Nuclear spin noise at room temperature

Volume 159, number 5,6

NUCLEAR

CHEMICAL PHYSICS LETTERS

SPIN NOISE AT ROOM TEMPERATURE

21 July 1989

*

M.A. MCCOY and R.R. ERNST Laboratorium fir Physikalische Chemie. Eidgenijssische Technische Hochschule, CH 8092 Zurich, Switzerland Received 8 May I989

It is demonstrated that it is possible to observe nuclear spin noise at room temperature using a conventional NMR spectrometer. The noise spectrum shows the impedance characteristics of the receiver circuit modified by the presence of the nuclear spins and manifests spontaneous emission of spin noise. The experimental findings are compared with analytical results derived from a simple electronic circuit model. The role of radiation damping in determining the features of spin noise is discussed.

1. Introduction

phenomena: the absorption of circuit noise by the spins and spontaneous emission of spin noise.

Recently, Sleator, Hahn, Hilbert and Clarke [ 1,2 ] observed, for the first time, random noise emission from nuclear spins. A superconducting SQUID was used to detect weak current fluctuations due to quadrupolar resonance of a sample maintained at 1.5 K. The spectrum of the noise in the detection circuit, observed under equilibrium conditions, shows a characteristic absorption dip at the nuclear resonance frequency which can be understood by the equivalent circuit impedance of the spins coupled to the detection circuit. By previous radio-frequency saturation of the nuclear spins it is possible to sup press this absorption signal and to detect an emission signal which can be attributed to incoherent nuclear magnetic dipole precession, that is called spin noise. It is shown in this paper that an analogous experiment can be performed even at room temperature provided that a high-resolution and high-sensitivity NMR spectrometer is used. The much reduced line width of liquid samples and the sensitivity gain with higher resonance frequencies more than compensate the disadvantages of lower Q values and increased amplifier noise. This enables the detection of both

* This work has been presented, in part, at the XXIVth Congress Amp&e on Magnetic Resonance and Related Phenomena, abstract D-8 1,Pozna& Poland, September 1988.

0 009-2614/89/$ ( North-Holland

03.50 0 Elsevier Science Publishers Physics Publishing Division 1

2. Spectral density of random noise We analyze the random noise in the simple receiver circuit of fig. la with the circuit elements L, and C,. An equivalent circuit diagram including the relevant noise voltage sources is indicated in fig. lb where R,is the series resistance of the receiver coil, L, (w ) is the effective inductance caused by the presence of the sample, R,(w) the corresponding series resistance, and U, and C&(w) are the noise voltage sources generated by R,and R,(co), respectively. The noise voltage source U, represents the noise characteristics of the amplifier(s) . We loosely follow in our analysis the procedure described by Sleator et al. [2]. All equations are written in SI units. The noise voltage U at the amplifier terminals is given by (iwC,)-‘( V,+&) (ioC,)-‘+i&+io.&(w)+R,+R,(w)

‘= +

u. .

(1)

L, ( w ) and R,( w ) canbe expressed by the suscep tibility x(w) =x’ (w) -ix” (w) of the sample that modifies the inductance of the receiver coil [ 31,

L:(w) =L,[l+tl~(w)]=L,+L,(w)f&R,(w), B.V.

(2) 587

Volume 159, number 56

CHEMICAL PHYSICS LETTERS

A

d(Ao)

=

21 July 1989

A0 (A,)‘+ (Aw)~ ’ A2 (A.,)‘+ (Aw)’ ’

a(Aw)=

The half width at half height is denoted by L2= 1/ T2 and the resonance offset is Aw. The voltage spectral densities of the two Nyquist noise sources at temperature T are given by WY=

?k,TR,, Tc

W:(o)=

;kBTR:(o).

(6)

It should be noted that the spin noise source W p ( w ) is determined by the product TR p (co) where R:(w) is the equivalent spin resistance at thermal equilibrium with equilibrium magnetization M, = MO = nyfi2aJ4kB T and number density of spins n. This product and therefore the spin noise power are temperature-independent. The reason for this astonishing fact is that the fluctuation of transverse magnetization components depends only on the total number of spins but not on their polarization in the magnetic field. This leads to the observable noise voltage spectral density at the amplifier terminals Fig. 1. (a ) Receiver circuit with an inductor LCand capacitor C, and (b) the equivalent circuit that includes the loss resistance R, and the effect of the spins that modify the circuit inductance and resistance by L, and R,, respectively. The circuit and the spin resistances generate the Nyquist noise voltages U, and U,. Noise sources external to the circuit, e.g. from tbe pre-amplifier, are represented by U..

with the filling factor q of the receiver coil, leading to L(O)=LW’(~),

K(w)=LWx”(w).

The susceptibility can be obtained equations in the form [ 31

(3) from the Bloch

x’ (w) = IhcaYM,d(Aw), x” (QJ) = t~o~M,a(Aw)

(4)

involving the vacuum permeability A, the z component of the magnetization M,, and the normalized dispersion and absorption signals (for notation, see ref. [4]) 588

w”(o)=(ocC)-2

wy + WP(w) ’

[R,+R,(w)]2+[wLc-(wC,)-‘+wL,(co)]2

+wy_

(7)

The additional noise sources WY, such as amplifier noise, do not interact with the sample and merely raise the noise level, It has been shown by Sleator et al. (21 that the coupling between sample and coil and the phenomena of spin noise are connected with radiation damping [ 5-71. The coupling between the spins (with resonance frequency wo) and the receiver coil (with resonance frequency o, = (L,C, ) - ‘/‘), leads to a radiation reaction field that drives the magnetization rapidly towards its equilibrium state and gives rise to a line broadening I, by radiation damping [ 57] which, for small deviations of the magnetization from the magnetic field direction, is given by

The quality factor of the receiver circuits is Q= w&J R,. By substituting L, into eq. (7)) we obtain for the voltage spectral density, W”(o)=

ikBT-$-

W”(w,)-

(9)

A more convenient form is obtained by introducing the equivalent parallel resistance R, of the circuit which can be expressed with the series resistance R, by R,= (~e-L)2/&=Q/ 1 kBTRp 1 +a(Aw)l2; [d(Aw)&

+ (w-w,)/LJ2

+wy.

(10)

This equation is also valid for a partially or fully saturated spin system where M, # MO. For a system in thermal equilibrium, one has &=A: with M,=Mo. IngeneraIA,=2~Mz/Mo. Eqs. (9) and (10) assume limited detuning of the receiver coil, w-We < w; +w,/2Q is the half width of the tuned circuit at the 3 dB point. For exact tuning of the receiver coil, w % o. eq. ( 10) can be simplified, 2 WU(w)=$JTR~

(A0)~+12(& (j12+~,)2+(Aw)2

W”(m)=

+A:)

+

wu a

zk,TR$.

(15) 2

(w-%)/&l2

+wy.

x [ 1 +a(Aw)&]*+

Thus the voltage spectral density shows a dip resonance with an amplitude that is given by the fractional contribution of radiation damping (1: ) to the full linewidth 1,+1:. On the other hand, for a saturated spin system with M,=O, A,=O, we have

c

1 +a(Aw)A: x wC,[l +a(Aw)&12+ [d(Aw)&+

W”(w)=

21 July 1989

CHEMICALPHYSIC8LETTERS

Volume 159,number 5,6

This represents a positive bump at resonance. For weak equilibrium radiation damping, ky =%A,, the amplitude of the dip and bump are equal while for strong equilibrium radiation damping, 1: %A,, the dip reaches a constant amplitude and the bump can grow without limits. For a comparison of low temperature experiments with SQUID detection and conventional high temperature experiments, the relative amplitude of the noise bump with respect to the noise level is relevant, i.e. the ratio JF/n,, disregarding the amplifier noise WY. Collecting the various terms in this ratio, we find the following quality factor

‘=

v2woQn I,T

.

We compare the situation of Sleator et al. [ 21 (~~0.35, y( 35C1)=0.098 y(‘H), oo/2x=30 MHz, Q=4000, n=1.1X102* spins/m3, l,/n=640 Hz, T= 1.5 K) with the present situation of proton resonance of benzene at 300 MHz (4=040, w,/2n = 300 MHz,Q=200,n=3.8X1028spins/m3,J.2/n:=0.2Hz, T= 300 K) and find a ratio

*

(11) Far from resonanke, Aw % AZ+&, we have W”(m)=

;k.TR,+

W,” I

(12)

while on-resonance 2 W”(wo)=

$JR,

J2(A2 +m (312+A,)2

(13)

+W:.

In the case of an equilibrium spin system with M,=M,, and I, =A:, one finds for the deviation of the spectrum at resonance,

W”(w,)-W”(m)=-

zk.TR&.

(14) 2

Amplifier noise is thereby neglected. But is is obvious that even with an amplifier with a poor noise figure the room temperature experiment is much easier to perform than the low temperature measurement.

r

3. Measurement spins

of noise absorption by nuclear

Noise absorption measurements for a spin system in thermal equilibrium were made in order to verify the dip in the noise spectrum, eq. ( 14), and its tun589

Volume 159,number 5,6

CHEMICAL. PHYSICS LETTERS

ing dependence predicted by eq. ( 10). The proton resonance experiments were performed on a 300 MHz spectrometer (home-built from Bruker subunits and further commercial equipment). The 5 mm sample contained 90% benzene with 10% benzene-d6 added for the deuterium field-frequency lock. The sensitive volume of the sample comprised approximately 8 X 102’ equivalent proton spins. Without the application of any radio frequency, 4 K data points were acquired over 1.364 s within a spectral bandwidth of 1500 Hz. After each acquisition, the noise trace was Fourier-transformed and the voltage spectral density computed. 500 spectra were co-added for five different tuning conditions of the receiver circuit with its center frequency offset at 0, ? 500, and * 1000 kHz, respectively, from the nuclear Larmor frequency (as measured with a spectrum analyzer). The 3dB points of the receiver circuit were at * 750 kHz. The results are collected in fig_ 2. It can be seen that on-resonance an absorption dip with a linewidth of approximately 1.7 Hz occurs. The amplitude amounts to 25% of the full noise level. The width of the dip agrees, within experimental accuracy, with the width of the benzene line obtained with a small flip angle excitation, measured to be 1.4 Hz. The measured line width in the absence of radiation damping is about 0.2 Hz. This confirms that the dip in the noise spectrum is broadened by radiation damping to the same extent as in a conventional NMR signal in agreement with eq. ( 10). Fig. 2 shows that detuning of the receiver coil leads to a distortion of the dip that resembles a phase change, although the voltage spectral density has been recorded. In order to appreciate this feature, some shapes of the dip have been simulated based on eq. ( 10) for different detuning conditions and represented in fig. 3. It is apparent that increasing detuning shifts the positive bump towards resonance (in the figure from the right side), graduaIIy narrows and finally for strong detuning overides the dip and leads to a narrow bump approaching the resonance frequency. With detuning, radiation damping also decreases and the bump for strong detuning assumes its natural width in the absence of radiation damping. At the same time, amplitudes of the bump and of the background noise decrease. It should be noted that the amplifier noise WY is not affected by detuning. 590

21 July 1989

coo-~

c

=

+5OOKHi!

mo- “, =

0 KHZ

= - 500 KHZ

wo- 61 = - 1000 KHZ c

t -24

I

:

-18

I

I

-12

I

I

-6

:

:

I

0 FREQUENCY

I

I

6

:

12

:

I

18

I

I

24

(HZ)

Fig. 2. Dependence of the absorption dip in the noise voltage spectral density on the tuning of the receiver coil. The circuit resonauce was varied from + 1 to - 1 MHz, relative to the nuclear spin Larmor frequency, by varying the tuning capacitor. 500 spectra of 90% benzene were co-added as described in the text.

For a more complicated spin system, the noise spectrum shows several dips that match the regular absorption spectrum of the sample. The noise spectrum of fig. 4 has been recorded for a sample of 90% ethanol and 10% acetone-d,. The spectral region of the CH3 triplet is shown with intensities that approach the expected 1: 2 : 1 ratio (for LF adz). It is the result of 8000 co-added noise spectra.

4. Measurement of the spontaneous emission of spin noise In order to detect the emission of spin noise, it is necessary to suppress the absorption of circuit noise, discussed in the previous section. This can be achieved by complete saturation of the spins, leading to M,=O and A,=O. It turns out that it is quite dif-

-2.5

21 July 1989

CHEMICAL PHYSICS LETTERS

Volume 159, number 5,6

-2.0

-1.5

I .o

-0.5

0

0.5

1.0

1.5

2.0

2.5

FREQUENCY (HZ)

Fig. 3. Shape of the noise spectrum with a sample in thermal equihbrium, M,=M,, for different detuning of the receiver coil mcasurcd by the offset wO--0, from the circuit resonance in terms of the half width A, of the circuit. It should be noted that the noise levels on the left and right of resonance are in general different even for infinite offset. On-resonance, a symmetric radiation-damped dip occurs while for strong detuning there is a narrow positive signal that is not affected by radiation damping. The intermediate peak shape is strongly asymmetric.

A

B

-48

-32

-16

0

16

32

48

FREQUENCY (HZ)

Fig. 4. Proton resonance of the methyl group in ethanol. (a) Standard absorption spectrum. (b) rms noise snectrum. 8000 spectra were co-added. The expected 1:2: 1 intensities are approximated.

ticult to fully saturate the spin system in the presence of strong radiation damping and to suppress all coherent remainders of the saturation process. It should be noted that the bump in the noise voltage spectral density is a factor 10’ smaller than the signal that would be obtained by z/2 pulse excitation. The saturation of the benzene sample with a longitudinal relaxation time r, = 20 s was achieved by an alternating sequence of rf pulses (2 s duration, 2 kHz amplitude) and magnetic field gradient pulses (0.25 s duration, 700 Hz/cm field gradient). In order to eliminate any remaining coherence, the experiment was repeated with the rf phases O”, 90°, 180’) 270” and the four signal traces were co-added. Saturation without the phase cycle allowed a suppression of the signal by a factor of about 10’ while the phase cycle causes a further reduction by a factor of up to 10. Following the saturation sequence, the noise was acquired in the form of 8 consecutive traces, each 2.73 s long. The traces were individually Fouriertransformed and the noise spectra added to the corresponding file. Eight consecutive noise spectra, to591

CHEMICAL PHYSICS LETTERS

Volume 159, number 5,6

21 July 1989

;

I

0

2.73

5.46

8.19

10.92

13.65

lb.38

19.11

____+

21.84

TIME (SEC)

Fib 5. (a) Spin noise spectra of 90% benzene recorded after initial full saturation of the proton spins as described in the text. The corresponding equilibrium absorption spectrum, reached after a long equilibration time, is also shown. The full amplitude of the noise voltage spectral density W“(co) is indicated on the right. The contribution of the amplifier noise W’t to this spectrum is W’F ~0.33 W”(m). (b) Theoretical ueak intensity as a function of elapsed time after saturation for assumed values 1,/2x=0.1 HZ, 1~/21t=0.6 , Hz and r, = 20 s that approach the experimental situation. I

._

gether with an equilibrium spectrum are shown in fig. 5. Fig. 5 shows that for a short time after saturation a positive signal occurs at the expected resonance position. The signal decays non-exponentially with the zero crossing of the peak intensity at x 12 s. It is apparent that the positive signals for short delay are narrower, due to small M, and reduced radiation damping, than the later traces that experience progressively increasing radiation damping as M, recovers towards A&. The preamplifier, with a noise figure of 1.8 dB, provides a frequency-independent contribution that amounts to 33% of the total noise W”(c0) far from resonance. (See eqs. (9)-(13).) The theoretical recovery curve of the peak intensity W”(tq) - W” (CO) as a function of time elapsed after saturation, shown in fig. 5b, has been computed based on eqs. ( 12) and ( 13) using for A, the time dependence n,(t)=n~[l-exp(-l/T,)]. The parameter values ,&/2x = 0.1 Hz, A:/2x= 0.6 Hz and T, =20 s approximate the experimental situation. The general tendencies of the data are repro592

duced. However the zero-crossing of the peak intensity occurs at an earlier time in the theoretical curve than in the experimental data. It should also be remembered that the experimental peak intensities reflect radiation broadening averaged over the acquisition time of 2.73 s used for each trace. In the absence of radiation damping, detuning of the receiver coil does not lead to a distortion of the bump as has been observed for the dip. Detuning merely reduces the amplitude but leaves the shape invariant. This follows directly from eq. ( 10).

5. Conclusions The absorption of circuit noise by nuclear spins does not differ significantly from conventional spectroscopic experiments. The external noise senses the circuit impedance modified by the presence of polarized spins through the inductance of the receiver coil that depends on the spin susceptibility. On the other hand, the possibility to directly observe the spontaneous emission of nuclear spin noise is more spectacular. Nuclear spin noise has its origin in the

Volume 159,number 54

CHEMICALPHYSICSLE’l-rERS

incoherent precession of nuclear spins. It leads to randomly fluctuating transverse fields that can be picked up by the receiver coil. All spins participate irrespective of temperature and polarization. Obviously, in thermal equilibrium, emission and absorption occur simultaneously. In equilibrium, both processes contribute constructively to the dip in the noise spectrum, although to different extents. The emission of spin noise is weighted by 11: while the absorption of circuit noise contributes with a weight d2 to the dip. Obviously both contributions are radiation-broadened. For a saturated sample, the circuit impedance becomes insensitive to the presence of the spins. Absorption by the spins ceases and the spontaneous emission of spin noise manifests itself now as a positive excursion in the noise spectrum due to the modified impedance of the circuit. It is unlikely that nuclear spin noise will be of practical importance for analytical NMR. It can only be detected for extremely strong resonance lines and in these cases conveys no structural information that could not be obtained in easier ways. The relevance of spin noise phenomena is rather to be found on the conceptual side. As has been pointed out by Sleator et al. [ 21, spin noise is formally connected with the phenomena of spontaneous emission and with the coupling of the spins to the cavity modes of the receiver coil [ 8 ]. Conceptually interesting questions concerning spontaneous emission at these very low frequencies arise in this context which have not been touched upon in this paper, the main purpose of which was to demonstrate the possibility of observ-

21 July 1989

ing spin noise phenomena even with commercial NMR equipment at ambient conditions. Acknowledgement This research has been supported by the Swiss National Science Foundation. The first experiments to demonstrate spin noise phenomena at room temperature were preformed in early 1988 by Dr. Christian Radloff in our laboratory [ 91. The authors are grateful for enlightening discussions with Professor Erwin Hahn. In the flnal stages of finishing this manuscript, we obtained from Professor Maurice Gueron two preprints that describe related experiments and considerations of spin noise and radiation damping at room temperature [ 10,111. Helpful correspondence with Professor Gutron is acknowledged. References [ 11T. Sleator, E.L. Hahn, C. Hilbert and J. Clarke, Phys. Rev. Letters 55 (1985) 1742. [2] T. Sleator, E.L. Hahn, C. Hilbctt and J. Clarke, Phys. Rev. B36 (1987) 1969. [3] A. Abragam, The principles of nuclear magnetism (Clarendon Press, Oxford, 196 1). [4] R.R. Ernst, G. Bodenhausen and A. Wokaun, Principles of NMR in one and two dimensions (Clarcndon Press, oxford, 1987). [ 5 ] N. Bloembergen and R.V. Pound, Phys. Rev. 95 ( 1954) 8. [6] C.R. Bruce, R.E. Norberg and G.E. Pake, Phys. Rev. 104 (1956) 419. [ 71 S. Bloom, J. Appl. Phys. 28 (1957) 800. [ 81 R.H. Dicke, Phys. Rev. 93 (1954) 99. [9] C. Radloff, unpublished results (1988). [lo] M. G&on and J.L. Leroy, preprint (1989). [ 111M. GuCron, preprint ( 1989).

593