Low density spectrum of transverse NMR in molecular gases. Application to hydrogen

Chemical Physics 32 (1978) 23-33 0 North-Holland Publishing Company

LOW DENSITY SPECPRDM OF TRANSVFitiE NMR IN MOLECULAR GASES. APPLICATION TO HYDROGEN F.R. MCCOURT * FOMInstituut voor Atoom- en Molecuuifisica, Amsterdam&m.,

The Netherlands

and G. TENT1 Guelph-Waterloo Centre for Graduate Work in Chemistry and Department of Chemistry, University of Waterloo, Waterloo, Ontario, anada N2L 3GI Received 8 August 1977 Revised manuscript received

21 December

1977

The low density behaviour of the spectral lineshape function associated with the relaxation of the transverse components of the nuclear magnetization of a homonuclear diatomic gas of spin-$ nucIei is studied from the point of view of a recent kinetic theory approach to NMR in molecular gases. It is shownthat as a critical density is approached (from above), the relaxation (in time-domain) passes over from exponential to nonexponential behaviour. For the special case in which the spin-rotation relaxation mechanism predominates, an analytic solution to the problem has been given, while for molecular hydrogen which has, in addition to the aforementioned mechanism, dipolar relaxation, the behaviour of the lineshape function has been obtained numerically. The critical density at which the relaxation passes from exponential to nonexponential in Hs is found to be of the order of 3 X KS3 amagats which lies, at present, still outside experimental accessibility. One important consequence of this result is that the traditional Abraganrformula for the transverse relaxation time Tz is clearly seen to be invalid below =5 X 10T3amagats.

1. Introduction IR a recent paper [l] (referred to as I in the following), a new version of the kinetic theory of nuclear magnetic relaxation (NMR) in dilute molecular gases was proposed. The main features of such a theory can briefly be summarized as follows. Firstly, in keeping with most of the work done in non-equilibrium statistical mechanics over the past two decades, the analysis was carried out in frequency

language, with a deriva-

tion and discussion of the fluctuation spectrum for both the longitudinal and the transverse magnetization. Secondly, and more importantly, it was shown that the * Permanent address: Guelph-Waterloo Centre for Graduate Work in Chemistry and Departments of Chemistry and Applied Mathematics,University of Waterloo,Waterloo,Ont.,

Canada.

low frequency components of the fluctuation spectrum of the transverse magnetization contain a non-vanishing dynamical shift, which modifies both the resonant fre_quency and the relaxation rate. Consequently, the transverse relaxation time, which is a function of the density among other things, goes through a minimum as the number density is decreased and tends to infmity as the density tends to zero. Thirdly, it was also pointed out that at very low densities, the behaviour of the fluctuation spectrum for the longitudinal magnetization is very different from that for the transverse part. In fact, while the low frequency part of the former consists of just a single (lorentzian) line at all densities, the latter shows a more complicated structure, so that the concept of ordinary Tyrelaxation (i.e., exponential time decay of.the transverse magnetization) breaks down near some critical density. However, the critical density was found to lie well below the then extant experimen-

24

RR. iUcCourr, G. TentilLowdensity spectrumof tmnwetse NMR in mohdargases

tal accessibility. Hence the analysis of the details of the spectrum at such low densities and for a realistic system was not carried out in I, which concentrated instead on a theoretical model gas where the spinrotation relaxation mechanism is the only one available. Recently, Kalechstein and Armstrong [2] have made measurements of T2 at room temperature at densities’below that at which T1 attains its characteristic minimum. Tl-& work has been carried out in the hope that the existence of a detailed theoretical study of the fluctuation spectrum of the transverse magnetization, in particular at low densities, will encourage the extension of T2 measurements further towards the critical region where, as this study shows, non-exponential decay sets in. In’the next section the analysis of the model gas with only the spin-rotation mechanism is continued, starting from where I left off. The reason for this is two-fold. On the one hand, the polynomial equation which determines the dynamical shift is simple enough that it can be studied analytically to a great extent. This allows a discussion of the finer details of the spectrum, thereby giving a clear picture of its physical interpretation. On the other hand, the complications introduced by the inchrsion of the dipolar relaxation mechanism in the case of H2, while changing the spectrum quantitativeiy, do not alter its basic qualitative features and physical meaning. Section 3 of this paper thus reviews the procedure established in I, whtfe extending it to allow for the additional presence of the dipolar relaxation mechanism. Further, in order not to obscure the main points of this paper the assumption will be made throughout either that the system has a single occupied level (e.g., o - H2 at 77 K) or thaf a single-relaxation-time approximation (SRTA) can meaningfully be applied. As has been shown [3], such an approximation gives excellent agreement with experimental T1 data in room temperature (296 K) hydrogen. The results of a numerical study of both the lineshape and the dynamical shift are reported and discussed in section 4 for the algebraically complicated case of H2 gas. The qualitative picture presented in section 2 is confirmed and at the same time, a realistic estimate is made of the density range which characterizes the transition region. A brief surnrnary of this work is presented in the final section, along with some

comments on the possibility of an experimental test of the theory in the transition region.

2. Detailed analysis and meaning of the spectrum In order to make this paper more self-contained, it is useful to review briefly the basic steps in the derivation of the spectrum. The starting pot& is the equation of motion for the transverse magnetization Am,(t) which is obtained from the kinetic equation and which has the form [l] t ~Am+(t)/Clt=iwIAm,(t)

dt’K,(t - t’) Am+@‘) ,

-s 0

where or is the (angular) Larmor frequency of the proton and the kernel K,(f) is given, for example, by eq. (15) below. Introducing the Laplace-transformed magnetization Am,(z) by Am+(z) = s 0

dt exp(izt) Am,(t)

(z = w f ie) ,

where e > 0 is a small quantity which is set equal to zero at the end of the calculation, the solution of the Laplace-transformed equation of motion is Am+(z) = [-iz - iwI f i ImK,(z) f ReK,(z)] -l Am,(O) , where Am,(O) denotes the initial (t = 0) value of the transverse magnetization. As is well known [4,5] the spectrumI+ can immediately be obtained from a knowledge of the complex response function. However, for the sake of completeness, the argument is reviewed in the appendix. For the normalized spectrum the result is

I,(m) =

ReU.4 [~+~~-ImIrn~(w)]*+ [ReKL(w)]*’

(1)

where the explicit expression for KL(o) depends upon assumptions made about the form of the interaction. When only the spin-rotation mechanism is considered, the real and imaginary parts of KL(F) are given by eqs. (1.40) and (1.41), respectively. The corresponding terms for the dipolar mechanisms are given in the next section. Leaving aside for a moment the specific form of

25

F.R. McCourt, G. TentijLow dens@ spectrum of transverseNMR in moleculargases

the kernel K,(o), it is useful first to make a few remarks of a general nature about the spectrum I+(w), even though these have been made many times before in connection with analogous problems of non-equilibrium statistical mechanics [4] Depending upon the theory used for calculating K,(o), there will be one or more values of the frequency w. for which, at a given value of the density, the equation w. + wr - ImKL(ag)

=0

(2)

is satisfied. If the quantity ReKJw) is small and slowly varying near wO, then in the neighbourhood of such a root of (2) the spectrum I+(o) will exhibit a peak which is, in general, non-lorentzian in shape. However, when the peak is well-defined and for a small range of frequencies, it is usually a good approximation to represent it with a lorentzian lineshape. Then, of course, the half-width at half-height can be interpreted as a relaxation rate and its inverse as a relaxation time. Thus the physical interpretation of the spectrum which emerges from this type of analysis is the familiar one in terms of modes, since it is close to the picture of a collection of coupled, damped harmonic osciilators [S] . It is therefore convenient to analyze the spectrum in terms of these modes, i.e., in terms of collective properties of the many-body system obeying simple macroscopic laws of motion such as the equation of motion for the transverse magnetization Am+(t) defined in I. In order to do so, the solution of eq. (2) is first studied under the assumption of a model gas in which only the spin-rotation relaxation mechanism is present. Then, after introducing the dynamical shift CTby writing w. = wr + o, it is easy to check that eq. (2) becomes eq. (1.61), viz., a=Q2

-- U [ v,2+02

A-U $+(A-u)~

1 ’

(3)

where the symbols have the same meaning as in I, except for having introduced the definitions A=UJ-Wr,

02 G 5 (J2) a2sr .

(4)

Eq. (3)‘is a polynomial equation of the fifth order in o, and can therefore only be solved approximately. However, because of the physical meaning of the dynamical shift, it is known that (Tis very small at high density. At the other extreme of ve_y low densities,

an asymptotic analysis of eq. (3) shows that u 2 Q. But under ordinary conditions; it is also true, as pointed out in I, that the relations wr % W, and [Al = lull hold. Thus, if the temperature (which enters the equilibrium average of J2) is not too high, IAl % !G!and therefore lul< IAl. This implies that eq. (3) reduces, to a very good approximation, to the following cubic equation a3 +a2u2 +alo+aO=O,

(5)

where a2 = -;

CPA

zl=vc- 2

Q2;

a0=v$a2.

(6)

$+A2

In order to find the location of the modes in the spectrum I+(o), the real solutions of eq. (5) are studied. The discriminant of (5) is [6] q3 + r2 = ($)3(vz - Q2)3 + 3vf[f @A/($ + [$ !$A/@;

+ A2)J4

+ A’)] 2 ($v; - 8 Q4 -I-;z@~)

, (7)

and, of course, there will be one or three real roots depending upon whether q3 + r2 is positive defiiite or negative semi-definite, respectively, with the transition from one to three real roots occurring when $ + r2 is zero. Since this happens for values of the number density such that vc Q IAI, eq. (7) can be rewritten in the form q3 + 9 = (93 n6{-[

1+ $(a/A)2]

+ (u,iQ)2 [3 + 5(S2/A)2 + (WA)4] f (~$2)~

I-3 + 2(W#l

+ (v,/G~~) ,

(8)

in which terms of the order of (v,/A)~ have been neglected. From this result, it is easy to find the value of the collision frequency for which q3 f r2 is zero, i.e., $)’ = fi [ 1 _ (6)1/3 (a/A)?/31 I/z ,

(9)

namely of order fi. For vc 5 viol, there is one dynamical shift only, which is approximately given by =0

= - 5 S22A/(v,2+ A’)

.

w

Thus, at higher densities, there is just one resonant frequency o. = wr t u and hence one well-defined peak in the spectrum. This is the usual Bloch mode, which has been the subject of all the previous work done in the study of the behaviour of the transverse magnetiza-

26

F.R. McCourt. G. TentifLowdensityspectrumof transverseNMR in moleculargases

tion Am,(t) (cf. references given in I). For v < v.$‘),however, eq. (5) has three real roots and, in ge liiit vc + 0, they are given by ao=-@/a

(11)

and o+=tsZ-;@/A_

is now given by eq. (1 l), may reappear with a shape close to a delta function. This region is of little interest, however, since the half-widths associated with the modes are practically zero. Naturally this means, in time language, that relaxation has stopped and the dynamical variables merely oscillate without damping.

(12)

It should be noticed that eq. (11) is just the zero density limit of eq. (lo), and is much smailer than the other two roots (hence the designation a,). Once the value of the dynamical shift has been found, the details of the variations of the spectral shape can be studied as a function of the density. More precisely, the spectrum has three characteristic features corresponding to three regions of the density. For values of the density such that vc 5 Vet),Z+(W) shows only one well-defied peak, centered at WZ+ oo_ This is the ordinary Bloch mode, which was referred to in I as the single-line-approximation (SLA), and its half-width at half-height gives the ordinary relaxation rate TF’_ Since in this region all other modes are heavily damped, it is referred to as the region where the relaxation of the transverse magnetization can properly be characterized by the T2-concept. Thus, thrs is the region of valrilty of the traditional theories (cf. references cited in I). When the density decreases and vc + VP) the spectrum changes drastically. The Bloch peak broadens continuously with decreasing density, and when vc is close to the critical collision frequency vi’), most of the structure in Z+(w) disappears. This region is therefore referred to as the transition region. Here the traditional T2-relaxation concept breaks down since no well-defined peaks are present. However, as the spectrum Z+(w) retains its physical meaning, what happens can be described by saying that, while the Bloch mode becomes further damped, the side modes, centered at wr + o, and wr f u_, become less and less so. The reduction in their damping makes them more and more prominent in the spectrum, until they predominate the lineshape when vc
3. Transverse relaxation in HZ_ Formulation of the theory The presence of the dipolar mechanism in the relaxation of the nuclear magnetization does not alter the formal aspects in the derivation of the equation of motion for the transverse magnetization. Thus, by following the procedure of I, the equation

aa~,(t)/at=i~~Am+(r)/-jdrfK1(t-

t’) Am&‘)

0

(13)

is still obtained. The difference appears only in the form of the kernel KL(t), which now consists of two terms, one for each relaxation mechanism &(t) = ICY(t) -I-&)

.

(14)

Since u - H2 at 77 K is the classic example of a singlelevel system, and since the SRTA has quite successfully explained [3] the 7-r versus number density data for H2 at room temperature, it will be assumed here +&at it will also give a good description of the behaviour of the transverse magnetization. Under these restrictions then, the spin-rotation contribution to the kernel (14) is given by eq. (L19), viz., KY(t) = 5 (52) &$ exp (-us, t) X [exp (iwJ t) + exp (iwIt)] _

WI

The

contribution of the dipolar mechanism to the kernel (14) can be worked out in complete analogy with the spin-rotation case. Thus, after some tedious, but straightforward, algebra the result Z&r) = & (412 - 3) @/(4fi X {3 exp(ioZt)+3

- 3)) w$ e-Vdr

exp(iwJt)

(16) + 2 exp [i(2wJ - wZ) t] + 2 exp [i(2wr - wJ) t] }

is obtained. Here ad is the strength of the dipolar coupling, and vd is the collision frequency characterizing

F.R McCourt. G. TentiiLow

density .+ectmm

the effect of reorientation collisions on the polarization [S](2) = [.I] c2)/(452 - 3), with [Jl c2)representing a second order irreducible (symmetric-traceless) tensor in J. Finally, the quantity Z2 = Z(Z + 1) = 2 for o-H,. With the form of the kernel explicitly known, it is then an easy matter to derive the lineshape expression. The formal part of it is the same as that given in I, namely, the spectrum is still given by eq_ (l), while the difference appears, naturally, in the real and imaginary parts of the kernel. Introducing the definitions s2,2,= $ @> a$;

CL; = & @/(4J2

eqs. (15) and (16) give for&(f) imaginary parts:

- 3)) IX; ,

(17)

the following real and

Re e’(w) 1 v,2,+ (w + cL$2

zJ&+ (w + WJ)2I ’

(18)

3

3

ReZ&w)=Qivd

[ vi + (w + GJr)2+2Vd+ (CJ+ WJ)2 2

f

1

+

2

+

V~+(w+2GJJ-C@

u,2+(w+2CI$-CzJ#

I- ’ (1%

of transverse NMR in mokcuiargases

A-U

vf +(A -

3@J+w[) [ vi + (w + or)2

2(w+2wJ-6_$ -Iv~+(w+2wJ-wr)2

+

+

v;+(A-v)~ _

2(2A-a)

2(A+u) +2v~+(A+v)~

~;+(2A-v)~

Vd”f (w + WJ)2

(22)

’

A v$ f A2 A $+A2

1 4A

)

ui t4A2 I

(23)

or, equivalently,

2(w+2wI-wJ) u;+(w+2w~-wJ)2

3

i.e., a thirteenth order polynomial equation, whose analysis is hopelessly complicated. It is also clear from this example that the more relaxation mechanisms that are included, the more complicated the behaviour of v may be. This is easy to understand in a qualitative way. In fact, the inclusion of more mechanisms of relaxation implies the presenceof extra terms in the internal hamiltonian and this, in turn, brings into play more dynamical variables. Thus, it is not surprising that, in principle, more modes may well appear in the spectrum_ In order to obtain a quantitative understanding of the problem under study, first notice that the order of eq. (22) is greatly reduced if o can be neglected with respect to A. As has already been discussed in section 2, this is a good approximation, which leads to

3v -_-__ v;+u2

3(w tw.r)

I

u)2

3(A--a)

-_- u & + a2 ImK,d(w)=sL~

27

I (21)

The analysis of the spectrum Z+(w) in terms of modes can now be performed in the same fashion as in section 2. Thus, eq. (2) is solved once again, including in addition to (20), the contribution (21) to the imaginary part of the kernel. Then, in the regions of the spectrum where there are well-pronounced peaks, the relaxation rates of the modes can be identified by taking the low frequency limit of the real parts (18) and (19) of the kernel. After introducing v to denote the dynamical shift, the counterpart of eq. (3) now reads

u~+~~04~ff~u3~~2v.2~(y~v+~yo=~,

(24)

where the coefficients are given by the following expressions:

%?r + ai +4n’

u& + A2

,,++A2

v;+4A2 I

A

’

(25) (26) (27) (28)

cl-0 = VsrVdL%4

Unfortunately,

-

(29) the resulting eq. (24) is still of high

28

F.R. McCcurt,

G. Tenti/Low

density spectrum of transverse NMR in molecular-gases

order and hence exact algebraic solutions car&t be obtained. A detailed numerical study of this equation is given in the next section. Here only an order-of-magnitude estimate of the values of u in the two extremes of very high and very low densities will be presented. Under the assumption that the density is so high that both co&ion frequencies vsr and vd are Of order A or higher, an asymptotic analysis of eq. (24) shows that

This result is the counterpart of eq. (IO). Thus, as before, the spectrum in the BIoch region consists of one well-defmed peak, whose haIf-width at haIf_heightis the ordinary TF’ relaxation rate. At the opposite extreme of very low densities, there are three dynamical shifts, each of which tends to a constant value as the density goes to zero. The two largest shifts are found to be

which are, obviously,

the counterparts

of eqs. (12).

4. Transverse relaxation in H,. Numerical results In this section, the results of a numerical study of both the spectrum I+(w) and of the polynomial equation for u h presented for H2 gas at 77 K and at 296 K. In order to find the values of the dynamical shifts, it is convenient to utilize the fact that the roots of eq. (24) are at most of the order of [S2&+ 3G$] l12. This latter quantity involves the temperature and the coupling constants and is found to be of the order of IO6 for the cases of interest here: in fact, it has the values 09656 X IO6 rad s-l at 77 K and 1.055 X lo6 rad s-l at 296 K. Because of this, it is useful to change the scale of u by putting

o-=10-60,

(32)

after which, eq. (24) becomes (u-)5 +&(a*)4 t&(u*)s

t~2(u*)+31u*

‘&J = 0,

(33) where the coefficients pi are explicit functions of the densityAs explained earlier, when the density is above the

Fig. 1. Behaviour of I-2 as a function of number density P in the density region for which it is unambiguously defied @ 2 0.005 amagats): 296 K; - - - 77 K. critical value, there is only one shift and hence only the Bloch mode in the spectrum. In this case, the shift

does not appreciably change the relaxation time T2 which is then given by the traditional (Abragam) theory to an extremely good approximation. For completeness, fig. 1 shows the actual behaviour of T2 versus p (the density in amagat units) for 77 K and 296 K. Notice that T2 is essentiahy undefined below 5 X 1O-3 amagats. In the remainder of this section, the numerical analysis of eq. (33) is limited to the region of densities bounded above by p 5 10m2 amagats. Then the coefficients pi are given (as functions of p) by the following expressions: &=-2.737X1O-3,

/33=2.084X105p2-l.113,

p-J = -570.4p2, pl= [1.0769X 105p2-1.252]105p2, PO= -2.948 X lo7 p4 ,

(34)

where the values shown in table 1 have been used for the parameters occurring in the calculation. The critical density at which eq. (33) passes over from one real root to three real roots is 2.85 X 10e3 amagats at 77 K and 3.35 X lop3 amagats at 296 K. Fig. 2 shows the behaviour, as functions of p, of the two most important low density roots o+ and o_ of eq. (33) and of the root u. corresponding to the behaviour of the Block mode at low density. Notice, in particular, that the root u. (with scale on the right-hand ordinate axis) has been

F.R. McCourt, G. TentijLowdensityspectrumof transverseNMR in moleculargases Table 1 Parameters employed in the calculation of the lineshape functionl+(w) for molecular hydrogen lo* Ws (tad s-l)

IO+ Wd (rad s-l )

10-7 WI (rad 5-l)

10-VWJ (rad s-l )

o-7157

5.435

38.330

4.57

T 00

10-S Vsr (s-l amp-‘)

1o-’ Vd 10% n,, (s-’ amg-’ ) (rad s-’ )

lo-6 ad (rad s-’ 1

77 296

4.89 p 3.32 P

2.91 p 3.04 p

0.4438 0.4348

0.5843 0.739 1

Fig. 2. Variation of the ical roots of the characteristic quintic equation with density p at 296 K and 77 K in molecular hydrogen_Note the expanded scale of the root co*:- 296 K, ---77K.

plotted on an expanded scale. Starting from the lower density end, the shifts-u& are practically constant at the asy.mptotic values given by eq. (3 1) while 00 is essentially zero. Only when p reaches LOW3amagats do the roots become appreciably sensitive to the density and decrease asp is further @creased. This is indicative of the onset of the transition region where the modes

29

corresponding to u+ and o_ become progressively damped as p + pc Z=3.35 X 10S3 amagats at 296 K. As o, and o_ decrease upon p approaching pc from below and vanish for p above pC, o. grows progressively more negative asp approaches p; (u. = r~_), undergoes a discontinuity at pz (u. = u+), finally decreasing progressively for p above pc_ By the time p has reached 4 X 10m3 amagats only one root, uo, is present and it is very small, indicating an entry into the Bloch region. It should be noticed that even though the roots of eq. (33) can be followed through so that the behaviour of each mode can be followed in detail, this is only meant as an intuitive representation of what happens in the transition region. The only quantity which is physically meaningful in this region is the spectrum, and the spectrum does not have well-pronounced peaks, so that no approximations of it (or parts of it) by lorentzians should be attempted. A full numerical computation of I+(o) has been performed for densities below 10m2 amagats both at 296 K and at 77 K using the parameters appearing in table 1. Only the results for H2 at 296 K are displayed in the figures. Figs. 3 and 4 show the behaviour of Re KL(w) and w + wI - Im KL(~) over the density range 10-2-10-3 amagats and 10-3-10-4 amagats, respectively, while fig. 5 shows the behaviour of I+(w) over the density range low4 amagats

up in the wings as the density is lowered can easily be understood (see, in particular, fig. 4). On the scales of figs. 3 and 4 the variation of the central zero with den-

30

F.R. McCourt.

G. TentijLow

density spectrum of transverse NMR in moleculargases

_/------___

_.I=__I--

---..

;p+‘-, r’s--

‘-.

‘N

%-<. ‘k.. A_______-____________.‘?:-.I. __.<,.”_ _____------,, -----LA____-.. .__ =.-ix ._A:. ,_l_L_*_l_._._*_.-l -1-1-.-..c~..“-.-...-.-.-.-.-.-.-* -.-.-,-.-. _‘x,S.S-_~~__x_ IzI____“‘“‘=_ ..-. -.-‘--,-‘-.--‘-‘--‘--‘-~--‘-~~~~-s~.-~ -.__*__,_ _.b~ --%___ ‘, ~_I~_ _/-_,+x=‘-’ ______I__-----__c

L L

‘.

-._J-.,

/*

_

.__..-m ,_,-:,z___.___---~ ,,_y&5~.r-*_ ,.-:“. --a _.A-+

\ =\ i

*_..--a--

-10

i

._. ‘T+ __/-/_’ _.’

-

-._______-.--

.-’ ;\ ..’ =\ / ,..’ ‘\ -.\ .* .C._______

.-’

._

Fig. 3. Behaviour of ReKl(w) and of the resonant functionw + WI - ImKl(w) at 296 K as functions of the radial frequency w fordensitiesintheregionl X 10m3 amagats~p C 8 X low3 at-nag&:--r--SX 10M3;-*-6 x 10s3;-..-4x 10s3;.--.-2 x 10-S; -..1 x 10-3. 1 1

n

106ReK,~w~:106Lw.w,-ImK,Ol

14 12 1 t 6

;i : ; ‘.i fi 15, i :_ ii :; :_ . I ,.q \_: ..

Y

296K

I

:E .:* r%

-P 4

__--y ___-zc.= -----~_____ _^._-__-..-y

2

J

xl00

-6

-$

lo-‘w

(rcld-s-‘)

Fig. 4. Behaviour of ReKl(w) arzdof w + WI - ImKl(w) at 296 K as functions of radial frequency w fo! densities-in the range 2 x lO* amagats G p G 8 X IO4 amagats: 8 X lo-; --6 X lo*; ---- 4 X 104; ..- 2 X lo*. Notice that the scales in the -_ wings are magnified by a factor of hundred so as to show the outer crossings.

F.R McCourt. G. TentilLow dens@

spectrum of ttansvetse NMR in moleculargases

31

-16

16-

13-

:i @. ;i il ii I; 1 ,I

12-

f'

-12

ll-

.i: : i ::: ; : ::

-11

15lb-

lo9c t 3 2

; i

-15 -14 -1s

_

10

-9

81fi5432-

10-7w

(rad-s-')

Fig. 5. The spectral lineshape function I+(w) at 296 K as a function of radial frequency w for densities in the transition region 2 X lo* amagats G p d 8 X 10q3 amagats. Legend same as in tigs. 3 and 4.

sity cannot be discerned and all the curves appear to possess a common crossing point. These results indicate that, should it be possible to extend the T2-type pulse measurements to densities of the order of 10-S amagats, there should occur a fairly sharp change from simple exponential decay of the magnetization in the time domain to a marked nonexponential decay over a change in density of, say, a factor of five from about 4 X 10B3 amagats to 8 X 1OA amagats.

5. Summary and discussion One of the basic results of this paper has been an analysis of the density dependence of the spectral lineshape function I+(o) associated with the behaviour of the transverse components of the nuclear magnetization in a polyatomic gas with special emphasis being placed on its behaviour at very low densities. It has been shown that as a critical value of the number den-

sity, pc, is approached from above, the lineshape function progressively broadens from a single, sharp (lorentzian) line and then splits at pc into a two-peaked function. With further decrease in density the two peaks shift apart, reaching their asymptotic splitting within about one decade, sharpening as they shift outwards from o = --wI; with still further decrease in density, the peaks simply continue to sharpen. In the region of PC, the root au characterizing the Bioch mode changes rapidly and at p = pc suffers a discontinuity. One consequence of this is that the resonant frequency of the Bloch mode becomes significantly different (by about 10%) from -wr as p approaches pc. Unfortunately, even this shift will likely be difficult to detect experimentally because the width of the line associated with the Bloch mode becomes of the same order of magnitude as the shift, due to the rapid sharpening up of ReKL(w) at w = -or with decreasing density (see fig. 4) Detection of the effect of the emergence of thetwo non-Bloch modes offers greater promise since in a pulse experiment, the ap-

32

F-R. McCourt, G. TentilLow density spectncm of transverseNMR in molecularga&

pear&e of these modes for p < pc will result in a non-exponential decay of the transverse components of the nuclear magnetization in time domain. The onSet of this non-exponential decay should be quite rapid, i.e., it tiould occur over a density change of around a factor of five. Moreover, since at densities of the order of lo4 amagats the two peaks are separated by about 2 M rad s-1, there should be in principle a possibility of observing the decay associated with the two nonBloch modes separately by s&ding in pulses at the shifted frequencies, i.e., by means of an “off-resonance” experiment (in the sense that the pulse is centered at a frequency different from the nuclear Lannor frequency). Aqualitative picture of why the spectrum splits into two peaks at low densities may be constructed by the following considerations. For p of the order of 10m4 amagats, the time between two successive reorientation collisions is about 5 X 10m5 s while the time scale associated with the intramolecular interactions Tiatra is about 5 X lop6 s. If there is only aHo field, then the Lannor frequency of the nuclei in H2 will be wI- At these low densities, however, J on the _ average remains fmed in orientation (but precessing around Ho) for times long enough (= 10 qntra) to establish a local field& in competition with I$,. On the average, there will be two predominating orientations, parallel and antiparallel to Ho giving rise to two resultant fields H = Ho + HL that can be seen by the nuclear spins on the H2 molecules, one for those molecules with J precessing parallel to HO and the other for those molecules with J precessing antiparallel to Ho

and there will thus be two I&nor frequencies for H2 molecules at sufficiently low densities and the NMR spectrum will therefore split into two peaks. The effects discussed here are purely of a bulk-gas nature associated with the solution of an initial value problem tid, as such, no consideration has been given to the influence of wall-relaxation processes. Of course, wall relaxation processes will have to be considered whenever the density is so low that the average time between the successive reorientation collisions is of the order of the time required for? free molecule to traverse the shortest distance’across the NMR cell. This would occur for a cell of diameter 1 cm in the neighbourhood of 5 X 10m4 amagats, so that it is imperative that a sufficiently large diameter cell be used for the observation of these effects. The major experi-

mental problem is likely to reniain one of sufficient signal intensity.

Acknowledgement One of the authors (F.R.M.) is grateful for the kind hospitality extended to him by the FOM Instituut voor Atoom- en Molecuulfysica in Amsterdam and especially by Professor J. Kistemaker, Professor J. Los and Dr. A. Tip during the sabbatical year spent in their institute. We are grateful also to Prqfessor R.L. Armstrong and Mr. W. Kalechstein for keeping us informed about their Tz measurements in H, and for their interest in our work. To Mr. M.A.F. Schell we express our thanks forhis interest and participation in part of the earlier aspects of this project. Financial support for this work h$ come from the National Research Council of Canada and from the ZWO of the Netherlands through the FOM. Appendix. Connection between the spectrum and the complex response function Consider the response function G,(t) function field

to a step-

H(t)=HeEr6(t),

(A-1)

where e(t) is the Heaviside function which equals unity for t < 0 and zero for t > 0. The transverse magnetization AK-Z+(~) can be written in terms of $+(t) as Am+(t) = i _i di xy+(f - f’) iY(t’) ,

(A-2)

and, if the integral representation of the response function dw xT,(t - t’) = _S, 2n (t-f) $+(GJ) , (A-3)

+m e-iw

is employed, then for t > 0, Am,(t) becomes Am,(t) iw(t-t’) xI+(W) E-+0_=

=H

-00

dw $ix= x esiwr x~+(w)/w . -D3

1

He”’

(A.41

E.R.

G. Tend/Low dens&v spectrum of transverse NMR in moleculargases

Mccom,

Now, taking the Laplace-transform

Am+(z=iw+c)=HJ

of eq. (A-4) gives

&(o)/w

dtekf n

* =HS

--03

!Iriw’[(w’-w)-L]

’

(A.3

and, since x1+(0’) is a smooth function of u’, this equation can be solved for x” with the help of the

identity

1

h

q-a-1

E+o w’-w-k

w-w

+ lri 6(w - 0’) , (A-6)

to give Am+(z = w + iOf)/H

(A-7) -m

is given in terms of &n+(z)

= 2 Re [Am+@ = w + iO+)/H] ,

as

(A-8)

and this is directly proportional to the spectrum since it represents physically the energy dissipation per unit frequency [4, p. 465; 5, p. 441.

x:+(~‘)

dw’

from which &(w)

33

References [1] G. Tentiand

F-R. McCourt, J. Chem. Phys. 65 (1976) 623.

[2] W. Kalechstein and R.L. Armstrong, Chem.Phys. 28 (1978) 125. [3] F.R. McCourt, TX. Raidy, T. Rude&y and A.C. Levi, Can. I. Phys. 53 (1975) 2463. [4] L.P. Kadanoff and PC Martin, Ann. Phys. (NY) 24 (1963) 419. [5] PC. Martin, in: Les Houches lectures, eds. Dewitt and Balian (Gordon and Breach, 1967) p. 39. [6] M. Abramowitz and A.I. Stegun, eds., Handbook of mathematical functions (Dover Publications, New York, 1966).