Torsional Spectroscopy by Nuclear Spin Polarization Losses in the Rotating Frame BY MILANM. PINTAR Physics Department University of Waterloo Waterloo Ontario N2L 3G1 Canada Received 30th January 1979 In some solids with moderately hindered torsional oscillators such as NH4 and CH3 groups it is possible to excite nuclear Zeeman-torsional transitions (double resonance). This type of resonance makes possible observations of resonant magnetization loss torsional specific heat torsion-phonon relaxation time and torsional energy spectrum. A brief description of these experiments is presented. In 1977 the Zeeman-torsional double resonance spectroscopy was introduced.' It was shown that when the nuclear Zeeman energy in the rotating frame is brought into resonance with the energy of the torsional oscillator on which the nuclear spins reside a semi-equilibrium between the two systems is established in a time of the order of the nuclear spin-spin relaxation time.The nuclear Zeeman energy states can be quickly polarized in the rotating frame. Since some of this order can be transferred in a few ps to the torsional oscillators (while the two systems are essentially isolated from the lattice) several experiments were made possible. In the magnetization loss experiment the proton Zeeman polarization was estab- lished in the rotating frame by the spin-locking sequence. In the rotating frame therefore Zeeman populations were characterized by a spin temperature which was some three orders of magnitude smaller than the lattice temperature.Since initially the torsional oscillators were in equilibrium with the lattice the polarizations of the two energy systems were drastically different. This difference vanished in a time of the order of 100 ,us if the matching between the two sets of levels was good. The equilibration time increased to several ms if matching was off by z 10 G. Thus if the two systems were allowed to evolve towards the semi-equilibrium for a fixed time of 100ps at different strengths of the r.f. field H, the degree of mixing varied. At those H where matching between the two energies was good as much as 40% of the proton magnetization was lost. If matching was off by z 10 G almost no magnetiza- tion was lost. A proton magnetization loss during a 100ps long spin-locking pulse of variable strength H in polycrystalline NHJ at 67 K is shown in fig.1 upper curve. The magnetization loss is more than z20 per cent at H between 5 and 20 G. Such a proton magnetization loss spectrum provides information on the energy range in which the nuclear Zeeman and the torsional oscillator energy spectra match. With a similar experiment' the specific heat of the torsional spectrum was deter- mined. If enough time is allowed for mixing Zeeman and torsional reservoirs a semi-equilibrium between the two is established over a wide range of r.f. fields. In such a case the magnetization loss dependence on H takes on a shape known from the dipolar local field e~periment.~ The torsional specific heat is derived from the r.f. field at which the proton magnetization decays to half its initial value; in complete analogy with the case of mixing the Zeeman and dipolar energies.Clearly the TORSIONAL SPECTROSCOPY torsional specific heat does not give the torsional spectrum. However it does provide for an independent evaluation of a proposed spectrum. The torsional specific heat was determined from the proton M dependence on Hl in polycrystalline NHJ at 67 K under the condition of a torsional-Zeeman semi-equilibrium. The spin- 16 12 I /a 8 4 0 10 20 30 40 Hi/G FIG.1.-Proton magnetization of polycrystalline NHJ as a function of HIat 50 K. The magnetiza- tion was spin-locked in an r.f. field for 100ps (upper graph) and for 4 ms (middle graph). The lowest graph shows the SPOTS magnetization observed after the pulse sequence n/2 FP1(400 ps) delay of 1 ms FP2(100 ps).locking field was 1 ms long (fig. 1 middle graph). M reached its half value at Hl = 11 & 2 G. At this field the torsional specific heat equals the Zeeman specific heat. In semi-equilibrium cz ~ (1) M = Mo CZ + CT’ where Cz = aHi and CT = aH$. The effective torsional field HT equals 11 & 2 G. C can be derived more accurately from the dispersion of the proton relaxation time in the rotating frame. In Zeeman-torsional semi-equilibrium the familiar spin thermometric equation holds (Cz + CT)Tib = CzTyi + CTTi;. (2) TlPis the common Zeeman-torsional relaxation time and TI and TITare the Zeeman and torsional relaxation times respectively.To a good approximation TTi = cz/H with a a constant. Eqn (2) is valid if the spin-locking pulse is at least 1 ms long. Eqn (2) was computer-fitted to the experimental data fig. 2. The value HT = 14 & 1 M. M. PINTAR G was obtained for NH,I at 72 K. The torsional relaxation time was determined in an independent experiment as described below. The same information can be derived also by a modified spin-locking pulse se- quence n/2,FP1 (400 ps) T( 1 ms) FP (1OOps). With the spin-locking pulse pair the Zeeman system is polarized and some of its order transferred to the torsional system. The delay T of 1 ms was much shorter than the torsion-phonon relaxation time of 20 10 I I I I 0 FIG.2.-Dependence of the proton Ti;of the polycrystalline NHJ on H ;'.The computer fitted curve corresponds to an effective torsional field HT = 14 & 1 G. T = 72 K. 60 ms. With FP (100 ps) a Zeeman system with no polarization is introduced while the torsional system has all the order it received while interacting with the Zeeman reservoir during FP (400ps). During FP1 the torsional inverse temperature becomes PTL =Pzl H H' + H; During FP2 the Zeeman and torsional systems reach a semi- equilibrium with the final inverse temperature pz2-pzl Hi + H;' H; As a result a Zeeman magnetization appears Its maximum value is Mo/4at H = HT. This magnetization is plotted at the bottom of fig. 1. The maximum is at H z 12 G in agreement with HT of 11 + 1 G derived from the specific heat experiment.The order transfer experiment was tested with resonantly coupled Zeeman and dipolar energies. The measurements were done at room temperature on NH,CI at 17 MHz fig. 3. The pulse sequence was n/2 FP (200 ps) t,FP (200 ps). The magnetization after the pulse FP depends on H as in eqn (1) with HT replaced by the dipolar local field in the rotating frame HL. This equation was fitted to the experi- mental points (fig. 3 upper curve). The derived value of the local field Hkwas 1.O 0.2 TORSIONAL SPECTROSCOPY G. The magnetization Mi following the pulse FP depends on H as in eqn (3). The computer fit of eqn (3) to the experiment gives the local field HL = 1.1 & 0.2 G. Both fields are in good agreement with the calculated HL of 1.08 -+ 0.05 G. The dependence of Mi on the delay z between FP and FP was found to be exponential with the characteristic time equal to 24 & 3 ms.With the Jeener-Brockaert pulse sequence the proton dipolar relaxation time TI of 27 & 2 ms was obtained. This agreement confirmed that the order was transferred from the Zeeman to the dipolar system since it relaxed to the lattice with T,,. 1.0 0.8 8 0.6 \ 22 0.4 0.2 0 2 4 8 HI/G FIG.3.-Proton magnetization of polycrystalline NH,CI in arbitrary units against the r.f. field at 273 K. The top line is the computer fit to M = M,N:/(H -H:); the experimental values of MXIMO are represented by circles. The bottom line is the computer fit to Mi = MoN;fH:/(H:+ H;f)’. The resonance frequency was 17 MHz.An interesting result of this double resonance technique was the development of a method for measuring the torsional oscillator relaxation time TIT. This torsion- phonon relaxation time is measured with the following pulse sequence 42,FP (I ms) z FP (100 ps). The first part is as above the spin-locking pulse sequence with the field H set close to the resonance. The torsional spectrum is polarized during the pulse FP (1 ms). As the Zeeman spectrum is removed by removing FP (1 ms) the torsional oscillator is allowed to relax undisturbed towards equilibrium with the lattice bath during the time z. Any torsional polarization left after the time zis then shared with the Zeeman spectrum which is reestablished by the second field pulse FP (100 ps). Since the FP pulse is applied without a preceeding 42 pulse there is no initial Zeeman polarization in FP,.During the pulse FP (100 ps) the torsional order is shared with the Zeeman system resulting in a small Zeeman signal M which depends exponentially on z. With this method Ty; was measured in polycrystalline NH41 as a function of temperature between 4 and 60 K fig. 4. The most interesting feature of the Ti; temperature dependence is the linear region 4-40 K. This dependence M. M. PINTAR implies a one-phonon process between two torsional levels separated by no more than z 12 K. However it is known from neutron spectroscopy that the 1st excited state is ~430 It is possible to speculate that a 24 fold degener- K above the ground ~tate.~ ate ground state of the NH4 oscillator is split into two sets of lines separated by less than z 12 K.Each rnultiplet has energy splittings no larger than z 100 kHz. At temperatures above 40 K multi-phonon processes contribute to Ti;causing an ex- ponential rise in Ti;with temperature. At z75 K this rate becomes so fast that the tunnelling spectrum disappears because of the torsional lifetime broadening. At essentially the same temperature the dipolar line narrows. Thus at this temper- ature TITis the correlation time for the NH4 group reorientation. -1 I 0 10 20 30 40 50 r/ K FIG.4.-Relaxation rate of the NH torsional oscillator in polycrystalline NH41 plotted against temperature. During the search for a better resolution of the magnetization loss spectrum it was soon realized that the non-equilibrium regime had to be explored.In this time regime the magnetization undergoes damped oscillations which were believed initially to have a frequency ~co, as in the case of Zeeman and dipolar mixing. When the field pulse was set to a few ,us duration and varied in 1 ,us steps beats appeared in the time evolution of the magnetization indicating at least two freq~encies.~ The 71/2 FP1(t) pulse sequence was applied with the FP,(t) duration varied from 1 to 200 ,us in 1 ,us steps. The magnetization M,(t) was recorded for each t. As the field pulse was made progressively longer the magnetization evolved towards the semi-equilibrium in the rotating frame. The analysis showed that in this evolution the frequencies 204 1204 & wTiI and mTi are present.By a Fourier transformation of M,(t) a spectrum was arrived at fig. 5 and 6. A brief summary of the perturbation calculation is as follows. The Hamiltonian of Zeeman (Z) torsional (R) and dipolar (D) energies in the rotating frame at exact resonance reads if = ifz + ZR-3Xg' + (3/8)"2(Xg'+ XL-"). (4) TORSIONAL SPECTROSCOPY 1.8 1.6 CI 0 1.4 32>- s v 1.2 1.01 0.8 I I I l l l r l l l 1 1 1 1 20 40 60 80 100 120 140 t/ 16%i 10 20 30 40 50 v/4kHz FIG.5.4~) Proton magnetization time evolution of polycrystalline CH3CD21in the rotating frame at 12.5 G measured at 17 MHz. M was recorded 9 psafter the end of the spin-locking pulse. The duration of the r.f. pulse was varied from 1 to 148 ps in 1 ps steps.(b) Fourier transform of the time evolution in (a) (the SPOTS spectrum). Note that only frequencies 20 + oTand 2wl -wTappear. Frequency is in units of 4 kHz. M. M. PINTAR 1 .o n 0.6 \ 0.4 0.2 0 10 20 30 40 50 v/4 kHz FIG.6.-SPOTS spectrum of the polycrystalline NHJ at 30 K. Proton magnetization was observed at 17 MHz with window position at 24 ,us. Time increments were in 1 ps steps the r.f. pulse was varied from 1 to 200 ,us. Observations were made at 2H1 of 26.9 and 29.5 G. The two spectra are shown as full and dotted lines respectively. Those terms in XDwhich are time dependent with frequencies mo = yHoor 2m0,have been dropped and the z-axis has been chosen along the direction of the r.f.field HI. In this case the non-diagonal spin operators in the dipolar interaction connect only states whose spin quantum numbers M differ by AM = 0 *2. Since the operators Xzand XRcommute we choose as a basis their common eigenfunctions (<M) (%R + xZ)l<M) = EcMIM) = (5 -fimM)lcM)-(5) We now examine the time evolution of the spin-torsional system due to the part D of the dipolar Hamiltonian in the rotating frame which does not commute with ZZ or ZR. Starting at time t = 0 the non-equilibrium density matrix in the high temper- ature approximation reads P(0) = Q-’(l -EoZz -PO~R). (6) Here Q = Tr(1 -&VR}and uo and Poare the initial inverse temperatures. The zero of energy may be chosen such that the trace of XRalso vanishes. 1 is the unit operator.In the interaction picture defined by U(t)= exp [i(Xz + %&/ti] the density matrix obeys the following equation of motion ap -= -i/h[D(t) p”(t)] (7) at in which D(t) = U(t)DUt(t). The solution of this equation is p”(t)= p”(0)-i/k ds[D(t) p”(O)] -1/h2[ dr [dr’{D(r) [D(r‘),p”(O)]}. 0 The expectation value of the Zeeman energy is then given by (XZ(t))= Tr{sz,5(t)). TORSIONAL SPECTROSCOPY The selection rule imposed by the operator D(t) allows frequencies 12w 5 wTl and 20,. The SPOTS spectrum of CH3CG21,fig. 5(b) consists of ]2w1 + uTland 120 -mTl lines of approximately equal intensity. The 2w line is not observed. This is not surprising considering that the ratio of Zeeman to dipolar specific heat is only a few percent.In solid NH41 the SPOTS spectrum of NH4 exhibits a very /-A2 / (11-N’ /- 3f2 (9)-/’ / //- E2 (4)-:I (9)-_ / .- El 3FI f F - -f E F f -FF F FIG.7.-Ground state splitting of the NH- torsional oscillator in various crystal fields (a)octahedral (6)octahedral with a small tetrahedral perturbation (c)tetrahedral (d)trigonal and (e)no symmetry. intense 20>,line in addition to several other lines fig. 6. In a trigonal crystal field6*’ four tunnelling frequencies are allowed fig. 7. The NH41spectrum shows more than seven lines. A possible origin of these lines is the existence of two nonequivalent oscillators in the NH41lattice. Pulse sequences with finer time increments are needed to resolve and assign the spectrum. Experimental and theoretical work of this nature is now being carried out in Waterloo.R. S. Hallsworth D. W. Nicoll J. Peternelj and M. M. Pintar Bull. Amer. Phys. SOC.,1977 22 550. ’R. S. Hallsworth D. W. Nicoll J. Peternelj and M. M. Pintar Phys. Rev. Letters 1977 39 1493 R. S. Hallsworth Ph.D. Thesis (University of Waterloo 1977 unpublished). M. Goldman Spin Temperature and Nuclear Magnetic Resonance in Solids (Oxford University Press London 1970) chap. 2. G. Venkataraman J. Phys. Chent. Solids 1966 27 1103. D. W. Nicoll and M. M. Pintar Phys. Reu. Letters 1978 41 1496 D. W. Nicoll Ph.D. Thesis (University of Waterloo 1978 unpublished). T. Nagamiya Progr. Theor. Phys. 1951 6 702. A. Huller and J. Raich Torsional Ground State Splitting for Tetrahedral Molecules to be pub- lished.