magnetic properties of the one-dimensional heisenberg antiferromagnet tetraphenylverdazyl

ELSEVIER Journal of Magnetism and Magnetic Materials 150 (1995) 165-174

~ Journal of mn~g netlsm magnetic

~ I ~ materials

Magnetic properties of the one-dimensional Heisenberg antiferromagnet tetraphenylverdazyl

Bernd Pilawa *, Thomas Pietrus Physikalisches lnstitut, Universitiit Karlsruhe (TH), POB 6980, D-76128 Karlsruhe. Germany

Received 14 April 1994; in revised form 14 February 1995

Abstract

(1,3,5,6)-tetraphenylverdazyl (Ph-TPV) is an organic Heisenberg antiferromagnet with quasi-one-dimensional properties. We measured the specific heat and electron spin resonance and determined an intrachain exchange constant of J ~ j / k B = - 17 K and the onset of magnetic long-range order at Tr~ = 0.92 K.

1. Introduction

Organic materials with interesting magnetic properties are an area of intense current research activity. Recently, different verdazyls were characterised and it was found that (l,3,5,6)-tetraphenylverdazyl (Ph- TPV) is a quasi-one-dimensional Heisenberg antiferromagnet. By an analysis of the measured static magnetic susceptibility an intrachain exchange constant of J i d / k B = - 1 6 K was determined [1,2]. However, no transition to a magnetically ordered phase was observed for temperatures above 2 K. Therefore we measured the specific heat between 0.1 and 3 K and found a N6el temperature of T N = 0.92 K. The small ratio between T N and Jld indicates that the magnetic properties of Ph-TPV can be well described by one-dimensional isotropic Heisenberg exchange interaction. In order to study one-dimensional spin correlation functions and the interaction between the delocalised magnetic moments of the

* Corresponding author.

Ph-TPV molecules, we measured the electron spin resonance (ESR) of small single crystals between 1.6 K and room temperature. Finally, we tried to estimate the exchange constants between the molecules by combining McConnell 's formula [3] with molecu- lar orbital calculations.

2. Experimental details

The organic radical Ph-TPV is obtained by adding one phenyl ring to the well-known free radical (1,3,5)-triphenylverdazyl (TPV) [4,5]. The spin density of the unpaired w-electron is mostly located on the central verdazyl ring (80%). The remaining spin density is distributed over the four phenyl rings [1].

Ph-TPV crystallises in an orthorhombic structure ( a × b × c = 11 .548x 12 .969× 13.737 ~3, space group Pmna). The molecules form stacks with strong exchange interaction along the crystallographic a-direction. More details on atomic positions, crystal structure and sample preparation are given by Dor- mann et al. [2]. The typical size of the crystals was

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166 B. Pilawa, T. Pietrus / Journal of Magnetism and Magnetic Materials 150 (1995) 165-174

100

10 -~

o

1 0 -2

' i . . . . . . . i d 0 s

0 0 ROe ~ ~°~° A 30 kOe (~ 8°~ C 60 kOe

Y

, ~ . . . . . . . . ,

0.1 1 T / K

Fig. 1. Specific heat of Ph-TPV between 0.1 and 3 K for 0, 30 and 60 kOe magnetic field strengths.

0.12 × 0.12 × 0.25 mm 3. The biggest ones were about 1 mm long and 0.2 mm thick.

The specific heat was measured between 0.1 and 3 K. 13.09 mg polycrystalline Ph-TPV was placed between two Si plates, which were covered with Apiezon N grease. Care was taken not to damage the crystals. The measurements were carried out on a dilution refrigerator [6]. Magnetic fields up to 60 kOe could be applied.

X-band ESR (9.5 GHz) was recorded with a Bruker ESP300E spectrometer. An Oxford ESR900 cryostat was used between room temperatures and 4.5 K. Lower temperatures could be achieved with a home-built cryostat.

to describe the spin correlations. Since at low temperatures the correlation length is by a factor of two larger for the X - Y model [7] than for the Heisenberg model [8], T N increases under an applied magnetic field. The observed shift is appreciably smaller than the value expected for classical spins.

Below T N the specific heat at zero magnetic field decreases proportional to T '~ ( a = 2.5 ___ 0.05) with decreasing temperature. This is just intermediate between a = 3 and 2, which may be expected for three- and two-dimensional antiferromagnetic order- ing, respectively. A Schottky anomaly due to the hyperfine and Zeeman interaction of the protons leads to an increase in the specific heat at low temperatures.

At temperatures just above T N the magnetic contribution to the specific heat of one-dimensional antiferromagnetic chains [9] c M = 0.35NAkaT/IJid I and the specific heat of the lattice c L = 234NAkB(T/OD) 3 (e.g. Ref. [10]) have to be consid- ered. Fig. 2 shows the specific heat together with a fit of c L and c M. We obtained 0 D = 5 3 K and I Jld I/kB = 17 K, which agrees roughly with t Jld I determined by magnetic susceptibility measurements.

The measurements of the specific heat prove that Ph-TPV is an extremely one-dimensional compound. From (kaTN) 2 = I J I d ' J ' l [11,12] the small ratio between the inter- and intrachain exchange J'/J~d = 0.003 can be estimated. The ratio of r = kaTN/2] JId I S(S + 1) = 0.036 for Ph-TPV is even

3. Specific heat

Fig. 1 shows a double logarithmic plot of the specific heat measured at magnetic field strengths of 0, 30 and 60 kOe. For zero field, a smooth anomaly at T N = 0.92 K indicates the transition from the one-dimensional antiferromagnetically correlated behaviour to a magnetically ordered phase. The transition shifts to higher temperature, if a magnetic field is applied. At 30 and 60 kOe the temperature of the anomaly is 1.06 and 1.12 K, respectively. A simple picture may explain this. A magnetic field introduces an additional anisotropy, so that with increasing magnetic field, the X - Y model becomes more and more appropriate than the isotropic Heisenberg model

1 . 0 . . . . i . . . . i . . . . i

0.8 e ~ 0.6

o E u 0.4 (a)

0.0 ~ 0 1 2 3

T/K

Fig. 2. Specific heat of Ph-TPV between 0.1 and 3 K at zero magnetic field. Solid lines: (a) lattice contribution; (b) sum of lattice and magnetic contributions to the specific heat.

B. Pilawa, T. Pietrus / Journal of Magnetism and Magnetic Materials 150 (I 995) 165-174 167

smaller than the corresponding ratios found for one- dimensional spin S = 1 /2 Heisenberg antiferromagnets containing Cu 2÷ ions (e.g. for Cu(NH3)4SO 4 • H20 r = 0.091 and for CuCI z • 2NCsH 5 r-~ 0.056 [12]).

4. ESR

4.1. Intensity of the ESR resonance curve

The area under the ESR resonance curve is proportional to the static magnetic susceptibility. The ESR intensity was obtained by numerical integration of the signal. Fig. 3 shows the experimental data. The intensity is normalised at 200 K, assuming a Curie-Weiss law with O = - 1 7 K. The maximum of the measured curve and the extrapolated value of the intensity at T = 0 K agrees well with the predic- tions of Bonner and Fisher [9] with Jld = --17 K: X . . . . = O.07346NA(gtxB)2/IJIa I = 6.46 X 10 -3 emu/mol and x ( T = 0 K) = O.05066NA(gI~B)2/ I J~a I = 4.46 × 10 -3 emu/mol , respectively. Fig. 3 also contains the numerically calculated susceptibility of a one-dimensional Heisenberg antiferromagnet [9]. I J]a I/kB was 17 K and the magnetic field 3365 Oe. Down to 10 K there is a very good agreement between experiment and calculation. Owing to the periodic boundary condition used in the calculation, the magnetic susceptibility diverges at zero tempera-

. . . . I . . . . I ' ' ' ' I

o 0 1 1 o • • • •

8 , ~ , , I , , , L I J , , , I

0 100 208 300 T /K

Fig. 4. Difference of the resonance field HII-H ± at 9.44 GHz compared with two calculations. Solid lines: (a) H± applied parallel to the b-direction; (b) H± applied parallel to the c-direction.

ture if an odd number of spins is used and ap- proaches zero for an even number of spins. The calculation was carried out with periodic boundary conditions of 8 and 9 spins, respectively. The large increase in the measured susceptibility below 3 K indicates the presence of broken spin chains, due to crystallographic defects or Ph-TPV molecules that have lost their unpaired electrons. From the increase we estimate that there are ~ 1% of paramagnetic defect spins. Therefore the mean number of one-dimensionally correlated spins is ~ 100.

4.2. Shift of the resonance field with temperature

. . . . i . . . . i . . . .

- - o N = 9 8 t . . . . . . t

E

~ 4

}.

0 100 290 380 T / K

Fig. 3. Intensity of the ESR absorption signal. Solid lines: calculations of the static magnetic susceptibility with periodic boundary conditions of 8 and 9 spins, respectively. Inset: intensity of the ESR signal at low temperatures.

The magnetic resonance field H 0 depends on the direction of the applied magnetic field, due to the anisotropy of the g-factor, the demagnetisation, the interactions between the radicals (e.g. magnetic dipole interaction, anisotropic exchange interaction) and the onset of magnetic short-range order. Fig. 4 shows the difference between the magnetic resonance field applied either parallel (HIt) or perpendicular ( H ± ) to the chain direction in the temperature range between 4.5 K ( n i l - H±) = 0.33 Oe and 300 K (Hll - H_L) = 1.81 Oe.

From the difference H I I - H± and its temperature-dependent variation the anisotropy of g and the anisotropy of the susceptibilities XI I - X ± can be determined. The magnetic resonance field is given by the first moment of the resonance line f(oo)


f~_~e- i~ ' (S+(t )S - ) dt:

1 <[H, S+]S -> Ho = % / 3 ' ° =

~,o h ( S + S - )

With ([H, S+]S - ) = kaT([S +, S - ] ) , the susceptibilities perpendicular and parallel to the magnetic field,

1 ( g± /Za ) 2 glItZB(S z)

X-+= 2 kBTV (S+S-)' XII= VH o '

the difference (HII - H ± ) / H o is [ 13]

HII - H ±

H0

{ 3 XII-XJ- } _ g ±g-gll x ( N ± -Nil ) 2 X "

(1)

S -+ denotes the total spin S + = ~ i s i 5: , and H 0, g and X the respective mean values. The Hamiltonian H = Hex + Hz¢ ~ + HdiooJe and its parameters are given in the Appendix. Assuming Nil = 0 and N± = 2at for a needle-shaped crystal, the anisotropy in g at room temperature is (g ± - g l l ) / g = (1.16 _ 0.08) × 10-3, and down to about 50 K there is no anisotropy in X. Below 50 K a small anisotropy develops with the value XII - X± = (1.26 -t- 0.I) × 10 -6 emu/mol at 4.5 K. This small difference can be attributed to the magnetic dipole-dipole interaction between one-dimensionally correlated spins. For a quantita- tive analysis of the anisotropy of the magnetic resonance field strength the first moment of f ( w ) has to be calculated. However, this is difficult, since the first moment f(¢o) is given by a sum of three spin correlation functions. Instead of this, it is much easier to calculate the first moment of F (oJ )= wf (w) . Nagata [14] found that the first moment of F(¢o) is given by the static two-spin correlation functions:

<[[H, S+]S-]>

([s+s-]>

= g t x a I H l + 3 h w a ~.~Ai ° ( s Z s f - s X s f ) i , j 2 (SZ) (2)

The geometrical factors Aqj and oJ d = 16.86 × 108 rad s-] are given in the Appendix. The connection

between the first moment of F ( w ) and the first moment of f(o~) is [13]:

([[ H, S+ ] S - ] ) ( [ S + S - ] ) ( ~ ' ~ 2 ) =hO, o

Therefore Eq. (2) can be used to calculate the difference H u H " , when the second moment,

0 2 = h2(S+S _ )

= ] - s ( s + 1)o~ ff ~ l A ° i l z I

+10E I a'l,12 + E I m~,12), i i

is small compared with ~o~. With the geometrical coefficients for Ph-TPV, Eil A°il 2 + 10Eil Allil 2 + Ei l A~I 2 = 4.5, we find that the ratio g2Jwo 2 = 2 × 10 -3 is really small, and a comparison of Eq. (2) with experiment is possible. Due to the properties of the factors A~ ° , the shift of the resonance line is essentially given by the static nearest-neighbour correlation function. We calculate the static nearest- neighbour correlation function (sZsZ+ ] - s~' si+ j ) / (s/z) numerically, and use periodic boundary conditions of 8 and 9 spins, in order to determine the eigenvalues and eigenfunctions of Hex. Fig. 5 shows ( s Z s ~ + l - s / ' s i + l ) / ( s ~) and (s/Z). Above 50 K, there is nearly no difference between the two func-

o o00tl .0.002I~ ~ m ~

-0.006 i i 0 1 ~ 2O0 30O

TIE

Fig. 5. Temperature dependence of expectation values for a magnetic field of 3365 Oe, calculated with periodic boundary conditions of 8 and 9 spins, respectively. Solid lines: (a) ( s [ ) ; (b) ( sfsL, - s?sL, )/¢, sl).

B. Pilawa, T. Pietrus / Journal of Magnetism and Magnetic Materials 150 (1995) 165-174 169

tions, and at lower temperatures the nearest- neighbour correlation deviates from ( s z). From Eq. (2) we get for the difference HII - H±

• g~B HI,- H± --- H o g ± g - g'-~--21 + (N± -NIl) ~ ( s : )

- - _ _ o ~ _AO't)(s z) 3 w d E (All li + 2 Ye /,(inter)

+ 3 w d 0 ~L 011 (s(s - si% (s()

(3) The first term on the left gives the anisotropy in g with H 0 = 3365 Oe. The second term accounts for the influence of the demagnetisation with (N±

-Nii)[g/xB/(VcelJ4)] ~ 226 Oe (Nil = 0 and N± - 2"rr). The contributions of the interchain and intrachain interaction are represented by the third and last

.011 0 ± terms, respectively. The factors ,4,ij and Aij were calculated for magnetic fields applied parallel and perpendicular to the direction of the magnetic chains, respectively. The sum ]~i(a°/~ - a ° l ) depends on the direction of H ± . It includes all molecules on the neighbouring chains of molecule 1. When the magnetic field is applied parallel to the crystallographic

0 ± 0 II b- or c-direction the values of Y'.i(Ali - A l l ) are -2 .01 + 0.03 and -3 .41 5: 0.03, respectively. The intrachain contributions of each of the two nearest-

0 • -- zlO II neighbour molecules (AI2 . .12z= 1.4. Fig. 4 shows that the experimental results are well approximated by the two calculations for H± applied either

1.0

,¢: 0.S • • 0 • • 0 0 0 0 0 •

0.8

, I I I I i I

-50 0 50 100 150 200 angle In degree

Fig. 6. Angular variation of the line w i d t h at room temperature. 0 °

corresponds to H II a.

2.0

1.6

.¢

1.2

0.8 •

0 100 200 300

T / K

Fig. 7. Temperature dependence of the line width (dots) between 4.5 and 270 K compared with 1/xT (solid line, arbitrary units).

parallel to the crystallographic b- and c-directions, respectively.

4.3. Line shape and line width

4.3.1. Experimental results For Ph-TPV a Lorentzian ESR line is observed at

all temperatures. Fig. 6 shows the anisotropy of the line width at room temperature. When the magnetic field is turned in a plane containing the crystallographic a-direction, the line width (peak to peak) is maximal (0.99 + 0.01) Oe for H ]l a, and minimal (0.89-I-0.01) Oe at the magic angle ( H / _ a ~-54°). A second maximum (0.93 + 0.01) Oe is observed for H _1_ a. So the line width is nearly the same for all field directions, but with a small anisotropy, which can be well approximated by a ( 1 - 3 cos2(O)) z law. Fig. 7 shows the temperature dependence of the line width, with the magnetic field is applied at the magic angle. The line width decreases from 0.89 Oe at room temperature to 0.80 Oe at about 140 K. At lower temperatures the line width increases, when the temperature is further reduced. At 1.6 K the line width is 2.98 Oe.

4.3.2. Discussion of the line width A general description of an ESR line f(o~) is

M ' ( w ) f ( o J ) =

( w - - w 0 + M " ( w ) ) 2 + M ' ( . , ) 2 ' (4)


In many cases the function M(o~) is given by the Laplace transformation of the total spin torque correlation function [15]:

M(w) = M'(6o) + iM"(~o)

= f e-i~°t(eiHt/h[Hss,S+]e -im/h ~0

× I s - , Hss ]) dt/h2(S+S-),

with /-/o~ = H - Hze e. The real part of M(to) gives the width A H = A t o / % of the ESR line. With Hs~ =/-/dipole and si±(t) = eilt*~t/hsi±e -H`'t/h, one gets

'(:o A Hdipol e = ~ R e dte-i(o~- .,o)t %

(s~-(t)sT(t)s'~s[> x E E 2(s+s_>

i~=j k~l

3 Ai jAk! + 2 A i j Akl o o + , - ,

× (2ei,~0t + 3 e - i ~,o,)

From ESR measurements of the Ph-TPV molecule solved in toluol [1], the isotropic hyperfine constant Akj/h = 10 s rad s -1 ( 5 . 8 0 e ) is obtained. With Y'.k½i(i + 1)1Akgl 2 = 2.78 × 1016 s -2 ( i = 1) the ratio Ek½i(i + 1)1 2 2 Akj[ / % -~ 10 - 2 is small, and the contributions of hyperfine interactions to the line width can be neglected in comparison with the dipole-dipole interaction of the electrons.

The shape and the width of an ESR line depend essentially on the correlation functions fi /kl(t)= ( s+(t)sT(t)s~ s[ ). If there is three-dimensional exchange narrowing, f iy(t) decays within times t << (IJld I/h) -I. The correlation function can be a.p- proximated by fijkl(t) = 2(2S(S + 1))28ik aile -(2J,dt)',

I and the Fourier transform for spin s = ~ is

fjkt( W) = f~fijkt( t) e-~'°t dt

-~- ~ ~ ~ / 4 e-I/4(°J/2JId/h)2 ik JI2Jid/h

Since 21Jld I/h>> % , M'(o~) is constant in the range A to = 3,~A Hdipol, around o% and the line shape is Lorentzian. The line width becomes

_+2_-2 -2,oo,)} Ai i Akl e (5)

In order to estimate the contribution of hyperfine interactions to the line width AHhyperfin e, we calculate the isotropic hyperfine interaction between the nuclear spins of the four nitrogen atoms and the spin of the unpaired electron. This hyperfine interaction is the most important one, since 80% of the spin density is located at the nitrogen atoms. With Hss = H r = ~k..iAkiik$/, the contribution to the line width is

A Hhyperfine

(sT( t )s ] - > =lSe / f ° dte-i('°-~°o)t} '', (S+S_) v~ ~ j,k

--31 i(i + l ) lAkj [2(1 +e - i '% ' )} (6)

m ndipol e

1 _/--¢ 3s( + l) o,2 I

3/e V 4 4 2JId/h

×{Y'/lA°il2+lOY'lAlli l2+ }-' I Al2i I 2). • i i

(7)

For H applied parallel to the magnetic chains, the value of {Y'-i I A°i 1 2 + 10~ I a11i 1 2 + E i I a ~ I 2} iS 4.5, and for H applied perpendicular to the chains 4.2. With 2 I JJd I/h = 4.2 × 1012 rad s-1 the calculated line width of A H = 0.08 G is only one-tenth of the whole line width. So there is some broadening.

ESR lines of quasi-one-dimensional antiferromagnets can be broadened due to spin diffusion along the chains [16]. If there is spin diffusion the correlation functions decay over long times t >> (I Jid ]/h)-I


according to l/v~" [17], and the line width becomes [15,18]

A Hdipole

= - - r l 3/2 Re A2,intra + Ao.inter) Ye

1 2 × ~/i(to- to0) + 2 r + (al'intra+a2'inter)

4 6 × +

~/i( ,o - 2 to0) + 2 r ¢ i to+ 2 r

' 1 +(2 , ( 8 )

Az'intra + A2'inter) ~/i( to+ too) + 2 F

with

3 s ( s + 1) to2 "1-/3/2

4 4 ( 2 f i - 0 ~ '

r /= 4.0 × 107 rad s- 1

and the geometrical coefficients A2qintra,imer = I E~A~kl 2. D is the diffusion constant D =

z 1.151Jlal/h. For Aq,intr a the sum runs over all molecules of chain 1. A2q,inter includes all interchain contributions. The cut-off F terminates the one-dimensional spin diffusion and avoids the divergence of the line width. In nearly perfect one-dimensional magnets, the cut-off is simply given by the line width F = y~AHaipol~ [15]. The width and the shape of the ESR line can be determined by solving Eq. (8). When the magnetic field is applied parallel to the a-direction, 2 2 (Ao,intr a + Ao,inte r) = 10, whereas the other geometric coefficients can be neglected. The line width A Haipom~ of 8.4 G for H II a is ten times larger than the experimental value. The line width can be reduced by increasing the cut-off. Interchain interactions, impurities or other defects of the crystal increase the cut-off [12,18]. With F~- 16 × 10 9 rad s -~ the line width is reduced to 0.8 Oe for H applied parallel to the chains. Considering this high value of the cut-off, it becomes obvious that M'(to) is constant at to = to o and no deviations from the Lorentzian line shape can be expected. However, two observations indicate that the theory of one-dimensional spin diffusion is not appropriate to de-

scribe the ESR line of Ph-TPV at high temperatures. First, the interchain exchange interaction estimated from the cut-off I J ' / J l d I = ( F / [ Jla I)3/4 =0.03 [12] is considerably smaller than the estimate based on the Ntel temperature I J'/JId I = 0.003. Second, when the magnetic field is not applied parallel to the chain direction, the calculated line width decreases continuously to 0.3 Oe for H Za. This large anisotropy of the line width predicted by spin diffusion contradicts the experiment.

When neither the short time range, nor the long time range is applicable to describe the experiment, the decay of the correlation functions during intermediate times t = (I Jld I/h)-~ has to be consid- ered. Since no analytical formulae are available for the correlation functions in this time range, they have to be calculated numerically.

From the correlation functions j~y ( to ) = f~fi jkt(t)e -i~' dt we calculated the nearest- neighbour intrachain correlation function

EJ~i+ lkk+ 1(O.)) ik

oo = E f e-i~t(s+(t)s++l(t)SkSk+l) dt

ik - o0

at to = 0 and compared its temperature dependence with experiment [19,20]. The calculation is carried out as follows:

ik

(1 I n m ik

with Anm(i)= ( n l s + s + + l l m ) and Hex In ) = Enln). Hcx is solved with a periodic boundary condition of 8 and 9 spins s = 1/2, respectively. Strictly, the sum should be restricted through the condition E~ =Era, but in order to get a better average all matrix elements with ] E~ - E m ] < 0.2 JId were included. The finite number of matrix elements due to the boundary condition of 8 or 9 spins avoids the divergence expected for the correlation functions of ideal spin chains at to = 0.

The variation of A H with temperature is determined by the temperature dependence of both the static correlation function (S+S - ) cxxT and the

172 B. Pilawa, T. Pietrus // Journal of Magnetism and Magnetic Materials 150 (1995) 165-174

dynamic correlation functions j~jkt(to). The comparison of AH with 1/xT (see Fig. 7) shows that there is a considerable variation of the ~jkt(to) with temperature. In order to compare the variation of J~jkt(t°) with experiment, Eq. (5) can be rewritten as follows:

3 AHxTct E ~ , , - s ( s + 1)toE( 0 0 ~ A i j A k l f i j k l ( O )

i,~i k~t 4 +1 - I ~ + lOAij A~/fijkt(tOo)

+2 - 2 ~ +aij ak/ f,.jk,(2tOo)). (9)

The contribution of the nearest-neighbour intrachain interaction to the right side of Eq. (9) is

+ 1) ik

xtO (I ° I Aii+l + 10l Aii l I + I A i i+ , I 2) if one assumes

EJ~,+,k~+l(0) -- EJ~i+,kk+t(tO0) ik ik

--- E f i + , k k + l(2to0). ik

Fig. 8 shows the comparison of the experimental AHxT and ~,i~'fii+lkk+l(tO = 0). For temperatures below 200 K, the behaviour of ~ikfi+lkk+l(tO = O) is similar tO that of the experimental AHxT data. Above 200 K the increase in AHxT may be caused by a small decrease in the exchange constant Jzd due

(s)

o., o° o.,

0 . 0 , . , i . . . . i , I

0 1 O 0 2 0 0 3 0 0

T / K

Fig. 8. Temperature dependence of A H x T (dots) compared with the nearest-neighbour intrachain correlation function. The calculations are carried out with boundary conditions of 8 (a) and 9 (b) spins, respectively.

to a thermal expansion of the lattice. The angular variation of ([ 0 2 l [2 2) Aii+ 11 + 10 [ Aii+ 1 + [ 2 Aii+ l [ cannot explain the anisotropy of the line width. Therefore further correlation functions (e.g. the nearest-neighbour interchain correlation), have to be analysed in order to account quantitatively for the observed line width. However, the comparison between the experiment and the calculated temperature dependence o f ~~'ikJ~i+ I kk + 1 ( to = 0 ) shows that one- dimensional correlation influences the dynamic spin correlation functions of Ph-TPV.

5. Estimate of exchange integrals

A simple formula to predict the exchange constant between neighbouring molecules would facilitate the search for ferromagnetic order in organic solids. McConnell [3] proposed the Hamiltonian

Hi,y = - - s i s j E ~al(i'j)r*(i)r'(J),fl P'o' Pfl (a,/3)

to estimate the interaction between two molecules i and j with delocalized 7r-spin densities. We applied this Hamiltonian to calculate the exchange constants between neighbouring molecules of Ph-TPV. In order to approximate the interatomic exchange constants l(i.j) we replaced the atoms a and fl by methyl radicals. The orbitals of the unpaired w-electrons were adjusted perpendicular to the plane formed by the nearest-neighbour atoms. The dependence of the exchange constant l(i'J) • "~,,t3 on the distance and the relative orientation of the atomic w-orbitals was estimated by calculating the energy difference between the singlet and triplet state of pairs formed by two methyl radicals. An unrestricted Hartree-Fock calculation was carried out applying the program Turbomole of Ahirichs et al. [21]. The exchange constant between the nearest-neighbour molecules in the a-direction is ( -1 .61 _ 0.08) K. In agreement with the experiment there is an antiferromagnetic interaction but the experimental value is roughly ten times larger than the estimated one. All other exchange constants are negligibly small, with the ex- ception of the nearest-neighbour molecule in the b-direction. We obtained for this exchange constant (0.684 + 0.001) K. Contrary to experiment, this pre-


dicts a two-dimensional magnetic structure of Ph- TPV. There may be two reasons why the estimation of exchange constants fails. First, the approximation of the wave function of the Ph-TPV radical by w-orbitals of methyl radicals may not be adequate. Second, small changes in the intermolecular dis- tances and orientations can seriously change the values of the interatomic exchange constants.

6. Conclusions

Both specific heat and ESR confirm that Ph-TPV is a quasi one-dimensional Heisenberg antiferromagnet with an one-dimensional exchange constant J~d/k B = --17 K. The static magnetic susceptibility follows the behaviour of one-dimensionally correlated spins for temperatures down to 3 K. At lower temperatures there is the paramagnetism of defect spins due to broken spin chains. The angular variation of the magnetic resonance field confirms the high isotropy of the magnetic susceptibility. Its small anisotropy below 50 K is caused by the magnetic dipole-dipole interaction of one-dimensionally correlated spins. The anisotropy of the line width could not be explained by the slow l /x /7 decay of the dynamic spin correlation function, which is expected for one-dimensional antiferromagnets. The interchain interaction estimated from the N~el temperature ( J ' / J I d ~ 0.003) is too small to cause an effective cut-off of the one-dimensional spin diffusion. Proba- bly, the slow decay of the dynamic correlation is disturbed by defects of the spin chains. The one-dimensional character of the spin dynamic of Ph-TPV shows up in the temperature dependence of the line width. This is confirmed by a numerical calculation of the dynamic nearest-neighbour correlation function.

Acknowledgements

We thank H. Naarmann for sample preparation and R. Ahlrichs, E. Dormann, H. von LiShneysen and H. Winter for stimulating discussions. One of the authors (B.P.) is especially indebted to J.B. Boucher and H. Benner for many discussions and the opportu-

nity to prepare a part of this work in their laborato- ries. This work was supported financially by the Bundesministerium fiir Forschung und Technologie as part of the project 03M4067-6.

Appendix

In this Appendix we put together all the Hamilto- nians used in this work.

The one-dimensional Heisenberg exchange Ham- iltonian: Hex = - -2J id~ , iS iS i+ I with 2 J i d / h = --4.2 × 1012 rad s - l .

The Zeeman Hamiltonian: H z e e = hTeHS = g txBHS with the gyromagnetic ratio 7e = 1.76 × 1 0 7

rad s -l Oe -l and the total spin S = E~s i. The magnetic dipole-dipole interaction: Hdipol~ =

h~,2= _ 2dM, with

1 = Z i j ( 3 3 i s j - s i s j ) , do E o ,z

i4=j

3 d++_l = + s?'s;) ,

i 4=.j

3 d + 2 = - - - f f w d E A~ 2s±Is±I

- i J '

i*i

to d = hT~Z/(a/2) 3 = 16.86 × 10 s s - i and the lattice parameter a = 1 i.548 ,~. The geometrical factors are

(a/2)3,-Ai)~(J) A,°i = 2 ( 1 - 3 c o s 2 ( O ~ j ) ) ) ~ ~ ' , ~ ' ~ ,

o~, B " , ~

A±'i.i = E sin(O~i~ j,) o:.[3

Mcos(O(;J) )eWiO,~ '~(a/ /2)3 .~( i ) , . ( . i ) t" a t<~ ,

()' i, ( O~ij))e ~- izq',Td' a / 2 ,(,),,,i) a .~ ' o43

where a and /3 denote the individual atoms of the molecules i and j, and 0 ~ j) is the angle between the direction of the magnetic field and the vector r(iJ) connecting the atoms c~ and /3 of molecules i t3 and j, respectively. The ~,,~-(i) and p~ denote the


rr-spin densi ty at the atoms. The factors Ai ° , A~ l and Ai~ 2 were ca lcula ted with the a tomic coordi-

nates and spin densi t ies g iven by Dormann et al.

[1,2].

R e f e r e n c e s

[I] E. Dormann, W. Dyakonow, B. Gotschy, A. Lang, H. Nat- mann, 13. Pilawa, N. Walker and H. Winter, Synth. Met. 55-57 (1993) 3273.

[2] E. Dormann, H. Winter, W. Dyakonow, B. Gotschy, A. Lang, H. Narmann and N. Walker, Ber. Bunsenges. Phys. Chem. 96 (1992) 922.

[3] H.M. McConnell, J. Chem. Phys. 39 (1964) 1910. [4] R. Kuhn and H. Trischmann, Angew. Chem. 75 (1963) 294. [5] R. Kuhn and H. Trischmann, Monatsh. Chem. 95 (1964) 457. [6] K. Albert, H. von LiShneysen, W. Sander and H.J. Schink,

Cryogenics 22 (1982) 417. [7] F. Wegner, Z. Phys. 206 (1967) 465. [8] M.E. Fisher, Am. J. Phys. 32 (1964) 343.

[9] J.C. Bonner and M.E. Fisher, Phys. Rev. 135 (1964) A640. [10] N.M. Ashcroft and N.D. Mermin, Solid State Physics (Holt-

Saunders, 1981). [11] M. Steiner, J. Villain and C.G. Windsor, Adv. Phys. 25

(1976) 87. [12] M.J. Hennessy, C.D. McEIwee and P.M. Richards, Phys.

Rev. B 7 (1973) 930. [13] J.P. Boucher, J. Magn. Magn. Mater. 15-18 (1980) 687. [14] K. Nagata and Y. Tazuke, J. Phys. Soc. Jpn. 32 (1972) 337. [15] J.P. Boucher, M. Ahmed Bakheit, M. Nechtschein, M. Villa,

G. Bonera and F. Borsa, Phys. Rev. 13 13 (1976) 4098. [16] H. Beaner and J.P. Boucher, in: Magnetic Properties of

Layered Transition Metal Compounds, ed. L.J. De Jongh (Kluwer, Dordrecht, 1990) p. 323.

[I 7] P.M. Richards, Scuola Internazionale di Fisica Enrico Fermi LIX Corso (1976) 539.

[18] A. Lagendijk, Phys. Rev. 13 18 (1978) 1322. [19] M.J. Hennessy and P.M. Richards, Phys. Rev. B 7 (1973)

4084. [20] F. Carboni and P.M. Richards, Phys. Rev. 177 (1969) 889. [21] R. Ahlrichs, M. BSr, M. H~iser, H. Horn and C. K/Jlmel,

Chem. Phys. Lett. 162 (1989) 165.

magnetic properties of the one-dimensional heisenberg antiferromagnet tetraphenylverdazyl

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