VOLUME 31, NUMBER 20 PHYSICAL REVIEW LETTERS 12 NOVEMBER 1973

1831 (1970). " R . C. Davidson and N. A. Krall, Phys. Fluids 13, 9W„ E. Drummond, J. R. Thompson, and H. V. Wong, 1543 (1970).

U.S. Air Force Office of Scientific Research Report No. 12B. Bernstein and I. Smith, IEEE Trans. Nucl. Sci. I-ARA-73-U-38 (unpublished), Sect. III. 20, No. 3, 294 (1973).

10H. V. Wong, M. L. Sloan, J. R. Thompson, and A. T. 13W. L. Kruer, J. M. Dawson, and R. N. Sudan, Phys. Drobot, Phys. Fluids 16, 902 (1973). Rev. Lett. 23, 838 (1969).

Scaling Function for Critical Scattering

Michael E . F i she r and Amnon Aharony Baker Laboratory, Cornell University, Ithaca, New York 14850

(Received 27 August 1973)

The zero-field, two-point correlation function of an w-vector system i n d ~ 4 - e dimen­sions is calculated to order €2 for T^TC. The scaling function is obtained as a closed, cutoff-independent integral. As t = (T - Tc)/Tc~~ 0 at fixed wave vector q, the leading vari­ation is Ejn'<\q)ti~a+E2

n*d{q)t, where a is the specific-heat exponent; thence the maxi­mum in the scattering above Tc is located, in good agreement with high-T series-expan­sion estimates.

As a c r i t i ca l point is approached in ze ro field (i .e. , along a line of symmet ry ) the co r re la t ion length diverges1 , 2 a s ^(T)~f1atmV where t =(T -Tc)/Tc, and a is the lat t ice spacing. The exponent ^depends on <2, the dimensional i ty of the la t t ice , and on n, the number of "sp in" components . Similar ly , the sca t te r ing intensity, which is propor t iona l to the F o u r i e r t r ans form, G(q, T), of the two-point c o r r e l a ­t ion function becomes la rge a t low wave numbers q.

According to the scal ing hypothesis1"4 for the c r i t i ca l cor re la t ions , one can wr i te

G(q, T)^cryb(q2e\ (i)

for t«1 and qa«l. On adopting the normal iza t ions D = - dD/dy = 1 at y = 0, the constant C becomes the ampli tude of the (reduced) init ial suscept ibi l i ty or ze ro-ang le sca t te r ing function XotF) = G(0, T), which d iverges with exponent y = y(n, d). Similar ly , | = ^ mus t then be identified a s the s e c o n d - m o ­ment cor re la t ion length.1*4 The ampli tudes C a n d / x must depend on the deta i ls of the in terac t ions , the la t t ice s t r uc tu r e , e tc . , but the form of the scal ing function Dix2) is expected to be "un ive r sa l , " depend­ing only on n and <i.5,6

In this note we p resen t , for the f i r s t t ime, an analyt ic calculation of the scal ing function £){x2) exact to o rde r c2, where d = 4 - €,7 '8 for a sy s t em of genera l n with i sot ropic ( s - s ' ) in te rac t ions of finite r ange . Our r e su l t may be wri t ten

l/D(x2) = t+x2-4pnx4Q(x2)e2+0(e3), (2)

where pn = | ( n + 2)/(n + 8)2 and Qix2) is defined by the fully convergent in tegral

Q(y) = y-2rdz[z(l + iz)]1/2in[(l + {z)1/2+±zl/2]

X{(1 + z) - 1 - y( 1 + z) " 3 - \y 'xz "x [1 + y + z - (1 + 2y + 2z + y2 - 2yz + z2)l/2 ]}. (3)

The cutoff independence of this in tegra l (see below) confirms the expected universa l i ty of D(x2). In the l imi t s e-~0 and n*°° the resu l t r educes to the Orns t e in -Zern ike (OZ) form, D = (l+x2)~1, a s may be expected. (A sepa ra t e exact calculation to o rde r 9 \/n has been undertaken.1 0) To leading o r d e r in e the deviation from the OZ form is propor t ional to the exponent7 , 8 t\ - r}{n, d), which de t e rmines the c r i t i ca l -poin t decay of cor re la t ion . 1 , 2

In the low-momentum l imit (q — 0 a t fixed T> Tc), expansion of (3) in powers of y leads to the s m a l l ­er expansion l/b(x2) = 1 + x2 -I:4x

4 + Z6x6 - . . . , the form of which has been anticipated on genera l grounds.1 '

Numer ica l evaluation of the definite in tegra l s der ived from (3) yields I,2k = 2b2kpne2 +0(e 3 ) , with 54

1238

VOLUME 31, NUMBER 20 P H Y S I C A L R E V I E W L E T T E R S 12 NOVEMBER 1973

^ 0.007 520 and bG ^ 0.000 191 9. The surprisingly small values of the coefficients b2k explain the ex­perimentally observed fact11"13 that the OZ form for D is a rather good approximation for small x (i.e., low q at fixed t) when d = 3. As evident from Table I, these results also confirm closely the compara­tively imprecise estimates of S 4 which can be derived from high-temperature series.4,14"16

The finite-momentum behavior, i.e., T-~TC at fixed q >0 (but qa « 1), following from (5) is of con­siderable interest. In contrast to the scaling functions previously calculated for the equation of state,17,18 the correlation scaling function itself is expected to display intrinsic, critical, (n, ^ -depen­dent singularities for large x. This follows from the conclusion19 that the scattering in the finite mo­mentum limit should vary as

G(q, T) *Dj{aqT^E1{q)t1~a+E2(q)t+ - , (4)

where a = a(n, d) is the specific-heat exponent. This form has played a valuable role in the interpreta­tion of critical-resistivity experiments.20 It is consistent with the series-expansion data4'14"16 although the coefficients Ex and E2 cannot be determined with any precision. By comparison with (1) one con­jectures19 that '

A(^)« i> 0 7^^+^i7^ 2 ^ + ( 1 " a ) / v +i> 2 7^ 2 " 7 7 + 1 / 1 , + ... (5)

a s x - ° o , where DQ^{n, d)= if2" v/C)D0 = (pc71/(l +^r}(pc

2), in which cpc is the universal parameter enter­ing the Fisher-Burford approximant for Dix2).4 From (5) one obtains the scaling prediction E^^iq) ~^-2+Tj-(i-a)/i/ a n c | a n a n a iogous prediction for E2

n>d(q)B19'15 The conjecture (5) may at last be checked

analytically by studying the behavior of Q(y) for large y. A somewhat tricky calculation21 leads to

Q(y)«Q0/ lny/y-Q0/y +Qi/,(lny)2/y2 + Qi' l n y / / -Q1/y

2 + . . . , (6)

withQ0 '=-£, Qi ' ^ f , Q / ^ 0 , Qo^0.51776, and 0<QX <0.013. A careful comparison with the € expan­sion of (5) to second order, using the known exponent expansions,7'8 then reveals that the values of Q0' and Q0" are precisely consistent with the anticipated singularity structure. (Some remarkable alge­braic cancellations occur!21) Furthermore we obtain

(7) 6 _(w+2)(7w + 20)

4 - w (n. + 8)(4-n)2 JJ2 ~~A_.~ / „ _ , Q \ / 4 _ M \ 2 e + C ' ( . € ;.

Table I compares the truncated expansion for D0" with series expansion estimates for d= 3. The ap­parent precision is about 5%, but it must be borne in mind that the uncertainties attached to the series estimates may not fully reflect the exponent uncertainties. It is remarkable that although the deviation

TABLE I. Comparison of numerical estimates for three dimensions.

P a r a m e t e r n

1 3 1 3 1 3 1 3

Order €2

0 o 019 a

0.021 a

2.8 x i o " 4

3.1X10"4

0.962 0.957

1.3 x i o - 2

1.8X1Q"2

High-X s e r i e s

0 . 0 5 6 ± 0 . 0 0 8 b

0.043 ±0.014 c

~ 6 . 3 x i o " 4 d

~ 2 . 6 x i 0 ' 4 d

0.90 ±0.01 c

0.91 ±0.02 c

( 1 . 0 - 2 . 3 ) x i o " 2 c

( 1 - 4 ) x i o " 2 c

aRef. 8. bRef. 4. cRef. 16. Using the Fisher-Burford approximant (Refs. 4 and 16), one has £4

- i w c2 ; direct series estimates for 24 are less precise (Ref. 21).

1239

VOLUME 31, NUMBER 20 P H Y S I C A L R E V I E W L E T T E R S 12 NOVEMBER 1973

of Dix2) from the OZ form is only of order e2, the coefficients Dt°° and D2°° have contributions of both first order and zeroth order. [As e - O o r w - * , one has r\ - 0 and 1 -OL-~2V SO that the two correction terms in (5) combine to give the OZ result - 1/x4.] The divergence of the coefficients D^ and D2°° at n = 4 is associated with the vanishing of a near n = 4 as € — 0, which, in turn, corresponds to a logarith­mic singularity in the specific heat. A closer investigation confirms this interpretation and yields the logarithmic amplitude.

By putting t/{f1aq)1,v = r, the finite-momentum results can be cast, to order e2, in the transparent form

Gte ,T) /G c (^« l + ( y - l ) T ( T " « - l ) / a - r + . . . , (8)

where the limit a — 0 correctly yields a logarithmic factor. From this expression, by neglecting the higher-order powers of r, one may locate a maximum in the scattering above Tc at rrmx^ [(1 - a)(y - 1 ) / (y - 1 +a)]1/a. This maximum was first predicted numerically by series expansions,4,16 but its precise location was not at all accurately revealed. The good agreement between this estimate of rmax and those derived from series (see Table I) is very gratifying and considerably strengthens the interpre­tation of the maxima seen experimentally in the same vicinity.12*13 For a >0, the existence of a maxi­mum can already be predicted from (4), but its continued presence for a <0 (for all finite n) could not previously be understood theoretically.

The derivation21 of the result (2) uses the graphical e -expansion technique8,17 for the standard, con­tinuous spin, US,4 Hamiltonian8 with a sharp cutoff at |q| = 7iA (a being se_t equal to unity). The correla­tion function is expanded in powers of u with propagator G0(q, r)= (r + q2)~l, where r is chosen to be the inverse susceptibility l /x 0~^ y /C- Using Wilson's fixed-point value8 for7^0c(e), the result is

G=G0[l+4pne2q2G0K+O(e3)], (9)

where

K(q,r)=h-4q-2fd4q' Jd4qff Goiq^^Go^+ q^, r)[G0(qf-q,r)-G0(q

r,r)]. (10)

For small r one finds K(0, r)«-£ln(r/A2) -k +0(r lnr) , where the constant ^^0.090 857 depends on the form of the cutoff. It may be used in deriving the e expansion of the nonuniversal amplitude C and / x . Note that this is a first step towards the ultimate calculation of the magnitude of the corrections to scaling for specific model systems, (A sharp cutoff is appropriate in this context.) Next one sets22

q = S /4 = crl/(2m v) x, where c=C1/(2~ v)/fl9 and, by comparison with (1), calculates D(x2) from the r - 0 limit of rG(%cr1/(2m ^ r). Making e expansions of D and c, comparing coefficients with those arising from (9), and using the adopted normalization then leads to

Qix2) = x-2lim[K(xy/7,r) - # ( 0 , r ) ] . (11)

This may finally be transformed into the form (3). In the process the result is shown to be cutoff in­dependent.

It is a pleasure to acknowledge discussions with Professor Kenneth G. Wilson. The support of the National Science Foundation, in part through the Materials Science Center at Cornell University, is gratefully acknowledged. One of us (A.A.) is indebted to the Fulbright-Hays Committee for a scholar­ship.

*M. E. Fisher, J. Math. Phys. (N.Y.) _5, 944 (1964). Numerical values of / j can be obtained from M. E. Fisher and R. J. Burford, Phys. Rev. j_56, 583 (1967); D. S. Ritchie and M. E. Fisher, Phys. Rev. B 5, 2668 (1972); H. B. Tarko and M. E. Fisher, Phys. Rev. Lett. SJ1, 926 (1973).

2M. E. Fisher, Rep. Progr. Phys. ^0, 615 (1967). 3L. P. Kadanoff, Physics 2, 263 (1966). 4Fisher and Burford, Ref. 1. 5P. G. Watson, J. Phys. C: Proc. Phys. Soc , London 2, 1883, 2158 (1969). 6L. P. Kadanoff, in Midwinter Conference on Phase Transitions, Newport Beach, California, January 1970 (unpub­

lished), and in Critical Phenomena, Proceedings of the International School of Physics "Enrico Fermi," Course No, LI, edited by M. S. Green (Academic, New York, 1973).

1240

VOLUME 31, NUMBER 20 P H Y S I C A L R E V I E W L E T T E R S 12 NOVEMBER 1973

7K. G. Wilson and M. E. Fisher, Phys. Rev. Lett. .28, 240 (1972). 8K. G. Wilson, Phys. Rev. Lett. 28, 548 (1972). 9See S. Ma, Phys. Rev. A _7, 2172 (1973); R. Abe, Progr. Theor. Phys. 49, 113 (1973).

10A. Aharony, to be published; R. Abe and S. Hikami, Progr. Theor. Phys. 49, 442 (1973). "See references cited in Fisher, Refs. 1 and 2; Kadanoff, Ref. 3; by P. Heller, Rep. Progr. Phys. ^0, 731 (1967);

J. Als-Nielsen and O. Dietrich, Phys. Rev. 153, 706, 711, 717 (1969); J. Als-Nielsen, Phys. Rev. 185, 664 (1969); D. Bally, M. Popovici, M. Totia, B. Grabcev, and A. M. Lungu, Phys. Lett. 26A, 396 (1968); J. Als-Nielsen, Phys. Rev. Lett. ,25, 730 (1970); M. Popovici, Phys. Lett. 34A, 319 (1971).

12Als-Nielsen and Dietrich, Ref. 11; Als-Nielsen, Ref. 11. 13J3ally, Popovici, Totia, Grabcev, and Lungu, Ref. 11; Als-Nielsen, Ref. 11; Popovici, Ref. 11. 14M. A. Moore, D. Jasnow, and M. Wortis, Phys. Rev. Lett. 22, 940 (1972). 15M. Ferer , M. A. Moore, and M. Wortis, Phys. Rev. Lett. 22, 1382 (1969) [the apparent failure of scaling found

here for E\(r) may well be due to the relatively small values of r/a sampled], and Phys. Rev. B _3, 3911 (1971); M. Ferer , Phys. Rev. B 4, 3964 (1971); M. Ferer and M. Wortis, Phys. Rev. B£, 3426 (1972).

l6Ritchie and Fisher, Ref. 1; Tarko and Fisher, Ref. 1. 17E. Brezin, D. J. Wallace, and K. G. Wilson, Phys. Rev. Lett, 29, 591 (1972), and Phys. Rev. B 7_, 232 (1973);

E. Brezin and D. J. Wallace, Phys. Rev. B T_, 1967 (1973). 18G. M. Avdeeva and A. A. Migdal, Pis 'ma Zh. Eksp. Teor. Fiz. _16, 253 (1972) [JETP Lett. Ij3, 178 (1972)]. 19See Refs. 1, 4, 15, and 16 in M. E. Fisher, in Critical Phenomena, edited by M. S. Green and J. V. Sengers, Na­

tional Bureau of Standards Miscellaneous Publication No. 273 (U.S. GPO, Washington, D.C., 1966), p. 108; M. E. Fisher and J. S. Langer, Phys. Rev. Lett. 2 )̂, 665 (1968). A general argument can also be based on the operator-product expansion: L. P. Kadanoff, Phys. Rev. Lett. 23, 1430 (1969); K. G. Wilson, Phys. Rev. 1/79, 1499 (1969), and Phys. Rev. D 2;, 1473 (1970). Recently E. Brezin, D. J. Amit, and J. Zinn-Justin, to be published, have pre­sented a formal derivation based on the operator-product expansion and the Callan-Symanzik equation which shows, particularly, the origin of the term linear in t.

20Fisher and Langer, Ref. 19. 21Full details of the calculation will be published elsewhere. 22This step assumes the scaling relation {2 — f])v=y which, however, is explicitly confirmed through the coeffi­

cient 4 in Eq. (24). See also D. J. Amit, Phys. Lett. 42A, 299 (1972); Abe and Hikami, Ref. 10.

Experimental Observation of the Nonoscillatory Parametric Instability at the Lower-Hybrid Frequency

R. P . H. Chang Bell Laboratories, Murray Hill, New Jersey 07974

and

M. Porkolab Plasma Physics Laboratory, Princeton University, Princeton, New Jersey 08540

(Received 7 August 1973)

The nonoscillatory parametric instability is observed when the "pump" frequency is near the lower-hybrid frequency. Strong heating of both the ions and electrons has also been measured.

Although predic ted by theory , 1 to our knowledge the nonosci l la tory p a r a m e t r i c instabil i ty (the o s ­cil lat ing t w o - s t r e a m instabil i ty or the pure ly growing mode) has not yet been verified e x p e r i ­mental ly . Some of the r e a s o n s for the difficul­ty of observing such modes a r e the following: (a) The osc i l la tory decay instabi l i ty has a lower threshold in mos t c a s e s of in t e re s t than the non­osc i l la tory instabil i ty; thus it tends to "cover u p " the poss ib le p r e s e n c e of the l a t t e r . 1 (b) Spec­

t r a l ana lys i s and wavelength m e a s u r e m e n t s a r e very difficult s ince the r ea l p a r t of the frequency for the purely growing mode is ze ro . Recently it has been shown, however, that for pump f r e ­quencies very near the lower -hybr id frequency the threshold for the osc i l la tory p a r a m e t r i c d e ­cay instabil i ty i n c r e a s e s rapidly. 2 The r eason for such i n c r e a s e in the threshold i s that the wave-vec tor matching condition p roduces con­s ide rab le e lec t ron Landau damping of the ion

1241