moment in the z-direction and a small staggered magnetization parallel to y. The change to the paramagnetic phase proceeds by tilting the spins more and more towards B until at B = 0.25 mT, AF3 and P are the same.

57

states, phase boundaries could be determined. First, with increasing S. a similar AXi vs. B pattern than in Fig. 61was observed, then the AF3 phase disappeared: all susceptibilities reached the maximum simultaneously, and finally only AFI was encountered.

The magnetic field dependence of ×i(t) can be seen in detail from Fig.6. The small increase in all AXi between the AFI and AF2 phases indicates a wide transition region. A large jump in A×z and a small one in AX. were observed around B = 0.17 mT. This marks ~he transition between AF2 and AF3. According to our susceptibility data

The entropy S below B = 0.1 mT was obtained by adding the measured nonadiabaticity AS to S.. A l shadowed region indicates that a first order phase change in proceeding in this area; during transitions, neighboring phases coexist as macroscopic domains. The critical field B = 0.27 c mT was found by extrapolation. The relation between entropy and temperature could not be determined in these experiments because, owing to eddy current shielding in the high-

AFI, AF2, and operations and,

conductivity single-crystal specimen, we were unable to find the correct calibration to 6Q.

AF3 have thus the

different symmetry transitions between

them are of first order.

0.3 Fig. 6 First,

I

•

y

.

z

The B-S phase diagram of copper was measured by changing the initial entropy S./Rin4 between l 0.10 and 0.50 before final demagnetization from I mT to lower fields. The B-S diagram is shown in Fig. 7. S i (equal to S above B = 0.1 mT) was found by varying the demagnetization procedure: We simply waited at I mT for different lengths of time. The entropy increase during and before the waiting period was carefully calculated using the experimental spin-lattice relaxation rate. The demagnetization at different S then i corresponds to vertical lines in the B-S diagram. By measuring X (t) and × (t) and by y z using the known characteristics of the ordered

I

'

I

AF3: ~,/ ~[ AF2: ~

shows additional interesting features: the quantity (AX~ + A× 2 + AX2) I/2, which

should increase with increaslng sublattice polarization, is smaller in AFI than in AF2. Further, the At vs. B plot shows that the paramagnetic phase boundary is reached sooner than expected (see the At vs B plots in Fig. 6). These observations indicate that the AFI phase is reached with an increased entropy due to the nonadiabatic demagnetization observed below B = 0.1 mT. The increase AS 0.12 R~n4 can be estimated from the measured relaxation rate of the dipolar energy in the paramagnetic state during a 3 min period. This AS is the same as observed earlier in the polycrystalline sample. 6 Second interesting feature is that the transverse susceptibilities, x~(t) and x°(t), are different. However, this is probably dee to the slab shape of the specimen.

'

A0.2

AFI:

4-

I,--

E 0.1 AF1 0

I

0

0.2

0.4 S/RI.n4

0.6

Fig. 7. The external field vs. entropy diagram for nuclear ordering in copper. 2 Different symbols denote measured data points along the respective phase boundaries. Shadowed bands indicate regions in which first order transitions take place. The end of the phase boundary in zero field at S = 0.65 R£n4 was determined from our earlier data on the polycrystalline sample. 6

5. 5.1

DISCUSSION Comparison of the experimental S vs. T data with theories

The measured critical temperature T = 60 nK is c smaller than predicted by theoretical calculations applied to copper so far. The mean field estimate is Tmf = 230 nK, which is almost

58

M.K

Huiku /,'\:uclear magnetism in copper

four times higher than that o b s e r v e d in copper. it has been shown, using the spherical modet (SM),19 that spin density wave fluctuations,

spectrum

which

done

are not taken approximation,

field

temperature 105 by

into account in the mean reduce the

critical

considerably;

nK.

The p r o b l e m

the

linked

the p r e d i c t i o n

has

cluster

is TSM

been a p p r o a c h e d mode]

also

(LCE). ?O

model

used

sl~ould

states

were in

Hamiltonian

observations paramagnetic

have

been

used for e a J c u l a t i n g

tile

entropy as a f u n c t i o n of temperature. C o m p a r i s o n of theories w i t h the e x p e r i m e n t a l data is shown -2 8; the leading T term of HTE is also in Fig. drawn.

Tile

mean

/

-0.5

~

field

experimental

[t may

fit,ld

field the

o.,

I

,

10

1

~ L

SM

,

100

I

,

1.0

,

1000 T(nK)

Fig.

8.

Reduction

shown as logarithmic scales. phase,

of the entropy

in zero

a function of temperature (right) and s e m i l o g a r i t h m i c

Open circles triangles for

field T on (left)

are for the p a r a m a g n e t i e the ordered phase, and

crosses for the old data. denoted by T -2 SM, e x p l a i n e d in the text.

The RA,

theoretical curves LCE, and }{TE are

Both the Pad6 [0,2] approximant and the LCE calculation (with identical results down to T ) c fit to the e x p e r i m e n t a l curve s u r p r i s i n g l y well. The SM model caused

by

suffers the

from n u m e r i c a l

insufficient

difficulties

eigenstate

density

£ur

in


cxtended

to

a

Lo~

[,has<

Therefor<, there seem to be the MF prediction and tiie di~{gram. be instructive

state

%o coflsid~r

if': t~om% ucbai

can

b~

~im.

1,3. [[] Yerc

was done k. ~' For

expressed

in th.~ copper hy

the

wave

:

]' dl e°s(k] ° ~'1 i

to

,)$,

:Jw~rage of' tile spin the s p i n

is

the wave

vector

ve:]tor

kl, .rod r'i is

the lattice reeler; k i r e p r e s e n t s the spatial variation of
oZ+~']~"~

weli

the n u m e r i c a l c a l c u l a t i o n space f o r 512 d i : : e r e t e

corresponding

T -~'

th,:

f[ with ?,E describes

only one o r d e r e d

where is the thermal 1 lattice site ~, iI HTE

* ~

!Mr)/2

prediction

ordered

0.01

1.0

w,~s ti!~t

diagram

predicts

phase

spin-density

,oy'~

~

theory

field,

it

in ::ere field.

phase

below 0.27 roT. conflict between

Fourier'

~,J'~

o

,¢

o"~"~°°~

0,001

However,

approximation:

[{ b

of

-is was

for copper. However', thr~se different states wet e o b s e r v e d in the presence of the external field B

mean

oo-

these

suffici( nil!/ stat~: and

Experilnental

Therefore,

0.0 -

analysis. }( =

'?he spheric~l

if the density

into account

me ,.sured

[].2

magnetic

to LCE

correctly

within

experimentally

The

expansion,

LCE

shown

critical properties of LCE are e s s e n t i a l l y the same as those of SM except that quantum spins i 3/2 are used in the analysis. This theory estimates T = 120 nK. These approaches (SH, c LCE) together with the high temperature e x p a n s i o n (HTE) 18 t e c h n i q u e to 5th degree and the Pad@ [0,2] a p p r o x i m a t i o n (PA)21of the series

taken

the

clearly the

in tr~e ua!eu]ations. fit, better'

k in the class

2rF/r(.

The p r e d i c t e d s t r u c t u r e has three c h a r a c t e r i s t i c 22 properties: First, k is at the boundary of the first BrillouJn zone. The analysis of' td<: eigenvalue

spectrum

interaction [ongest k.

shows that the RK force prefers the In bee reoip:'oeal lattice thes,

vectors zone

are

the

R u d e r m a n - K i t tel

in the <1 0 3. d i r e c t i o n s

boundary.

interaction k=O state.

of

The

ferromagnetie

and iT

ti~{

dipolar

in fee lattice tends to order to th:~ Therefore, ,'opper a l r e a d y is in the

region where the RK interaction is stronk: enodL~h to win the competition {gainst the dipolar force. The

second

the

spin

interesting vector

and

feature the

in Eq.

4 is th:kt

corresponding

wave

vector are perpendicular, i.e. k I -d] = O.T[ is is a consequence of the a n i s o t r o p i o dipolar interaction. Thu s t r u c t u r e is highly degenerate, because d has rotational freedom in the plane perpendicular to k.

M.T. Huiku / Nuclear magnetism in copper

59

24 Finally, when we require permanent structures, in which the length of the spin vector does not

rather than a constant entropy process. The supercooled state relaxes after demagnetization

depend on the lattic~ site, the dl'S perpendicular, i.e. dl,- dl= 0, l~l'. How trustworthy predict io n 23 was

in a constant field towards equilibrium by increasing its entropy by AS. In the AFI phase, AS corresponds the measured entropy gain of 12%. In AF2, AS is much smaller and it could not be seen in our experiments.

The discrepancy between theory and experiments may arise from an error in the Hamiltonian. When

In a conclusion, both the static and dynamic susceptibility measurements on copper have given important information on the nuclear ordering in metals. Our results include I) the measurements of the Ruderman-Kittel exchange parameter, which is essential for placing copper with respect to

is reduced by about 30% (to ~ = 0.52) the predicted spin arrangement changes gradually to states which are described by shorter k-vectors in the (~0,0) direction, as was shown by Oja dD and Kumar. However, the reduction is too large to be explained by errors in the experimental RK-strength. In addition, the conflict may arise from the incomplete Hamiltonian, but additional terms have not been proposed so far. Further, the rather good fit of the LCE calculation to

other metals and for knowing the Hamiltonian, 2) measurements of the susceptibility vs. temperature and the entropy vs. temperature curves in zero field and in the paramagnetic state, I which were used for comparisons with theoretical models, 3) observations of the two different antiferromagnetic resonance frequencies in the ordered state 26 which, in addition to the static susceptibility experiments reported in this paper, give information about

the experimental corrections to

results does not suggest such the Hamiltonian in the

the ordered structures and, finally, 4) measurements of the B-S phase diagram, 2 which shows the transitions between the three ordered phases in an external field. However, in spite

However, more important is the change in the eigenvalues close to k = 2~/a(I,0,0). It has been recently pointed out by Oja and Kumar 25 that a small reduction in the nearest neighbor interaction can give rise to a devil's-staircase-like sequency of k-states, with a locked state between, as a function of the external field. The change in the NN-interaction influences the eigenvalues of the RK-interaction close to the maximum k-vector k but does not

of our extensive studies of copper by magnetic measurements, detailed structures of the ordered states are not known. The only way to obtain this information is to apply neutron diffraction techniques. Such experiments are planned and found feasible in copper.

are

is the MF analysis? The calculated at T where the c fluctuations are large. In fact, this may make the prediction doubtful. However, fluctuations are not expected to change the ordered state, even though they influence the estimation of T . c

paramagnetic state.

-

O

ACKNOWLEDGEMENTS Over

the

years

since

1974

many

persons

in

change the k = 0 state. Even though the calculation has to be analyzed in more detail,

Otaniemi

we regard it probable that the emerged ordered states are sensitive to the form of the

nuclear magnetism in copper. G.J. Ehnholm, in particular, should be mentioned. Expecially I

RK-interaction.

want to thank J. Soini and M. Loponen for making me acquainted with the work, when I came to the group (1980). I am greatly indebted to O.V. Lounasmaa for his continuous support and interest in the experiments. The present experimental group T. Jyrkki6, J. Kyyn~r~inen, and A. Oja deserves my sincere thanks. Theory has been an essential part of these studies. In particular, I have benefited from discussions with P. Kumar, J. Kurkij~rvi, and M. Salomaa.

The the

observed nonadiabacity, 2 AS = 0.12 R~n4 in transition to AFI, can be explained by

supercooling. During a rapid change of magnetic field, when a first order phase boundary is reached, depending on the initial entropy, either AF3 or P supercools. This is due to the long nucleation time of the AFI and AF2 states. The slow nucleation is probably caused by the first order transition which must allow a transfer of a spin from one coexistent phase to another without a change in the energy of the spin. This tends to be a constant temperature

have

participated

in

the studies of

60

M. T. Huiku / Nuclear magnetism in copper

REFERENCES I. M.T. Huiku, T.A. Jyrkki6, and M.T. Loponen, Phys. Rev.Lett. 50, 1516(1983). 2. M.T. Huiku, T.A. Jyrkkid, J.M. Kyyn~r[inen, A.S. Oja, and O.V. Lounasmaa, Phys. Rev. Lett. 53 (to be published). 3. K. Andres and O.V. Lounasmaa, Progress in Low Temperature Physics, Vol.8, 222 (Ed. D.F. Brewer, North Holland Publishing Company 1982). 4. M. Goldman, Spin Temperature and Nuclear Magnetic Resonance in Solids, Clarendon Press, Oxford (1970). 5. J. Korringa, Physica 16, 601 (1950). 6. M.T. Buiku, T.A. Jyrkki6, M.T. Loponen, and O.V. Lounasmaa, in Quantum Fluids and Solids1983, eds. E.D. Adams and G.G. Ihas (AIP Conf. Prec. No. 103), 441(1983). 7. R.G. Gylling, Construction a n d operation of a nuclear refrigeration cryostat, in Acta Polytechnica Scandinavica series No.81, Helsinki (1971). 8. A. Abragam and M. Goldman, Nuclear Magnetism: order and disorder, Clarendon Press, Oxford (1982). 9. J. Babcock, J. Kiely, T. Manley, and W. Weyhman, Phys. Rev. Lett. 4__33, 380(1979); M. Kubota, H. R. Folle, Ch. Buchal, R.M. Mueller, and F. Pobell, Phys. Rev. Lett. 45, 1812(1980); see also K. Andres and O.V. Lounasmaa, in Progress in Low Temperature Physics, Vol VIII, edited by D.F. Brewer, 222(1982). 10. W.A. Halperin, C.N. Archie, F.B. Rasmussen, R.A. Buhrman, and R.C. Richardson, Phys. Rev. Lett. 32, 927(1974); M. Roger, J.H. Hetherington, and J.M. Delrieu, Rev. Mod. Phys. 55, No.I(1983). 11. M.A. Ruderman and C. Kittel, Phys. Rev. 96, 99(1954). 12. J.P. Ekstr6m, J.F. Jacquinot, M.T. Loponen, J.K. Soini, and P. Kumar, Physica 98B, 45 (1979). 13. E.R. Andrew and W.S. Hinshaw, Phys. Lett. 43A, 113(1973). 14. K.J. Niskanen, L.H. Kj~idman, and J. Kurki j~rvi, J. Low Temp. Phys. 49, 241(1982). 15. G.J. Ehnholm, J.P. Ekstr6m, J.F. Jacquinot, M.T. Loponen, O.V. Lounasmaa, and J.K. Soini, J. Low Temp. Phys. 39, 417(1980). 16. J. Peterseim, G. Thunner, and H.H. Mende, Z. Metallkde 70, 266(1979); M.T. Huiku, M.T. Loponen, T.A. Jyrkki6, J.M. Kyyn~r~inen, A.S. Oja, and J.K. Soini, to be published in LT-17 Conf. Proc.(1984). 17. Copper Material supplied b y Materials Research Corporation, New York 10962, U.S.A.. 18. K.J. Niskanen and J. Kurkij~rvi, J. Phys. C 14, 5517(1981).

19. P. Kumar, M.T. Loponen, and L.H. Kj~idman, Phys. Rev.Lett. 44, 493(1980); L.R. Kj~ldman, M.'I Loponen, and P. Kumar, Phys.Rev. B 23, 2051 (1981) 20. K.J. Niskanen and J. Kurkij~rvi, J.Phys.A 16 1491(1983). 21. G.A. Baker, Jr., in The Pade Approximant in Theoretical Physics, ed. by G.A. Baker, Jr., and J.L. Gammel, Acad. Press, New York and London (]970). 22. P. Kumar, d. Kurkij~rvi, and A.S. (ja, to be published (1984). 23. L.H. Kj~idman and ,]. KurKl.j~rvi, Phys. L,~t1. 71A, 454 (1979). 24. M. Goldman, Phys. Rep. 32C, I (1977). 25. A.S.J. Oja and P. Kumar, to be published ~n LT-17 Conf. Prec. (1984). 26. Huiku et al., to be published ',1964).

POSTCRIPT by N i e h o f a s

Kurt i

(~xI'or'd;,

The d o g did bark. At LT-Io hp±d at t.h~ Unlversity of 'd{~if:)rln ~, Los Angeles in August 1981 i gaw) an anecdotal htsbory "From the

of

relrigeratio~i

first

mist

o~

uHder liquid

i,nc

oxygen

",h~, ! [~ !~ to

r]u(,IL,~r,

cooling" (Physiea !09B a n d i I O B p p . 1 7 3 7 - 1 7 5 1 1 j . I c o n c l u d e d my t a l k w i t h t h e f o l l o w i n g remarks:

"SHERLOCK

HOLMES'

ViEW £N NUCLEAR

COOLIN(}

But, to f'inish, i should like to return to tile question of nuclear oo]ing in whi
M. T. Huiku / Nuclear magnetism in copper

had gone into this experiment. But one can always rely on Professor Abragam to come up with an apt quotation to console disappointed colleagues. In this particular instance, he has taken a passage from Conan Doyle's "The Silver Blaze" in which the Scotland Yard Inspector condescendingly asks Sherlock Holmes whether he has any comment on the case. "Is there any point to which you would wish to draw my attention?" "To the curious incident of the dog at the nighttime." "The dog did nothing in the night-time." "That was the curious incident." One is tempted to speculate about the outcome of

future

experiments

61

in

Otaniemi

or elsewhere.

Will the dog bark eventually? Will the nuclear spins in metallic copper be eventually knocked into ordered lifelessness as happens in most localized spin systems, or will they follow the example of the surrounding conduction electrons - with whom they interact, albeit weakly - and drift gradually into peaceful slumber at absolute zero?" The Conference Proceedins were published in August 1982 and a couple of months later came the announcement by Huiku and Loponen of the clear experimental proof of nuclear antiferromagnetism in metallic copper. The copper nuclei do behave after all as they were expected to.