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Low temperature orbital dynamics and NMR
Physica 90B (1977) 11-19 © North-Holland Publishing Company
LOW TEMPERATURE ORBITAL DYNAMICS AND NMR R. COMBESCOT Groupe de Physique des Solides de l'Ecole Norrnale Sup~rieure, 24 rue Lhomond, 75231 Paris Cedex 05, France; and Bell Laboratories ,, Murray Hill, N.J. 07974, USA
Orbital dynamics for zero wavevectorq = 0 is studied ,nt low temperature. The consequences for low temperature NMR are investigated.
1. Introduction
enough since we still have to face Landau damping. This can be avoided by working at very long wavelengths (wavevector q = 0). But this is not yet sufficient to allow a study of the reactive forces acting on ~. The reason is that, as indicated by Leggett and Takagi [ 1], if the quasiparticles do not relax at all, they still produce a restoring force on ~ because moving ~ increases their energy. Since a quasiparticle energy is of order k T at low temperature (the gap is zero for kll ~), ~ will move only if the input energy 60 is at least of order of kT. Otherwise, if 6o < < kT, ~ will be essentially locked by the quasiparticles. In other words, only the condition 60 > > k T (which ensures ~utomatically 60r > > 1) allows elimination of all the effects due to quasiparticles. However, as will be seen, pair-breaking damping also becomes important in this regime, because as noted in ref. 2, pair-breaking occurs whenever 60 > > (q. ~)VF which is always satisfied for q = 0. Nevertheless, this is the only regime, at q = 0, where orbital dynamics exhibits anything other than straight collisional damping or locking. Actually, the best way to test reactive orbital dynamics seems to be the study of orbital waves in the regime 60 > > kT, where the dispersion relation is 60 " q z VF as considered in ref. 2 In this paper we will study orbital dynamics at q = 0 and its consequence on the NMR properties. The case q = 0 is interesting because it exhibits most of the features of orbital dynamics, and also because experiments with (quasi) uniform systems are rather easy to perform; for just the same reason, NMR experiments are very conve)aient for spin dynamics. We will consider only the collisionless regime since the hydrodynamic regime
There is presently much interest [1,2] in the low temperature dynamics of the anisotropy axis of 3He A, the so-called orbital dynamics. The reason for the interest in the low temperature behaviour is that, at "ordinary" temperature, the motion of the anisotropy axis is dominated by relaxation processes [3,4]: a change in the direction of ~ produces a change in the energy of a quasiparticle. The quasiparticles then tend to relax to a new equilibrium position corresponding to this energy change. Therefore, in the motion, ~ drags along the quasiparticles (the "normal fluid") which gives rise to a viscous force acting on ~. In essentially all the domains presently accessible to experiment, orbital dynamics is dominated by this force. However, one would like to be able to eliminate this damping force in order to study the reactive forces which act on £. Such reactive forces are precisely the ones which make spin dynamics of superfluid 3He so rich and so interesting. To avoid the damping, one has to eliminate the quasiparticles. This is done by going to a low enough temperature [5]. A first result obtained by lowering the temperature is to remove one important relaxation mechanism, namely quasiparticle collisions. For T/Tc ~ 0.1, a typical collision time [6] r should be of order 10 -3 s, which means that any experiment at frequencies 60 higher than 1 kHz will be in the collisionless regime. However, the condition 60r > > 1 is not
Summer visitor at Bell Laboratoriesl 11
12
R. Combescot / Low temperature orbital dynamics and NMR
has already been studied in ref. 3. We will restrict ourselves to the usual domain co < < max IAk(T)I because, as is well known, one has to consider degrees of freedom other than the ~ motion when this condition is not satisfied, and then the physical picture becomes much more complicated. Although the general equations will be derived without this restriction, we will only study the case kT < < max IAk(T)I which implies at least moderately low temperature. The reason is that, for kT ~ max Iak(T)l, ~ is essentially locked, except if co ~ max lak(T)l which is beyond our scope. And anyway, the lower the temperature, the easier it is to see the unlocking of ~. On the other hand, we do not put any restriction on the ratio ~o/kT. Actually, one will have co < < k T or co ~ kT in most experimental situations and especially when one uses NMR to excite the motion of ~. Because NMR is probably the easiest way to study orbital dynamics at q = O, we will consider in detail what modifications appear at low temperature to NMR properties.
In eq. (1) ~k is the kinetic energy measured with respect to the Fermi level and, in eq. (2), Vke' is the pairing interaction which is taken as Vee, = V k. k' for p-wave pairing. In eq. (1)we have already performed the Hartree-Fock factorization of the interaction and kept only the anomalous averages. In the A phase, the order parameter is believed to be
2. Orbital dynamics at q = 0 and low temperature
nk=(n~
In order to study orbital dynamics we use kinetic theory. This theory can actually be cast, for orbital dynamics, in a form which allows us to treat in the same framework the hydrodynamic and the collisionless regime, as has already been done for spin dynamics [7,8]. This will be published elsewhere. However, in the present case where we are only interested in the collisionless regime, we do not need such a form and we will use straight kinetic theory. Naturally, we will use the BCS model to treat the superfluidity of 3He, as was done in most of the theoretical work to date. The justification is, as usual, that although strong coupling effects are known to bring corrections, these are not expected to be qualitative and may hopefully be not too important quantitatively. For p-wave pairing, the BCS hamiltonian is [9]
At equilibrium
a , ~ ( k ) = [Oy o. d]~0 [it- (A' + i A")],
(3)
where IA'l = [K'l = 1 and A' • A " = 0. By definition = A' × A". In orbital dynamics we are only concerned by the motion of ~., and d can be taken as constant (the motion of d comes in spin dynamics). If d is chosen along the y axis aat3(k ) : d[k. (A' + iA")lSao.
(4)
Therefore the two spin populations are completely decoupled and we can omit the spin indices. Let us define the matrix quasiparticle distribution
(5)
=
jf=~ k, ol
+
~kCk, o~Ck,o: +
~
+ + .[ck,,~c_k,#A:~a(k) + h.c.}
$
k
(1)
where aa~(k) = k~, Vkk' (C_k,c~Ck,~)
(2)
is the order parameter; in eqs. (1) and (2), ck, a and + Ck,c, are respectively the destruction and creation operators of a helium-3 atom in plane wave k with spin a.
ntJ
((C_kC k)
+¢:o/G'
(C_kC+_k>/
co= ~k ak
at --~d '
~k = --~-tanh(-~ ~Ek).
(6)
The equation of motion for n k is easily obtained from the Heisenberg equation of motion for operators. From eq. (1)we have
(7)
i - ~ n k = [nk, ek] ,
where e k is given by eq. (6), but with A k taking its instaneous value Ak(t). Eq. (7) is the kinetic equation. From eq. (2) the order parameter is self-consistently given by
Ak = ~k' Vkk'n;' "
(8)
We will actually consider only the linear orbital dynamics: the order parameter is assumed to have only small deviations from its equilibrium position. Let us take ~ parallel to the z axis at equilibrium, with A' along the x axis and A" along the y axis. Then the order parameter will be of the form Ak = Aok + 6Ak ,
Aok = d(kx + i k y ) ,
R. Combescot / Low temperature orbital dynamics and NMR
5Ak : - d kz(8~ x + i 6~y),
(9)
where 8~ is the deviation of £ from equilibrium. In the same way nk = n o + 6nk, and after linearizing and Fourier transforming, eq. (7) becomes
13
to be not modified by the dipole interaction. Now the calculation goes by solving eq. (10) for 8n~ and substituting in eq. (1 I), which gives the equation of motion for 8~x and 6~y. A problem arises because eq. (11) gives in general six independent equations: if we write 5Ak = k • 84, eq. (11) becomes
coSn k = [ 6 n k , 601 + [n o , 8 e k ] ,
10
8/' 1
(10)
8 k=\SAk 0 "' while eq. (8) goes into
8 A k = ~k Vkk'bn-~',
8A~ = k~Vkk, Sn~'. (11)
Up to now we have not considered the dipole interaction. However, this force is all important since it is the only one which couples the orbital dynamics to the NMR. The dipole interaction can actually be handled by fairly simple considerations. But here we will use a slightly more complicated way to show how it can be systematically included in the framework of kinetic theory. We treat the dipole interaction in the spirit of most of the theoretical work in spin dynamics: we do not consider its effect on the equilibrium order parameter and take it into account only as a small restoring force when the order parameter is away from equilibrium. In other words, we do not include the dipole term in e °, but only in 6e k. This is the way which has been used in ref. 3 to calculate the viscous term in the hydrodynamic regime. If d is taken along the y axis, the dipole interaction gives rise to an additional term in Jf which can be written
84 = V ~
kSn~,
~{'D - ~ 1~
+ + , g*k , [(ckc-,) c-,'ck' + h.c.],
( - d ) (8~x + iS~y) = V ~ kzSn-~, k
(15)
( - d ) (8~ x - i 8 ~ y ) = V ~ ]¢zSn~¢. k
In the case q 4= 0, as explained in ref. 3, only three parameters, 8 ~x, 6 ~y and the phase of the order parameter, are hydrodynamic: in the regime co < < d, the three other parameters are very small in comparison with these. Three of the equations (14) allow the calculation of those non-hydrodynamic parameters (and check that they are indeed small). The three other equations give the equations of motion for the hydrodynamic parameters. One of these turns out to be the mass conservation.law, the two others are eq. (15). As we said, in the case q = 0, the phase decouples from 8~x and 8~y so that we are left only with eq. (15). Solving eq. (10) gives
4AE~¢- 2wB~g
(~k
4E~ - 602
X Ek--' (16)
AkSn~+ A~Sn~
where again the Hartree-Fock factorization has been performed, and only anomalous averages have been retained. ~fo is actually of the same form as the pairing interaction, the only difference being the replacement of Vkk' by gkk" Therefore, the only modification brought by the dipole interaction in the kinetic equation (10) is, in the definition of 8ek, the replacement of6A k by 6Ak + 6Dk where (I 3)
k'
On the other hand, eq. (11) is unchanged because it corresponds to the definition of Ak which we assume
(14)
which gives indeed six equations. This is quite natural since the complex vector 8 A corresponds to six independent parameters. Actually, in the case q = 0, the equations for 5~x and 8~y decouple from the equations for the other parameters, and the only equations one has to retain from eq. (14) are
(12)
p
6Dk = ~ g k k ' S n k ' •
]c 5n~:,
k
Akfn~ - A~Sn~-=
as
8A* = V ~
k
4B~I~- 2coA~k ¢k = 4E 2 _ 6°2 X Ek--'
where A = Ak(SA~ + 5D~) - A~(SAk + 5Dk), (17) B = Ak(SA~ + 8D~) + A~(6Ak + 8Ok). When, in eq. (13), we replace 8n~, by its value taken from eq. (16), we can make co = 0 because keeping w 4= 0 would only produce corrections of order cold at least [actually they would be of order (w/d) (T¢/EF) and (w/d) 2 log (co/d)]. Such terms would correspond to finite frequency corrections to the dipole energy
R. Combescot / Low temperature orbital dynamics and NMR
14
term. Moreover, gkk' provides no cut-off [9] in eq. (13) for the integration over k', so that the main contribution comes from regions where ~k ~ Ek and therefore we have
so that expression (21) is merely -(8~x +-i6~y)VE'd'/d. In deriving eq. (24) we have only retained the dominant terms in the region ~k > > Ak as we did before. The equations of motion for ~ are finally [10]:
/
8Dk=~k, gkk'~,&Xk', (18) × (5~x -+iS~y) = 0 ,
6D~ = ~g, gkk' Ok_£'5A~' Ek,
where we have made SAg + 5Dg "" 8Ak since we are working to lowest order in dipole energy. To obtain the equations of motion for 8~, we substitute 8n~ from eq. (16)into eq. (15). We simplify the result by using the gap equation, eq. (8), when A a and n~- take their static equilibrium value:
&= v ~k' k.k'0k' ~k' Ak'"
(19)
By taking the derivative of eq. (19)with respect to small deviations of ~ from its equilibrium value, we obtain (-d) (8~x + iS~y) =
v 52 k
['¢k
LEk
+ aok, - -
a
t%l
2Ek aEa~-k)J
where ¢' = 80/aE. The term (co~/2E 2) is actually an approximation for 2co~/(4E 2 - co2): this }erm gives a zero contribution except if we take into account particle-hole asymmetry, and in this case the main contribution comes from regions where ~k > > d, co which justifies the approximation (the imaginary is always negligible compared to the first term imaginary part). Therefore we find from eq. (25) two eigenmodes, non-degenerate, circularly polarized, whose frequencies are obtained by equating to zero the brackets in eq. (25). The various terms entering eq. (25) are explicitly evaluated in the appendix with the further approximation kT<
(kT)2 - -g - [log---d * ~ * g 8a
.
(20)
Finally the part coming from 6Dk and 6D~ can again be simplified by taking co = 0 and ~k ~- Ek for exactly the same reasons as before. The contribution of these terms to the right-hand side of eq. (15) is therefore
^,Ok (-d) (8~x +-iSQy) V ~ gkk'kzkz - - ~Ok' . k,k' Ek Ek' -
(25)
(21)
+ ~ irr tanh (4_~)] ~ codZN'0 coc /~a) 6No t o g - ~ + Noo ]
X (5~x +-i6~y) = 0 ,
(26)
where
f--
I(1 ;)
But, from eqs. (12) and (6), the dipole energy Ed is given by
1 g(x) = ~
Ed(~) = ~ ~ gkk'OkOk' ~ A~___:'.
The calculation ofg(x) is discussed in the appendix. Eq. (26) displays all the characteristic frequencies of orbital dynamics. Let us first consider the case T = 0. Eq. (26) becomes
k, k'
(22)
lZk Ek'
This energy is minimal when ~ is parallel to the z axis, and for small deviations of ~ from this position we have Ed(0-) = E~d+ } E~ [(8 ~x)2 + (8 ~y)2] .
(23)
From eq. (22) we have "' Ok Ok' E d" = ~ggk'd 5^kzkz k, k' Ek Ek'
(24)
dp log cosh 2 P
o
---~-T
•
(27)
co2 2d ,qi g- ogG÷h÷ °logcocd
c~d /V',, 6No
tt
Ea No
-0.
(28)
If we assume particle-hole symmetry (N~ = 0), this
R. Combescot / Low temperature orbital dynamics and NMR 3. E f f e c t o f orbital dynamics on the N M R
reduces to ~2~ ~_ n]_E J - ~ og +~+½i -'N00'
(29)
which shows a characteristic frequency of order (E~/No) 1/2. This is of the same order of magnitude as the longitudinal NMR frequency ~o, which is given with these notations by ~o~ = 4E~'(1 + bo)/No. Actually the motion described by eq. (29) is analogous to longitudinal NMR: under the action of the dipole force oscillates around its equilibrium position, the modes being linearly polarized and degenerate. However, there is one main difference: this mode is rather heavily damped due to pair-breaking effects. At the resonance frequency the imaginary part of the frequency is of the order of one-tenth of the real part. On the other hand, the second term in eq. (28) which is due to particle-hole asymmetry gives rise to a precession of ~ around its equilibrium position. Such a motion has already been considered by Cross and Anderson [4] in the hydrqdynamic regime. Without the dipole force, one would obtain a precession frequency of order d2/EF which turns out to be also of order of 60o. Therefore no term in eq. (28) is negligible and the actual modes are circularly polarized, non-degenerate, with frequencies of order of 60o and an important damping. At finite temperature the characteristic frequency kT/~ appears. For k T ~ hWo, nothing is changed qualitatively with respect to the case T = 0. Naturally k T ~ h6oo is presently far beyond experimental reach, since it corresponds to T/Tc ~ 5 X 10 -3. In the situations experimentally accessible, h6oo < < k T and we can neglect the dipole interaction and the precession term, which corresponds to keeping only the first two terms in eq. (26):
Up to now we have only considered the motion of with fixed d, which was the convenient way to study orbital dynamics. However, in the real motion, ~ and d are both moving because they are coupled by the dipole interaction. Since the motion o f d is coupled to the NMR, orbital dynamics modify the standard resuits of NMR in 3He A since it is usually assumed that is fixed. We want to study these modifications, essentially at low temperature. Let us assume, as we did before, that the equilibrium position o f d is along the y axis, with a magnetic field H along the z axis. Under a small rotation 0, d will have a small deviation from the y axis and the components dx and dz will be non-zero. They are related to 0 by Ox = ~lz and Oz = -~tx. The first equation for the NMR is the conservation law for the total spin S of the system [8,9] S i --~ = ([S, ~1) = iS × 7 / / + ([S, ~fo]),
if2
9
(kr) ~ .
(30)
We see that the situation is rather analogous to the case T = 0 in eq. (29), but with the dipole force replaced by a force due to quasiparticles as indicated by Leggett and Takagi [1]. Again, as for T = O, the motion is strongly damped by pair-breaking effects, and there is no sharp resonance but rather a broad structure in the spectrum of the response of ~ to an excitation.
(31)
where 3' is the gyromagnetic ratio for the 3He nucleus. Since S is the generator of the spin rotations in the system, the last term in eq. (31) is directly related to the change of the dipole energy Ea under spin rotation [9]: ([S, ~CDI>= - i ~ d / ~ 0 •
(32)
At equilibrium, ~ is along the y axis and if we assume that it does not move, we have for small deviations of d 1 L"t r/',?,/2 Ed = E~d + ~-~d~.Ux + 32).
(33)
But if £ is moving, we have ed = ~ = E0
-6°2~ ( o g -2d+ ~ rI~ + g (~-~) + 7~ in tanh(4-~) ]
-
15
1
tt
^
+ ~ ~a [(dx - ~ ~x)5 + (~tz - ~ ~ ) ~ ] I ,, +]Ea[(Oz+6~x)2 +(Ox-6~z)2].
(34)
The second equation of the NMR says [8] that, in a reference frame moving with d (i.e. deduced from the fixed frame by the rotation 0), there is a "Josephson field" proportional to the instantaneous velocity of the moving frame. In the linear regime, this field is (1/7)dO/dt. The magnetization adjusts to the total field according to the dynamical susceptibility ~'(t): t
7Sm(t ) = f
d t ' ~ ( t - t')[Itm(t' ) + (1/7)dO/dt') . (35)
_ o o
Here Sm and Hm are the spin and the field in the mov-
16
R. Combescot / Low temperature orbital dynamics and NMR
ing frame. In the linear regime they are related to their values S and H in the fixed frame by Sm=S+SXO
,
Hm=H+HXO.
(36)
In an NMR experiment, H is the sum of the static field H o and the r.f. field H', and $ is the sum of the static spin S O = ×0//0/7 and the oscillating spin S'. After linearizing and Fourier transforming, we obtain from eq. (35): 7S' + XoHo ×0 = ~(co)[H' + H o ×0 -(ico/')')O] .(37) Finally, we have to modify slightly eq. (25). First, the equilibrium position of ~ is along the x axis and not along the z axis as before. Then, in the dipole term, it is naturally the departure of ~ from d which comes in, and not the departure from the equilibrium position. Therefore we have Y,±(8~ + iS~x) + E~[(5~z - dz) +-i(5~x -/tx)] = 0 , (38) where ~_+ represents the sum over k in eq. (25). To~ gether with eqs. (32) and (34), eqs. (31), (37) and (38) allow one to solve for the NMR. If we solve eq. (38) for 5~z +-iS~x, we obtain (5~z + iS~x) -
E'd (dz+idx)=R+(co)(~lz+i~lx). E~' + ~_ -
(39) Eq. (39) is interesting in itself because it shows the response of fi to an excitation through d. Except when co, k T ~ coo, this response is always very small, essentially because ~ is excited through the weak dipole force. In any case, Im R(co) never shows a sharp resonance because of the importance of pair-breaking damping, as discussed before. In the practical situations where k T > > coo, Z. can be approximated as in eq. (30) Noo
9 (kT)2 - --6
+ ~- in tanh (4--~)
og
+ ~ +g (40)
In this regime, Im R(co) has a broad bump with a maximum for co ~ k T and a width of the same order. An interesting qualitative modification of the NMR by the motion ~ is that the longitudinal and the transverse resonances are coupled, due to the precession term in eq. (26). At T = 0, if the A phase could be
stabilized without magnetic field, this coupling would be very strong. In the strong magnetic field (~5 kG) which seems necessary to stabilize the A phase, the coupling would be only of order COo/COL where coL = 3'Ho is the Larmor frequency. In the longitudinal resonance, the coupling makes the magnetization precess around the field in addition to its usual oscillation along the field. In the transverse resonance, the magnetization has a small oscillation out of the X Y plane in addition to the usual precession in the X Y plane. However, at presently available temperatures, k T > > coo, the coupling is only of order (coo/kT)4(coo/coL), and seems very difficult to detect. Therefore we will neglect this effect and forget about the precession term. With this approximation R+ = R_ -- R and substitution of eq. (39) into eq. (31) shows immediately that the only effect of orbital dynamics on the NMR is the replacement of E~' by E~' [1 - R(co)]. Now we consider the practical consequence of this result on the NMR in the usual temperature range k T > > coo, where R(co) can be approximated as in eq. (40). The longitudinal NMR frequency is given by 602 = co~ [1 - R(coo)],
(41)
where co(~_~)2 I1 + 3i [coo~3] , (42) 16rr ~k-T/A which gives rise to a longitudinal resonance frequency
R(coo)
1
9
1 +----~o21r~
~
1
co~ = coo
9.__9__[~o, ~2-]
1 + F~ 47r2~kT} .]
and a full linewidth 1 27 ~coo~s Aw~- 1 + F~ 327r3 \k-T] w°"
(43)
(44)
These results must be compared to the shift and the width coming from spin relaxation effects [8], because all these effects add up. The shift is due to the difference between the susceptibility in the collisionless regime and in the hydrodynamic regime. From eq. (86) of ref. 8,the shift at low temperature is cool(T)/2(1 + F~) where f(T) = (rr2/9)(kT/d) 2 so that, taking into account eq. (43) ~(z3=
~o(Z3
9
+ 18(1 + F D \
[Wo]27
-4zr2(1 + F~)~-k-f] _]"
d !
(45)
R. Combescot / Low temperature orbital dynamics and NMR
From eq. ( 8 7 ) o f ref. 8, the linewidth due to relaxation is f(T)/r(1 + F~)where r is the relaxation time, so that at low temperature the linewidth will be
cot -~ (co2 + coo2)i/2(1
-ACOQ- 9(I + F~)) r \ d /
Acot ~ o ImR(coL). - - -COL
-~
32rr 3 (i + F~) ~ " ~ ]
e°°"
(46) In the shift, the orbital dynamics effect becomes dorainant when (81/2rr4)(cood/k2T2) 2 >~ 1. With the BCS value d ( T = O) "~ 2kTc, kTc/h "" 3.5 × 108 at melting pressure and [11] coo(T = 0) -~ 7 X l0 s, this gives 6 X 10-6(To/T) 4 >~ 1, which means that this effect will be observable only at very low temperature Te/T >~40. At this temperature the shift would be of order 1 kHz, which is rather large. We must note that coo(T) has also a temperature dependence which probably gives corrections of order (kT/d) 2 in the spin susceptibility term, but with the opposite sign, so that it would be impossible to see the difference between these two terms experimentally. But the orbital dynamics effect will lead to a decrease of the longitudinal frequency with a different power law and should therefore be easy to separate from the two other terms. In the same way, the part due to orbital dynamics will dominate inthe linewidth if (243/32rrs)(coo/kT)S(d/kT)2coor ~1. The relaxation time grows [6] like 1/T 4 at low temperature and a rough (over)-estimate is r(T) ~ r(Te) (Tc/T) 4 with r(Tc) ~ 5 × 10 -a. Therefore we obtain the condition 10-16(Tc/T)I l ~> 1, which again implies a very low temperature Tc/T >~ 30. But below such a temperature, the linewidth (due to orbital effects) is of order 4 × 10-1°(To/T) s Hz and Tc/T ~ 102 is necessary to reach a linewidth of order 1 Hz. But it is interesting to note that the linewidth grows like 1/Ts in this range, which is very fast. Turning to the transverse resonance, we have co2 = co2 + co2 [1 - R(co)] (47) • The transverse resonance is markedly different from the longitudinal only if COL> > COO,and anyway a field of order 5 kG seems necessary to stabilize the A phase at low temperature. Therefore we have a transverse resonance frequency
17
co~ Re [R(coL)])
(48)
- ~ and a linewidth (49)
From eq. (40), Re [R(coL)] will be of order [coo/max(kT, coL)]2 and the correction to the usual resonance frequency is of order 10 -8 and therefore impossible to observe experimentally. Since coL = k T i n a field of 5 kG gives a temperature T/Tc ~ 0.25, we have actually always k T ~ co L at low temperature, so we can only consider the case k T < < coL. In this case we obtain Acot = cooloao,.. ,3 3rr 1 \COL! 2(1 + F]) [log(2d/coL)] 2 '
(50)
which gives numerically ACOt ~ (10/H3G) Hz. This seems again far too small to be observable. It is interesting to compare the linewidth, eq. (50), with the linewidth coming from spin relaxation effects. We have to use eqs. (89) and (91) from ref. 8 in the collisionless regime COL'/"> > 1, which gives Acol ___coo(coo/a ( k T ) 2 7ra \COL! - --g- (1 + F~) 2 ,
(51)
or numerically, Acol ~ (0.2/Hac,)(T/Tc) 2 Hz. We see that the width is essentially due to the orbital effect, but unfortunately the high field necessary to stabilize the A phase makes the linewidth too small. In conclusion, the orbital effects on the NMR are very small in the range of temperature currently obtained experimentally• The main reason is the smallness of the dipole force: all the effects are obtained through the dipole term in the NMR and moreover the coupiing to f~is through the dipole force. In other words, ~ is essentially locked, and this is the small departure from locking that one is trying to observe. The effects can be increased rather drastically in the longitudinal resonance by going low enough in temperature, but the required temperature does not seem presently within reach•
R. Combescot / Low temperature orbitaldynamics and NMR
18 Appendix
With N O being the density of states of the Fermi surface for both spin populations, we have -
~ k z ~ [A[2 O' - No +o. //2 d 2 sin 2 0 1 k ~-2 -8---~ f d~ dO sin 0 cos 2 0 E2 cosh 2 (½fiE)' --oo
(A.1)
0
where E 2 = ~2 + d 2 sin 2 0. At low temperature kT < < d, the main contribution comes only from 0 < < ~-n because of the l/cosh 2 (~/~E), so that the integral over 0 can be extended to +oo and we can take sin 0 ~ 0. We make the change of variables 0 = (2kT/d) p sin c~, ~ = 2kTp cos a, and obtain -
k
=
~'2
o;
d
0
"
cosh 2 p
0
-No
9-
- -
.
(A.2)
The imaginary part of 2E2-IAI2
J= ~ k2z ' ~ 2 - - - ~ k
¢
(A.3)
2E 3
is easily obtained as ImJ = -
16
dO sin 0 cos 2 0 0
d~ 2E2 E4 - IAI2 tanh
E
5(6o - 2 E ) .
(A.4)
--oo
1 With the approximation co < < d where the main contribution comes from 0 <:< ~Tr, we obtain
N°Tr2 tanh (4-~T) Im J = - 12d
(A.5)
The real part of J is first calculated at T = 0 2~2 + d2 sin2 0 1 Re JT=O = X 4 N o / " +~ d~ fa / 2 dO sin 0 cos 2 0 J 4~ 2 + 4d 2 sin 2 0 - co2 (~2 + d 2 sin 2 0)3]2 . --oo
(A.6)
0
With the change of variables ~ = do cos a, sin 0 = 19 sin a, this becomes
No//2
~ / s i n ¢~
0
P
d P 4 p 2 _co2/d 2 [(x/1 - p2 sin2 c~- 1)+ 1].
da sin a (1 + cos2 a)
Re JT=o = - ~-~
(A.7)
0
In the/term with x/1 - p2 sin 2 a - l, we can take o3 = 0 if we neglect contributions which go to zero when
cold --->O. The integral is then easily performed to give (No/6d2)(1 - log 2). The other term in eq. (A.7) is also easy to calculate in the same approximation and the final result is
=No
Re JT=o - ~
Then Re J = Re (J - JT=o) + Re JT=o and we are left with the calculation of 2E 2 - I A I 2 1 tanh(-~fE)) S e ( J - Jr=o) = ~ k2z - ~ -_ - ~ - ~ (1 -
(A.9)
k
Again the main contribution at low temperature comes from the regions near the nodes of the gap (0 < < ~r) and with the change 0 = (2kT/d)p sin a, ~ = 2kTp cos a, we obtain N0 R e ( J - JT=O) = ~
dp log O2 cosh2----~
I
co 2 - l ° g 4 - - ~
(A.10)
0
By expanding the log in powers of 4kTp/¢o, Re(J - Jr=o) can easily be put in the form of a series expansion in
R. Combescot / Low temperature orbital dynamics and NMR
19
powers o f (kT/~o) which is particularly convenient for 60 >~ kT. The first t e r m gives R e ( J - J r = o ) ~ - ( N o / 1 2 d 2) (rr2/12)(4kT/w) 2 for w > > kT. In the case co ~ k T i t is b e t t e r to calculate the integral in eq. (A.10) b y residue which gives ~rt - C , cosh 2 p = an [Otn + (6o/4kT) 2] ~ log l o where C is the Euler constant and ctn = (-~70 (2n + 1). Finally
~kz k
(A.11)
~¢) N'° log d C 2E 3 ~ - ~ '
where N~ is the derivative o f the density o f states at the F e r m i surface and ~ c is the BCS cut-off.
References
[1] A.J, Leggett and S. Takagi, Phys. Rev. Letters 36 (1976) 1379. [2] M. Combeseot and R. Combescot, Phys. Rev. Letters 37 (1976) 390. [3] R. Combescot, Phys. Rev. Letters 35 (1975) 1646. [4] M.C. Cross and P.W. Anderson, Proc. 14th Conf. on Low Temp. Physics, M. Krusius and M. Vuorio, eds. (NorthHolland, Amsterdam, 1975): M.C. Cross, to be published. [5] The A phase does not exist actually at low temperature in zero magnetic field. However, it is presently believed that a field of order of 5 kG is enough to stabilize the A phase, with respect to the B phase at T = 0. [6] R. Combescot, Phys. Rev. Letters 35 (1975) 471. [7] R. Combescot, Phys. Rev. A 10 (1974) 1700. [8] R. Combescot, Phys. Rev. B 13 (1976) 126.
[9] For a review paper on the theory of superfluid 3He, see A.J. Leggett, Rev. Mod. Phys. 47 (1975) 331. [10] Naturally some of the terms appearing in eq. (25) have been found by various authors working on collisionloss kinetic theory, starting probably with P. W~lfle, Phys. Rev. Letters 30 (1973) 1169, although they are not mentioned explicitly in his paper. Referring to the author's work, the third term of eq. (25) is the limit, for q = 0, of the left-hand side of eq. (7) in ref. 3. This third term (multiplied by co/(c~ + i/r) to take collisions into account), together with the fourth one, was used in ref. 3 to calculate the quasiparticle damping of ~ in the hydrodynamic regime. The first term, at T = 0, is the q = 0 limit ofeq. (1) in ref. 2, and the second one is the lefthand side of eq. (5) in ref. 2, and gives rise to the quadratic dispersion relation of orbit waves at low frequencies for kT << to. [ 11] For an experimental review on superfluid 3He, see J.C. Wheatley, Rev. Mod. Phys. 47 (1975) 415.
LOW TEMPERATURE ORBITAL DYNAMICS AND NMR R. COMBESCOT Groupe de Physique des Solides de l'Ecole Norrnale Sup~rieure, 24 rue Lhomond, 75231 Paris Cedex 05, France; and Bell Laboratories ,, Murray Hill, N.J. 07974, USA
Orbital dynamics for zero wavevectorq = 0 is studied ,nt low temperature. The consequences for low temperature NMR are investigated.
1. Introduction
enough since we still have to face Landau damping. This can be avoided by working at very long wavelengths (wavevector q = 0). But this is not yet sufficient to allow a study of the reactive forces acting on ~. The reason is that, as indicated by Leggett and Takagi [ 1], if the quasiparticles do not relax at all, they still produce a restoring force on ~ because moving ~ increases their energy. Since a quasiparticle energy is of order k T at low temperature (the gap is zero for kll ~), ~ will move only if the input energy 60 is at least of order of kT. Otherwise, if 6o < < kT, ~ will be essentially locked by the quasiparticles. In other words, only the condition 60 > > k T (which ensures ~utomatically 60r > > 1) allows elimination of all the effects due to quasiparticles. However, as will be seen, pair-breaking damping also becomes important in this regime, because as noted in ref. 2, pair-breaking occurs whenever 60 > > (q. ~)VF which is always satisfied for q = 0. Nevertheless, this is the only regime, at q = 0, where orbital dynamics exhibits anything other than straight collisional damping or locking. Actually, the best way to test reactive orbital dynamics seems to be the study of orbital waves in the regime 60 > > kT, where the dispersion relation is 60 " q z VF as considered in ref. 2 In this paper we will study orbital dynamics at q = 0 and its consequence on the NMR properties. The case q = 0 is interesting because it exhibits most of the features of orbital dynamics, and also because experiments with (quasi) uniform systems are rather easy to perform; for just the same reason, NMR experiments are very conve)aient for spin dynamics. We will consider only the collisionless regime since the hydrodynamic regime
There is presently much interest [1,2] in the low temperature dynamics of the anisotropy axis of 3He A, the so-called orbital dynamics. The reason for the interest in the low temperature behaviour is that, at "ordinary" temperature, the motion of the anisotropy axis is dominated by relaxation processes [3,4]: a change in the direction of ~ produces a change in the energy of a quasiparticle. The quasiparticles then tend to relax to a new equilibrium position corresponding to this energy change. Therefore, in the motion, ~ drags along the quasiparticles (the "normal fluid") which gives rise to a viscous force acting on ~. In essentially all the domains presently accessible to experiment, orbital dynamics is dominated by this force. However, one would like to be able to eliminate this damping force in order to study the reactive forces which act on £. Such reactive forces are precisely the ones which make spin dynamics of superfluid 3He so rich and so interesting. To avoid the damping, one has to eliminate the quasiparticles. This is done by going to a low enough temperature [5]. A first result obtained by lowering the temperature is to remove one important relaxation mechanism, namely quasiparticle collisions. For T/Tc ~ 0.1, a typical collision time [6] r should be of order 10 -3 s, which means that any experiment at frequencies 60 higher than 1 kHz will be in the collisionless regime. However, the condition 60r > > 1 is not
Summer visitor at Bell Laboratoriesl 11
12
R. Combescot / Low temperature orbital dynamics and NMR
has already been studied in ref. 3. We will restrict ourselves to the usual domain co < < max IAk(T)I because, as is well known, one has to consider degrees of freedom other than the ~ motion when this condition is not satisfied, and then the physical picture becomes much more complicated. Although the general equations will be derived without this restriction, we will only study the case kT < < max IAk(T)I which implies at least moderately low temperature. The reason is that, for kT ~ max Iak(T)l, ~ is essentially locked, except if co ~ max lak(T)l which is beyond our scope. And anyway, the lower the temperature, the easier it is to see the unlocking of ~. On the other hand, we do not put any restriction on the ratio ~o/kT. Actually, one will have co < < k T or co ~ kT in most experimental situations and especially when one uses NMR to excite the motion of ~. Because NMR is probably the easiest way to study orbital dynamics at q = O, we will consider in detail what modifications appear at low temperature to NMR properties.
In eq. (1) ~k is the kinetic energy measured with respect to the Fermi level and, in eq. (2), Vke' is the pairing interaction which is taken as Vee, = V k. k' for p-wave pairing. In eq. (1)we have already performed the Hartree-Fock factorization of the interaction and kept only the anomalous averages. In the A phase, the order parameter is believed to be
2. Orbital dynamics at q = 0 and low temperature
nk=(n~
In order to study orbital dynamics we use kinetic theory. This theory can actually be cast, for orbital dynamics, in a form which allows us to treat in the same framework the hydrodynamic and the collisionless regime, as has already been done for spin dynamics [7,8]. This will be published elsewhere. However, in the present case where we are only interested in the collisionless regime, we do not need such a form and we will use straight kinetic theory. Naturally, we will use the BCS model to treat the superfluidity of 3He, as was done in most of the theoretical work to date. The justification is, as usual, that although strong coupling effects are known to bring corrections, these are not expected to be qualitative and may hopefully be not too important quantitatively. For p-wave pairing, the BCS hamiltonian is [9]
At equilibrium
a , ~ ( k ) = [Oy o. d]~0 [it- (A' + i A")],
(3)
where IA'l = [K'l = 1 and A' • A " = 0. By definition = A' × A". In orbital dynamics we are only concerned by the motion of ~., and d can be taken as constant (the motion of d comes in spin dynamics). If d is chosen along the y axis aat3(k ) : d[k. (A' + iA")lSao.
(4)
Therefore the two spin populations are completely decoupled and we can omit the spin indices. Let us define the matrix quasiparticle distribution
(5)
=
jf=~ k, ol
+
~kCk, o~Ck,o: +
~
+ + .[ck,,~c_k,#A:~a(k) + h.c.}
$
k
(1)
where aa~(k) = k~, Vkk' (C_k,c~Ck,~)
(2)
is the order parameter; in eqs. (1) and (2), ck, a and + Ck,c, are respectively the destruction and creation operators of a helium-3 atom in plane wave k with spin a.
ntJ
((C_kC k)
+¢:o/G'
(C_kC+_k>/
co= ~k ak
at --~d '
~k = --~-tanh(-~ ~Ek).
(6)
The equation of motion for n k is easily obtained from the Heisenberg equation of motion for operators. From eq. (1)we have
(7)
i - ~ n k = [nk, ek] ,
where e k is given by eq. (6), but with A k taking its instaneous value Ak(t). Eq. (7) is the kinetic equation. From eq. (2) the order parameter is self-consistently given by
Ak = ~k' Vkk'n;' "
(8)
We will actually consider only the linear orbital dynamics: the order parameter is assumed to have only small deviations from its equilibrium position. Let us take ~ parallel to the z axis at equilibrium, with A' along the x axis and A" along the y axis. Then the order parameter will be of the form Ak = Aok + 6Ak ,
Aok = d(kx + i k y ) ,
R. Combescot / Low temperature orbital dynamics and NMR
5Ak : - d kz(8~ x + i 6~y),
(9)
where 8~ is the deviation of £ from equilibrium. In the same way nk = n o + 6nk, and after linearizing and Fourier transforming, eq. (7) becomes
13
to be not modified by the dipole interaction. Now the calculation goes by solving eq. (10) for 8n~ and substituting in eq. (1 I), which gives the equation of motion for 8~x and 6~y. A problem arises because eq. (11) gives in general six independent equations: if we write 5Ak = k • 84, eq. (11) becomes
coSn k = [ 6 n k , 601 + [n o , 8 e k ] ,
10
8/' 1
(10)
8 k=\SAk 0 "' while eq. (8) goes into
8 A k = ~k Vkk'bn-~',
8A~ = k~Vkk, Sn~'. (11)
Up to now we have not considered the dipole interaction. However, this force is all important since it is the only one which couples the orbital dynamics to the NMR. The dipole interaction can actually be handled by fairly simple considerations. But here we will use a slightly more complicated way to show how it can be systematically included in the framework of kinetic theory. We treat the dipole interaction in the spirit of most of the theoretical work in spin dynamics: we do not consider its effect on the equilibrium order parameter and take it into account only as a small restoring force when the order parameter is away from equilibrium. In other words, we do not include the dipole term in e °, but only in 6e k. This is the way which has been used in ref. 3 to calculate the viscous term in the hydrodynamic regime. If d is taken along the y axis, the dipole interaction gives rise to an additional term in Jf which can be written
84 = V ~
kSn~,
~{'D - ~ 1~
+ + , g*k , [(ckc-,) c-,'ck' + h.c.],
( - d ) (8~x + iS~y) = V ~ kzSn-~, k
(15)
( - d ) (8~ x - i 8 ~ y ) = V ~ ]¢zSn~¢. k
In the case q 4= 0, as explained in ref. 3, only three parameters, 8 ~x, 6 ~y and the phase of the order parameter, are hydrodynamic: in the regime co < < d, the three other parameters are very small in comparison with these. Three of the equations (14) allow the calculation of those non-hydrodynamic parameters (and check that they are indeed small). The three other equations give the equations of motion for the hydrodynamic parameters. One of these turns out to be the mass conservation.law, the two others are eq. (15). As we said, in the case q = 0, the phase decouples from 8~x and 8~y so that we are left only with eq. (15). Solving eq. (10) gives
4AE~¢- 2wB~g
(~k
4E~ - 602
X Ek--' (16)
AkSn~+ A~Sn~
where again the Hartree-Fock factorization has been performed, and only anomalous averages have been retained. ~fo is actually of the same form as the pairing interaction, the only difference being the replacement of Vkk' by gkk" Therefore, the only modification brought by the dipole interaction in the kinetic equation (10) is, in the definition of 8ek, the replacement of6A k by 6Ak + 6Dk where (I 3)
k'
On the other hand, eq. (11) is unchanged because it corresponds to the definition of Ak which we assume
(14)
which gives indeed six equations. This is quite natural since the complex vector 8 A corresponds to six independent parameters. Actually, in the case q = 0, the equations for 5~x and 8~y decouple from the equations for the other parameters, and the only equations one has to retain from eq. (14) are
(12)
p
6Dk = ~ g k k ' S n k ' •
]c 5n~:,
k
Akfn~ - A~Sn~-=
as
8A* = V ~
k
4B~I~- 2coA~k ¢k = 4E 2 _ 6°2 X Ek--'
where A = Ak(SA~ + 5D~) - A~(SAk + 5Dk), (17) B = Ak(SA~ + 8D~) + A~(6Ak + 8Ok). When, in eq. (13), we replace 8n~, by its value taken from eq. (16), we can make co = 0 because keeping w 4= 0 would only produce corrections of order cold at least [actually they would be of order (w/d) (T¢/EF) and (w/d) 2 log (co/d)]. Such terms would correspond to finite frequency corrections to the dipole energy
R. Combescot / Low temperature orbital dynamics and NMR
14
term. Moreover, gkk' provides no cut-off [9] in eq. (13) for the integration over k', so that the main contribution comes from regions where ~k ~ Ek and therefore we have
so that expression (21) is merely -(8~x +-i6~y)VE'd'/d. In deriving eq. (24) we have only retained the dominant terms in the region ~k > > Ak as we did before. The equations of motion for ~ are finally [10]:
/
8Dk=~k, gkk'~,&Xk', (18) × (5~x -+iS~y) = 0 ,
6D~ = ~g, gkk' Ok_£'5A~' Ek,
where we have made SAg + 5Dg "" 8Ak since we are working to lowest order in dipole energy. To obtain the equations of motion for 8~, we substitute 8n~ from eq. (16)into eq. (15). We simplify the result by using the gap equation, eq. (8), when A a and n~- take their static equilibrium value:
&= v ~k' k.k'0k' ~k' Ak'"
(19)
By taking the derivative of eq. (19)with respect to small deviations of ~ from its equilibrium value, we obtain (-d) (8~x + iS~y) =
v 52 k
['¢k
LEk
+ aok, - -
a
t%l
2Ek aEa~-k)J
where ¢' = 80/aE. The term (co~/2E 2) is actually an approximation for 2co~/(4E 2 - co2): this }erm gives a zero contribution except if we take into account particle-hole asymmetry, and in this case the main contribution comes from regions where ~k > > d, co which justifies the approximation (the imaginary is always negligible compared to the first term imaginary part). Therefore we find from eq. (25) two eigenmodes, non-degenerate, circularly polarized, whose frequencies are obtained by equating to zero the brackets in eq. (25). The various terms entering eq. (25) are explicitly evaluated in the appendix with the further approximation kT<
(kT)2 - -g - [log---d * ~ * g 8a
.
(20)
Finally the part coming from 6Dk and 6D~ can again be simplified by taking co = 0 and ~k ~- Ek for exactly the same reasons as before. The contribution of these terms to the right-hand side of eq. (15) is therefore
^,Ok (-d) (8~x +-iSQy) V ~ gkk'kzkz - - ~Ok' . k,k' Ek Ek' -
(25)
(21)
+ ~ irr tanh (4_~)] ~ codZN'0 coc /~a) 6No t o g - ~ + Noo ]
X (5~x +-i6~y) = 0 ,
(26)
where
f--
I(1 ;)
But, from eqs. (12) and (6), the dipole energy Ed is given by
1 g(x) = ~
Ed(~) = ~ ~ gkk'OkOk' ~ A~___:'.
The calculation ofg(x) is discussed in the appendix. Eq. (26) displays all the characteristic frequencies of orbital dynamics. Let us first consider the case T = 0. Eq. (26) becomes
k, k'
(22)
lZk Ek'
This energy is minimal when ~ is parallel to the z axis, and for small deviations of ~ from this position we have Ed(0-) = E~d+ } E~ [(8 ~x)2 + (8 ~y)2] .
(23)
From eq. (22) we have "' Ok Ok' E d" = ~ggk'd 5^kzkz k, k' Ek Ek'
(24)
dp log cosh 2 P
o
---~-T
•
(27)
co2 2d ,qi g- ogG÷h÷ °logcocd
c~d /V',, 6No
tt
Ea No
-0.
(28)
If we assume particle-hole symmetry (N~ = 0), this
R. Combescot / Low temperature orbital dynamics and NMR 3. E f f e c t o f orbital dynamics on the N M R
reduces to ~2~ ~_ n]_E J - ~ og +~+½i -'N00'
(29)
which shows a characteristic frequency of order (E~/No) 1/2. This is of the same order of magnitude as the longitudinal NMR frequency ~o, which is given with these notations by ~o~ = 4E~'(1 + bo)/No. Actually the motion described by eq. (29) is analogous to longitudinal NMR: under the action of the dipole force oscillates around its equilibrium position, the modes being linearly polarized and degenerate. However, there is one main difference: this mode is rather heavily damped due to pair-breaking effects. At the resonance frequency the imaginary part of the frequency is of the order of one-tenth of the real part. On the other hand, the second term in eq. (28) which is due to particle-hole asymmetry gives rise to a precession of ~ around its equilibrium position. Such a motion has already been considered by Cross and Anderson [4] in the hydrqdynamic regime. Without the dipole force, one would obtain a precession frequency of order d2/EF which turns out to be also of order of 60o. Therefore no term in eq. (28) is negligible and the actual modes are circularly polarized, non-degenerate, with frequencies of order of 60o and an important damping. At finite temperature the characteristic frequency kT/~ appears. For k T ~ hWo, nothing is changed qualitatively with respect to the case T = 0. Naturally k T ~ h6oo is presently far beyond experimental reach, since it corresponds to T/Tc ~ 5 X 10 -3. In the situations experimentally accessible, h6oo < < k T and we can neglect the dipole interaction and the precession term, which corresponds to keeping only the first two terms in eq. (26):
Up to now we have only considered the motion of with fixed d, which was the convenient way to study orbital dynamics. However, in the real motion, ~ and d are both moving because they are coupled by the dipole interaction. Since the motion o f d is coupled to the NMR, orbital dynamics modify the standard resuits of NMR in 3He A since it is usually assumed that is fixed. We want to study these modifications, essentially at low temperature. Let us assume, as we did before, that the equilibrium position o f d is along the y axis, with a magnetic field H along the z axis. Under a small rotation 0, d will have a small deviation from the y axis and the components dx and dz will be non-zero. They are related to 0 by Ox = ~lz and Oz = -~tx. The first equation for the NMR is the conservation law for the total spin S of the system [8,9] S i --~ = ([S, ~1) = iS × 7 / / + ([S, ~fo]),
if2
9
(kr) ~ .
(30)
We see that the situation is rather analogous to the case T = 0 in eq. (29), but with the dipole force replaced by a force due to quasiparticles as indicated by Leggett and Takagi [1]. Again, as for T = O, the motion is strongly damped by pair-breaking effects, and there is no sharp resonance but rather a broad structure in the spectrum of the response of ~ to an excitation.
(31)
where 3' is the gyromagnetic ratio for the 3He nucleus. Since S is the generator of the spin rotations in the system, the last term in eq. (31) is directly related to the change of the dipole energy Ea under spin rotation [9]: ([S, ~CDI>= - i ~ d / ~ 0 •
(32)
At equilibrium, ~ is along the y axis and if we assume that it does not move, we have for small deviations of d 1 L"t r/',?,/2 Ed = E~d + ~-~d~.Ux + 32).
(33)
But if £ is moving, we have ed = ~ = E0
-6°2~ ( o g -2d+ ~ rI~ + g (~-~) + 7~ in tanh(4-~) ]
-
15
1
tt
^
+ ~ ~a [(dx - ~ ~x)5 + (~tz - ~ ~ ) ~ ] I ,, +]Ea[(Oz+6~x)2 +(Ox-6~z)2].
(34)
The second equation of the NMR says [8] that, in a reference frame moving with d (i.e. deduced from the fixed frame by the rotation 0), there is a "Josephson field" proportional to the instantaneous velocity of the moving frame. In the linear regime, this field is (1/7)dO/dt. The magnetization adjusts to the total field according to the dynamical susceptibility ~'(t): t
7Sm(t ) = f
d t ' ~ ( t - t')[Itm(t' ) + (1/7)dO/dt') . (35)
_ o o
Here Sm and Hm are the spin and the field in the mov-
16
R. Combescot / Low temperature orbital dynamics and NMR
ing frame. In the linear regime they are related to their values S and H in the fixed frame by Sm=S+SXO
,
Hm=H+HXO.
(36)
In an NMR experiment, H is the sum of the static field H o and the r.f. field H', and $ is the sum of the static spin S O = ×0//0/7 and the oscillating spin S'. After linearizing and Fourier transforming, we obtain from eq. (35): 7S' + XoHo ×0 = ~(co)[H' + H o ×0 -(ico/')')O] .(37) Finally, we have to modify slightly eq. (25). First, the equilibrium position of ~ is along the x axis and not along the z axis as before. Then, in the dipole term, it is naturally the departure of ~ from d which comes in, and not the departure from the equilibrium position. Therefore we have Y,±(8~ + iS~x) + E~[(5~z - dz) +-i(5~x -/tx)] = 0 , (38) where ~_+ represents the sum over k in eq. (25). To~ gether with eqs. (32) and (34), eqs. (31), (37) and (38) allow one to solve for the NMR. If we solve eq. (38) for 5~z +-iS~x, we obtain (5~z + iS~x) -
E'd (dz+idx)=R+(co)(~lz+i~lx). E~' + ~_ -
(39) Eq. (39) is interesting in itself because it shows the response of fi to an excitation through d. Except when co, k T ~ coo, this response is always very small, essentially because ~ is excited through the weak dipole force. In any case, Im R(co) never shows a sharp resonance because of the importance of pair-breaking damping, as discussed before. In the practical situations where k T > > coo, Z. can be approximated as in eq. (30) Noo
9 (kT)2 - --6
+ ~- in tanh (4--~)
og
+ ~ +g (40)
In this regime, Im R(co) has a broad bump with a maximum for co ~ k T and a width of the same order. An interesting qualitative modification of the NMR by the motion ~ is that the longitudinal and the transverse resonances are coupled, due to the precession term in eq. (26). At T = 0, if the A phase could be
stabilized without magnetic field, this coupling would be very strong. In the strong magnetic field (~5 kG) which seems necessary to stabilize the A phase, the coupling would be only of order COo/COL where coL = 3'Ho is the Larmor frequency. In the longitudinal resonance, the coupling makes the magnetization precess around the field in addition to its usual oscillation along the field. In the transverse resonance, the magnetization has a small oscillation out of the X Y plane in addition to the usual precession in the X Y plane. However, at presently available temperatures, k T > > coo, the coupling is only of order (coo/kT)4(coo/coL), and seems very difficult to detect. Therefore we will neglect this effect and forget about the precession term. With this approximation R+ = R_ -- R and substitution of eq. (39) into eq. (31) shows immediately that the only effect of orbital dynamics on the NMR is the replacement of E~' by E~' [1 - R(co)]. Now we consider the practical consequence of this result on the NMR in the usual temperature range k T > > coo, where R(co) can be approximated as in eq. (40). The longitudinal NMR frequency is given by 602 = co~ [1 - R(coo)],
(41)
where co(~_~)2 I1 + 3i [coo~3] , (42) 16rr ~k-T/A which gives rise to a longitudinal resonance frequency
R(coo)
1
9
1 +----~o21r~
~
1
co~ = coo
9.__9__[~o, ~2-]
1 + F~ 47r2~kT} .]
and a full linewidth 1 27 ~coo~s Aw~- 1 + F~ 327r3 \k-T] w°"
(43)
(44)
These results must be compared to the shift and the width coming from spin relaxation effects [8], because all these effects add up. The shift is due to the difference between the susceptibility in the collisionless regime and in the hydrodynamic regime. From eq. (86) of ref. 8,the shift at low temperature is cool(T)/2(1 + F~) where f(T) = (rr2/9)(kT/d) 2 so that, taking into account eq. (43) ~(z3=
~o(Z3
9
+ 18(1 + F D \
[Wo]27
-4zr2(1 + F~)~-k-f] _]"
d !
(45)
R. Combescot / Low temperature orbital dynamics and NMR
From eq. ( 8 7 ) o f ref. 8, the linewidth due to relaxation is f(T)/r(1 + F~)where r is the relaxation time, so that at low temperature the linewidth will be
cot -~ (co2 + coo2)i/2(1
-ACOQ- 9(I + F~)) r \ d /
Acot ~ o ImR(coL). - - -COL
-~
32rr 3 (i + F~) ~ " ~ ]
e°°"
(46) In the shift, the orbital dynamics effect becomes dorainant when (81/2rr4)(cood/k2T2) 2 >~ 1. With the BCS value d ( T = O) "~ 2kTc, kTc/h "" 3.5 × 108 at melting pressure and [11] coo(T = 0) -~ 7 X l0 s, this gives 6 X 10-6(To/T) 4 >~ 1, which means that this effect will be observable only at very low temperature Te/T >~40. At this temperature the shift would be of order 1 kHz, which is rather large. We must note that coo(T) has also a temperature dependence which probably gives corrections of order (kT/d) 2 in the spin susceptibility term, but with the opposite sign, so that it would be impossible to see the difference between these two terms experimentally. But the orbital dynamics effect will lead to a decrease of the longitudinal frequency with a different power law and should therefore be easy to separate from the two other terms. In the same way, the part due to orbital dynamics will dominate inthe linewidth if (243/32rrs)(coo/kT)S(d/kT)2coor ~1. The relaxation time grows [6] like 1/T 4 at low temperature and a rough (over)-estimate is r(T) ~ r(Te) (Tc/T) 4 with r(Tc) ~ 5 × 10 -a. Therefore we obtain the condition 10-16(Tc/T)I l ~> 1, which again implies a very low temperature Tc/T >~ 30. But below such a temperature, the linewidth (due to orbital effects) is of order 4 × 10-1°(To/T) s Hz and Tc/T ~ 102 is necessary to reach a linewidth of order 1 Hz. But it is interesting to note that the linewidth grows like 1/Ts in this range, which is very fast. Turning to the transverse resonance, we have co2 = co2 + co2 [1 - R(co)] (47) • The transverse resonance is markedly different from the longitudinal only if COL> > COO,and anyway a field of order 5 kG seems necessary to stabilize the A phase at low temperature. Therefore we have a transverse resonance frequency
17
co~ Re [R(coL)])
(48)
- ~ and a linewidth (49)
From eq. (40), Re [R(coL)] will be of order [coo/max(kT, coL)]2 and the correction to the usual resonance frequency is of order 10 -8 and therefore impossible to observe experimentally. Since coL = k T i n a field of 5 kG gives a temperature T/Tc ~ 0.25, we have actually always k T ~ co L at low temperature, so we can only consider the case k T < < coL. In this case we obtain Acot = cooloao,.. ,3 3rr 1 \COL! 2(1 + F]) [log(2d/coL)] 2 '
(50)
which gives numerically ACOt ~ (10/H3G) Hz. This seems again far too small to be observable. It is interesting to compare the linewidth, eq. (50), with the linewidth coming from spin relaxation effects. We have to use eqs. (89) and (91) from ref. 8 in the collisionless regime COL'/"> > 1, which gives Acol ___coo(coo/a ( k T ) 2 7ra \COL! - --g- (1 + F~) 2 ,
(51)
or numerically, Acol ~ (0.2/Hac,)(T/Tc) 2 Hz. We see that the width is essentially due to the orbital effect, but unfortunately the high field necessary to stabilize the A phase makes the linewidth too small. In conclusion, the orbital effects on the NMR are very small in the range of temperature currently obtained experimentally• The main reason is the smallness of the dipole force: all the effects are obtained through the dipole term in the NMR and moreover the coupiing to f~is through the dipole force. In other words, ~ is essentially locked, and this is the small departure from locking that one is trying to observe. The effects can be increased rather drastically in the longitudinal resonance by going low enough in temperature, but the required temperature does not seem presently within reach•
R. Combescot / Low temperature orbitaldynamics and NMR
18 Appendix
With N O being the density of states of the Fermi surface for both spin populations, we have -
~ k z ~ [A[2 O' - No +o. //2 d 2 sin 2 0 1 k ~-2 -8---~ f d~ dO sin 0 cos 2 0 E2 cosh 2 (½fiE)' --oo
(A.1)
0
where E 2 = ~2 + d 2 sin 2 0. At low temperature kT < < d, the main contribution comes only from 0 < < ~-n because of the l/cosh 2 (~/~E), so that the integral over 0 can be extended to +oo and we can take sin 0 ~ 0. We make the change of variables 0 = (2kT/d) p sin c~, ~ = 2kTp cos a, and obtain -
k
=
~'2
o;
d
0
"
cosh 2 p
0
-No
9-
- -
.
(A.2)
The imaginary part of 2E2-IAI2
J= ~ k2z ' ~ 2 - - - ~ k
¢
(A.3)
2E 3
is easily obtained as ImJ = -
16
dO sin 0 cos 2 0 0
d~ 2E2 E4 - IAI2 tanh
E
5(6o - 2 E ) .
(A.4)
--oo
1 With the approximation co < < d where the main contribution comes from 0 <:< ~Tr, we obtain
N°Tr2 tanh (4-~T) Im J = - 12d
(A.5)
The real part of J is first calculated at T = 0 2~2 + d2 sin2 0 1 Re JT=O = X 4 N o / " +~ d~ fa / 2 dO sin 0 cos 2 0 J 4~ 2 + 4d 2 sin 2 0 - co2 (~2 + d 2 sin 2 0)3]2 . --oo
(A.6)
0
With the change of variables ~ = do cos a, sin 0 = 19 sin a, this becomes
No//2
~ / s i n ¢~
0
P
d P 4 p 2 _co2/d 2 [(x/1 - p2 sin2 c~- 1)+ 1].
da sin a (1 + cos2 a)
Re JT=o = - ~-~
(A.7)
0
In the/term with x/1 - p2 sin 2 a - l, we can take o3 = 0 if we neglect contributions which go to zero when
cold --->O. The integral is then easily performed to give (No/6d2)(1 - log 2). The other term in eq. (A.7) is also easy to calculate in the same approximation and the final result is
=No
Re JT=o - ~
Then Re J = Re (J - JT=o) + Re JT=o and we are left with the calculation of 2E 2 - I A I 2 1 tanh(-~fE)) S e ( J - Jr=o) = ~ k2z - ~ -_ - ~ - ~ (1 -
(A.9)
k
Again the main contribution at low temperature comes from the regions near the nodes of the gap (0 < < ~r) and with the change 0 = (2kT/d)p sin a, ~ = 2kTp cos a, we obtain N0 R e ( J - JT=O) = ~
dp log O2 cosh2----~
I
co 2 - l ° g 4 - - ~
(A.10)
0
By expanding the log in powers of 4kTp/¢o, Re(J - Jr=o) can easily be put in the form of a series expansion in
R. Combescot / Low temperature orbital dynamics and NMR
19
powers o f (kT/~o) which is particularly convenient for 60 >~ kT. The first t e r m gives R e ( J - J r = o ) ~ - ( N o / 1 2 d 2) (rr2/12)(4kT/w) 2 for w > > kT. In the case co ~ k T i t is b e t t e r to calculate the integral in eq. (A.10) b y residue which gives ~rt - C , cosh 2 p = an [Otn + (6o/4kT) 2] ~ log l o where C is the Euler constant and ctn = (-~70 (2n + 1). Finally
~kz k
(A.11)
~¢) N'° log d C 2E 3 ~ - ~ '
where N~ is the derivative o f the density o f states at the F e r m i surface and ~ c is the BCS cut-off.
References
[1] A.J, Leggett and S. Takagi, Phys. Rev. Letters 36 (1976) 1379. [2] M. Combeseot and R. Combescot, Phys. Rev. Letters 37 (1976) 390. [3] R. Combescot, Phys. Rev. Letters 35 (1975) 1646. [4] M.C. Cross and P.W. Anderson, Proc. 14th Conf. on Low Temp. Physics, M. Krusius and M. Vuorio, eds. (NorthHolland, Amsterdam, 1975): M.C. Cross, to be published. [5] The A phase does not exist actually at low temperature in zero magnetic field. However, it is presently believed that a field of order of 5 kG is enough to stabilize the A phase, with respect to the B phase at T = 0. [6] R. Combescot, Phys. Rev. Letters 35 (1975) 471. [7] R. Combescot, Phys. Rev. A 10 (1974) 1700. [8] R. Combescot, Phys. Rev. B 13 (1976) 126.
[9] For a review paper on the theory of superfluid 3He, see A.J. Leggett, Rev. Mod. Phys. 47 (1975) 331. [10] Naturally some of the terms appearing in eq. (25) have been found by various authors working on collisionloss kinetic theory, starting probably with P. W~lfle, Phys. Rev. Letters 30 (1973) 1169, although they are not mentioned explicitly in his paper. Referring to the author's work, the third term of eq. (25) is the limit, for q = 0, of the left-hand side of eq. (7) in ref. 3. This third term (multiplied by co/(c~ + i/r) to take collisions into account), together with the fourth one, was used in ref. 3 to calculate the quasiparticle damping of ~ in the hydrodynamic regime. The first term, at T = 0, is the q = 0 limit ofeq. (1) in ref. 2, and the second one is the lefthand side of eq. (5) in ref. 2, and gives rise to the quadratic dispersion relation of orbit waves at low frequencies for kT << to. [ 11] For an experimental review on superfluid 3He, see J.C. Wheatley, Rev. Mod. Phys. 47 (1975) 415.