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Nuclear spin relaxation in hydrogen
Bloom, Myer 1957
Physica X X I I I 767-780
NUCLEAR SPIN RELAXATION IN HYDROGEN I I I T H E SOLID NEAR T H E MELTING POINT by MYER BLOOM Communication No. 307c from the Kamerlingh Onnes Laboratorium, Leiden, Nederland
Synopsis The longitudinal and the transverse relaxation times tl and t2 have been measured at 301VLHzusing the pulse method for the proton magnetic resonances in solid He and HD in the region near the melting point where appreciable self-diffusion takes place, The logt2 versus lIT plots are linear for both solids over a wide range of temperatures, indicating that the coefficients of self-diffusion may be written in the form D0exp (-- Ea/Rt). The values of Ea obtained for He and HD are 380 4- 20 calorie/mole and 600 -4- 40 calorie/mole respectively, while the approximate values obtained for Do are l0 -3 cmZ/sec and 0.1 cm2/sec respectively, tl in He is caused by the effect of reorientations of the ortho-H2 molecules, and is found to be roughly temperature independent at 0.22 second, in agreement with the theory of M o r i y a and M o t i z u k i . tl in HI)is caused by the presence of a small amount oJ~1-I8impurity. A quantitative interpretation of the effect of the He impurity shows that the activation energy for diffusion of He in HD is approximately the same as in He and that Do ~- 10-4 cmZ/sec. I. Introduction. T h e first t w o articles in this series r e p o r t e d experim e n t a l m e a s u r e m e n t s on tl, the longitudinal or " s p i n - l a t t i c e " r e l a x a t i o n time, of p r o t o n s in h y d r o g e n gas a n d liquid. I t was shown in article I 1) t h a t m e a s u r e m e n t s on t h e gas can yield i n f o r m a t i o n on those interactions b e t w e e n pairs of h y d r o g e n molecules which reorient the molecular axes of rotation. T h e m a i n features of the tl m e a s u r e m e n t s in the liquid, discussed in article I I 2), are t h a t the usual m o d e l for i n t e r p r e t i n g tl in liquids b r e a k s down in h y d r o g e n , t h o u g h the results are q u a l i t a t i v e l y u n d e r s t a n d a b l e in t e r m s of t h e b e h a v i o u r of tl in the gas. I t is well k n o w n t h a t , for m o s t liquids and gases, the t r a n s v e r s e relaxation time, t2, is equal to tl 3). I n the gas this e q u a l i t y was directly verified, the spin echo technique 4) used for all m e a s u r e m e n t s r e p o r t e d in this series of articles m a k i n g it possible to m e a s u r e t2 even w h e n it is so long t h a t o r d i n a r y " s t e a d y - s t a t e " m e t h o d s c a n n o t easily be applied. " S t e a d y - s t a t e " m e t h o d s are not easily applicable to the m e a s u r e m e n t of tz w h e n the n a t u r a l line w i d t h of the resonance is m u c h smaller t h a n the line w i d t h associated w i t h the i n h o m o g e n e i t y of the applied m a g n e t i c fie]d, a situation t y p i c a l l y o b t a i n e d in liquids and gases. R o u g h l y speaking, t2 can be defined 767
768
MYER BLOOM
as the reciprocal of yAH, where y is the gyromagnetic ratio of the nucleus and AH is the line width in oersted for the resonance line when a perfectly homogeneous external magnetic field is applied. The equality or near equality of tl and t~ in gases and liquids is due to the rapid motion of the molecules relative to one another, which causes an averaging of the interactions between neighbouring spins. This is not true for/solids at temperatures low enough to inhibit self-diffusion appreciably, and there one usually observes a broad line., t2 being m u c h shorter than tl. It has, however, been observed that, even in solids, sufficiently rapid self-diffusion m a y occur to lengthen t2 over the value obtained for a rigid lattice. The study of t2 as a function of temperature can be interpreted to yield accurate values of the coefficient of self-diffusion 5). The e f f ~ t of diffusion on tl is also often observable and aids greatly in the understanding of the diffusion processes 5). In normal (75~/o ortho) solid H2 the p r o t o n line width, AH, has been found to be constant at about 5.4 oersted between 2.3°K and 10°K, b u t decreases rapidlsy about 10°K 6). R o l l i n and Watsonhave interpreted this rapid decrease in AH in terms of selfdiffusion of the Hg. molecules, the activation energy for self-diffusion, Ea, being found to be about 350 calorie/mole. One interesting feature of their results, as they point out 6), is that Ea is greater than the energy required to evaporate an H2 molecule from the surface of hydrogen. One limitation on the accuracy of R o l l i n and W a t s o n ' s results is that the diffusion increases so rapidly with increasing temperature that AH has decreased below the line width associated with the inhomogeneity of their applied magnetic field for temperatures above 12°K. Using the spin echo method, we have repeated their measurements and extended them up to ithe melting p o i n t at 14°K. With ~the accuracy available using the spin echo method and the wider temperature range accessible to direct measurement, the results of R o l l i n and W a t s o n have been verified and much narrower limits placed on Ea. Similarly, measurements on the proton resonance in H D have yielded Ea for self-diffusion of the H D molecules. The self-diffusion should also contribute to tl. At the frequencies employed here, however, this contribution is negligible relative to another effect; namely, that due to the intra-molecular interactions experienced b y the protons fluctuating in time as a result of reorientation of the molecules due to the artisotropic part of the H2--H2 interaction. As we shall point out, tl measurements at lower frequencies should reveal tl contributions due to self-diffusion. tl measurements in H D have also been carried out and are of unusual interest. As pointed out in I and II, reorientation of the H D molecules contributes much more weakly to tl than in ortho-H~ molecules since the lowest rotational state of H D corresponds to J = o, while for ortho-H2 the lowest state is the J = 1 state. At the temperatures of the solid, it
NUCLEAR
SPIN RELAXATION
769
IN H Y D R O G E N
should be negligible compared with the self-diffusion contribution. The measurements show, however, a much shorter tl than expected from selfdiffusion of the H D molecules, and, also, a completely different temperature dependence than predicted b y that mechanism. We can explain the observed tl, both as to magnitude and temperature dependence, as arising from the 2.5% Hg. molecules known to be present as impurities, in our H D sample 2). The H2 molecules have a very short h ; the protons on the H2 molecules can then interact with the H D protons, the Am(HD) = -t- 1, Am(H2) = :}: 1 transition having a "relaxation time" comparable with t2. The tl measurements in H D can be interpreted to give information on the diffusion of H2 molecules in HD. 2. Experimental results. The measurements of tl and t2 reported here were made using the pulse method. As discussed in more detail in article II, tl m a y be measured from the difference in the amplitudes of the induction signals immediately following the apphcation of two successive pulses of r.f. power at the Larmor frequency as a function of the separation of the pulses. All the measurements of tl quoted here were made using this method. O.S
SEC
0.2
O.1 70
Fig. l. P l o t of logtx
versus
IOOO/T
~
80
90
t e m p e r a t u r e for protons in H2 of a p p r o x i m a t e l y 66% ortho-molecules. :
Figure 1 shows tl for a sample of H2 as a function of temperature. The observation of a temperature independent tl is to be expected theoretically from the effect of reorientation of the ortho-H2 molecules 7) and will be discussed in section 3B. As will be shown in section 3A, the contribution to tl from self-diffusion effects is negligible at the frequency of 30 MHz used here. No direct measurement of the ortho-H2 concentration was made, but the sample used for tl measurements had been i n t h e cryostat, in liquid form, for about 10 hours when the measurements were begun. From the orthopara conversion rate in the liquid s), the sample had a b o u t 69% ortho-Hz molecules when the measurements were begun. The measurements were Physica x x n I 49
770
MYER BLOOM
made over a period of about 3 hours So that about 5~/o conversion took place during the measurements s) 9). The remarks regarding ortho-par.a concentration apply t o the t2 measurements in H2 shown in figure 2, except that measurements were begun very soon after hquefaction in this case, the amount of ortho-H2 at the start of measurements probably being greater than 70%. The t2 measurements lasted about 5 hours so that the ortho-H,, concentration at the end of the measurements was about 8% lower than at the start. The measurements of t2 were made in the usual w a y using spin echoes 2) 4) over most of the temperature range. For pulses at times 0 and r, the amplitude of the echo occurring at 2r was found to be proportional to exp(--Xr/t2) for all of the echo measurements made. This exponential decrease probably inElicates that the spin system is well described b y the Bloch phenomenological equations lO) while motional narrowing takes place. 10000
i£tec
\
lo0o
1o o
\
tO TO tOOO/T80
90
$
•
\
tOO
110
120
Fig. 2. Plot of logt~ v e r s u s temperature for protons in H2 having approximately 65% ortho-H2 molecules. - Ea = 380 cal/mole, the horizontal lines gives t2 for a rigid lattice. (D measurements using decay of echo. • measurements using decay of free induction signals following pulses. As the temperature was lowered, t2 became so short that it was impossible to measure t2 using echoes. Instead, the rate of the decay of the induction signal following the pulse (the "tail") was used, applying appropriate corrections for the inhomogeneity of the externally apphed field. Over a small range of temperatures, both the echo and "tail" types of t2 measurements were possible and, as seen from figure 2, the two types of measurements agree very well. In fig. 3, the tl measurements for the protons in solid H D are plotted as a function of temperature. It should be emphasized that the sample used contained approximately 2.5% ortho-H2 impurity as measured in the
NUCLEAR SPIN RELAXATION IN HYDROGEN
77 ]
liquid 2). In 3A, we show that the observed tl behavior cannot be explained b y considering only interactions between the'nuclei on neighboring H D molecules. The impurity plays a very vital role as we show in detail in section 3C.
5oo.
mSeC 200 100
5O
60.. 1OOO/T; 6S
70
75
Fig. 3. P l o t of ] o g l l v e r s u s t e m p e r a t u r e for protons in H D molecules. Sample of H D c o n t a i n e d a b o u t 2 . 5 % H2 i m p u r i t y . A n a d d i t i o n a l m e a s u r e m e n t , n o t p l o t t e d h e r e , g a v e tl = 40.5 millisecond for T ~- l l.2°K. 10
I'nS~C S
,Q5
t~[ Q2 Ol 60
IO00/T
65
70
75
Fig. 4. P l o t of logt2 v e r s u s t e m p e r a t u r e for p r o t o n s in H D molecules. S a m p l e of H D c o n t a i n e d a b o u t 2.5~/o H2 i m p u r i t y . A n a d d i t i o n a l m e a s u r e m e n t , n o t p l o t t e d here, g a v e to ---- 10 millisecond for T = 11.2°K. Ea = 600 cal/mole.
Figure 4 shows t2 for the protons in H D as a function of temperature. All measurements were made using spin echoes except the very lowest temperature (ll.2°K) measurement where the "tail" was used. It should be remarked that the reason why only one measurement is quoted below 14°K for H D is that we had difficulty in establishing thermal contact between the thermal bath and the HD sample when the H2 bath
772
MYER
BLOOM
was solid (see article II for cryostatic details). This difficulty was not present in the H~ experiments since, in that case, the coil was immersed directly in the bath. For all the H D measurements, a separate check on the temperature of the sample was made b y monitoring the vapour pressure of the HD. Only in one of our runs did we succeed in getting the temperature of the H D to come to thermal equilibrium with the H2 bath below 14°K. This enabled us to make the measurements quoted for l l.2°K.
3. Theory and interpretation o/ the experimental results. A. T h e e f f e c t of s e l f - d i f f u s i o n : In a system of nuclear spins, the effect of self-diffusion tends to average the dipole-dipole interaction between neighbouring spins, hence making the resonance line width smaller (or t2 longer) than that corresponding to a rigid lattice. The fluctuations in the dipole-dipole interaction are usually discussed in terms of the correlation time ze, which is defined as the mean time taken for the interaction between a pair of spins to change appreciably due to molecular motion. ~Expressed in terms of the coefficient of self-diffusion of the molecules D, rc for a pair of interacting identical nuclear spins separated b y a distance r can be written as 3): re = r2/12D.
(1)
For a system of identical nuclei of spin I undergoing "appreciable" selfdiffusion (re ~ t2), one can write 3) 5):
1/t2 = (3/2~)~4~2I(I + 1)J0(0)
(2)
where J0(co) is the Fourier transform of the correlation function of (1 - - 3 cos 20)/r3 summed over all neighboring nuclei, r being the vector joining a pair of nuclei and making an angle 0 with the direction of the external field. N o r b e r g and H o l c o m b 5) assume a homogeneous distribution of nuclear dipoles approximating the sum over all neighbors b y an integral, and obtain, making the usual assumptions about the form of the correlation function 3) : 1/t2 = 474h2I(I + 1)N/5Da (3) where N is the number of nuclear dipoles per cm3 and a is the distance of closest approach of pairs of nuclei. For the case of H~, N is taken as the density of ortho-H2 molecules since para-H~ has I = 0. The two protons on the ortho-H~ molecule behave like a single nucleus having the proton gyromagnetic ratio and I = 1. The fluctuations induced in the dipole-dipole interaction b y the diffusion also cause spin-lattice relaxation; therefore, tl also depends on the diffusion rate, a principal characteristic of this dependence being that tl has a minim u m value when re is of the order of the Larmor period of the nuclei in the applied magnetic field. The general formula for tl is 3) n):
l/t1 = (3/2h,4~2I(I + 1)[J1(~0) + J2(2o~0)]
(4)
N U C L E A R S P I N R E L A X A T I O N IN H Y D R O G E N
773
where Jl(CO), Js(oJ) are the Fourier transforms of the correlation functions of sin 0 cos 0 exp i9/r8 and sins 0 exp 2i9/r8 respectively. The factor of 2 difference between the term involving Js(2co0) in (4) and that originally derived s) has been pointed out b y A b r a g a m 11). Again, assuming a homogeneous distribution of nuclear dipoles, one obtains 8)5): .
1 tl
.
8z~ ~'4hz/( I + 1)N . . 5
rc 92 ~ 1 + O~orc
+
4rc 1 + 4rOot ~ o2
(5)
where o~0 is 2~ times the Larmor frequency. It should be emphasized that the detailed behaviour of (5) in the region of the minimum should not be taken too seriously. One should introduce diffusion in a more rigorous manner as was done b y T o r r e y is), b u t this will not be necessary for our purpose. The procedure we follow is to evaluate D from the t2 measurements using equation (3). The tl contribution due to self-diffusion is then calculated according to equation (5). From the straight lines obtained over a wide temperature region for the log t2 versus reciprocal temperature plots for both Hs and H D in figs. 2 and 4 respectively, we can write for the self-diffusion coefficient: D = Do exp(-- Ea/RT).
(6)
The activation energies obtained are: for H2, Ea ~---380 + 20 calorie/mole and for HD, Ea ~ 600 4 - 4 0 calorie/mole. The first result confirms the value of Ea previously obtained 6) for H2, though considerably greater accuracy has been obtained here due to the wider temperature range accessible to the spin-echo method. It should be kept in mind that the total time involved in making the t2 measurements in H2 (5 hours) is of some importance since ortho-para conversion was probably occurring at the rate of about 2% per hour 8). Since all the measurements were made while decreasing the temperature, and since t2 should become shorter with greater ortho-H2 concentration, a systematic error was probably introduced such as to make our value of Ea slightly too low. The values of Do m a y now be evaluated from equations (3) and (6) using the values of Ea obtained above and the measured value of t2 at any given temperature on the straight line part of the log t2 versus 1/T plots. The volume of 1 mole of solid hydrogen is about 23 cm 3 13) which gives a density of 2.6 × 1029'molecules/cm 8. We have taken N as equal to this number for HD, while, assuming an average ortho-H2 concentration of about 65% we calculate N ~ 1.7 × 1029'/cm8 for the H2 sample. The separation of nearest neighbours in hydrogen (3.75 A) is taken as "a". We then find Do ~ 1.4 × 10-8 cm2/sec for H2 and Do ~ 0.17 cm2/sec for HD. The values of Do obtained are only valid insofar as order of magnitude
774
MYER BLOOM
is Concerned because of the approximations involved in the derivation of equation (3). Usilig the values of D So obtained, we pow plot tl as a function of tem.perature for Hg. and HD, as calculated b y formula (5), in figs. 5 and 6 1o0o
S~C
/
100
t
,o/ / ~
/
,/ ,Y
/ -
0.1 70
I000/T80
90
IO 0
Fig. 5. Plot of |ogtl v e r s u s temperature according to equation (5) for the protons in 1-12. 1000 s~c
1oo
1o
I
// /
/
o.1
60 1000/T~O
80
90
Fig. 6. Plot of logtl v e r s u s temperature according to equation (5) for the protons in HD.
respectively. We see that the values of tl obtained experimentally (figs. 1 and 3) are much shorter than predicted for 30 MHz b y equation (5) over the temperature range studied and have very different temperature dependences. The mechanism for t h e t l observed in Ha will be discussed in B. As will be
N U C L E A R S P I N R E L A X A T I O N IN H Y D R O G E N
775
seen in C, the tl values observed in HD are due to the effect of the 2.5% H2 impurity. In a pure sample of HD, one should,observe the temperature dependence shown in figure 6. B. E f f e c t of m o l e c u l a r r e o r i e n t a t i o n . As discussed in articles I and II, the relaxation time of the protons in gaseous and liquid H2 is due to intramolecular interactions, i.e. the dipole-dipole interaction between protons on the same ortho-H2 molecule and the interaction between the protons' magnetic moments and the magnetic moment associated with the rotational angular momentum of the ortho-Ha molecule. In the gas and liquid 1) 2), measurement of tl gives information on the rate of reorientation of the angular momentum axis of the molecule. The s a m e mechanism for tl is also possible in the solid. Whereas the reorientation rate of the molecules in the gas is determined by collisions between pairs of Ha molecules and hence is expressible in terms of the two-body intelmolecular interaction, the reorientation rate of the ortho-H2 molecules in the solid is determined by interactions with all neighbouring molecules. A detailed calculation of tl for tiffs mechanism in solid Ha and D2 in terms of the intermolecular interactions has recently been carried out 7) and we quote the result for Ha, in which tl was predicted to be temperature independent in the temperature range of these experiments. The result is: tl = 0.361v/K second
(7)
where K is the fraction of ortho-Ha molecules. The results of figure 1 were obtained for a sample in which K ~ 0.66, so that (7) gives tl ~ 0.29 second. Considering the accuracy with which the intermolecular interactions are known, this is in very satisfactory agreement with the approximate temperature independent value of 0.22 second shown in figure 1. We are very grateful to Drs. M o r i y a and Motiz u k i for communicating their calculation to us before publication. They have also pointed out that the tl values in Ha hquid a) can also be understood in terms of their calculation, the fact that the liquid tl is lower than the solid being due to the correlation time for translational motion of the molecules becoming shorter than the correlation time for reorientation of the molecules in a static lattice. The above mechanism gives a much weaker contribution to tl in solid HD because the lowest rotational state of HD is the J = 0 state, in which case the interactions necessary for the above relaxation mechanism vanish completely. Just as in the gas 1), the molecule would have to be excited to the f = 1 state in order to relax b y the above mechanism and the resulting tl would be too long at these temperatures to compete with the self-diffusion effect discussed in A or, as in the case of the present experiment, with the effects of a small amount of Ha impurity.
776
MYER BLOOM
C. E f f e c t of He i m p u r i t y in HD. As we have mentioned earlier, the presence of a small amount of He (2.5~) as impurity in our H D sample has a very profound effect on the spin-lsttice relaxation of the protons in HD. This is especially true of the solid in the temperature region studied here and, probably, at lower temperatures as well. Before proceeding with a quantitative derivation of the contribution of the He impurity to tl, we can explain qualitatively w h y even a small amount of impurity can, in this case, play so important a role. Firstly, tl for the protons in ortho-H2 molecules is determined b y the mechanism discussed in B. We have seen that tl from that mechanism is independent of temperature and not too different from that observed in the hquld 8). In the H D sample studied here, the protons of the He impflrity had tl ~ 10 millisecond in the liquid. The tl predicted for the H D protons due to self-diffusion processes in the solid is more than two orders of magnitude greater than this over the temperature range studied. Secondly, all our experiments in the solid were done in a temperature region such that t2 ~ tl insofar as diffusion effects are concerned. The reason for this is that we always satisfy the inequality re >~ 1/(o0 so that J0(0) >~ fl(o~0), J2(2ooo);i.e. a Fourier analysis of the fluctuating dipoledipole interaction gives mucl~ larger intensities near zero frequency than near the Larmor frequency. Now, one of the terms in the dipole-dipole interaction between nuclei, labelled a and b, say, has the effect of causing the transition Area = 4- 1, Arab = ~: 1, where ma, mb are the components of I of nuclei a and b respectively in the direction of the applied magnetic field. For identical nuclei these transitions do not contribute to tl since they do not change the total energy of the spin system in the external field, and their effect is incorporated in equations (2) and (3). For non-identical nuclei, however, these terms contribute to tl since they affect the spin temperatures of both the a and b systems. The point is that, in our H D sample, we have the unusual situation of two different groups of nuclear spins having the same gyromagnetic ratio; namely, the protons of the H D molecules and the protons of the He impurity. Processes such as those described above will require no exchange of energy insofar as the total energy of the two spin systems is concerned and will, therefore, occur at a rate proportional to Jo(0). At low enough temperatures they will, therefore, occur very quickly in spite of the small percentage of the impurity and, coupled with the short relaxation time of the impurity, can provide the main relaxation mechanism for the protons o'n the H D molecules. Proceeding to a quantitative derivation of these effects, we define two interacting spin systems, a and b. Defining: ~a = 7ahn/kT~, 2b = ~bhH/kT~ ~a = r ~ H / k T L , 6b = rb~H/kTL
NUCLEAR S P I N R E L A X A T I O N I N H Y D R O G E N
777
where T~, T~ are the spin temperatures of the a and b systems respectively and TT, is the lattice temperature, it m a y be shown 11) that: d2a [__@__ 1 d----t-+ + (~b)---~ + (t~b)2 +
-d---/- +
1 (t~)0
[
-+ (t~)--~ + (t[~)~-7 +
=
((~)0
(a~- ~)= 1 (~b) 2 ~(25-
~b)
(;tb -- ~b) =
(t~a)~ (a~-~a)
(Sb)
where ~a, t[b are the relaxation times which one obtains in the a and b systems respectively in the absence of coupling between the systems, while (~b)l, (~b)2 and (t~b)0 represent relaxation processes due to coupling between the systems.
1/(ttb)l = I/(t~b)2 = 1/(t~b)0 ----
(3/2)7~7~,h~.ib(15 + ! )]~ab (~o~)
(ga)
(3/4)y~y~,hg"Ib(Ib + l)f~b(wa + wb)
(95)
(1/12)r~r~a2Ib(Ib + 1)fab(coa -- cob),
(9c)
j~b(@, j~b(co) and j~b(w) are the same as defined in A and elsewhere a) but pertain only to interactions between nuclei of types a and b. Analogous expressions are obtained for (tba)l, (t~a)2 and (t~a)0. From the considerations of section A and equations (9) it is obvious that if Na, Nb are the densities of nuclei a and b respectively: l/(tb~)l = //(t~b)l
( 10a)
where
/ = (NdNb)Ia(Ia + 1)/Ib(Ib + l)
(lOb)
if coa = o~b. Equations (8a) and (8b) m a y be easily expressed in terms of a single second order differential equation whose solution is straightforward. We consider only the special case satisfying the following approximations:
(~)
(t~b)o < (t~~) ~, ( ~ )
which is equivalent to Jr~btw 0 ~ a - ~ b ) > > J i ~ r~broj~~, , ] ~~b (~+~b), (5) / < 1 which is satisfied for Na ~ Nb. In our system, the "a" nuclei represent the pair of protons on the orthoH2 molecules, while the "b" nuclei represent the protons on the HD mole-
778
MYER BLOOM
cules. W i t h the above approximations the solutions to (8a) and (8b) m a y be written as 2a = ha + / ' [ 2 a ( 0 ) - - ha] exp {-- (1/t bb + [F/t~a)t} + (1 la) + [(la(0) -- ha) -- F(2b(0) -- ~b)] exp {--(1/t~ ~ + 1/(t~b)0)t} Ib = ~b + [2b(0) -- ~b] exp { , ( 1 / t bb + [F/t~)t}
(1 lb)
where ,la(0), ;tb(0) are the values of 2a, ,lb respectively at t = 0
and
F = 1/(t~b)oE1/(t~b)o + I/t~ a -
1/tlbb3.
U n d e r conditions (a) and (b) therefore, system "b" returns to thermal equilibrium with a time constant tle b ff given b y llt ef
= llt +/rlt = -= 1/t bb + flbb/{tblb[(t~b)o + t~ ~] -- (t~b)0t~~}
(12)
For tlbb>~ (t~b)o, t~ a 1/tbetf = 1/t~ b + [/[(t~b)o + t~a].
(13)
Putting N b / N a ~- 50, Ia = 1, Ib = ½ corresponding to our identification of the "a" system as the 2.5% H2 i m p u r i t y in the H D sample and the " b " system as the protons in HD, we obtain [ ~___1/19 from equation (10b). F r o m figure 6, which gives the predicted tlbb as a function of temperature, we see that, to a good approximation, the term involving t bb in (13) m a y be dropped. Furthermore, noting t h a t ~Oa = COb,Ya = yb = y, equation (9c)
gives: 1/(t~b)0 = (1/16)),4h2j'~b(0).
(14)
FoUowing t h e same procedure used in deriving equation (3), one obtains:
I/(t~b)0 ----~ryah2Nb/15[D(HD) + D(H2)Ja
(I5)
where D ( H D ) is the coefficient of self-diffusion for the H D molecules which has been shown in A to be 0.1 exp (-- 600/RT) cm2/sec. D(H2) is the diffusion coefficient for H2 molecules in H D . Our procedure is to use the experimental results of figure 3 to evaluate (t~b)0 as a function of temperature using equation (13) and then use equation (16) to evaluate D(H~) as a function of temperature. To do this, we m a k e a further assumption ; namely, that at T = 11.2°K, where we measured tl to be 40.5 millisecond, (t~b)0 ~ t~a. Then, equation (13) gives
t~a ~ 40.5/msec = 2.1 msec. Since this relaxation time is due to the mechanism discussed in B, which has been predicted to be temperature independent 7), a fact which has been verified for H2, we assume that t~" is independent of temperature. The value of 2.1 millisecond obtained above for t~a should be calculable
779
NUCLEAR SPIN RELAXATION IN HYDROGEN
in terms of the detailed interaction between H2 and H D molecules using the method of M o r i y a and M o t i z u k i 7). Using equation (13) again, (t~b)0 is now calculated as a function of temperature from the results of figure 3 to be: (t~b)0 = /tl(HD) -- 2.1 msec.
(16)
Using the curve drawn through the experimental points of figure 3 as tl(HD) in equation (16), we have plotted the calculated values of log (t~b)0 as a function of the reciprocal temperature in figure 7. To a good approximation the plot is a straight line, the slope of which indicates an activation energy of 360 calorie/mole. Extrapolating (t~b)0 to l l.2°K, one obtains 0.13 millisecond, which is consistent with our approximation (t~b)0 ~ t~a at l l.2°K made above. m sec
20
10
60
I O 0 0 / T ~ 65
70
75
Fig. 7. P l o t of log(tlab)0 as a f u n c t i o n of t e m p e r a t u r e . C a l c u l a t e d f r o m e q u a t i o n (17) u s i n g line d r a w n t h r o u g h d a t e of figure 3.
If, as one might expect, D(HD) ~ D(H2), this indicates that the activation energy for self-diffusion of H2 in H D is, within experimental error, the same as in H2. Equation (15) would then indicate the magnitude of Do to be approximately 10-4 cm2/sec. I wish to thank Professor C. J. G o r t e r for his stimulating interest and encouragement during the course of this work and for his warm hospitality during m y stay in Leiden. The work could not have been completed in the time available without the enthousiastic and expert technical assistance of Mr. D. de J o n g . Many others of the technical staff gave valuable assistance, especially Mr. A. R. B. G e r r i t s e and Mr. C. Le P a i r . I wish to thank Dr. J. B e e n a k k e r and Mr. F. H. V a r e k a m p , nat. phil. cand., for their preparation of HD, and Dr. N. J. P o u l i s for useful suggestions on the design of the cryostatic apparatus. Received 24-7-57.
780
NUCLEAR
SPIN . R E L A X A T I O N IN H Y D R O G E N
REFERENCES 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13)
B l o o m , M., Commun. Kamedingh Onnes Lab., Leiden No. 307a; Physica 23 (1957) 237. B l o o m , M., Commun. No. 307b; Physica 23 (1957) 378. B l o e m b e r g e n , N., P u r c e l l , E. N. and P o u n d , R. V., Phys. Rev. 73 (1948) 679. H a h n , E. L., Phys. Rev. 80 (1950) 580. H o l e o m b , D. P. and N o r b e r g , R. E., Phys. Rev. 98 (1954) 1074. R o l l i n , B. V. and W a t s o n , E., Suppl. Bull. Inst. int. du Froid, Annexe 1955-3, 474. M o r i y a , T. and M o t i z u k i , K., Progr. theor. Physics (to be published). F a r k a s , A., Light and heavy Hydrogen (Cambridge University Press, 1935). M o t i z u k i , K. and N a g a m i y a , T., J. phys. Soc. Japan 11 (1956) 93. Bloch, F., Phys. Rev. 70 (1946) 460. A b r a g a m , A., Notes, Nuclear magnetic Resonance. T o r r e y , H. C., Phys. Rev. 92 (1953) 962. W o o l l e y , H. W., S c o t t , R. B. and B r i c k w e d d e , F. G., J. Res. Nat. Bur. Stand., Washington 41 (1948) 379.
Physica X X I I I 767-780
NUCLEAR SPIN RELAXATION IN HYDROGEN I I I T H E SOLID NEAR T H E MELTING POINT by MYER BLOOM Communication No. 307c from the Kamerlingh Onnes Laboratorium, Leiden, Nederland
Synopsis The longitudinal and the transverse relaxation times tl and t2 have been measured at 301VLHzusing the pulse method for the proton magnetic resonances in solid He and HD in the region near the melting point where appreciable self-diffusion takes place, The logt2 versus lIT plots are linear for both solids over a wide range of temperatures, indicating that the coefficients of self-diffusion may be written in the form D0exp (-- Ea/Rt). The values of Ea obtained for He and HD are 380 4- 20 calorie/mole and 600 -4- 40 calorie/mole respectively, while the approximate values obtained for Do are l0 -3 cmZ/sec and 0.1 cm2/sec respectively, tl in He is caused by the effect of reorientations of the ortho-H2 molecules, and is found to be roughly temperature independent at 0.22 second, in agreement with the theory of M o r i y a and M o t i z u k i . tl in HI)is caused by the presence of a small amount oJ~1-I8impurity. A quantitative interpretation of the effect of the He impurity shows that the activation energy for diffusion of He in HD is approximately the same as in He and that Do ~- 10-4 cmZ/sec. I. Introduction. T h e first t w o articles in this series r e p o r t e d experim e n t a l m e a s u r e m e n t s on tl, the longitudinal or " s p i n - l a t t i c e " r e l a x a t i o n time, of p r o t o n s in h y d r o g e n gas a n d liquid. I t was shown in article I 1) t h a t m e a s u r e m e n t s on t h e gas can yield i n f o r m a t i o n on those interactions b e t w e e n pairs of h y d r o g e n molecules which reorient the molecular axes of rotation. T h e m a i n features of the tl m e a s u r e m e n t s in the liquid, discussed in article I I 2), are t h a t the usual m o d e l for i n t e r p r e t i n g tl in liquids b r e a k s down in h y d r o g e n , t h o u g h the results are q u a l i t a t i v e l y u n d e r s t a n d a b l e in t e r m s of t h e b e h a v i o u r of tl in the gas. I t is well k n o w n t h a t , for m o s t liquids and gases, the t r a n s v e r s e relaxation time, t2, is equal to tl 3). I n the gas this e q u a l i t y was directly verified, the spin echo technique 4) used for all m e a s u r e m e n t s r e p o r t e d in this series of articles m a k i n g it possible to m e a s u r e t2 even w h e n it is so long t h a t o r d i n a r y " s t e a d y - s t a t e " m e t h o d s c a n n o t easily be applied. " S t e a d y - s t a t e " m e t h o d s are not easily applicable to the m e a s u r e m e n t of tz w h e n the n a t u r a l line w i d t h of the resonance is m u c h smaller t h a n the line w i d t h associated w i t h the i n h o m o g e n e i t y of the applied m a g n e t i c fie]d, a situation t y p i c a l l y o b t a i n e d in liquids and gases. R o u g h l y speaking, t2 can be defined 767
768
MYER BLOOM
as the reciprocal of yAH, where y is the gyromagnetic ratio of the nucleus and AH is the line width in oersted for the resonance line when a perfectly homogeneous external magnetic field is applied. The equality or near equality of tl and t~ in gases and liquids is due to the rapid motion of the molecules relative to one another, which causes an averaging of the interactions between neighbouring spins. This is not true for/solids at temperatures low enough to inhibit self-diffusion appreciably, and there one usually observes a broad line., t2 being m u c h shorter than tl. It has, however, been observed that, even in solids, sufficiently rapid self-diffusion m a y occur to lengthen t2 over the value obtained for a rigid lattice. The study of t2 as a function of temperature can be interpreted to yield accurate values of the coefficient of self-diffusion 5). The e f f ~ t of diffusion on tl is also often observable and aids greatly in the understanding of the diffusion processes 5). In normal (75~/o ortho) solid H2 the p r o t o n line width, AH, has been found to be constant at about 5.4 oersted between 2.3°K and 10°K, b u t decreases rapidlsy about 10°K 6). R o l l i n and Watsonhave interpreted this rapid decrease in AH in terms of selfdiffusion of the Hg. molecules, the activation energy for self-diffusion, Ea, being found to be about 350 calorie/mole. One interesting feature of their results, as they point out 6), is that Ea is greater than the energy required to evaporate an H2 molecule from the surface of hydrogen. One limitation on the accuracy of R o l l i n and W a t s o n ' s results is that the diffusion increases so rapidly with increasing temperature that AH has decreased below the line width associated with the inhomogeneity of their applied magnetic field for temperatures above 12°K. Using the spin echo method, we have repeated their measurements and extended them up to ithe melting p o i n t at 14°K. With ~the accuracy available using the spin echo method and the wider temperature range accessible to direct measurement, the results of R o l l i n and W a t s o n have been verified and much narrower limits placed on Ea. Similarly, measurements on the proton resonance in H D have yielded Ea for self-diffusion of the H D molecules. The self-diffusion should also contribute to tl. At the frequencies employed here, however, this contribution is negligible relative to another effect; namely, that due to the intra-molecular interactions experienced b y the protons fluctuating in time as a result of reorientation of the molecules due to the artisotropic part of the H2--H2 interaction. As we shall point out, tl measurements at lower frequencies should reveal tl contributions due to self-diffusion. tl measurements in H D have also been carried out and are of unusual interest. As pointed out in I and II, reorientation of the H D molecules contributes much more weakly to tl than in ortho-H~ molecules since the lowest rotational state of H D corresponds to J = o, while for ortho-H2 the lowest state is the J = 1 state. At the temperatures of the solid, it
NUCLEAR
SPIN RELAXATION
769
IN H Y D R O G E N
should be negligible compared with the self-diffusion contribution. The measurements show, however, a much shorter tl than expected from selfdiffusion of the H D molecules, and, also, a completely different temperature dependence than predicted b y that mechanism. We can explain the observed tl, both as to magnitude and temperature dependence, as arising from the 2.5% Hg. molecules known to be present as impurities, in our H D sample 2). The H2 molecules have a very short h ; the protons on the H2 molecules can then interact with the H D protons, the Am(HD) = -t- 1, Am(H2) = :}: 1 transition having a "relaxation time" comparable with t2. The tl measurements in H D can be interpreted to give information on the diffusion of H2 molecules in HD. 2. Experimental results. The measurements of tl and t2 reported here were made using the pulse method. As discussed in more detail in article II, tl m a y be measured from the difference in the amplitudes of the induction signals immediately following the apphcation of two successive pulses of r.f. power at the Larmor frequency as a function of the separation of the pulses. All the measurements of tl quoted here were made using this method. O.S
SEC
0.2
O.1 70
Fig. l. P l o t of logtx
versus
IOOO/T
~
80
90
t e m p e r a t u r e for protons in H2 of a p p r o x i m a t e l y 66% ortho-molecules. :
Figure 1 shows tl for a sample of H2 as a function of temperature. The observation of a temperature independent tl is to be expected theoretically from the effect of reorientation of the ortho-H2 molecules 7) and will be discussed in section 3B. As will be shown in section 3A, the contribution to tl from self-diffusion effects is negligible at the frequency of 30 MHz used here. No direct measurement of the ortho-H2 concentration was made, but the sample used for tl measurements had been i n t h e cryostat, in liquid form, for about 10 hours when the measurements were begun. From the orthopara conversion rate in the liquid s), the sample had a b o u t 69% ortho-Hz molecules when the measurements were begun. The measurements were Physica x x n I 49
770
MYER BLOOM
made over a period of about 3 hours So that about 5~/o conversion took place during the measurements s) 9). The remarks regarding ortho-par.a concentration apply t o the t2 measurements in H2 shown in figure 2, except that measurements were begun very soon after hquefaction in this case, the amount of ortho-H2 at the start of measurements probably being greater than 70%. The t2 measurements lasted about 5 hours so that the ortho-H,, concentration at the end of the measurements was about 8% lower than at the start. The measurements of t2 were made in the usual w a y using spin echoes 2) 4) over most of the temperature range. For pulses at times 0 and r, the amplitude of the echo occurring at 2r was found to be proportional to exp(--Xr/t2) for all of the echo measurements made. This exponential decrease probably inElicates that the spin system is well described b y the Bloch phenomenological equations lO) while motional narrowing takes place. 10000
i£tec
\
lo0o
1o o
\
tO TO tOOO/T80
90
$
•
\
tOO
110
120
Fig. 2. Plot of logt~ v e r s u s temperature for protons in H2 having approximately 65% ortho-H2 molecules. - Ea = 380 cal/mole, the horizontal lines gives t2 for a rigid lattice. (D measurements using decay of echo. • measurements using decay of free induction signals following pulses. As the temperature was lowered, t2 became so short that it was impossible to measure t2 using echoes. Instead, the rate of the decay of the induction signal following the pulse (the "tail") was used, applying appropriate corrections for the inhomogeneity of the externally apphed field. Over a small range of temperatures, both the echo and "tail" types of t2 measurements were possible and, as seen from figure 2, the two types of measurements agree very well. In fig. 3, the tl measurements for the protons in solid H D are plotted as a function of temperature. It should be emphasized that the sample used contained approximately 2.5% ortho-H2 impurity as measured in the
NUCLEAR SPIN RELAXATION IN HYDROGEN
77 ]
liquid 2). In 3A, we show that the observed tl behavior cannot be explained b y considering only interactions between the'nuclei on neighboring H D molecules. The impurity plays a very vital role as we show in detail in section 3C.
5oo.
mSeC 200 100
5O
60.. 1OOO/T; 6S
70
75
Fig. 3. P l o t of ] o g l l v e r s u s t e m p e r a t u r e for protons in H D molecules. Sample of H D c o n t a i n e d a b o u t 2 . 5 % H2 i m p u r i t y . A n a d d i t i o n a l m e a s u r e m e n t , n o t p l o t t e d h e r e , g a v e tl = 40.5 millisecond for T ~- l l.2°K. 10
I'nS~C S
,Q5
t~[ Q2 Ol 60
IO00/T
65
70
75
Fig. 4. P l o t of logt2 v e r s u s t e m p e r a t u r e for p r o t o n s in H D molecules. S a m p l e of H D c o n t a i n e d a b o u t 2.5~/o H2 i m p u r i t y . A n a d d i t i o n a l m e a s u r e m e n t , n o t p l o t t e d here, g a v e to ---- 10 millisecond for T = 11.2°K. Ea = 600 cal/mole.
Figure 4 shows t2 for the protons in H D as a function of temperature. All measurements were made using spin echoes except the very lowest temperature (ll.2°K) measurement where the "tail" was used. It should be remarked that the reason why only one measurement is quoted below 14°K for H D is that we had difficulty in establishing thermal contact between the thermal bath and the HD sample when the H2 bath
772
MYER
BLOOM
was solid (see article II for cryostatic details). This difficulty was not present in the H~ experiments since, in that case, the coil was immersed directly in the bath. For all the H D measurements, a separate check on the temperature of the sample was made b y monitoring the vapour pressure of the HD. Only in one of our runs did we succeed in getting the temperature of the H D to come to thermal equilibrium with the H2 bath below 14°K. This enabled us to make the measurements quoted for l l.2°K.
3. Theory and interpretation o/ the experimental results. A. T h e e f f e c t of s e l f - d i f f u s i o n : In a system of nuclear spins, the effect of self-diffusion tends to average the dipole-dipole interaction between neighbouring spins, hence making the resonance line width smaller (or t2 longer) than that corresponding to a rigid lattice. The fluctuations in the dipole-dipole interaction are usually discussed in terms of the correlation time ze, which is defined as the mean time taken for the interaction between a pair of spins to change appreciably due to molecular motion. ~Expressed in terms of the coefficient of self-diffusion of the molecules D, rc for a pair of interacting identical nuclear spins separated b y a distance r can be written as 3): re = r2/12D.
(1)
For a system of identical nuclei of spin I undergoing "appreciable" selfdiffusion (re ~ t2), one can write 3) 5):
1/t2 = (3/2~)~4~2I(I + 1)J0(0)
(2)
where J0(co) is the Fourier transform of the correlation function of (1 - - 3 cos 20)/r3 summed over all neighboring nuclei, r being the vector joining a pair of nuclei and making an angle 0 with the direction of the external field. N o r b e r g and H o l c o m b 5) assume a homogeneous distribution of nuclear dipoles approximating the sum over all neighbors b y an integral, and obtain, making the usual assumptions about the form of the correlation function 3) : 1/t2 = 474h2I(I + 1)N/5Da (3) where N is the number of nuclear dipoles per cm3 and a is the distance of closest approach of pairs of nuclei. For the case of H~, N is taken as the density of ortho-H2 molecules since para-H~ has I = 0. The two protons on the ortho-H~ molecule behave like a single nucleus having the proton gyromagnetic ratio and I = 1. The fluctuations induced in the dipole-dipole interaction b y the diffusion also cause spin-lattice relaxation; therefore, tl also depends on the diffusion rate, a principal characteristic of this dependence being that tl has a minim u m value when re is of the order of the Larmor period of the nuclei in the applied magnetic field. The general formula for tl is 3) n):
l/t1 = (3/2h,4~2I(I + 1)[J1(~0) + J2(2o~0)]
(4)
N U C L E A R S P I N R E L A X A T I O N IN H Y D R O G E N
773
where Jl(CO), Js(oJ) are the Fourier transforms of the correlation functions of sin 0 cos 0 exp i9/r8 and sins 0 exp 2i9/r8 respectively. The factor of 2 difference between the term involving Js(2co0) in (4) and that originally derived s) has been pointed out b y A b r a g a m 11). Again, assuming a homogeneous distribution of nuclear dipoles, one obtains 8)5): .
1 tl
.
8z~ ~'4hz/( I + 1)N . . 5
rc 92 ~ 1 + O~orc
+
4rc 1 + 4rOot ~ o2
(5)
where o~0 is 2~ times the Larmor frequency. It should be emphasized that the detailed behaviour of (5) in the region of the minimum should not be taken too seriously. One should introduce diffusion in a more rigorous manner as was done b y T o r r e y is), b u t this will not be necessary for our purpose. The procedure we follow is to evaluate D from the t2 measurements using equation (3). The tl contribution due to self-diffusion is then calculated according to equation (5). From the straight lines obtained over a wide temperature region for the log t2 versus reciprocal temperature plots for both Hs and H D in figs. 2 and 4 respectively, we can write for the self-diffusion coefficient: D = Do exp(-- Ea/RT).
(6)
The activation energies obtained are: for H2, Ea ~---380 + 20 calorie/mole and for HD, Ea ~ 600 4 - 4 0 calorie/mole. The first result confirms the value of Ea previously obtained 6) for H2, though considerably greater accuracy has been obtained here due to the wider temperature range accessible to the spin-echo method. It should be kept in mind that the total time involved in making the t2 measurements in H2 (5 hours) is of some importance since ortho-para conversion was probably occurring at the rate of about 2% per hour 8). Since all the measurements were made while decreasing the temperature, and since t2 should become shorter with greater ortho-H2 concentration, a systematic error was probably introduced such as to make our value of Ea slightly too low. The values of Do m a y now be evaluated from equations (3) and (6) using the values of Ea obtained above and the measured value of t2 at any given temperature on the straight line part of the log t2 versus 1/T plots. The volume of 1 mole of solid hydrogen is about 23 cm 3 13) which gives a density of 2.6 × 1029'molecules/cm 8. We have taken N as equal to this number for HD, while, assuming an average ortho-H2 concentration of about 65% we calculate N ~ 1.7 × 1029'/cm8 for the H2 sample. The separation of nearest neighbours in hydrogen (3.75 A) is taken as "a". We then find Do ~ 1.4 × 10-8 cm2/sec for H2 and Do ~ 0.17 cm2/sec for HD. The values of Do obtained are only valid insofar as order of magnitude
774
MYER BLOOM
is Concerned because of the approximations involved in the derivation of equation (3). Usilig the values of D So obtained, we pow plot tl as a function of tem.perature for Hg. and HD, as calculated b y formula (5), in figs. 5 and 6 1o0o
S~C
/
100
t
,o/ / ~
/
,/ ,Y
/ -
0.1 70
I000/T80
90
IO 0
Fig. 5. Plot of |ogtl v e r s u s temperature according to equation (5) for the protons in 1-12. 1000 s~c
1oo
1o
I
// /
/
o.1
60 1000/T~O
80
90
Fig. 6. Plot of logtl v e r s u s temperature according to equation (5) for the protons in HD.
respectively. We see that the values of tl obtained experimentally (figs. 1 and 3) are much shorter than predicted for 30 MHz b y equation (5) over the temperature range studied and have very different temperature dependences. The mechanism for t h e t l observed in Ha will be discussed in B. As will be
N U C L E A R S P I N R E L A X A T I O N IN H Y D R O G E N
775
seen in C, the tl values observed in HD are due to the effect of the 2.5% H2 impurity. In a pure sample of HD, one should,observe the temperature dependence shown in figure 6. B. E f f e c t of m o l e c u l a r r e o r i e n t a t i o n . As discussed in articles I and II, the relaxation time of the protons in gaseous and liquid H2 is due to intramolecular interactions, i.e. the dipole-dipole interaction between protons on the same ortho-H2 molecule and the interaction between the protons' magnetic moments and the magnetic moment associated with the rotational angular momentum of the ortho-Ha molecule. In the gas and liquid 1) 2), measurement of tl gives information on the rate of reorientation of the angular momentum axis of the molecule. The s a m e mechanism for tl is also possible in the solid. Whereas the reorientation rate of the molecules in the gas is determined by collisions between pairs of Ha molecules and hence is expressible in terms of the two-body intelmolecular interaction, the reorientation rate of the ortho-H2 molecules in the solid is determined by interactions with all neighbouring molecules. A detailed calculation of tl for tiffs mechanism in solid Ha and D2 in terms of the intermolecular interactions has recently been carried out 7) and we quote the result for Ha, in which tl was predicted to be temperature independent in the temperature range of these experiments. The result is: tl = 0.361v/K second
(7)
where K is the fraction of ortho-Ha molecules. The results of figure 1 were obtained for a sample in which K ~ 0.66, so that (7) gives tl ~ 0.29 second. Considering the accuracy with which the intermolecular interactions are known, this is in very satisfactory agreement with the approximate temperature independent value of 0.22 second shown in figure 1. We are very grateful to Drs. M o r i y a and Motiz u k i for communicating their calculation to us before publication. They have also pointed out that the tl values in Ha hquid a) can also be understood in terms of their calculation, the fact that the liquid tl is lower than the solid being due to the correlation time for translational motion of the molecules becoming shorter than the correlation time for reorientation of the molecules in a static lattice. The above mechanism gives a much weaker contribution to tl in solid HD because the lowest rotational state of HD is the J = 0 state, in which case the interactions necessary for the above relaxation mechanism vanish completely. Just as in the gas 1), the molecule would have to be excited to the f = 1 state in order to relax b y the above mechanism and the resulting tl would be too long at these temperatures to compete with the self-diffusion effect discussed in A or, as in the case of the present experiment, with the effects of a small amount of Ha impurity.
776
MYER BLOOM
C. E f f e c t of He i m p u r i t y in HD. As we have mentioned earlier, the presence of a small amount of He (2.5~) as impurity in our H D sample has a very profound effect on the spin-lsttice relaxation of the protons in HD. This is especially true of the solid in the temperature region studied here and, probably, at lower temperatures as well. Before proceeding with a quantitative derivation of the contribution of the He impurity to tl, we can explain qualitatively w h y even a small amount of impurity can, in this case, play so important a role. Firstly, tl for the protons in ortho-H2 molecules is determined b y the mechanism discussed in B. We have seen that tl from that mechanism is independent of temperature and not too different from that observed in the hquld 8). In the H D sample studied here, the protons of the He impflrity had tl ~ 10 millisecond in the liquid. The tl predicted for the H D protons due to self-diffusion processes in the solid is more than two orders of magnitude greater than this over the temperature range studied. Secondly, all our experiments in the solid were done in a temperature region such that t2 ~ tl insofar as diffusion effects are concerned. The reason for this is that we always satisfy the inequality re >~ 1/(o0 so that J0(0) >~ fl(o~0), J2(2ooo);i.e. a Fourier analysis of the fluctuating dipoledipole interaction gives mucl~ larger intensities near zero frequency than near the Larmor frequency. Now, one of the terms in the dipole-dipole interaction between nuclei, labelled a and b, say, has the effect of causing the transition Area = 4- 1, Arab = ~: 1, where ma, mb are the components of I of nuclei a and b respectively in the direction of the applied magnetic field. For identical nuclei these transitions do not contribute to tl since they do not change the total energy of the spin system in the external field, and their effect is incorporated in equations (2) and (3). For non-identical nuclei, however, these terms contribute to tl since they affect the spin temperatures of both the a and b systems. The point is that, in our H D sample, we have the unusual situation of two different groups of nuclear spins having the same gyromagnetic ratio; namely, the protons of the H D molecules and the protons of the He impurity. Processes such as those described above will require no exchange of energy insofar as the total energy of the two spin systems is concerned and will, therefore, occur at a rate proportional to Jo(0). At low enough temperatures they will, therefore, occur very quickly in spite of the small percentage of the impurity and, coupled with the short relaxation time of the impurity, can provide the main relaxation mechanism for the protons o'n the H D molecules. Proceeding to a quantitative derivation of these effects, we define two interacting spin systems, a and b. Defining: ~a = 7ahn/kT~, 2b = ~bhH/kT~ ~a = r ~ H / k T L , 6b = rb~H/kTL
NUCLEAR S P I N R E L A X A T I O N I N H Y D R O G E N
777
where T~, T~ are the spin temperatures of the a and b systems respectively and TT, is the lattice temperature, it m a y be shown 11) that: d2a [__@__ 1 d----t-+ + (~b)---~ + (t~b)2 +
-d---/- +
1 (t~)0
[
-+ (t~)--~ + (t[~)~-7 +
=
((~)0
(a~- ~)= 1 (~b) 2 ~(25-
~b)
(;tb -- ~b) =
(t~a)~ (a~-~a)
(Sb)
where ~a, t[b are the relaxation times which one obtains in the a and b systems respectively in the absence of coupling between the systems, while (~b)l, (~b)2 and (t~b)0 represent relaxation processes due to coupling between the systems.
1/(ttb)l = I/(t~b)2 = 1/(t~b)0 ----
(3/2)7~7~,h~.ib(15 + ! )]~ab (~o~)
(ga)
(3/4)y~y~,hg"Ib(Ib + l)f~b(wa + wb)
(95)
(1/12)r~r~a2Ib(Ib + 1)fab(coa -- cob),
(9c)
j~b(@, j~b(co) and j~b(w) are the same as defined in A and elsewhere a) but pertain only to interactions between nuclei of types a and b. Analogous expressions are obtained for (tba)l, (t~a)2 and (t~a)0. From the considerations of section A and equations (9) it is obvious that if Na, Nb are the densities of nuclei a and b respectively: l/(tb~)l = //(t~b)l
( 10a)
where
/ = (NdNb)Ia(Ia + 1)/Ib(Ib + l)
(lOb)
if coa = o~b. Equations (8a) and (8b) m a y be easily expressed in terms of a single second order differential equation whose solution is straightforward. We consider only the special case satisfying the following approximations:
(~)
(t~b)o < (t~~) ~, ( ~ )
which is equivalent to Jr~btw 0 ~ a - ~ b ) > > J i ~ r~broj~~, , ] ~~b (~+~b), (5) / < 1 which is satisfied for Na ~ Nb. In our system, the "a" nuclei represent the pair of protons on the orthoH2 molecules, while the "b" nuclei represent the protons on the HD mole-
778
MYER BLOOM
cules. W i t h the above approximations the solutions to (8a) and (8b) m a y be written as 2a = ha + / ' [ 2 a ( 0 ) - - ha] exp {-- (1/t bb + [F/t~a)t} + (1 la) + [(la(0) -- ha) -- F(2b(0) -- ~b)] exp {--(1/t~ ~ + 1/(t~b)0)t} Ib = ~b + [2b(0) -- ~b] exp { , ( 1 / t bb + [F/t~)t}
(1 lb)
where ,la(0), ;tb(0) are the values of 2a, ,lb respectively at t = 0
and
F = 1/(t~b)oE1/(t~b)o + I/t~ a -
1/tlbb3.
U n d e r conditions (a) and (b) therefore, system "b" returns to thermal equilibrium with a time constant tle b ff given b y llt ef
= llt +/rlt = -= 1/t bb + flbb/{tblb[(t~b)o + t~ ~] -- (t~b)0t~~}
(12)
For tlbb>~ (t~b)o, t~ a 1/tbetf = 1/t~ b + [/[(t~b)o + t~a].
(13)
Putting N b / N a ~- 50, Ia = 1, Ib = ½ corresponding to our identification of the "a" system as the 2.5% H2 i m p u r i t y in the H D sample and the " b " system as the protons in HD, we obtain [ ~___1/19 from equation (10b). F r o m figure 6, which gives the predicted tlbb as a function of temperature, we see that, to a good approximation, the term involving t bb in (13) m a y be dropped. Furthermore, noting t h a t ~Oa = COb,Ya = yb = y, equation (9c)
gives: 1/(t~b)0 = (1/16)),4h2j'~b(0).
(14)
FoUowing t h e same procedure used in deriving equation (3), one obtains:
I/(t~b)0 ----~ryah2Nb/15[D(HD) + D(H2)Ja
(I5)
where D ( H D ) is the coefficient of self-diffusion for the H D molecules which has been shown in A to be 0.1 exp (-- 600/RT) cm2/sec. D(H2) is the diffusion coefficient for H2 molecules in H D . Our procedure is to use the experimental results of figure 3 to evaluate (t~b)0 as a function of temperature using equation (13) and then use equation (16) to evaluate D(H~) as a function of temperature. To do this, we m a k e a further assumption ; namely, that at T = 11.2°K, where we measured tl to be 40.5 millisecond, (t~b)0 ~ t~a. Then, equation (13) gives
t~a ~ 40.5/msec = 2.1 msec. Since this relaxation time is due to the mechanism discussed in B, which has been predicted to be temperature independent 7), a fact which has been verified for H2, we assume that t~" is independent of temperature. The value of 2.1 millisecond obtained above for t~a should be calculable
779
NUCLEAR SPIN RELAXATION IN HYDROGEN
in terms of the detailed interaction between H2 and H D molecules using the method of M o r i y a and M o t i z u k i 7). Using equation (13) again, (t~b)0 is now calculated as a function of temperature from the results of figure 3 to be: (t~b)0 = /tl(HD) -- 2.1 msec.
(16)
Using the curve drawn through the experimental points of figure 3 as tl(HD) in equation (16), we have plotted the calculated values of log (t~b)0 as a function of the reciprocal temperature in figure 7. To a good approximation the plot is a straight line, the slope of which indicates an activation energy of 360 calorie/mole. Extrapolating (t~b)0 to l l.2°K, one obtains 0.13 millisecond, which is consistent with our approximation (t~b)0 ~ t~a at l l.2°K made above. m sec
20
10
60
I O 0 0 / T ~ 65
70
75
Fig. 7. P l o t of log(tlab)0 as a f u n c t i o n of t e m p e r a t u r e . C a l c u l a t e d f r o m e q u a t i o n (17) u s i n g line d r a w n t h r o u g h d a t e of figure 3.
If, as one might expect, D(HD) ~ D(H2), this indicates that the activation energy for self-diffusion of H2 in H D is, within experimental error, the same as in H2. Equation (15) would then indicate the magnitude of Do to be approximately 10-4 cm2/sec. I wish to thank Professor C. J. G o r t e r for his stimulating interest and encouragement during the course of this work and for his warm hospitality during m y stay in Leiden. The work could not have been completed in the time available without the enthousiastic and expert technical assistance of Mr. D. de J o n g . Many others of the technical staff gave valuable assistance, especially Mr. A. R. B. G e r r i t s e and Mr. C. Le P a i r . I wish to thank Dr. J. B e e n a k k e r and Mr. F. H. V a r e k a m p , nat. phil. cand., for their preparation of HD, and Dr. N. J. P o u l i s for useful suggestions on the design of the cryostatic apparatus. Received 24-7-57.
780
NUCLEAR
SPIN . R E L A X A T I O N IN H Y D R O G E N
REFERENCES 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13)
B l o o m , M., Commun. Kamedingh Onnes Lab., Leiden No. 307a; Physica 23 (1957) 237. B l o o m , M., Commun. No. 307b; Physica 23 (1957) 378. B l o e m b e r g e n , N., P u r c e l l , E. N. and P o u n d , R. V., Phys. Rev. 73 (1948) 679. H a h n , E. L., Phys. Rev. 80 (1950) 580. H o l e o m b , D. P. and N o r b e r g , R. E., Phys. Rev. 98 (1954) 1074. R o l l i n , B. V. and W a t s o n , E., Suppl. Bull. Inst. int. du Froid, Annexe 1955-3, 474. M o r i y a , T. and M o t i z u k i , K., Progr. theor. Physics (to be published). F a r k a s , A., Light and heavy Hydrogen (Cambridge University Press, 1935). M o t i z u k i , K. and N a g a m i y a , T., J. phys. Soc. Japan 11 (1956) 93. Bloch, F., Phys. Rev. 70 (1946) 460. A b r a g a m , A., Notes, Nuclear magnetic Resonance. T o r r e y , H. C., Phys. Rev. 92 (1953) 962. W o o l l e y , H. W., S c o t t , R. B. and B r i c k w e d d e , F. G., J. Res. Nat. Bur. Stand., Washington 41 (1948) 379.