PH YSIC AL REVIE% B VOLUME 12, NUMBER 11 1 DECEMBER 1975

Study of spin-cell transformations for renormalization-group equations of spin systems

Johan S. H@ye~Department of Mechanics, State University of New York, Stony Brook, New York 11794

(Received 4 November 1974)

A study is made of the behavior of the spin correlation function for spin systems and lattice gases under spin-;cell transformations in which the cell spin is chosen as a linear combination of the individual spins in the cells.It is found that the linear combination must satisfy certain conditions in order to give a satisfactorytransformation; if too "unsymmetric" it will not yield convergence to a fixed point. We work in the context ofthe one-dimensional Gaussian and spherical models with long-range forces. For these models the criticalindices can be found from our treatment since the connection between the interaction parameters and the spincorrelation function is known. Although this explicit connection is lost in other models, our derivationincludes certain general features that are independent of the underlying model and of dimensionality.

I. INTRODUCTION

Central to the success of the renormalization-group approach of Wilson in treating critical pro-perties of spin systems is the appropriate choiceof a spin-cell transformation. This is not only offundamental theoretical importance in Wilson'sapproach, but as the beautiful work of Niemeijerand van Leeuwen2 makes clear, a judicious choiceof transformation also enables one to use the re-normalization-group equation as a simple and ef-fective computational tool in evaluating criticalexponents with precision.

In this paper we introduce a one-parameter fam-ily of transformations and study the conditionsthat they must meet in order to yield a satisfactoryfixed point. We consider the transformationsmainly in the context of the one-dimensional spher-ical and Gaussian models with potential of asymp-totic inverse-power form. Thus our Hamiltonianis

-PB = g ~ Ks sg sg +As)

where in the spherical model, one has the spheri-cal constraint not present in the Gaussian model,

sg ——Q.2

Here s& is a continuous scalar spin variable, -~(s& &~, N is the number of spins, and g&& .is cho-sen to behave as an inverse power

These models offer several important advantagesas the starting point of a general study. First ofall, their exact properties are already available'4to guide us. At the same time they are highly non-trivial, and for a certain range of a, the sphericalmodel exhibits many of the same critical proper-

ties that Ising systems of higher dimensionalityshow. (We note in passing that most of the analysesof this paper can be generalized to arbitrary di-mensionality without difficulty. )

Moreover, there is a trick that enables us todirectly carry over some of our spherical-modelresults to the Ising problem. It consists of ob-serving' that the spin-spin correlation function ofthe spherical model with Hamiltonian given by Eqs.(1) and (2) is identical (for a given field h and in-verse temperature P) with that of the Ising modelwhose direct correlation function is E;~. At itscritical point, the direct correlation function of anIsing system is known to be of the form (3) for awide range of interaction potentials (including thenearest-neighbor potential in two dimensions, ) andcertain of our results can be applied immediatelyto the Ising model because of this. As might beexpected, they go over only in much weakenedform: In the spherical model the o in (3) is apotential parameter whereas on the Ising case atthe critical point it remains an undetermined ex-ponent. Nevertheless, important issues of self-consistency can be probed in the latter case.Moreover, the observation immediately reveals amuch closer connection than might otherwise beapparent between the criteria relevant to spin-celltransformations in the spherical-model case forlong-range potentials and those relevant to theIsing-model case for short mange potentials.

To our knowledge, our work represents the firstdetailed study of spin-cell transformation equa-tions in the case of long-range potentials. It isbased upon a brief note by Niemeijer and vanLeeuwen, 6 who suggested a special case of thetransformations that are treated here, and sketch-ed some anticipated results from its use in thespherical-model case. For certain reasons dis-cussed below, their transformation does not workwell and leads to difficulties unless it is suitably

5219

JOHAN S. HQYE

generalized.In Sec. II we will show that the spin-cell trans-

formation sr =qs„makes the fixed-point solutionfor the spin-correlation function both positive andnegative, which is physically untenable. Further„this transformation gives two temperaturelikeeigenvalues ~, & 1, which does not fit into the acceptedscaling picture concerning the relation between thetransformation of the Hamiltonian and the criticalindices, according to which there should be onlyone X, & 1."Owing to this we consider, inSec. III, aspin-cell transformation s,' = —,'q(s„+ s„,,) which issymmetric with respect to even- and odd-num-bered spins. One then finds that this transfor-mation gives a well-behaved fixed-point solution,and the second eigenvalue ~, &1 has changed toA., & 1. So the reason why the transformationsr = qs2; fails is related to its too unsymmetrictreatment of odd- and even-numbered spins, i.e.,

the cell spin is too far from being the averagespin of the cell. From this one can conclude thata spin-cell transformation cannot be chosen com-pletely at random if it is to work well.

In Sec. IV we consider a more general spin-celltransformation sl = —,'qj(1+a)s„+(1 —e)s„„]and

give in more detail proofs for which values of e itfails and for which it works well. It is shown thatin contrast to the case (i.e. , for values of e) inwhich it works well, the integrated-out part of thefree energy becomes singular and the fixed pointis not approached by repeated transformation whenit fails.

[it is understood that )((k) is a periodic function ink with period 2)(.J

At the fixed point jsince there K(0) =0],

p(z- fy312

Now the critical point has to do with the long-wave-length fluctuations, k-0. So the behavior of (5) inthis limit is most important. Equation (5) is sin-gular at 0 =0 at the fixed point, and as we primar-ily are interested in K;& -K&& we introduce thisquantity into (5). We get from (5)

K[-,'k J K[-',k + ~]K[-',k] +K[-,'k+((] (8)

Linear izing around the fixed point (for small k 0)we get from (8) [K(k) =1/)((k) =k'=0, K(k+(() =

1/)((k+)() is finiteJ

f)K' (k) = (2/q') ~[-.'k]. (9)

The k' in terms of k we obtain most simply via (4)and (5) utilizing the fluctuation theorem

m=)((0)k (m and k small),m' qm q)((0)k 2)((0)k

)(' (0} )(' (0) )(' (0) q[)((0)+X(&)]

1/g(h)=X(k)-Q(i-j( '"'e"' "-k'. (6)

This determines q since at the fixed pointX'(k) =X(k),

m' ={sf') =q{s, ) =qm

Xl l{I sz& = q'& 2 -&,f ) = q')(2i mf ~- (4)

II. CASE sI q~2

Let us first rederive the equations of Niemeijerand van Leeuwen. ' One notes that the integrandgiving the partition function both before andafter the spin-cell transformation must be somemultivariable Gaussian distribution, and such adistribution is fully determined by average valuesof and the correlations between the variables.Therefore a more detailed calculation can beavoided since only averages and correlations needto be considered. Kith the Niemeijer-van Leeuwenchoice of cell spins, sI =qs„. , one gets

Approaching the critical point, we find )((0)-~ and

)(()(}stays finite. From that

k' g2/q)h . (12)

.Putting ddt' (k) = )(, AK(k) and h' = X„k one findsfrom (9) and (12) the eigenvalues [with AK(k)= const. ],

)(, = 2/q = 2 ', )(„= 2/q = 2('+o) /' (13)

Instead of finding )(„ from (12}we might have found)(„=q from (4). From scaling arguments one thenhas )(.„=2/)(. = 2/q, which agrees with (13). Witha, =Ink, /ln2 =o and a„=ln)(„/In2 =-,'(1+a), thecritical indices for the Gaussian model are found(d is dimensionality}.

{)(=2 —1/ai =2 —1/o, v =1/dai=1/o',

Fourier transformation then leads to

iA {I-J') ~ ~ ail(/2( 2i-2f )XI-~ e ~ X.g~ -2g

Qg 1+0g1 —a„1—0' & +12 —R I 1y=(2-)))v=1, )3 =1+5 o

(14)

(5)

Strictly speaking the indices 5 and P (as well asa', v', and y') are not defined for the Gaussianmodel.

STUDY OF SPIN-CELL TRANSP GRMAVIONS FOR. ..To find the indices for the corresponding spher-

ical model one can no longer assume MC(k) orat

where t is deviation from critical temperature.Instead, one can make the ansatz ~(k) oo t~ since,by definition, X(0) =t " and accordingly E(k)=1/x(k) = tt +k'. This, combined with the spheri-cal constraint, then gives the critical indices ofthe spherical model [ddC'(k) = X'it~(k) -t'r =Aft&—t' =E, t -A,, =2' r].o' (This holds only for "non-classical" 6 & 3.)'

Let us now turn to the difficulties to which theuse of sI =qs2& leads. The fixed-point solution of(5), i.e. , X'(k) =X(k) =X*(k), is given by

coslkx*(k) =const ~

l =1,2, ~ o ~

(15)

which for k-0 gives X*(k)-1/k' as it should. Thismeans

const/~ i -j ~' ', i -j wOXa-g =

) 0, (16)

or

sinlkx (k)—

f =1e22 o ~ ~

(17)

is solution of the eigenvalue equation'

x X(k) =-'4'lX(-'k)+X(-'k+ )1 (18)

giving A., =2'. Another solution of this eigenvalueequation is'

cos(2' mk)X f (i —o).i

i =Os lo 2 ~ ~

Before making further comments upon the fixed-point solution we note that

coslkx(k) =

l 1222 ~ ~ ~

odd m (&0).Looking more closely at the fixed-point solution

(15) one will discover that it does not behave pro-perly. Integration of (15) gives

"x*(k)dk=O,

which obviously means that x*(k) is both positiveand negative making x*(k) physically untenable.Further (16) shows that (s, s, ) = X oo = 0, which isnot acceptable either. From this, one must con-clude that the spin-cell transformation sr = qs2&

does not lead to a renormalization-group trans-formation that behaves properly. This is furtherconfirmed by A., &1 from (21). According to thetheory for the spin-cell transformation and itsconnection with the critical indices which scale,there should be only one ~, &1.' ' Here we havetwo of them as there is also one A,, ) 1 given by(13). The solution (17) gives a continuum of. X, ,which also may be greater than 1. But on generalgrounds we can get rid of those making only thosewith A., & 1 valid solutions. The reason is asfollows: A solution with A., &1 for small k behavesas X(k)-1/k'+' (a&0), or X, f -I/(i —j ~' ' ' forlarge i —j. But such deviations from the fixedpoint are impossible for a )0 because then thedeviation decays more slowly for large i —j thandoes the fixed-point solution, and the fixed pointwould no longer be x(k)-1/k, but x(k)-1/k'".Thus we can disregard all ~, ) 1 from the continuum

spectrum.Niemeijer and van I eeuwen' were not aware of

this continuous spectrum, but they picked out ofit a discrete set, plus the eigenvalue given by (21),which was used to determine the critical indices.They found

or (19) A.i =x„=2 n=0, 1, 2, ... (22)

sin(2' mk)

l=0 ~ 1~ 2 ~ o o.o

(m is odd), with A., =(2/p)'Finally, (18) has the solution

(20)

III. CASE sr= ~q(s2. +s~. , )

Consider now the spin-cell transformation

1sf = 2'V(S2i +s2i+z) (23)

with

=21 (21)

The possibility of other solutions of (18) is still anopen question, but we assume we have got theessential ones.

Here we note that (19) is simply a subsum of(17) when p' is identified with 2' ' '. One fur-ther sees that (17) is just a sum over (19) for all

which treats odd- and even-numbered spins sym-metrically. Equation (12) giving k k' and thedetermination of q which gives (7) remain un-changed. In analogy with (4) and (5) we then get

J —(SJ Sg) —4IP((S2i S2f ) +(S2i+is2f)

+(S2i S2f+i) +(S2i+iS»+i) ) .(24)

5222 JOHAN S. HQYE 12

X' (k) =-,'q~Q( —,'k)[1+cos(—,'k)]

+X(—,'k+v)[1 —cos(,'-k)] J . (25)

Q*(k) = constZ ~ 2 ~ 0 0 4

l z+ coslk . (27)

Instead of (17) the solution of the correspondingeigenvalue equation becomes

l —Z ~ 2 ~ ~ o ~

~z+a+ a o Lk' (28)

with eigenvalues A. , =2'. The relation X(k) =Q(k)(1 —cosk) is equivalent to x„=Q„——,'(Q„„+Q„,),so (27) and (28) correspond to [ as do (15) and (17)],

X„*-1/n' ' for large n,X„-1/n' ' ' for large n.

(29)

(30)

Further this eigenvalue equation has the solution

x(k) =1,

Performing the steps corresponding to Eqs.(6)-(12) one again arrives at the same eigenvaluesas those given by (13), and the resulting criticalindices will be as before. A transformationX(k}= Q(k) (1 —cosk) of (25) will lead to

Q'(k) =mq'[Q(2k)+Q(~3k+~}] (26)

which is exactly of the same form as (5) except —,'

is replaced by —,'. Instead of (15) the fixed-pointsolution will be

(where e is a constant), thereby also investigatingthe restrictions that have to be put on e to get atransformation that behaves properly, and alsoanswering the question of why some transforma-tions fail.

Equation (32) leads to

g—(sg sg) =~q [(1+a) (ski s~i ) +(1+E)(1 e)

X ((Sgi+ iSgi) + (Sgi:Spy+i) )

+(1 -')'(s. .is.i.i)](33)

orX ( ) =-.'q'01+")[X(-.'k) X(-.'k")J

+ (1 —e')cos(ak)l X(kk) —X(2k+ii)J J, (34)

or

xn = &q [(1+~ )xa + a(1 ~ )(xgn+i+xpn-i)] ~ (35)

The value of q and the eigenvalues given by (13)will remain unchanged independent of e. The fixed-point solution of (34) and the solution of the cor-responding eigenvalue equation will asymptoticallybe as given by (29) and (30) [with X „=X„orx(-k) =x(k)J:

1 1Xn z- o ~ X.n z- a-a for large n

~n n

(36)

X*(k) 1/O', X(k) -1/k +' for small k,with eigenvalues ~, =2', and as before only a &0 isacceptable for this continuum spectrum. Finally(34) has a solution

n=0Xn

=

.0, n~0

A) =2

(31)

n=oX(k) =1 or X„=

0, ne0

with eigenvalue

(37)

IV. PROOFS IN THE MORE GENERAL SITUATION

Since we have not proven that there are no othersolutions than the ones we already have found, wecannot yet be quite sure that the transformationsz' =-,'q(s„+s„„)gives a, spin-cell transformationthat behaves properly in every respect, though itseems so. We therefore want to investigate thismore carefully, generalizing the situation some-what to a spin-cell transformation

sz ———,'q[(1 +e)s„+(1 —c)s„+,J (32)

Using the same arguments as before, one seesthat the continuum spectrum of (28) is valid onlyfor a &0, i.e., A., &I. Finally one sees that ~, of(31) is less than 1, thereby making all eigenvalues,except those given by (13), less than 1 as expected.Further the fixed-point solution now seems to be-have properly, i.e. , X*(k)& 0, and X o*= (s, s, ) & 0.

Xi =-,' q'(1+ ~') = (1+e&)2-' (38)

ff„-1/n" for large n, (39)

or

ff(k}-t +k' for small k . (40)

The parameter t describes the deviation from thecritical point, and the behavior of t-t' by a spin-cell transformation is described by Eq. (9}and theeigenvalues given by (13) which lead to the criticalindices. (We will briefly come back to t e 0 at the

We will now prove that the condition A., && 1 of (38}determines whether the spin-cell transformationgiven by (32) behaves properly or fails. The fol-lowing proofs are made under the assumption thatsolutions of the form (36) exist. [For X, = q'(1 +-')this does not hold as may be seen from (35).J

First we show that the transformation fails if ~,of (38) is greater than 1. The interaction param-eters are given by

STUDY OF SPIN-CELL TRANSFORMATIONS FOR. .. 5223

end of this section. ) We now focus our attention oncritical Hamiltonians, i.e. , K(k) -k'(for small k).All K(k) having the same o as leading power havethe same critical properties and are expected tohave the same fixed point K*(k) = 1/x*(k)- k' forsmall k.

Consider Eq. (38) for A., &1. Assume that in thiscase X*(k) has a minimum X;„=A&0. By estab-lishing a contradiction we will see that this is im-possible. Subtraction gives

gives no singular contribution to the free energy.This assumption turns out to fail here. To com-pute the Gaussian model one has to perform in-tegrals like

exp — A k

To perform the spin-cell transformation (32} ex-plicity, i.e. , perform integrations such that slremain fixed, one has to perform an integral like

x~(k) =x*(k) -A -0. (41)

One then puts expr ession (41) into (34) which via(38) gives

x~(k) =x*(k) —x&A . (42)

X„(k)= X(k) —A & 0 for all k . (43)

This we put into Eq. (34) repeating the transfor-mation mtimes. By that we get

o' X~(k) =X'(k) —xP& .Accordingly,

X'(k) & A,, A „=~ for all k,

(44}

(45)

Since A & 0 and A., & 1, X„'(k) of (42) must be lessthan 0 for some k. Inspecting Eq. (34) one seesthat if X(k}&0 for all k [i.e. , (41) holds] thenx' (k) &0 for all k also. Equation (42) contradictsthis. Thus for A., =(1+F2)2 &1, X*(k)&0 for somek. [We show later that minimum of X*(k) cannot bezero. j So in this case X*(k) will be physically un-tenable.

Next we show that with A, &1 the fixed point X*(k)will never be approached by repeated spin-celltransformation if one starts with a physically ac-ceptable critical Hamiltonian, i.e., x(k) 0. Ifx(k) &0 then (34) shows that x'(k) &0. Thereforex'(k) by repeated transformation never can ap-proach x*(k), since x*(k)&0 for some k. Insteadby repeated transformation X'(k)-~ for all k. Thisis shown as follows.

One has to start with x(k) & 0. From this one cansubtract a positive constant A &0 such that

+h(k)a)IIda, ,

x~(k) =x*(k) +& -0 (47)

Expression (47) is then put into (34) which via (38}then gives

X~ (k) =X*(k)+xg& . (48)

Since A & 0 and A., & 1, X„-' (k} of (48) must be lessthan 0 for some k when A is chosen appropriately.But this contradicts that x„'(k) has to be greater orequal to zero as can be seen by inspecting (34).Thus

which one would expect gives no singular contri-bution to the free energy. This also is true for thefirst transformation since f (k)K(-,') +g(k)K(-', + m) & 0for all k (also k =0) for a critical K(k). But thefact that this holds for the first transformationdoes not necessarily mean that it holds for anarbitrary number of repeated transformations.From (46) one sees that by repeated transformationthis integral becomes more and more singularsince, if X, =(1+a')2 '&1 then f(k)K'(2k)+g(k)K'(2k+ v) -0 for all k, thus making the assumption fail.

Consider now Eq. (38) for x, &1. We will showthat x*(k}& 0 for all k in this case, which meansthat X*(k) is physically acceptable. Let us assumethis is not true, but that x*(k)&0 for some k. Thenwe can add a positive constant A such that

X*(k) &0 for all k . (49)

K'(k) = 1/X'(k) „=0 for all k . (48)

Now we are in a position to see which basic as-sumption fails when e in (32) is such that A.,=(1+a')2 '&1. According to the theory for theconnection between the spin-cell transformationand the critical indices, this ~, should be lessthan 1 since we already have A. , given by (13)greater than 1.

One assumption under which the theory is de-rived is that the part of the free energy that is in-tegrated out during the spin-cell transformation

x*(k)& 0 for all k,when ~, &1.

In the same way one can show that

X~(k) &0 for all k,

(50)

(51)

where Xz(k)-1/k'" for small k is a solution of

Next we want to show that this x*(k) does notequal zero for any:k. For small k, x*(k)-1/k'.Thus X*(k)&0 for ~k~ &o where 5&0. If (49) holds,use of (34) then shows that X*(k)& 0 for ~k j

& 2"6(ns =1, 2, 3, . . . ), and by that

5224 JOHAN S. H /AYE

the corresponding eigenvalue equation with eigen-value ~ =, 2' and ~ is restricted to ~, & ~.

Any critical Hamiltonian which for small k be-haves .as K(k)-k', therefore, may be written as

X*(k) -AX) (k)'X(k) =1«(k) X*(k)+AX) (k), (52)

where A&0 and A., &X&1. Repeated use of (34) mtimes then gives

X*(k) -»"Xk(k)&X'(k)'X*(k)+» Xx(k)

Accordingly,

(53)

~(k) =X(k)/[X*(k)J' -t for small k, (55)

or

»m X'(k)/X" (k) =1 . (54)

Therefore if A., of (38}is less than one, K'(k) byrepeated transformation converges to the fixed-point solution K*(k)& 0 (k t 0), but not if h., & 1.[For A, , =1 there is no fixed point solution exceptx*(k) = const as can be seen from Eq. (35}, i.e. ,X„*=0 for nw0, and the transformation X(k}-X'(k)will not converge. ]

Here it should be pointed out that each eigenvalueof the continuum is degenerate with solutions sim-ilar to (19). This means that apart from a multi-plicative constant there will be an infinite numberof fixed points, not just one only. But all solutionssimilar to (19) are ruled out, They will not bephysically acceptable since lim„„n' ' 'g„-both 0and const' 0. The only fixed point a physicalHamiltonian can converge to is the one (and therecan be no more than one since the convergence isunique) for which lim„„n' 'X„*-const. Thus result(54} is not ambiguous since it is impossible towrite a physically acceptable Hamiltonian in theform (52) with an unphysical X*(k) and a A. &1.

Let us finally make some comments on ~, givenby (13). This X, describes deviations from a criti-cal Hamiltonian and as we have shown it is the only~, & I for a spin-cell transformation that workswell. To obtain it one has to introduce K(k) =1/x(k)into Eq. (34) and linearize around the fixed point.The resulting equation one gets by substituting X

in (34} with X*'b K. A solution of this equation isthen obviously EK(k) = X(k)/X*'(k). All AK(k)[or ax(k)J that keeps K(k) critical we have alreadytreated. The only variation which we have nottreated in more detail is deviation from a criticalone. A noncritical K(k) can always be written as acritical one plus a constant f, i.e. , K(k) =f +k +. . . .Thus the only possibility is

X(k}=tX*(k)'-t/k' for small k . (56)

The corresponding eigenvalue thus becomes(2c = o + a} X, = 2' = 2'& 1 as given by (13).

A conclusion of this analysis is that one mustkeep certain restrictions on the spin-cell trans-formation (32), i.e. , the cell spin must correspondto something that does not deviate too much frombeing the average spin of the cell. If it deviatestoo much, the basic assumption that the part whichis integrated out gives a regular contribution tothe free energy will fail.

The spin-cell transformation for the Gaussianmodel can be easily generalized to higher dimen-sions by proper extension of Eqs. (32)-(34).

Aa mentioned in Sec. I, the spin-cell trans-formation investigated here has a wider aspectnot limited to the Gaussian and spherical models.This is seen from the derivation of Eqs. (4) and(33)-(35). These equations are valid in generalwhen a spin-cell transformation given by (32) isused, since the spin correlation function entersthe equations in a way that is independent of theunderlying model, whether this is Gaussian, Ising,or something else; e.g. , if the Ising model has acritical spin correlation function that decays assome power, this function may be considered asbelonging to a Gaussian or spherical model, andthe results of our investigation concerning the be-havior of the spin correlation function under thespin-cell transformation is applicable (when gen-eralized to higher dimensionality if necessary).As long as (10) holds one notes that X„via thestandard scaling relations [as given by (14)] willquite generally give a self-consistent determina-tion (i.e. , give the correct value} of the criticalexponent q which is directly given by the decay ofthe spin correlation function.

ACKNOW LEDGMENTS

We are deeply grateful to T. Niemeijer for hisencouragement, and for the interest and patiencethat he has shown in the course of a number of veryhelpful discussions. We are also indebted to P.Hemmer for his aid via an extended correspondenceconcerning this work and to G. Stell for valuablecomments emphasizing the more general aspectsof this work. Acknowledgment is made to theNational Science Foundation, to the NorwegianResearch Council for Science and Humanities, andto the donors of the Petroleum Research Fund,administered by the American Chemical Society,for support of this research.

12 STUDY OF SPIN-CELL TRANSFORMATIONS FOR. .. 52Z5

*Present address: Institutt for Teoretisk Fysikk, 7034Trondheim-NTH, Norway.

K. G. Wilson, Phys. Rev. B 4, 3174 {1971);4, 3184(1971).

2T. Niemeijer and J. M. F. van Leeuwen, Physica (Utr. )71, 17 (1974).

T. H. Berlin and M. Kac, Phys. Rev. 86, 821 (1952).4G. S. Joyce, Phys. Rev. 146, 349 {1966).5G. Stell, Phys. Rev. 184, 135 (1969).

T. Niemeijer and J. M. F. van Leeuwen, Phys. Lett. A4]., 211 {1972). Also T. Niemeijer, lecture notes 1973(unpublished) .

tThis is because a regular nonscaiing part of JX(k)dk,which enters the analysis via the spherical constraint,dominates when 0& 0 & 2. See Refs. 4 and 5 for techni-cal details.

T. Niemeijer and P. C. Hemmer {private communica-tion).