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Magnetic ordering of the nuclear spins in metallic copperat nanokelvin temperatures and low applied fields
Physica 127B (tPM) 317-321 North..lqolla0d, Amsterdam
MAGNETIC ORDERING ,OF THE N U C L E A R SPINS IN M E T A L L I C COPPER AT NANO~LV]~
T]EMPIERATURES A N D L O W A P P ~
FIELDS
.Iuhani KURKIJA~RVI Department of Technlc¢~l Physles, Helsl!nki University of Technology. SF-02150 E.sooo IS. Finland Th~ theoretically predicted ordered ph~es at copper nuclei a~ r~nokelvlrl temperatures ate critically disctt.~sedand compared with ex0eriment.
Although nuclear magnetic ordering has been seen in metallic copper [1], the precise nature of the ordering is not known [2, 3], In order to learn more,, neutron diffraction experiments are planned [4] which should east light on the arr a n g e m e n t of the spins at the different lattice points of the metal. It would be exciting to be able to predict just what will be seen, and this article is a short review of what we know about the system at this moment. Nuclear magnetic ordering has been extensively studied in salts [5]. It is not quite as obvious that the nuclei in a metal, as well, shouid become sepa,'ated from the lattice at tow temperatures. Their coupling to conduction electron excitations is reduced by Fermi statistics to the point of making the relaxation time to mutual thermal equilibrium between the nuclei and the electrons inversely proportional to t h e temperature, -r~ = K/T~. At temperatures of iate~'est in the study of p h e n o m e n a depending on the interaction strength between the nuclei, ~r~, t:~rns out to be much longer than most rela~xation times within the nuclei themselves. I say most relaxation times since we cannot be quite sure about the decay times of eventual metastable states. In practice [3~, "r~ can be as long as three hours. A l t h o u g h the coupling o~ the nucle:i to conduction electron excitations varies with the temperature, the elect~'ons mediate a n interaction, the R u d e r m a n - K i t t e l (henceforth RK) indirect exchange interaction [6], which does not, It can be viewed as a modification of the conduction electron ground state energy as a function of the
nuclear spin configuration via the hyperfine interaction. It oscillates as a function of the distance as is typical in cases of a sharp cut-off at ~he Fermi surface HRK/const = 2kv cos(2kvr)/rS-sin(2k~r)/r 4
(1)
which form of the interaction has been calculated assuming plane wave s-electrons within a sperical Fermi surface of radius k~. Although the .strength of the R K interaction is known flora ~:xperiment [7], it is instructive to study the nuclear ordering letting it vary according to 0.26"0 (0.26 an arbitrary constant) as compared with the relative strength unity of the dipole-dipole interaction which depends only on natural con,,~tants. The full interaction between the nuclei ~:hus consiszs of the dipole--dipole interaction, which is exactly known, and the approximate R K interaction. It is clear without saying that the R K interaction makes the nuclei in a metal a much harder problem thar, in a salt. It is relatively easy to find the m e a n field predictions of the ordered phases. Ov.e writes the Hamiltonian of the system aporoximately in terms of the average field each spin sees. which consists of the spin--spin interaction s u m m e d over all the o t h e r spins assumed ordered accordinlz to the order parameter one is testing. T h e different order parameters can be classified by their spatial variations, i.e. by the wave vectors describing tho'.~e variations. The usual m e a n field expression for the average spin at a site ther: [cads, after a lineariza~ion of the Brillouin fuocti~n and a Fourier transform, to the eigenvalue
0378-4363/84/$03.00 l ~ Elsevier Science Publishers B.V. (North-Holland Physics Pt, blishing Division)
318
J,
Kurki]iirvi / Copper nucl¢'i at nanokelvin temperatures
problem [8]
x 0,"(t~)> = ~'~ A " " ( k ) ( r ( k ) > ,
(2)
i,
where A""(~) is the three by three spin-spin interaction in the spin c o m p o n e n t space. T h e eigenvalue gives the transition temperature according to
(31 where ~:/i~/4 is the strenEth of the dipolar interaction between the spins and a is the edge of t h e conventional cubic cell in copper. It is useful to take a look at a c o n t i n u u m distribution of spins first. Then the dipolar part of A "'(Ic) has off diagonal elements proportional to k,~k,,/k ~, On the diagonal it has the R K interaction 0.26~g(/¢) and the scalar part of the dipole-dipole interaction. All in all we have A " " ( k ) = (0.26r/g(k).- 4¢r/3) ~,,. + 4~lq,k,Jk :.
(4) Thus, if the distribution of spins were uniform instead of on a lattice, the eigenvatue problem would have the eigenvector Ik with th~ eigenvalue 0.26~g(k)+8~r/3 and two degenerate eigenvectors peper~dicular to k with the eigenvalue 0.26~lg(k)-~rt/3 which latter would give the positive transition temperature. We note that the positive eigenvalue of the dipole--dipole interaction is twice that of the negative degenerate one in absolute value. Since g(k) is always postfive for the uni~")rmly distributed system of spins, a strong R K ir~t;,.~ra,~tion would always end up suppressing the tr::asition altogether. It is easy to see that the above general struclure of eigcnvectors and eigenvalues survives the change to a latl:ice t~f spins if k is in one of the symmetry directions of the lattice. The symmetry directions seem to be the relevant ones anyway, acc(~rding to numerical calculations. There are some changes in the cigcnvalues. With growing k the positive dipolar contribution to the eigenvalue gets reduced a r d the negative one of lhe perpendicular elgenvectors stays half of the positive one
in absolute value. T h e second significant change is that t h e Fourier transform of the R K interaction on a lattice, fee in particular, goes negmive before the zone boundary. This again m e a n s that the transition temperatures to high k structures rise with growing strengtl~ of the R K interaction rl. T h e experimental value of rl is r I = 2.75 [7] and the highest transition temperature structure corresponds to /c = (2rtta)(1, O, 0> with the eigenvaIue - 2 . 1 7 - 1 . 1 3 r / a:nd two degenerate eigenvectors perpendicular to k in the same class <1,0,0). T h e second best structure would correspond to k = (3~r/2a)(1, 0, 0) with the same eigenvectors as above and the eigenvalue - 2 . 4 8 - 0.96Tr. T h e predicted transition temperalures to the best and the second best structures at r t - - 2 . 7 5 are 230 nK and 223 nK. it may be interesting to notice that the best k is at the zone boundary. In t h e (l, 1, 1) direction, where the Fermi surface osculates the zone boundary, the free electron approximation gives the degenerate eigenvalue - 3 . 6 1 + 0 . 6 9 r l and the transition temperature 75 nK at ~1 = 2-75. Such as structure therefore does not cornpete. T h e mos~ general spin structure corresponding to the best k above is given by (/~) = (0, a~, az!cos(k~ • ri) + (a3, 0, a4)eos(/~.~ • r,) + (as, a~,, 0)cos(k.~ • r,),
(5~
where kl = (2rr/a)(1, 0, 0), k~ = (2w/a)(0, 0, 1).
k~ = (2~'/a)(0, 1, 0), (6)
In principle one sh,~utd include sine and cosine variatior~s according to all /~ belonging to (2~-/a)
J+ Kurkijilrvi / Copper nutlei at nanokelvin teraperatur~"
p e n d e n t of i. Thl,s sets conditions on the quantities a~ ata~ = azaa = azas.
(7)
'll~en there are two possible structures
(0, at, a 2 ) c o s ( k t - r~) + ( a 3 , 0, 0)cos(k2 - r~)
(8) and (I,) - (0,, at, O)eos(k~ • r~) + (0, O, a,Ycos(k2 • r+)
-+"(as, 0, 0)cos(/q • r,).
(9)
T h e strr~cture of eq. (8) is displaved in fig. 1. A pexTnanent struetm-e wh:c]~ is stable at Tc is also the best structure at T = ( : [9]. Although nothing certain can be said about the range between T = 0 and T = "_r we know that the best structure will eventually emerge ~ffter demagnetization at a low enough entropy. How trustworthy are these predictions? As to the value of ~, the Otani6mi experimental group have an ingenious method of determining the strength ,c)f the R K interaction [7] based on the iT,terfereraee oJ~ the N M R absorptio~ lines on the isotopes '+3Cu and 6SCu. Their me, hod rests on the observation that although an exchange field cannot s~ift N M R lines coming from a single isotope, two iso-:epes with different resonance frequenees can influence the effective field seen by the other. Their result already quoted above is not in conflict with the entirely different measu r e m e n t of Andrew et al. [I0] with the high speed magic+angle specimen rotation technique. The value at which the best and the second
Fig. I. Spin arrangemem accordingto eq. (8).
319
best structures w e a l d cross is around 71 = 1.9., appreciably below the experimental finding. Therefore, as fat ~'as the predicted ~tructure is concerned, the strength of l:he R K interaction does not seem to be a problem. Otl'oer sources of error are included in the approximate form of the R u d e r m a n Kittel interaction and in the mean field procedure of course. T h e presence of two randomly distributed isotopes has not been taken into account either. Errors in the form of the R K interaction arise from the nonspherical Fermi surface on the o n e hand and from the contribution of the d electrons on the other. T h e elfccts of the nonspherical Fermi surface have beel:~ estimated [1].] to be on the order of 2 - 3 % on the eigenvalues of the R K interaction near the zone boundary. Eq. (1) was taken for the RK interaction with kF depending on the direction and the now anisotropic Fourier transforms were calculated. T h e s e were then used in the mean field calculation up to the 20th .shell of nearest neighbors. At that distance t h e interaction was cut off and the results were compared with those from the similarly abrogated isotropic R K interaction. For copper this calculation leads to an increased difference in the transitio~a temperatures of the two best phases quoted above. Niskanen et aL Ell] have also estimated the contribution of the d electrons to the R K interaction. This would come via the altered energi~, of s--d hybridized states because pure d states contribute nothing u n d e r the probably very satisfactory assumption of a contact interaction between ~:onduetion electrons and the nuclei. The chang..e brought about by the energy corrections decays faster than r -a. Therefore the contribution of the d electrons was imitated as a change in the strength of the nearest neighbor (nn) coupling. T h e most interesting thing found was that a 50% reduction of the nn interaction would make the second best phase stable. An increase by 50% would just push the transition temperature down by 8 % . It is hard to believe that such strong effects should come from the d electrons, and one is tempted to conclude 'that the phase corre~;pending to k = (2w/a)(1, 0, 0) is the best. T h e experimentally observed transitio~ temperature in copper is 60 nK as compared with the
320
./. Kurkijlirvi [ Copper nuclei at nanokch4n temp
m e a n field prediction 230 nK. Obviously fluctuations account for a part in the discrepancy [12, [1] but there seems to me to be a real discrepancy here. High temperature expansions suggest transition temperatures on t h e order of 180 nK [13], which is still m u c h higher than seen experimentally. It has been shown that n o n c o m m a t i n g spins reduce classically calculated transitior~ temperatures by up to 30q" [14] in the case of spin 1/2. In the present case of spin 3/2, q u a n t u m :spins can account only for a 10% reduction. Experimentally an antiferromagnetic ordered phase i.,; observed in the Otaniemi Low T e m p e r ature Laboratory with the tram.ition temperature of 6 0 n K [3], They use a two stage nuclear demagnetization cryostat. T h e natural variables of thin system are entropy and energy, and the determination of the temperature of t h e system is a non-trivial task. T h e most striking demonstrat~on of the new phases is therefore a measurem e n t of the static magnetic susceptibility of the system as a function of the ~Eemperature after demagnetizations to different end fields. The susceptibility behaves in a characteristic fashion. It rises first for a while, goes through a m a x i m u m and then slowly joins the pa~amagnetic, exponentially relaxing line. Different end fields of demagnetization can m e a n that the system ends up in differ.ent phases which ~,,hould show up as different behaviors of th,~ susceptibilitiez as functions of the end field. Such differences are displayed in fig. 2. T h e initial susceptibility after demagnetization wtries as a fnnction of the e n d field and so does the m a x i m u m the susceptibility goes through in its time development. T h e plot at the bottom of lig. 2 is the difference of the two. "l'wo distinct regions can be ide,~tified separated by the minimum in tht. difference of the m a x i m u m and minimum susceptibilities. In end fields abo~e 0.25 naT, the syst<;m remained in the par,'tmagnetic state. A charaeteri~.tic metastability accompanied these observations. When t h e field was swept to the paramagnetic region for a short time and the~ back during the relaxation after a demagnetization to a low end field, returning to the low field did not bring the system back to the original ph~Lse. The susceptibility often remained
o~' ,"
o
~.o
%=/\.
"~
Xin O. 1 :
0.0
"A"
'
I O.O
"B"
,
,' ~.~
r
"P"
I X¢,. 0.2
i 0.3
El (roT)
Fig. 2. I=iidal susceptibility ~'f~er demagnetizatioa and the maximum susceptibility reachvd in the course of time. Both are I~ivtn as functions of the ~nd field of the demagnetization. In the lower ~art of the figure, the difference of the two is pioUed. A and B denote two different phases. P is the paramagnetie pha,~e.
above the m a x i m u m value it would have reached in the course of time. O n th,~ scale on temperature, 0.1 naT on the end field scale corresponds to around 50 nK, Obviously, somcffling much more complicated than the theoreltical predictions is seen. [t seems that an external field favors some other antiferromagnetically ordered arrangement than predicted at zero field. A n o t h e r piece of evidence favors the presence of two different phases [15], an N M R shift from the internal fietd peak seen everywhere where the ordezq,~d phase is assumed present. At Io~ fields this shift is negative but positive m high fields, .An N M R shift as such is not unexpected [16] since alll the predicted spin orders have less than cubic syrrmaetry. It is diffio,tlt to use t h e N M R i~formatiov here to predict the o r d e ; as there is ~o strong field, such as the exchange field in solid aHe, to make an adiabatic calculation possible. Conceivably different ordered phases could arise through a spin flop transition. T h e type of ord~;r p a r a m e t e r predicted has a large freedom in choosing its configuration due to the degenerate eigenv¢lucs. Jt could therefore undergo a spin flt p tran:~itiorJ s:arting from some tree dimensional arrangement and line up all the spins in
Jr. KurJ~i]ilrvi / Copper nuclei at nanokelvin temperatures
the plane perpendicular to the field. It is not easy to understand, however, where the anisotropy energy required by the spin flop would come from. The necks of the Fermi surface, for one thing, would not do sine'= they preserve the symmetry of the (1, 0, 0) axi; and therefore the degeneracy. The metastability seen hnplies first order transitions at least in the presence of an external field. The part of the evolutio~ of the susceptibility after the maximum and heroine joining the paramagnetic curve is also interpreted as a first order transition [2] even in zero external field. On the other hand, all the theoretical calculations are in the spirit of second order transitions. As a conclusion I should suggest, nevertheless, that there is no strong experimental evidence against the predicted phase with the structure shown in eqs. (8) and (9) in a vanishing external field.
AekJowtedgemten~s I am indebted to Pradeep Kumar and members of the Otaniemi experimental group, Matti Huiku and Markku Loponen in particular, for countless discussions and argumet~ts.
32I
RetercnceS [1] M.T. Huiku and M,T. Loponen, Phys. Rev. LeR. 49 (1982} 1288. [2] M.T, Haiku, T.A. JyrkkiO and M.T. Loponen, Phys. Rev. Left. 50 (1983) 1516, I'3] M.T. Haiku, T.A. Jyrkkie. M.T. Loponen xnd O.V. Lotmasmaa, in: Quantum Fluids and Solids 1983, G.G. lhas, ed. (American InsL of Physics, New "York~1983). 1'4] K, Clausen, M.T. Hulku, T.A. Jyrkki6, M.T. Loponen, O.V. Lounesmaa, P,R. Roach and K. Sk61d, Report TKK-F-A529 (1983). [5] A, Abzagamand M, Goldman, in: Nuclear Ma/zaefism: Order and Disorder (Clarendon. Oxford, 1982). [6] M.A. Ruderman and C. Kittcl, Phys, Rev. 96 (1954} 99. 1'7] J.P. Ekstr6m, J.F. Jacquinoto M.T. Loponen, J,K. $oini and P. Kumar, Physica 98B (1979) 45, [8] L.FI. KjSldman and J. Kurki]iirvl. Phys. Let;, 71A (19'19) 454. [9] M. Goldman, Physi~ Reports 32 (1977} 1, [tO] E,R, Andrews :and W.S. Hinshaw. Phys, Lett, 43A (1973) 113. [11] K.J. Niskanen. L.H. Kj~ildmanand J, Kurkij~rvi, JLTP 49 (1982} 241. [12] P. Kumar, M.T. Loponen and L.H. Kj~ldman, Phys. Eev. Lc.tt. 44- (1980) 493. [13] R.J. Niskanen and J. Kurkij~rvi, .1, Phys. C 14. (1981} 5:317. [14] K~3. Niskanen and J. Kurkijfil'vLJ. Phys. A 16 (19531 1491. [15] M.T. Haiku, private communiear.ien(1984). [16] D.D~ Oshe~ff, M.C. Cross and D.S. Fisher. Phys. Rev. Len, 44 (1980) 792.
MAGNETIC ORDERING ,OF THE N U C L E A R SPINS IN M E T A L L I C COPPER AT NANO~LV]~
T]EMPIERATURES A N D L O W A P P ~
FIELDS
.Iuhani KURKIJA~RVI Department of Technlc¢~l Physles, Helsl!nki University of Technology. SF-02150 E.sooo IS. Finland Th~ theoretically predicted ordered ph~es at copper nuclei a~ r~nokelvlrl temperatures ate critically disctt.~sedand compared with ex0eriment.
Although nuclear magnetic ordering has been seen in metallic copper [1], the precise nature of the ordering is not known [2, 3], In order to learn more,, neutron diffraction experiments are planned [4] which should east light on the arr a n g e m e n t of the spins at the different lattice points of the metal. It would be exciting to be able to predict just what will be seen, and this article is a short review of what we know about the system at this moment. Nuclear magnetic ordering has been extensively studied in salts [5]. It is not quite as obvious that the nuclei in a metal, as well, shouid become sepa,'ated from the lattice at tow temperatures. Their coupling to conduction electron excitations is reduced by Fermi statistics to the point of making the relaxation time to mutual thermal equilibrium between the nuclei and the electrons inversely proportional to t h e temperature, -r~ = K/T~. At temperatures of iate~'est in the study of p h e n o m e n a depending on the interaction strength between the nuclei, ~r~, t:~rns out to be much longer than most rela~xation times within the nuclei themselves. I say most relaxation times since we cannot be quite sure about the decay times of eventual metastable states. In practice [3~, "r~ can be as long as three hours. A l t h o u g h the coupling o~ the nucle:i to conduction electron excitations varies with the temperature, the elect~'ons mediate a n interaction, the R u d e r m a n - K i t t e l (henceforth RK) indirect exchange interaction [6], which does not, It can be viewed as a modification of the conduction electron ground state energy as a function of the
nuclear spin configuration via the hyperfine interaction. It oscillates as a function of the distance as is typical in cases of a sharp cut-off at ~he Fermi surface HRK/const = 2kv cos(2kvr)/rS-sin(2k~r)/r 4
(1)
which form of the interaction has been calculated assuming plane wave s-electrons within a sperical Fermi surface of radius k~. Although the .strength of the R K interaction is known flora ~:xperiment [7], it is instructive to study the nuclear ordering letting it vary according to 0.26"0 (0.26 an arbitrary constant) as compared with the relative strength unity of the dipole-dipole interaction which depends only on natural con,,~tants. The full interaction between the nuclei ~:hus consiszs of the dipole--dipole interaction, which is exactly known, and the approximate R K interaction. It is clear without saying that the R K interaction makes the nuclei in a metal a much harder problem thar, in a salt. It is relatively easy to find the m e a n field predictions of the ordered phases. Ov.e writes the Hamiltonian of the system aporoximately in terms of the average field each spin sees. which consists of the spin--spin interaction s u m m e d over all the o t h e r spins assumed ordered accordinlz to the order parameter one is testing. T h e different order parameters can be classified by their spatial variations, i.e. by the wave vectors describing tho'.~e variations. The usual m e a n field expression for the average spin at a site ther: [cads, after a lineariza~ion of the Brillouin fuocti~n and a Fourier transform, to the eigenvalue
0378-4363/84/$03.00 l ~ Elsevier Science Publishers B.V. (North-Holland Physics Pt, blishing Division)
318
J,
Kurki]iirvi / Copper nucl¢'i at nanokelvin temperatures
problem [8]
x 0,"(t~)> = ~'~ A " " ( k ) ( r ( k ) > ,
(2)
i,
where A""(~) is the three by three spin-spin interaction in the spin c o m p o n e n t space. T h e eigenvalue gives the transition temperature according to
(31 where ~:/i~/4 is the strenEth of the dipolar interaction between the spins and a is the edge of t h e conventional cubic cell in copper. It is useful to take a look at a c o n t i n u u m distribution of spins first. Then the dipolar part of A "'(Ic) has off diagonal elements proportional to k,~k,,/k ~, On the diagonal it has the R K interaction 0.26~g(/¢) and the scalar part of the dipole-dipole interaction. All in all we have A " " ( k ) = (0.26r/g(k).- 4¢r/3) ~,,. + 4~lq,k,Jk :.
(4) Thus, if the distribution of spins were uniform instead of on a lattice, the eigenvatue problem would have the eigenvector Ik with th~ eigenvalue 0.26~g(k)+8~r/3 and two degenerate eigenvectors peper~dicular to k with the eigenvalue 0.26~lg(k)-~rt/3 which latter would give the positive transition temperature. We note that the positive eigenvalue of the dipole--dipole interaction is twice that of the negative degenerate one in absolute value. Since g(k) is always postfive for the uni~")rmly distributed system of spins, a strong R K ir~t;,.~ra,~tion would always end up suppressing the tr::asition altogether. It is easy to see that the above general struclure of eigcnvectors and eigenvalues survives the change to a latl:ice t~f spins if k is in one of the symmetry directions of the lattice. The symmetry directions seem to be the relevant ones anyway, acc(~rding to numerical calculations. There are some changes in the cigcnvalues. With growing k the positive dipolar contribution to the eigenvalue gets reduced a r d the negative one of lhe perpendicular elgenvectors stays half of the positive one
in absolute value. T h e second significant change is that t h e Fourier transform of the R K interaction on a lattice, fee in particular, goes negmive before the zone boundary. This again m e a n s that the transition temperatures to high k structures rise with growing strengtl~ of the R K interaction rl. T h e experimental value of rl is r I = 2.75 [7] and the highest transition temperature structure corresponds to /c = (2rtta)(1, O, 0> with the eigenvaIue - 2 . 1 7 - 1 . 1 3 r / a:nd two degenerate eigenvectors perpendicular to k in the same class <1,0,0). T h e second best structure would correspond to k = (3~r/2a)(1, 0, 0) with the same eigenvectors as above and the eigenvalue - 2 . 4 8 - 0.96Tr. T h e predicted transition temperalures to the best and the second best structures at r t - - 2 . 7 5 are 230 nK and 223 nK. it may be interesting to notice that the best k is at the zone boundary. In t h e (l, 1, 1) direction, where the Fermi surface osculates the zone boundary, the free electron approximation gives the degenerate eigenvalue - 3 . 6 1 + 0 . 6 9 r l and the transition temperature 75 nK at ~1 = 2-75. Such as structure therefore does not cornpete. T h e mos~ general spin structure corresponding to the best k above is given by (/~) = (0, a~, az!cos(k~ • ri) + (a3, 0, a4)eos(/~.~ • r,) + (as, a~,, 0)cos(k.~ • r,),
(5~
where kl = (2rr/a)(1, 0, 0), k~ = (2w/a)(0, 0, 1).
k~ = (2~'/a)(0, 1, 0), (6)
In principle one sh,~utd include sine and cosine variatior~s according to all /~ belonging to (2~-/a)
J+ Kurkijilrvi / Copper nutlei at nanokelvin teraperatur~"
p e n d e n t of i. Thl,s sets conditions on the quantities a~ ata~ = azaa = azas.
(7)
'll~en there are two possible structures
(0, at, a 2 ) c o s ( k t - r~) + ( a 3 , 0, 0)cos(k2 - r~)
(8) and (I,) - (0,, at, O)eos(k~ • r~) + (0, O, a,Ycos(k2 • r+)
-+"(as, 0, 0)cos(/q • r,).
(9)
T h e strr~cture of eq. (8) is displaved in fig. 1. A pexTnanent struetm-e wh:c]~ is stable at Tc is also the best structure at T = ( : [9]. Although nothing certain can be said about the range between T = 0 and T = "_r we know that the best structure will eventually emerge ~ffter demagnetization at a low enough entropy. How trustworthy are these predictions? As to the value of ~, the Otani6mi experimental group have an ingenious method of determining the strength ,c)f the R K interaction [7] based on the iT,terfereraee oJ~ the N M R absorptio~ lines on the isotopes '+3Cu and 6SCu. Their me, hod rests on the observation that although an exchange field cannot s~ift N M R lines coming from a single isotope, two iso-:epes with different resonance frequenees can influence the effective field seen by the other. Their result already quoted above is not in conflict with the entirely different measu r e m e n t of Andrew et al. [I0] with the high speed magic+angle specimen rotation technique. The value at which the best and the second
Fig. I. Spin arrangemem accordingto eq. (8).
319
best structures w e a l d cross is around 71 = 1.9., appreciably below the experimental finding. Therefore, as fat ~'as the predicted ~tructure is concerned, the strength of l:he R K interaction does not seem to be a problem. Otl'oer sources of error are included in the approximate form of the R u d e r m a n Kittel interaction and in the mean field procedure of course. T h e presence of two randomly distributed isotopes has not been taken into account either. Errors in the form of the R K interaction arise from the nonspherical Fermi surface on the o n e hand and from the contribution of the d electrons on the other. T h e elfccts of the nonspherical Fermi surface have beel:~ estimated [1].] to be on the order of 2 - 3 % on the eigenvalues of the R K interaction near the zone boundary. Eq. (1) was taken for the RK interaction with kF depending on the direction and the now anisotropic Fourier transforms were calculated. T h e s e were then used in the mean field calculation up to the 20th .shell of nearest neighbors. At that distance t h e interaction was cut off and the results were compared with those from the similarly abrogated isotropic R K interaction. For copper this calculation leads to an increased difference in the transitio~a temperatures of the two best phases quoted above. Niskanen et aL Ell] have also estimated the contribution of the d electrons to the R K interaction. This would come via the altered energi~, of s--d hybridized states because pure d states contribute nothing u n d e r the probably very satisfactory assumption of a contact interaction between ~:onduetion electrons and the nuclei. The chang..e brought about by the energy corrections decays faster than r -a. Therefore the contribution of the d electrons was imitated as a change in the strength of the nearest neighbor (nn) coupling. T h e most interesting thing found was that a 50% reduction of the nn interaction would make the second best phase stable. An increase by 50% would just push the transition temperature down by 8 % . It is hard to believe that such strong effects should come from the d electrons, and one is tempted to conclude 'that the phase corre~;pending to k = (2w/a)(1, 0, 0) is the best. T h e experimentally observed transitio~ temperature in copper is 60 nK as compared with the
320
./. Kurkijlirvi [ Copper nuclei at nanokch4n temp
m e a n field prediction 230 nK. Obviously fluctuations account for a part in the discrepancy [12, [1] but there seems to me to be a real discrepancy here. High temperature expansions suggest transition temperatures on t h e order of 180 nK [13], which is still m u c h higher than seen experimentally. It has been shown that n o n c o m m a t i n g spins reduce classically calculated transitior~ temperatures by up to 30q" [14] in the case of spin 1/2. In the present case of spin 3/2, q u a n t u m :spins can account only for a 10% reduction. Experimentally an antiferromagnetic ordered phase i.,; observed in the Otaniemi Low T e m p e r ature Laboratory with the tram.ition temperature of 6 0 n K [3], They use a two stage nuclear demagnetization cryostat. T h e natural variables of thin system are entropy and energy, and the determination of the temperature of t h e system is a non-trivial task. T h e most striking demonstrat~on of the new phases is therefore a measurem e n t of the static magnetic susceptibility of the system as a function of the ~Eemperature after demagnetizations to different end fields. The susceptibility behaves in a characteristic fashion. It rises first for a while, goes through a m a x i m u m and then slowly joins the pa~amagnetic, exponentially relaxing line. Different end fields of demagnetization can m e a n that the system ends up in differ.ent phases which ~,,hould show up as different behaviors of th,~ susceptibilitiez as functions of the end field. Such differences are displayed in fig. 2. T h e initial susceptibility after demagnetization wtries as a fnnction of the e n d field and so does the m a x i m u m the susceptibility goes through in its time development. T h e plot at the bottom of lig. 2 is the difference of the two. "l'wo distinct regions can be ide,~tified separated by the minimum in tht. difference of the m a x i m u m and minimum susceptibilities. In end fields abo~e 0.25 naT, the syst<;m remained in the par,'tmagnetic state. A charaeteri~.tic metastability accompanied these observations. When t h e field was swept to the paramagnetic region for a short time and the~ back during the relaxation after a demagnetization to a low end field, returning to the low field did not bring the system back to the original ph~Lse. The susceptibility often remained
o~' ,"
o
~.o
%=/\.
"~
Xin O. 1 :
0.0
"A"
'
I O.O
"B"
,
,' ~.~
r
"P"
I X¢,. 0.2
i 0.3
El (roT)
Fig. 2. I=iidal susceptibility ~'f~er demagnetizatioa and the maximum susceptibility reachvd in the course of time. Both are I~ivtn as functions of the ~nd field of the demagnetization. In the lower ~art of the figure, the difference of the two is pioUed. A and B denote two different phases. P is the paramagnetie pha,~e.
above the m a x i m u m value it would have reached in the course of time. O n th,~ scale on temperature, 0.1 naT on the end field scale corresponds to around 50 nK, Obviously, somcffling much more complicated than the theoreltical predictions is seen. [t seems that an external field favors some other antiferromagnetically ordered arrangement than predicted at zero field. A n o t h e r piece of evidence favors the presence of two different phases [15], an N M R shift from the internal fietd peak seen everywhere where the ordezq,~d phase is assumed present. At Io~ fields this shift is negative but positive m high fields, .An N M R shift as such is not unexpected [16] since alll the predicted spin orders have less than cubic syrrmaetry. It is diffio,tlt to use t h e N M R i~formatiov here to predict the o r d e ; as there is ~o strong field, such as the exchange field in solid aHe, to make an adiabatic calculation possible. Conceivably different ordered phases could arise through a spin flop transition. T h e type of ord~;r p a r a m e t e r predicted has a large freedom in choosing its configuration due to the degenerate eigenv¢lucs. Jt could therefore undergo a spin flt p tran:~itiorJ s:arting from some tree dimensional arrangement and line up all the spins in
Jr. KurJ~i]ilrvi / Copper nuclei at nanokelvin temperatures
the plane perpendicular to the field. It is not easy to understand, however, where the anisotropy energy required by the spin flop would come from. The necks of the Fermi surface, for one thing, would not do sine'= they preserve the symmetry of the (1, 0, 0) axi; and therefore the degeneracy. The metastability seen hnplies first order transitions at least in the presence of an external field. The part of the evolutio~ of the susceptibility after the maximum and heroine joining the paramagnetic curve is also interpreted as a first order transition [2] even in zero external field. On the other hand, all the theoretical calculations are in the spirit of second order transitions. As a conclusion I should suggest, nevertheless, that there is no strong experimental evidence against the predicted phase with the structure shown in eqs. (8) and (9) in a vanishing external field.
AekJowtedgemten~s I am indebted to Pradeep Kumar and members of the Otaniemi experimental group, Matti Huiku and Markku Loponen in particular, for countless discussions and argumet~ts.
32I
RetercnceS [1] M.T. Huiku and M,T. Loponen, Phys. Rev. LeR. 49 (1982} 1288. [2] M.T, Haiku, T.A. JyrkkiO and M.T. Loponen, Phys. Rev. Left. 50 (1983) 1516, I'3] M.T. Haiku, T.A. Jyrkkie. M.T. Loponen xnd O.V. Lotmasmaa, in: Quantum Fluids and Solids 1983, G.G. lhas, ed. (American InsL of Physics, New "York~1983). 1'4] K, Clausen, M.T. Hulku, T.A. Jyrkki6, M.T. Loponen, O.V. Lounesmaa, P,R. Roach and K. Sk61d, Report TKK-F-A529 (1983). [5] A, Abzagamand M, Goldman, in: Nuclear Ma/zaefism: Order and Disorder (Clarendon. Oxford, 1982). [6] M.A. Ruderman and C. Kittcl, Phys, Rev. 96 (1954} 99. 1'7] J.P. Ekstr6m, J.F. Jacquinoto M.T. Loponen, J,K. $oini and P. Kumar, Physica 98B (1979) 45, [8] L.FI. KjSldman and J. Kurki]iirvl. Phys. Let;, 71A (19'19) 454. [9] M. Goldman, Physi~ Reports 32 (1977} 1, [tO] E,R, Andrews :and W.S. Hinshaw. Phys, Lett, 43A (1973) 113. [11] K.J. Niskanen. L.H. Kj~ildmanand J, Kurkij~rvi, JLTP 49 (1982} 241. [12] P. Kumar, M.T. Loponen and L.H. Kj~ldman, Phys. Eev. Lc.tt. 44- (1980) 493. [13] R.J. Niskanen and J. Kurkij~rvi, .1, Phys. C 14. (1981} 5:317. [14] K~3. Niskanen and J. Kurkijfil'vLJ. Phys. A 16 (19531 1491. [15] M.T. Haiku, private communiear.ien(1984). [16] D.D~ Oshe~ff, M.C. Cross and D.S. Fisher. Phys. Rev. Len, 44 (1980) 792.