The ferromagnetically ordered state in the nuclear spin system of silver has been reached at negative absolute temperatures by adiabatic nuclear demagnetization at entropies below 0.82 ln2. The ordering, caused by the antiferromagnetic Ruderman-Kittel interaction, was observed below -1.9 nK as a saturation of susceptibility close to -1 and as an increase of the NMR frequencies. Comparison with recent mean-field calculations by Viertiö and Oja suggests a domain configuration. The phase diagram of silver nuclei at T<0 is outlined in the magnetic field versus entropy plane.Peer reviewe

Hakonen, Pertti J.

Lounasmaa, O. V.

Nummila, K. K.

Vuorinen, R. T.

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This is an electronic reprint of the original article.
This reprint may differ from the original in pagination and typographic detail.
Author(s): Hakonen, Pertti J. & Nummila, K. K. & Vuorinen, R. T. & Lounasmaa,
O. V.
Title: Observation of nuclear ferromagnetic ordering in silver at negative
nanokelvin temperatures
Year: 1992
Version: Final published version
Please cite the original version:
Hakonen, Pertti J. & Nummila, K. K. & Vuorinen, R. T. & Lounasmaa, O. V. 1992.
Observation of nuclear ferromagnetic ordering in silver at negative nanokelvin
temperatures. Physical Review Letters. Volume 68, Issue 3. 365-368. ISSN 0031-9007
(printed). DOI: 10.1103/physrevlett.68.365
Rights: © 1992 American Physical Society (APS). This is the accepted version of the following article: Hakonen,
Pertti J. & Nummila, K. K. & Vuorinen, R. T. & Lounasmaa, O. V. 1992. Observation of nuclear ferromagnetic
ordering in silver at negative nanokelvin temperatures. Physical Review Letters. Volume 68, Issue 3.
365-368. ISSN 0031-9007 (printed). DOI: 10.1103/physrevlett.68.365, which has been published in final form
at http://journals.aps.org/prl/abstract/10.1103/PhysRevLett.68.365.
All material supplied via Aaltodoc is protected by copyright and other intellectual property rights, and
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Powered by TCPDF (www.tcpdf.org)
VOLUME 68, NUMBER 3 PHYSICAL REVIEW LETTERS 20 JANUARY 1992
Observation of Nuclear Ferromagnetic Ordering in Silver at Negative Nanokelvin Temperatures
P. J. Hakonen, K. K. Nummila, R. T. Vuorinen, and O. V. Lounasmaa
Low Temperature Laboratory, Helsinki University of Technology, 02150 Espoo, Finland
(Received 15 October 1991)
The ferromagnetically ordered state in the nuclear spin system of silver has been reached at negative
absolute temperatures by adiabatic nuclear demagnetization at entropies below 0.82%'ln2. The order-
ing, caused by the antiferromagnetic Ruderman-Kittel interaction, was observed below —1.9 nK as a
saturation of susceptibility close to —1 and as an increase of the NMR frequencies. Comparison with
recent mean-field calculations by Viertio and Oja suggests a domain configuration. The phase diagram
of silver nuclei at T & 0 is outlined in the magnetic field versus entropy plane.
PACS numbers: 75.30.Kz, 75.90.+w
Nuclear magnetic ordering has been studied extensive-
ly at T) 0 in copper and silver. In these metals, com-
petition between the dipolar force and the antiferromag-
netic exchange interaction leads to interesting phase dia-
grams below the transition temperatures of 58 nK and
560 pK, respectively [1-3]. Fundamentally different be-
havior is expected when T & 0 [4,5].
At positive temperatures the distribution of spins
among the Zeeman levels is given by the usual Boltzmann
factor, exp( pB/kT)—. Since the energy spectrum of nu-
clear spins is limited, a population inversion is possible,
e.g. , by reversing the magnetic field quickly in a time
t & z2 10 msec, so that the spins do not have a chance
to redistribute themselves among the energy levels. This
state of inverted spin populations can also be described by
the Boltzman distribution but with a negative absolute
temperature. At T &0, an isolated spin system has the
highest possible Helmholtz free energy. In silver, which
is dominated by an antiferromagnetic Ruderman-Kittel
exchange interaction, this favors parallel alignment of the
spins and, consequently, leads to a ferromagnetic order-
ing, which provides the state of maximum energy at
T &0.
In our earlier experiments on silver [6], a clear tenden-
cy to ferromagnetism was seen when the measured sus-
ceptibility data were fitted with the Curie-Weiss law, but
the transition point was not reached. After vital irnprove-
ments in our experimental techniques, we are finally able
to report in this Letter the first observation of the fer-
romagnetically ordered nuclear spin state in a metal at
T &0. This ferromagnetic ordering, caused by an anti-
ferromagnetic interaction, is a concrete example of the
significance of spin temperatures. At T) 0, the very
same interactions produce antiferromagnetic order.
The Hamiltonian in silver can be written in the form
H =H41p+ Hex+ Hz
=Hd;t, —QjtiI; Il —hB gy;I;,
where B is the external magnetic field appearing in the
Zeernan term Hz. The dominating spin-spin energy is
the nearest-neighbor exchange interaction H,„, whose
strength can be estimated from earlier NMR experiments
[7,8] which yield J/1't = —26.5 Hz for the exchange in-
tegral; the dipolar energy H4'p is smaller roughly by a
factor of 3. Natural silver is composed of two isotopes
'o7Ag and 'o9Ag; both have spin I= —,' . Their gyromag-
netic ratios y are suSciently close so that, at low fields,
the NMR lines merge.
Our silver sample, of nominal purity 99.99+%, consist-
ed of 48 polycrystalline foils with dimensions 8x0.025
&65 mm, oriented along the x, y, and z directions, re-
spectively. The assembly of the specimen is described in
Ref. [9]. In order to facilitate the production of negative
temperatures by the rapid field reversal, the new coil set-
up and the radiation shields were carefully prepared to
prevent eddy currents which must have been the main
problem during our previous, less successful experiments
[6]. Two saddle-shaped static field coils of lengths 60 and
65 mm and one 70 mm long solenoid for B„all wound of
50-pm multifilamentary NbTi superconducting wire,
were located on a coil former (diameter of 14 mm) inside
the brass radiation shield (diameter of 16.5 mm) which
had four cuts symmetrically along its length. The astati-
cally wound pickup coil was oriented parallel to the z
axis.
The experiments were carried out in our cascade nu-
clear demagnetization cryostat [10]. Achieving a high
polarization in silver is a slow process: The spin-lattice
relaxation time ri =14 h at 200 pK. Details of the cool-
ing process and experimental techniques are described in
Ref. [9]. However, the demagnetization sequence was
modified: We employed degaussing of the second stage
magnet (+7.4 T —0.1 T +0.02 T 0) to guaran-
tee proper operation of the SQUID measurement system,
susceptible to vibrational noise in the presence of a small
remanent field. Fortunately, this procedure did not sub-
stantially decrease the largest initial polarization, p
=0.85, achieved in these experiments. Measurements of
the longitudinal adiabatic susceptibility were employed to
determine the remanent field which varied between 0 and
5 pT.
NMR spectra, measured at 200 pT, were used to cal-
culate the polarization from the equation [4] p
=AJZ"(f)df; the proportionality constant A was cali-
brated against the Pt-NMR temperature scale around 1
1992 The American Physical Society 365
VOLUME 68, NUMBER 3 P H YSICAL REVI E% LETTERS 20 JANUARY 1992
mK [9]. We also employed the frequency shifts of the
NMR lines, with the static field along the x and y axes,
to determine the ratio of the demagnetization factors
D:D~=-1:3 for our sample; on the basis of this we esti-
mate D„0.25, D~ =0.75, and D, =0.
The solenoidal magnetic field B, employed in the rapid
field reversal was typically 400 pT. Our best inversion
efficiency, 90%, was reached for small initial polarizations
on the order of p 0.30. At high polarizations, the re-
sults varied irregularly. Despite many efforts by chang-
ing the speed (0.5-5 msec), symmetry, and magnitude of
the field flip, the optimum inversion efficiency was only
60%-75% at p 0.85. Therefore, polarizations at T & 0
were limited to p=- —0.65. In copper, production of neg-
ative temperatures has not succeeded owing to the short
150 psec.
Figure 1 displays NMR line shapes measured in zero
field. Instead of absorption, as at T &0, the system is
emitting energy when T (0. The emission maximum
shifts towards higher frequencies with increasing polar-
ization. The solid curves are fits of Lorentzian line
shapes to the measured spectra, with the negative fre-
quency side included; the least-squares minimization was
done for the emission and dispersion curves, simultane-
ously. The static susceptibility was obtained from the fits
by using the Kramers-Kronig relation g'(0) =(2/z)
x fo (g"/f )df; the upper limit of integration was set to
200 Hz. The absolute value of the susceptibility was cali-
brated in the paramagnetic state by using the equation
[9] I/g (0) 8/popMsat+Dz D~, where—po~sat =45 1
pT is the field corresponding to saturation magnetization.
Figure 2 displays the static susceptibility g g'(0), in-
—1.2
—20
T (nK)
tegrated from the Kramers-Kronig relation, as a function
of polarization in zero field and at four different values of
8,; at 5 pT, data obtained in the parallel field B: are also
shown. At zero field and at 5 pT, there is first a monoto-
nous increase of g'(0) with p, which then saturates be-
tween p= —0.4 and —0.5. According to studies on
dielectric materials [4] and the mean-field calculations to
be discussed below, this behavior is caused by ferromag-
netic ordering; the susceptibility is governed by dipolar
interaction through the formation of ferromagnetic do-
mains which, in the ideal case, leads to g= —1 for our
sample.
In zero field and at small polarizations the susceptibili-
ty of silver is of the form I/@=i/g +oR+L D„wh—ere
@0=1.30p is the Curie-law susceptibility of the nonin-
teracting spin system (see Ref. [9]) and the Lorentz fac-
tor L = 3 . The uppermost curve in Fig. 2 is the best fit of
points with )p~ (0.45 by this equation, using R = —1.1
which differs quite a lot from the value R = —2.5 ~0.5
obtained from the NMR experiments [8]. This indicates
that substantial deviation from the mean-field behavior
takes place already at intermediate polarizations. %ithin
the scatter of the measured data, the same curve fits the
experimental results at 8=0 and at B 5 p T. As is typi-
cal for dipolar ferromagnetism [4], our data do not satu-
rate completely even when the polarization is high but,
nevertheless, we have described the results for
~p~ & 0.55
approximately by a constant line, g =g„.t = —1.05.
We identify the crossing of the two lines in Fig. 2 as
the transition point to the ferromagnetic state. Owing to
—1.0
-0.2
—0.8
-0.4
-0.6
-0.8
-1.0
-1.2
0 50 100 0
f (Hz)
50 100 150
—0.4
—0.2
00 —0.2 —0.4 —0.6 —0.8
FIG. 1. NMR absorption g" and dispersion g' curves (in di-
mensionless Sl units) of the nuclear spin system in silver at
T &0, measured in zero field at reduced entropies S/Aln2
equal to 0.87 (o), 0.80 (a), and 0.73 (rj), which correspond to
polarizations p —0.42, —0.51, and —0.59 in the paramag-
netic state, respectively. The zeros for the lower curves have
been shifted downwards by 0.25 and 0.55 in both frames.
FIG. 2. Static susceptibility g (0) vs polarization p of silver
nuclear spins at T &0, measured in a magnetic field B, of 0
(O), 5 (V), 20 (D), 50 (0 ), and 100 p T (~), and at Bz =5 p T(x). The fitted curves are discussed in the text. The scale on
top gives estimated temperatures based on the formula I9l
I/(p( —1 =0.55(T) (in nK), which applies only at B=O; values
to the right of —2 nK must be considered tentative.
366
VOLUME 68, NUMBER 3 PHYSICAL REVIEW LETTERS 20 JANUARY 1992
the rounding of g(p), we estimate conservatively, in zero
field and at 5 pT, for the critical polarization p, =0.49
~0.05, which corresponds to S,/Rln2=0. 82+'0.035.
This critical entropy is quite large when compared with
the values for antiferromagnetic ordering, S,/R ln4
=0.48-0.61 and S„/Rln2=0. 53, in copper [I] and silver
[2], respectively. By employing the linear relationship
between the temperature and inverse polarization, 1/
)p~ —1 =0.55[T~ (in nK), determined in earlier experi-
ments [9], we obtain T, = —1.9 ~ 0.4 nK; this value may
have a systematic error of 20%, owing to the uncertaihty
of the relationship between p and T at negative tempera-
tures.
We also tried to determine T, directly from the weight-
ed average (first moment) of our NMR data, which
display a clear minimum (see Fig. 3) at p =0.3; this sug-
gests a lower critical polarization than the saturation of g.
We think, however, that the analysis based on Fig. 2 is
more reliable since the behavior of the NMR frequencies
upon ordering is not very well understood.
According to the mean-field theory, T, =8= —5 nK in
silver; this is clearly larger than the observed value. Cal-
culations [11]on the spin- —,' Heisenberg model in an fcc
lattice yield T, =0.678= —3.4 nK, which exceeds our re-
sult by 80%; the computed value does not change much
even if the small next-nearest-neighbor interaction is tak-
en into account [12]. In contrast, the critical entropy ac-
cording to the Heisenberg model, S,=0.66% ln2, is clear-
ly lower than our experimental value. This is reasonable
since the long-range dipolar interaction acts ferromagnet-
ically at negative temperatures owing to domain forma-
tion, which increases S, in silver.
Monte Carlo calculations [13] predict T, = —1.7 nK,
which is close to our measured result. However, the com-
puted critical entropy S, =0.93%1n2 deviates substantial-
ly from the experimental value, which indicates that the 80
60
T)0
T(0
good agreement between the theoretical and experimental
data on T„may be fortuitous.
The saturation of susceptibility to g„.t= —1 in the or-
dered state can be explained only by the formation of
domains, since otherwise the susceptibility would diverge
at T,. Instead of needles, as at T&0, plate-shaped
domains are expected [4]. According to mean-field calcu-
lations [13], a ferromagnetic state with broken plate-
shaped domains yields g,,&= —I/(1 —Dsr); here Dsi is
the demagnetization factor along the spontaneous magne-
tization M. This is in agreement with our results at 8 0
and 8, 5 p T. In a small transverse field, however, M is
perpendicular to the plates and a much larger g should
result, but it was not observed at 8~ 0-10 pT. This
suggests the presence of a single-domain state which can
be stable in a transverse field. The mean-field theory [14]
then yields g„.& —1/(D» D, ) = ——1.33, which is fairly
close to the experimental value g„.&= —1.05. Moreover,
Monte Carlo simulations [13] have confirmed the mean-
field result in the ideal case when D~ D, = l. —
At higher fields, g'(0) is directly proportional to p (see
Fig. 2), i.e., the behavior is paramagnetic. The saturation
of susceptibility was lost in the g(r) sweeps already
around 10 p T and a reliable determination of the transi-
tion point became impossible. On the basis of the detect-
ed transitions at 0 and 5 pT (see Fig. 2), we have con-
structed a schematic B vs S/Rln2 phase diagram for
silver which is depicted in Fig. 4. We estimate B,
—ppN. ,i/g. ,i(T=O) =40 p T at zero temperature; this
value has been used when drawing the dashed transition
curve for the ferromagnetic phase.
38—
40
20
34—
A
V 32-
0-
0.4 0.5 0.6
S/%ln2
0.7
't
e
0.8 0.9
30—
0.0 —0.2 —0.4 —0.6
P
FIG. 3. First moment (f) of the NMR line shapes as a func-
tion of p, measured in zero field (O) and at B» =5 pT (V). The
curves are just to guide the eyes.
FIG. 4. Phase diagram in the B vs S/Rln2 plane for the nu-
clear spins in silver metal at negative absolute temperatures.
Solid circles represent our experimental data, obtained from
Fig. 2, and the dashed curve displays a theoretical phase bound-
ary between the ferromagnetic (at left) and paramagnetic re-
gions. The shape of this curve is based on the mean-field theory
[l 1].with the assumption of a linear relationship between 5 and
T The antiferromagneti. c phase transition curve at T) 0 [2],
marked by the solid line, is included for comparison.
367
VOLUME 68, NUMBER 3 PH YSICAL REVIEW LETTERS 20 JANUARY 1992
We searched for hysteresis in M(B) in the ordered
state by measuring g' at the fixed frequency of 10 Hz
while sweeping the transverse magnetic field between
+ 30 pT. The adiabatic susceptibility g'(10 Hz) dis-
played a small, 5% asymmetry in amplitude with respect
to zero field, but less than 0.5 pT of hysteresis between
opposite field sweeps. Since the hysteresis was so small,
we prefer to assign the observed asymmetry in g'(IO Hz)
to small changes in the phase of the signal during the
field sweep while keeping a constant phase setting in the
lock-in amplifier.
In summary, we have made the first observation of a
ferromagnetic state in a metallic nuclear spin system at
T &0. Comparison of the saturation values of suscepti-
bility with the mean-field theory and Monte Carlo calcu-
lations [13] indicate that in fields parallel to the sample
plates there is a multidomain state in silver, whi1e in a
transverse field a single-domain structure is preferred.
The hysteresis in M(B) was found to be small, less than
0.5 pT, which points to a ferromagnetic system without
anisotropy. The measured critical entropy, S, =0.82%
xln2, is larger than the value for the Heisenberg model,
S„=0.66%[n2, which indicates that, even though the
Ruderman-Kittel exchange is dominating, the dipolar in-
teraction substantially helps in the ordering process at
T & 0; this leads to a ferromagnetic phase diagram which
differs substantially from that of the antiferromagnetic
state at T&0.
Useful interaction with A. Oja and H. Viertio on
theoretical issues is gratefully acknowledged. This work
was supported by the Academy of Finland.
[I] M. T. Huiku and M. T. Loponen, Phys. Rev. Lett. 49,
1288 (1982); A. J. Annila, K. N. Clausen, P.-A.
Lindgard, O. V. Lounasmaa, A. S. Oja, K. Siemens-
meyer, M. Steiner, J. T. Tuoriniemi, and H. Weinfurter,
Phys. Rev. Lett. 64, 1421 (1990).
[2] P. J. Hakonen, S. Yin, and K. K. Nummila, Europhys.
Lett. 15, 677 (1991).
[3] For a recent review, see P. J. Hakonen, O. V. Lounasmaa,
and A. S. Oja, J. Magn. Magn. Mater. (to be published).
[4] See, e g , A. . Abragam and M. Goldman, Nuclear Magne
tism: Order and Disorder (Clarendon, Oxford, 1982).
[5] P. Bonamour, V. Bouff'ard, C. Fermon, M. Goldman, J. F.
Jacquinot, and G. Saux, Phys. Rev. Lett. 66, 2810
(1991).
[6] P. J. Hakonen, S. Yin, and O. V. Lounasmaa, Phys. Rev.
Lett. 64, 2707 (1990).
[7] J. Pointrenaud and J. M. Winter, J. Phys. Chem. Solids
25, 123 (1964).
[8] A. S. Oja, A. J. Annila, and Y. Takano, Phys. Rev. Lett.
65, 1921 (1990).
[9] P. J. Hakonen and S. Yin, J. Low Temp. Phys. (to be
published).
[10] G. J. Ehnholm, J. P. Ekstrom, J. F. Jacquinot, M. T.
Loponen, O. V. Lounasmaa, and J. K. Soini, J. Low
Temp. Phys. 39, 417 (1980).
[I I] See, e.g. , L. J. de Jongh and A. R. Miedema, Adv. Phys.
23, I (1974).
[12] This has been reviewed briefly, e.g. , by R. H. Swendsen,
Phys. Rev. Lett. 32, 1439 (1974).
[13] H. E. Viertio and A. S. Oja, J. Magn. Magn. Mater. (to
be published).
[14] H. E. Viertio and A. S. Oja (private communication).
368


English

Observation of nuclear ferromagnetic ordering in silver at negative nanokelvin temperatures

Aaltodoc Publication Archive

This is an electronic reprint of the original article.This reprint may differ from the original in pagination and typographic detail.Author(s): Hakonen, Pertti J. & Nummila, K. K. & Vuorinen, R. T. & Lounasmaa,O. V.Title: Observation of nuclear ferromagnetic ordering in silver at negativenanokelvin temperaturesYear: 1992Version: Final published versionPlease cite the original version:Hakonen, Pertti J. & Nummila, K. K. & Vuorinen, R. T. & Lounasmaa, O. V. 1992.Observation of nuclear ferromagnetic ordering in silver at negative nanokelvintemperatures. Physical Review Letters. Volume 68, Issue 3. 365-368. ISSN 0031-9007(printed). DOI: 10.1103/physrevlett.68.365Rights: © 1992 American Physical Society (APS). This is the accepted version of the following article: Hakonen,Pertti J. & Nummila, K. K. & Vuorinen, R. T. & Lounasmaa, O. V. 1992. Observation of nuclear ferromagneticordering in silver at negative nanokelvin temperatures. Physical Review Letters. Volume 68, Issue 3.365-368. ISSN 0031-9007 (printed). DOI: 10.1103/physrevlett.68.365, which has been published in final format http://journals.aps.org/prl/abstract/10.1103/PhysRevLett.68.365.All material supplied via Aaltodoc is protected by copyright and other intellectual property rights, andduplication or sale of all or part of any of the repository collections is not permitted, except that material maybe duplicated by you for your research use or educational purposes in electronic or print form. You mustobtain permission for any other use. Electronic or print copies may not be offered, whether for sale orotherwise to anyone who is not an authorised user.Powered by TCPDF (www.tcpdf.org)VOLUME 68, NUMBER 3 PHYSICAL REVIEW LETTERS 20 JANUARY 1992Observation of Nuclear Ferromagnetic Ordering in Silver at Negative Nanokelvin TemperaturesP. J. Hakonen, K. K. Nummila, R. T. Vuorinen, and O. V. LounasmaaLow Temperature Laboratory, Helsinki University of Technology, 02150 Espoo, Finland(Received 15 October 1991)The ferromagnetically ordered state in the nuclear spin system of silver has been reached at negativeabsolute temperatures by adiabatic nuclear demagnetization at entropies below 0.82%'ln2. The order-ing, caused by the antiferromagnetic Ruderman-Kittel interaction, was observed below —1.9 nK as asaturation of susceptibility close to —1 and as an increase of the NMR frequencies. Comparison withrecent mean-field calculations by Viertio and Oja suggests a domain configuration. The phase diagramof silver nuclei at T & 0 is outlined in the magnetic field versus entropy plane.PACS numbers: 75.30.Kz, 75.90.+wNuclear magnetic ordering has been studied extensive-ly at T) 0 in copper and silver. In these metals, com-petition between the dipolar force and the antiferromag-netic exchange interaction leads to interesting phase dia-grams below the transition temperatures of 58 nK and560 pK, respectively [1-3]. Fundamentally different be-havior is expected when T & 0 [4,5].At positive temperatures the distribution of spinsamong the Zeeman levels is given by the usual Boltzmannfactor, exp( pB/kT)—. Since the energy spectrum of nu-clear spins is limited, a population inversion is possible,e.g. , by reversing the magnetic field quickly in a timet & z2 10 msec, so that the spins do not have a chanceto redistribute themselves among the energy levels. Thisstate of inverted spin populations can also be described bythe Boltzman distribution but with a negative absolutetemperature. At T &0, an isolated spin system has thehighest possible Helmholtz free energy. In silver, whichis dominated by an antiferromagnetic Ruderman-Kittelexchange interaction, this favors parallel alignment of thespins and, consequently, leads to a ferromagnetic order-ing, which provides the state of maximum energy atT &0.In our earlier experiments on silver [6], a clear tenden-cy to ferromagnetism was seen when the measured sus-ceptibility data were fitted with the Curie-Weiss law, butthe transition point was not reached. After vital irnprove-ments in our experimental techniques, we are finally ableto report in this Letter the first observation of the fer-romagnetically ordered nuclear spin state in a metal atT &0. This ferromagnetic ordering, caused by an anti-ferromagnetic interaction, is a concrete example of thesignificance of spin temperatures. At T) 0, the verysame interactions produce antiferromagnetic order.The Hamiltonian in silver can be written in the formH =H41p+ Hex+ Hz=Hd;t, —QjtiI; Il —hB gy;I;,where B is the external magnetic field appearing in theZeernan term Hz. The dominating spin-spin energy isthe nearest-neighbor exchange interaction H,„, whosestrength can be estimated from earlier NMR experiments[7,8] which yield J/1't = —26.5 Hz for the exchange in-tegral; the dipolar energy H4'p is smaller roughly by afactor of 3. Natural silver is composed of two isotopes'o7Ag and 'o9Ag; both have spin I= —,' . Their gyromag-netic ratios y are suSciently close so that, at low fields,the NMR lines merge.Our silver sample, of nominal purity 99.99+%, consist-ed of 48 polycrystalline foils with dimensions 8x0.025&65 mm, oriented along the x, y, and z directions, re-spectively. The assembly of the specimen is described inRef. [9]. In order to facilitate the production of negativetemperatures by the rapid field reversal, the new coil set-up and the radiation shields were carefully prepared toprevent eddy currents which must have been the mainproblem during our previous, less successful experiments[6]. Two saddle-shaped static field coils of lengths 60 and65 mm and one 70 mm long solenoid for B„all wound of50-pm multifilamentary NbTi superconducting wire,were located on a coil former (diameter of 14 mm) insidethe brass radiation shield (diameter of 16.5 mm) whichhad four cuts symmetrically along its length. The astati-cally wound pickup coil was oriented parallel to the zaxis.The experiments were carried out in our cascade nu-clear demagnetization cryostat [10]. Achieving a highpolarization in silver is a slow process: The spin-latticerelaxation time ri =14 h at 200 pK. Details of the cool-ing process and experimental techniques are described inRef. [9]. However, the demagnetization sequence wasmodified: We employed degaussing of the second stagemagnet (+7.4 T —0.1 T +0.02 T 0) to guaran-tee proper operation of the SQUID measurement system,susceptible to vibrational noise in the presence of a smallremanent field. Fortunately, this procedure did not sub-stantially decrease the largest initial polarization, p=0.85, achieved in these experiments. Measurements ofthe longitudinal adiabatic susceptibility were employed todetermine the remanent field which varied between 0 and5 pT.NMR spectra, measured at 200 pT, were used to cal-culate the polarization from the equation [4] p=AJZ"(f)df; the proportionality constant A was cali-brated against the Pt-NMR temperature scale around 11992 The American Physical Society 365VOLUME 68, NUMBER 3 P H YSICAL REVI E% LETTERS 20 JANUARY 1992mK [9]. We also employed the frequency shifts of theNMR lines, with the static field along the x and y axes,to determine the ratio of the demagnetization factorsD:D~=-1:3 for our sample; on the basis of this we esti-mate D„0.25, D~ =0.75, and D, =0.The solenoidal magnetic field B, employed in the rapidfield reversal was typically 400 pT. Our best inversionefficiency, 90%, was reached for small initial polarizationson the order of p 0.30. At high polarizations, the re-sults varied irregularly. Despite many efforts by chang-ing the speed (0.5-5 msec), symmetry, and magnitude ofthe field flip, the optimum inversion efficiency was only60%-75% at p 0.85. Therefore, polarizations at T & 0were limited to p=- —0.65. In copper, production of neg-ative temperatures has not succeeded owing to the short150 psec.Figure 1 displays NMR line shapes measured in zerofield. Instead of absorption, as at T &0, the system isemitting energy when T (0. The emission maximumshifts towards higher frequencies with increasing polar-ization. The solid curves are fits of Lorentzian lineshapes to the measured spectra, with the negative fre-quency side included; the least-squares minimization wasdone for the emission and dispersion curves, simultane-ously. The static susceptibility was obtained from the fitsby using the Kramers-Kronig relation g'(0) =(2/z)x fo (g"/f )df; the upper limit of integration was set to200 Hz. The absolute value of the susceptibility was cali-brated in the paramagnetic state by using the equation[9] I/g (0) 8/popMsat+Dz D~, where—po~sat =45 1pT is the field corresponding to saturation magnetization.Figure 2 displays the static susceptibility g g'(0), in-—1.2—20T (nK)tegrated from the Kramers-Kronig relation, as a functionof polarization in zero field and at four different values of8,; at 5 pT, data obtained in the parallel field B: are alsoshown. At zero field and at 5 pT, there is first a monoto-nous increase of g'(0) with p, which then saturates be-tween p= —0.4 and —0.5. According to studies ondielectric materials [4] and the mean-field calculations tobe discussed below, this behavior is caused by ferromag-netic ordering; the susceptibility is governed by dipolarinteraction through the formation of ferromagnetic do-mains which, in the ideal case, leads to g= —1 for oursample.In zero field and at small polarizations the susceptibili-ty of silver is of the form I/@=i/g +oR+L D„wh—ere@0=1.30p is the Curie-law susceptibility of the nonin-teracting spin system (see Ref. [9]) and the Lorentz fac-tor L = 3 . The uppermost curve in Fig. 2 is the best fit ofpoints with )p~ (0.45 by this equation, using R = —1.1which differs quite a lot from the value R = —2.5 ~0.5obtained from the NMR experiments [8]. This indicatesthat substantial deviation from the mean-field behaviortakes place already at intermediate polarizations. %ithinthe scatter of the measured data, the same curve fits theexperimental results at 8=0 and at B 5 p T. As is typi-cal for dipolar ferromagnetism [4], our data do not satu-rate completely even when the polarization is high but,nevertheless, we have described the results for~p~ & 0.55approximately by a constant line, g =g„.t = —1.05.We identify the crossing of the two lines in Fig. 2 asthe transition point to the ferromagnetic state. Owing to—1.0-0.2—0.8-0.4-0.6-0.8-1.0-1.20 50 100 0f (Hz)50 100 150—0.4—0.200 —0.2 —0.4 —0.6 —0.8FIG. 1. NMR absorption g" and dispersion g' curves (in di-mensionless Sl units) of the nuclear spin system in silver atT &0, measured in zero field at reduced entropies S/Aln2equal to 0.87 (o), 0.80 (a), and 0.73 (rj), which correspond topolarizations p —0.42, —0.51, and —0.59 in the paramag-netic state, respectively. The zeros for the lower curves havebeen shifted downwards by 0.25 and 0.55 in both frames.FIG. 2. Static susceptibility g (0) vs polarization p of silvernuclear spins at T &0, measured in a magnetic field B, of 0(O), 5 (V), 20 (D), 50 (0 ), and 100 p T (~), and at Bz =5 p T(x). The fitted curves are discussed in the text. The scale ontop gives estimated temperatures based on the formula I9lI/(p( —1 =0.55(T) (in nK), which applies only at B=O; valuesto the right of —2 nK must be considered tentative.366VOLUME 68, NUMBER 3 PHYSICAL REVIEW LETTERS 20 JANUARY 1992the rounding of g(p), we estimate conservatively, in zerofield and at 5 pT, for the critical polarization p, =0.49~0.05, which corresponds to S,/Rln2=0. 82+'0.035.This critical entropy is quite large when compared withthe values for antiferromagnetic ordering, S,/R ln4=0.48-0.61 and S„/Rln2=0. 53, in copper [I] and silver[2], respectively. By employing the linear relationshipbetween the temperature and inverse polarization, 1/)p~ —1 =0.55[T~ (in nK), determined in earlier experi-ments [9], we obtain T, = —1.9 ~ 0.4 nK; this value mayhave a systematic error of 20%, owing to the uncertaihtyof the relationship between p and T at negative tempera-tures.We also tried to determine T, directly from the weight-ed average (first moment) of our NMR data, whichdisplay a clear minimum (see Fig. 3) at p =0.3; this sug-gests a lower critical polarization than the saturation of g.We think, however, that the analysis based on Fig. 2 ismore reliable since the behavior of the NMR frequenciesupon ordering is not very well understood.According to the mean-field theory, T, =8= —5 nK insilver; this is clearly larger than the observed value. Cal-culations [11]on the spin- —,' Heisenberg model in an fcclattice yield T, =0.678= —3.4 nK, which exceeds our re-sult by 80%; the computed value does not change mucheven if the small next-nearest-neighbor interaction is tak-en into account [12]. In contrast, the critical entropy ac-cording to the Heisenberg model, S,=0.66% ln2, is clear-ly lower than our experimental value. This is reasonablesince the long-range dipolar interaction acts ferromagnet-ically at negative temperatures owing to domain forma-tion, which increases S, in silver.Monte Carlo calculations [13] predict T, = —1.7 nK,which is close to our measured result. However, the com-puted critical entropy S, =0.93%1n2 deviates substantial-ly from the experimental value, which indicates that the 8060T)0T(0good agreement between the theoretical and experimentaldata on T„may be fortuitous.The saturation of susceptibility to g„.t= —1 in the or-dered state can be explained only by the formation ofdomains, since otherwise the susceptibility would divergeat T,. Instead of needles, as at T&0, plate-shapeddomains are expected [4]. According to mean-field calcu-lations [13], a ferromagnetic state with broken plate-shaped domains yields g,,&= —I/(1 —Dsr); here Dsi isthe demagnetization factor along the spontaneous magne-tization M. This is in agreement with our results at 8 0and 8, 5 p T. In a small transverse field, however, M isperpendicular to the plates and a much larger g shouldresult, but it was not observed at 8~ 0-10 pT. Thissuggests the presence of a single-domain state which canbe stable in a transverse field. The mean-field theory [14]then yields g„.& —1/(D» D, ) = ——1.33, which is fairlyclose to the experimental value g„.&= —1.05. Moreover,Monte Carlo simulations [13] have confirmed the mean-field result in the ideal case when D~ D, = l. —At higher fields, g'(0) is directly proportional to p (seeFig. 2), i.e., the behavior is paramagnetic. The saturationof susceptibility was lost in the g(r) sweeps alreadyaround 10 p T and a reliable determination of the transi-tion point became impossible. On the basis of the detect-ed transitions at 0 and 5 pT (see Fig. 2), we have con-structed a schematic B vs S/Rln2 phase diagram forsilver which is depicted in Fig. 4. We estimate B,—ppN. ,i/g. ,i(T=O) =40 p T at zero temperature; thisvalue has been used when drawing the dashed transitioncurve for the ferromagnetic phase.38—402034—AV 32-0-0.4 0.5 0.6S/%ln20.7'te0.8 0.930—0.0 —0.2 —0.4 —0.6PFIG. 3. First moment (f) of the NMR line shapes as a func-tion of p, measured in zero field (O) and at B» =5 pT (V). Thecurves are just to guide the eyes.FIG. 4. Phase diagram in the B vs S/Rln2 plane for the nu-clear spins in silver metal at negative absolute temperatures.Solid circles represent our experimental data, obtained fromFig. 2, and the dashed curve displays a theoretical phase bound-ary between the ferromagnetic (at left) and paramagnetic re-gions. The shape of this curve is based on the mean-field theory[l 1].with the assumption of a linear relationship between 5 andT The antiferromagneti. c phase transition curve at T) 0 [2],marked by the solid line, is included for comparison.367VOLUME 68, NUMBER 3 PH YSICAL REVIEW LETTERS 20 JANUARY 1992We searched for hysteresis in M(B) in the orderedstate by measuring g' at the fixed frequency of 10 Hzwhile sweeping the transverse magnetic field between+ 30 pT. The adiabatic susceptibility g'(10 Hz) dis-played a small, 5% asymmetry in amplitude with respectto zero field, but less than 0.5 pT of hysteresis betweenopposite field sweeps. Since the hysteresis was so small,we prefer to assign the observed asymmetry in g'(IO Hz)to small changes in the phase of the signal during thefield sweep while keeping a constant phase setting in thelock-in amplifier.In summary, we have made the first observation of aferromagnetic state in a metallic nuclear spin system atT &0. Comparison of the saturation values of suscepti-bility with the mean-field theory and Monte Carlo calcu-lations [13] indicate that in fields parallel to the sampleplates there is a multidomain state in silver, whi1e in atransverse field a single-domain structure is preferred.The hysteresis in M(B) was found to be small, less than0.5 pT, which points to a ferromagnetic system withoutanisotropy. The measured critical entropy, S, =0.82%xln2, is larger than the value for the Heisenberg model,S„=0.66%[n2, which indicates that, even though theRuderman-Kittel exchange is dominating, the dipolar in-teraction substantially helps in the ordering process atT & 0; this leads to a ferromagnetic phase diagram whichdiffers substantially from that of the antiferromagneticstate at T&0.Useful interaction with A. Oja and H. Viertio ontheoretical issues is gratefully acknowledged. This workwas supported by the Academy of Finland.[I] M. T. Huiku and M. T. Loponen, Phys. Rev. Lett. 49,1288 (1982); A. J. Annila, K. N. Clausen, P.-A.Lindgard, O. V. Lounasmaa, A. S. Oja, K. Siemens-meyer, M. Steiner, J. T. Tuoriniemi, and H. Weinfurter,Phys. Rev. Lett. 64, 1421 (1990).[2] P. J. Hakonen, S. Yin, and K. K. Nummila, Europhys.Lett. 15, 677 (1991).[3] For a recent review, see P. J. Hakonen, O. V. Lounasmaa,and A. S. Oja, J. Magn. Magn. Mater. (to be published).[4] See, e g , A. . Abragam and M. Goldman, Nuclear Magnetism: Order and Disorder (Clarendon, Oxford, 1982).[5] P. Bonamour, V. Bouff'ard, C. Fermon, M. Goldman, J. F.Jacquinot, and G. Saux, Phys. Rev. Lett. 66, 2810(1991).[6] P. J. Hakonen, S. Yin, and O. V. Lounasmaa, Phys. Rev.Lett. 64, 2707 (1990).[7] J. Pointrenaud and J. M. Winter, J. Phys. Chem. Solids25, 123 (1964).[8] A. S. Oja, A. J. Annila, and Y. Takano, Phys. Rev. Lett.65, 1921 (1990).[9] P. J. Hakonen and S. Yin, J. Low Temp. Phys. (to bepublished).[10] G. J. Ehnholm, J. P. Ekstrom, J. F. Jacquinot, M. T.Loponen, O. V. Lounasmaa, and J. K. Soini, J. LowTemp. Phys. 39, 417 (1980).[I I] See, e.g. , L. J. de Jongh and A. R. Miedema, Adv. Phys.23, I (1974).[12] This has been reviewed briefly, e.g. , by R. H. Swendsen,Phys. Rev. Lett. 32, 1439 (1974).[13] H. E. Viertio and A. S. Oja, J. Magn. Magn. Mater. (tobe published).[14] H. E. Viertio and A. S. Oja (private communication).368

P. J. Hakonen

K. K. Nummila

R. T. Vuorinen

O. V. Lounasmaa

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Physical Review Letters

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Observation of nuclear ferromagnetic ordering in silver at negative nanokelvin temperatures

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