258 Bulletin of Magnetic Resonance

NMR at Very Low Temperatures:Population Difference Thermometry1

Neil S. Sullivan

Department of PhysicsUniversity of Florida215 Williamson Hall

Gainesville, Florida 32611-8440

ContentsI. Introduction

II. Principles

III. Practical Examples

IV. Experimental Results

V. Current Studies

VI. Acknowledgments

VII. References

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I. IntroductionLow temperature physicists have employed mag-

netic resonance techniques to study the fundamen-tal properties of materials at very low tempera-tures from the earliest days following the discoveryof NMR. Many of the familiar degrees of freedomsuch as phonons and vacancy propagation are frozenout at temperatures below a few degrees Kelvin andmagnetic resonance provides one of the most pow-erful methods for studying the fundamental dynam-ics and local ordering at very low temperatures (1).With the development of a number of ultra-low tem-perature facilities to study fundamental phenom-ena at sub-millikelvin temperatures, such as nuclearmagnetic ordering and transport in highly polarizedquantum fluids, (2) there has arisen a need for areliable first principles primary thermometer to de-termine the absolute temperatures of the systemsunder study. The purpose of this review is to re-port studies of the use of the asymmetry of the in-tensities of multiplet magnetic resonance spectra for

1 Based upon lectures presented at the First CanadianNMR Summer School, July, 1993.

thermometry at millikelvin and sub-millikelvin tem-peratures.

Pt NMR thermometry has up to the presentbeen used as a secondary thermometer for ultra-low temperature experiments. The NMR absorp-tion intensity of Pt wire in a low field is used to de-termine the temperature assuming a nuclear Curielaw, x ^ T~l, for the nuclear susceptibility. Thismethod requires calibration against known stan-dards at relatively high (mK) temperatures and theextrapolation by two or three orders of magnitudeto the microkelvin range. For studies at even moder-ately large fields, the thermometer must be locatedsome distance from the experiment of interest andthis can lead to uncertainties in the temperature.The advantages of the proposed method of popu-lation difference thermometry (PDT) is that itis a true primary thermometer. Observation of theasymmetry of the intensities of the multiplet spectragives the nuclear spin temperature directly and thetechnique can be used in very high magnetic fieldsfor appropriately chosen samples.

Vol. 18, No. 3/4 259

II. Principles

Let us consider a spin system with three or moredistinct energy levels (Ei, E2,E^,...) that may bedegenerate. The thermal population of each level Eiis given by

where iV is the total number of spins, kB is Boltz-mann's constant, and T the absolute temperature.If there is an allowed magnetic transition betweenthe energy levels, magnetic resonance is observedby the absorption of radiofrequency (RF) power atfrequencies

Ui = (Ei+1 - Ei)/U (2)corresponding to the energy differences of the levels.If the RF power is sufficiently low that the inducedtransitions do not significantly perturb the popula-tions Pi, the intensity Mj of the absorption signalsis proportional to the population difference,

Mi = C(Pt - Pi+1)

where C is a constant that depends on the systemgeometry and the method of detection. At hightemperatures and in most applications of NMR forwhich the nuclear spin polarization is very low, wecan use a "high-temperature" approximation for Fand

This is the familiar nuclear "Curie law" dependenceand can be used as a secondary thermometer forcases where fiiOi -C k$T, but this application re-quires calibration against known standards at rela-tively high temperatures.

A more direct method for determining the tem-perature can be realized by measuring the ampli-tudes of two resonance lines in a multiplet, say Mjand Mk. Using the exact expressions for Pi and Pk,

(3)Mk Pk - Pk+l

exp(-Ej/kBT) - exp(-Ei+l/kBT)exp(-Ek/kBT) - exp(-Ek+1/kBT)

If we define the difference frequency ujik =Ek)/h,

The significance of this result, eqn. 4, is that themeasurement of the intensities of the componentsof the spectra, M% and Mk, and the frequencies ofthe lines Wj and uik is sufficient to determine theabsolute nuclear spin temperature directly withoutfurther calibration.

In order to use this method of population dif-ference thermometry, several conditions need tobe satisfied. First, the components of the spectramust be well resolved, i.e., the line widths must besmall compared to the separations LOik. Second, theratio of the frequencies should be close to unity tominimize any changes in the spectrometer gain asthe detector frequency is swept through the full ab-sorption spectrum. Third, the nuclear spin-latticerelaxation times need to be reasonably short (e.g.,less than about 1 h) to avoid saturation of the tran-sitions with very low RF power levels. Finally, formost cases, cryogenic spectrometers need to be usedthat can operate at very low power dissipations, be-cause the length of the refrigerator assemblies (e.g.,for nuclear demagnetization stages) and their needfor thermal isolation precludes the use of RF ca-bling to high temperatures. This latter feature isthe most challenging that we have had to considerin the development of prototype systems.

In the next section we will consider two very dif-ferent methods of using population difference ther-mometry; (1) the perturbation of nuclear quad-rupole resonance (NQR) spectra by small appliedmagnetic fields, and (2) the perturbation of NMRspectra by quadrupolar or dipolar interactions.These two examples serve to demonstrate the verygeneral nature of PDT and its application to meetvery different constraints. In section III we willpresent the results for a specific application ofmethod 2, and in section IV outline current plansto develop method 1 for thermometry at microkelvintemperatures.

M±

Mfc

hujlk \

kBT) { }

III. Practical Examples

(a) Field resolved NQR spectroscopy.As a first example we will consider field-resolved

NQR spectroscopy at very low temperatures. Fora nucleus with spin I > \ and bearing a nuclearelectric quadrupole moment Q, the Hamiltonian for

260

Figure la: Specifications of the orientation of theapplied magnetic field B with respect to the princi-pal axes reference frame (x,y,z) for the local electricfield gradient tensor q.

an axial symmetric crystalline electric field gradientq is

_ 1} (3JZ2 - 1(1 + 1)) + 7ftf • B (5)

where z is the axis of symmetry for the principalaxes reference frame associated with the tensor q,B is the applied magnetic field, and 7 the gyromag-netic ratio (see Figure la).

For small applied magnetic fields or small nuclearmagnetic moments we can treat the last term of eqn.5 as a perturbation. In this case the energy levelsEm where m = Iz (Figure lb), are given by

E±m = A[3m2 - 1(1 + 1)] + mhu;L cos 9 (6)

for m > \. UL is the nuclear Larmor frequency, and9 is the polar angle specifying the orientation of Bin the crystal field principal axes reference frame(Figure lb).

The case m = Jj is special because of the zerothorder mixing of the states m = ± j , and the energyeigenvalues are given by

E± = A[- - 1(1 + 1)] ± mfhu)L cos 6

where/ = 1 +[-(21 + 1) tan21

4-1

(7)

The fine structure splitting for the NQR lines re-sulting from the Zeeman perturbation is illustrated

Bulletin of Magnetic Resonance

-(m+1)

+ m

B = 0 B # 0Figure lb: Energy level diagram for a nuclear elec-tric quadrupole interaction for a nucleus with spin/ in the presence of a small magnetic field B. Thequantum number m = Iz.

- ( m + 1 )

—m

+mFigure 2a: Illustration of the splitting of nuclearquadrupole resonance (NQR) lines by a small ap-plied magnetic field B for m\ i .

in Figure 2a. The degeneracy of the NMR absorp-tion frequencies for the transitions ± m —> ± ( m + 1)is removed by the application of a small magneticfield, and the ratio of the intensities of the two com-ponents o;^ and CJ,^ for the transitions (m + 1 —f mand — (m + 1) —> —m, respectively, is given by

4) -(m+1) -P-r(m+1) -Pr,

(8)

sinh ( isinh l^

exp[-/?(2m

where ft, — LOLCOS9 and (3 — l/(fcsT). The expres-sion is exact and observation of the frequency spac-ing 2Q of the doublet and the ratio of the intensitiescan be used to determine the absolute temperature.

Vol. 18, No. 3/4 261

2fl

O)r0),

Figure 2b: Absorption spectrum of field-perturbedNQR spectrum at low temperatures for a single crys-tal and for transitions to levels \m\ > | .

Figure 2b illustrates the spectrum of the doubletfor a single crystal for low temperatures, kBT~hQ.For small field splittings Cl, this method is suitablefor the microkelvin temperature range. The methodcan also be used for a powder spectrum as indicatedin Figure 2c. Measurement of the ratio of the in-tensities at the two symmetrically related positionsshown by the arrows in figure 2c can be used witheqn. 8 to determine the temperature.

The analysis for transitions to levels m — ±Jjfollows in a similar manner. For a single crystal,four distinct transitions are observed for transitionsfrom ± | to the lowest levels shown in Figures 3a and3b. The ratio of the intensities for the transitionsat frequencies cua and uai is given by

= exp[-/?(3 - f)m/2]-sinn (hsinh [\

(9)

where / is given by eqn. 7. As illustrated in Figure3c, the asymmetry in the intensities of the powderspectrum follows from averaging over all field orien-tations 6, and the temperature can also be deter-mined from a powdered sample.

(b) Quadrupolar perturbation of Zeeman lines.As a second example of the use of population dif-

ference thermornetry we can consider the fine struc-ture of nuclear Zeeman lines resulting from nuclearquadrupolar interactions or intramolecular dipole-dipole interactions. The Hamiltonian for such sys-tems is given by

m

Figure 2c: Absorption spectrum of field-perturbedNQR spectrum at low temperatures for a powdersample and for transitions to levels \m\ > | .

where Bo is the applied magnetic field, A =e2qQ/4I(2I — 1) for an axially symmetric nuclearquadrupolar perturbation, P2(cos#) is the Legendrepolynomial, and 0 is the polar angle specifying theorientation of Bo with respect to the principal axesreference frame of the electric field gradient q. Fig-ures 4a and 4b show the fine structure splitting dueto the quadrupolar interaction. Two magnetic res-onance absorption lines occur at frequencies u± =uL ± u)Q for m = ±1 with UUJQ = (3e2qQ/4) P2{0)for / = 1. The ratio of the intensities of the transi-tions a>+ and w_ is given by

sinh (hsinh (£

H = A(3I2Z - (10)

(11)

As in the previous example, the low frequency com-ponent of the fine structure has the highest intensity.It is important to note in this case that the predom-inant temperature dependence is given by the expo-nential prefactor of eqn. 11. This factor dependson the Larmor frequency and NOT the frequencysplitting which can be very small. This method cantherefore be used for applications in very high mag-netic fields.

The intensity asymmetry predicted by eqn. 11was observed several years ago in studies of molec-ular ortho-H2 in high magnetic fields (7.5T) and atmoderately low temperatures (T ~ 70 mK) (3). Thetotal nuclear spin of the proton pair is I = 1 forortho-H2 and the perturbing interaction responsiblefor the fine structure is the intra-molecular dipole-dipole interaction. The secular component of thedipolar interaction can be written in the form

HDD = D(3I2Z - I2)(3J2

2 - J2)P2{9) (12)

262 Bulletin of Magnetic Resonance

| m | = |

CO. to,

+1

to.

Figure 3a: Energy level diagram for a field-perturbed nuclear electric quadrupole interactionfor transitions to magnetic quantum numbers ml2"

co. G),

Figure 3b: Absorption spectrum of field-perturbedNQR at low temperatures for transitions to levels\m\ > 2 for a single crystal.

where D is the strength of the dipolar interaction, Jthe molecular orbital angular momentum and 9 theorientation of the molecular axis relative to the ap-plied magnetic field. The perturbation given by eqn.12 is a second-rank tensor that transforms analo-gously to the quadrupolar perturbation of eqn. 10.

Studies of the Pake doublet given by the protonsof CaSO4.a;H2O (for x = 0.5, 1.0 and 2.0) have alsobeen carried out for high nuclear spin polarizations(4). The line shapes have the same asymmetries asthose observed for solid H2, with the low frequencycusp of the Pake doublet having a higher intensitythan the high frequency components. The ratio ofthe observed asymmetry is in good agreement witheqn. 11.

Figure 3c: Absorption spectrum of field-perturbedNQR at low temperatures for transitions to \m\ = \for a powdered sample.

IV. Experimental ResultsAs a prototype system for millikelvin tempera-

tures we have studied MnSb which is a ferromag-net with an internal hyperfine field of 32T. MnSbhas three magnetic resonance multiplets; one asso-ciated with 55Mn (/ = §), and the others with 121Sb(/ = | ) and 123Sb (J = \). The / = § nuclei give5-line multiplets due to the nuclear quadrupole per-turbation of the Zeeman levels.

A typical 55Mn ferromagnetic magnetic reso-nance (FMR) absorption spectrum observed at 258MHz and at 30 mK is shown in Figure 5. The inten-sities of the five lines develop a particularly strikingexponential dependence on position in the spectrumat low temperatures. This dependence is the pop-ulation difference effect given by eqn. 8. We havemade systematic measurements of the ratio of theasymmetries of two neighboring lines of the pentu-plet to determine the feasibility of using this systemfor millikelvin thermometry. The results are shownin Figure 6, and apart from some scatter at hightemperatures where the relative accuracy is poor,the comparison with temperatures measured inde-pendently using a 3He melting curve thermometeris very good.

In these studies, particular care was taken to de-velop a system with high sensitivity for small vol-umes at very low RF power levels. For this purposewe developed a high frequency ultra low power cryo-genic marginal oscillator using GaAs high electronmobility transistors (6). The total power dissipa-tion of the oscillator was less than 20 fxW and itwas operated affixed to the shields (T ~ 200 mK)of a dilution refrigerator and in fields up to 4.0T.

Vol.

Iz

18,

- 1

n

1

Mo.

u

a

3/4

±(m+l)

263

0),

Figure 4: Schematic representation of the fine structure splitting of NMR lines due to a quadrupolar or dipolarperturbation for nuclear spin 1 = 1; (A) level spacing, (B) absorption spectra.

nits)

ION

oCD*£

4.0

3.5

3.0

2.S

2.0

1.5

1.0

0.5

0.0

_

- \

\\

1

t\ i I

_

; ; A

1 1 I240 252 258 264 270

FREQUENCY (MHz)Figure 5: Observed quadrupolar-perturbed ferro-magnetic resonance of 55Mn in MnSb at 30 mK.The exponential dependence of the intensities of the5-line multiplet gives the nuclear spin temperaturedirectly.

The spectrum shown in Figure 5 was recorded using10 mg of MnSb dusted onto a Cu wire brush thatserved as the sample holder.

The development of these or similar cryogenic de-tectors is very important for the implementation ofsubmillikelvin PDT. The larger Cu nuclear demag-netization in the University of Florida MicrokelvinResearch Laboratory has a cooling capacity of theorder of nW below 100 /zK. The RF dissipation inthe detection coil must therefore be held below thenW level for thermometry in this range. This can beachieved with the GaAs marginal oscillators that wehave developed (6). The oscillators can be attachedto the heat shields or thermal posts extending fromthe mixing chamber of dilution refrigerators with asmall length of RF cable extending to the sample.

V. Current Studies

In order to extend this method of populationdifference thermometry to the microkelvin temper-ature range we are studying the perturbation of thenuclear quadrupole resonance of pure Sb. 121Sb with57.3% abundance has a nuclear spin / = | withan NQR resonance of 11.529 MHz for the m = ± |to 77i = ±2 transitions. The line widths are typi-cally 8 kHz and observation of the splittings in smallapplied fields will provide a direct reading ther-mometer for temperatures below 200 [iK. The nu-clear spin lattice relaxation times are relatively fast,T\T = 0.092sec-K at high temperatures, althoughdeviations are expected at high nuclear spin polar-izations. The isotope 123Sb with 42.7% abundanceand / = 5, has an NQR absorption at 6.9996 MHzfor the ±§ to ±\ transition and TXT = 0.156s-K.The principal difficulty in developing this applica-tion is the design of the spectrometer to measurethe spectra using RF levels below the nW level. Pre-liminary studies indicatp that pulse techniques usingsmall angle tipping pulses are the most effective, andthese studies are currently in progress.

In a related effort we have developed a microwavestripline spectrometer operating in a bridge config-uration at 2 GHz to observe NMR in pulsed mag-netic fields up to 50T at the National High Mag-netic Field Laboratory (7). This system is designedto have high sensitivity for ultra-small sample vol-umes, typically 10—13 mm3, and will be used tostudy new phenomena at very high fields such asthe spin dynamics in high-Tc superconductors andfield induced metal-insulator transitions.

264 Bulletin of Magnetic Resonance

0.005 0.010 0.015 0.020 0.025 0.030 0.035 0.040

INVERSE TEMPERATURE (1/mK)

Figure 6: Comparison of the nuclear spin temper-atures deduced from the ratio of the intensities ofneighboring components of the pentuplet of figure 5with the temperatures determined from a 3He melt-ing curve thermometer. The solid line is the theo-retical dependence.

VI. AcknowledgmentsThe studies of the fine structure of 55Mn FMR

at low temperatures has been carried out in col-laboration with Edgar Genio and Peter Andersenat the University of Florida, and the developmentof the applications for very high magnetic fieldswith Bill Moulton and Alfred Kleinhammes at theNational High Magnetic Field Laboratory. Thework has been supported by a grant from the Na-tional Science Foundation DMR9623536, and theNational High Magnetic Field Laboratory is sup-ported by the State of Florida and the National Sci-ence Foundation through NSF Cooperative GrantNo. DMR9016241.

(1992).6P.M. Andersen, N.S. Sullivan, L.W. Phelps and

J.B. Legg, Rev. Sci. Instr. 63, 157 (1992).7W.G. Moulton, N.S. Sullivan, A. Kleinhammes

and I. Suberbielle, Bull. Am. Phys. Soc. 38, 527(1993).

VII. ReferenceslA. Abragam and M. Goldman, Nuclear Mag-

netism: Order and Disorder, Clarendon Press, Ox-ford, (1982).

2K.S. Bedell, I.F. Silvera and N.S. Sullivan,M.R.S. Bulletin, XVIII, 38 (1993).

3N.S. Sullivan and R.V. Pound, Phys. Rev. A6,1102 (1972).

4P. Kuhns, O. Gonen and J. Waugh, J. Mag.Res. 72, 548 (1987).

5P.M. Andersen, N.S. Sullivan, B. Andraka, J.S.Xia and E.D. Adams, Physics Letters A172, 85