Electron paramagnetic and antiferromagnetic resonance in LiCuCl3.2H2O

Physica 90B (1977) 119-130 © North-Holland Publishing Company

ELECTRON PARAMAGNETIC AND ANTIFERROMAGNETIC RESONANCE IN LiCuCI3.2H 20 N. J. ZIMMERMAN* and A. J. Van DUYNEVELDT Kamerlingh Onnes Laboratorium der Ri]ksuniversiteit Leiden, The Netherlands

(Communication No. 429a) Received 10 February 1977

The 9.6 GHz antiferromagnetic and paramagnetic resonance spectrum of LiCuC13.2H20 is studied as a function of temperature for various directions of the external magnetic field. The easy axis, e, is found to be in the a - c plane at an angle of 70.5° to the a axis and 38.5° to the c axis. Four AFMR lines are detected in the case H//e below the N6el temperature of 4.4 K. Each of these lines shows the characteristic temperature and angular dependence, as predicted by the theory of Nagamiya, Keffer and Kittel. Three new resonance lines are detected below 2.5 K, these lines are suggested to be the resonances due to the occurrence of a second, two-sublattice system in the compound. The critical line broadening of the paramagnetic resonance line above TN is also noted; various descriptions for this broadening are presented.

1. Introduction

the three crystal axes are shown in fig. 1. The crystals are delisquescent, noticeable from the green layer appearing on the surface within a few minutes after exposure to the air. A plane of perfect cleavage exists perpendicular to the crystal b axis. The details of the crystallographic structure have been examined by Vossos et al. [7] and by Abrahams et al. [8]. The space group is P21/c; the monoclinic unit cell contains four molecules. The constants determining the unit

Lithium copper chloride dihydrate (LiCuC13.2H20 ) is a compound that orders antiferromagnetically below a N6el temperature of about 4.4 K [1 ]. The crystal structure as well as the magnetic behaviour of LiCuC13. 2 H 2 0 are known in some detail, but information concerning the paramagnetic and antiferromagnetic (AFM) resonance spectra is relatively sparse. In the present paper we will concentrate on the resonance behaviour near the transition from the AFM phase to the spinflop (SF) phase at a magnetic field of approximately 10 kG. In particular, we will determine whether the AFM resonance spectrum is described adequately by the theory of Nagamiya, Keffer and Kittel [ 2 - 5 ] . In addition to these studies we will consider the temperature dependence of the paramagnetic resonance line width in the so-called critical region above Ty . Brown-red crystals of LiCuC13.2H20 were obtained from a saturated aqueous solution of CuC12 and HC1 [6]. These crystals prefer to grow along the crystallographic a axis, forming elongated prisms. The typical form of such crystals and the orientation of

Cl

\

b

c

* Present address: Laboratorium voor Technische Natuurkunde, Technische Hogeschool, Delft, The Netherlands.

Fig. 1. Typical form of a LiCuC13.2H20 crystal and orientation of the crystallographic axes. 119

120 N. J. Zimmerman and A. J. Van Duyneveldt/Electron paramagnetic and antiferromagnetic resonance in LiCuCI 3. 2H20 cell are a = 6.078 A, b = 11.145 )~, c = 9.145 A and /3 = 108050 ' at 298 K, while b = 11.22 A and c = 9.07 A at 4.2 K if~ is assumed not to vary with temperature. The positions of the copper ions are drawn in fig. 2, projected along the a, b and c axes, respectively. The distance between nearest neighbouring copper ions is 3.46 A, the distance between next nearest neighbours is 3.84 A. The other copper ions are over 6 A apart. In the structure plane Cu2C14Li2C12 (fig. 3) groups attract attention, in which nearest neighbours of copper ions are in line with lithium ions. The static susceptibility of LiCuC13.2H20 was studied by Vossos et al. [9] and showed a maximum at 5.9 K, leading these authors to conclude that a magnetic ordering of the copper ions occurred near that temperature. Specific heat and NMR measurements were performed by Forstat and McNeely [ 1 ] ; they suggested a transition from the paramagnetic to the antiferromagnetic phase at 4.40 K. Another proton spin resonance experiment, performed by Klaassen [t0], led to a N~el temperature of 4.415 K. The magnetic space group of LiCuC13.2H20 was found to be P2'1/c; the dimensions of the magnetic and chemical unit cell being identical [8]. The antiferromagnetic structure of lithium copper chloride has been indicated in fig. 2, the white and black circles distinguish the anti-parallel spin directions. The four crystallographically nonequivalent copper ions behave as two magnetically nonequivalent sets of copper ions. The antiferromagnetic ordering direction e (easy axis) lies in the a - c plane. The magnetic phase diagram of LiCuC13.2H20 shows a transition from the antiferromagnetic to a spin-flop phase at a critical field H c of about 10 kG. The b axis appears to be the second preferred direction, the axis y=

~

to which the spins are parallel in the spin-flop phase [111. A.c. susceptibility measurements by Metselaar [12] have demonstrated that LiCuC13.2H20 cannot be considered as a simple two-sublattice antiferromagnet as the compound exhibits two more phase transitions. These transitions occur at T = 2 K, at magnetic fields of 30 and 55 kG, respectively. The presence of these phase transitions has been confirmed by the proton spin resonance experiments of Henkens and Diederix, mentioned in ref. [12].

2. The antiferromagnetic resonance fields of an orthorhombic antiferromagnet The classical theory of antiferromagnetic resonance, as formulated by Nagamiya, Keffer and Kittel from 1951 to 1956 [ 2 - 5 ] (the NKK theory), describes the positions of the AFMR lines as a function of temperature and external magnetic field (strength and orientation). This theory is based on the molecular field approximation and thus it is not expected to give all the details of the resonance behaviour. Nevertheless, the more realistic theoretical approaches that are possible in some cases [13] are in good agreement with the results from the NKK theory. In this section we will resume the expressions for the resonance frequencies of the modes which can be observed in an orthorhombic antiferromagnet, according to the NKK theory in which the anisotropy of the g value in the plane perpendicular to the easy axis has been taken into account [14, 15]. We define the following quantities:

x

-''

o- -4~------<> b

I;

0---o- ..... o--

...... 0-. "(3

i

~

Z

Fig. 2. Projections of the positions of the copper ions in LiCuCI3.2H20 along the a, b and c axis. Nearest neighbours are connected by a full line, next nearest neighbours by a dashed line. The antiparallel directions of the spins in the antiferromagnetic structure are indicated by white and black circles.

N. J. Zimmerman and A. J. Van Duyneveldt/Electron paramagnetic and antiferromagnetic resonance in LiCuCI3.2H20

121

G

@

.3st-~

~

a/" ~

o

Cu

®

Li

©o

Fig. 3. The surroundings of two nearest neighbours of copper ions in LiCuCIa.2H20. g~' gn and g~. are the g values in the directions ~, r/ and ~'; these directions are such that ~//e, ~7//secondeasy axis and ~'//hard axis [ 11 ]. /3 is the angle between the easy axis and the magnetic field H, i f H lies in the G-r/plane. 0 is the angle between ~ and H, i f H lies in the ~-~" plane. H e = eoh/g~la s is the paramagnetic resonance field for H//~. H C is the critical (spin-flop) field for H//~. X~ is the susceptibility for H//~ and H < H C, while the susceptibility in the case H//r7 is indicated as X n. An anisotropy parameter, a, is introduced, which is defined as o~ -~ 1 - (gn]g~ )2X~/Xn. The anisotropy energy of an orthorhombic antiferromagnet can be represented by E a = K 1 cos 2 cb1 + K2 c°s2 q~2 in which K 2 > K 1 > 0. The angles ¢1 and ~b2 are the angles between the direction of the spins and the second-easy axis and that between the direction of the spins and the hard axis, respectively. Another magnetic field value of interest is HB, which is equal to (K2/K1)~ H C. We shall also use an orthorhombicity parameter p = H2/H2c = K 2 / K 1. The experimental values for the susceptibilities and the spin-flop field are used in the above expressions in order to optimize the agreement between theory and experiment. We will distinguish between four different

cases that differ by the direction of the external magnetic field. (a) H//~. The antiferromagnetic resonance fields H 0 for the various modes are given by LF mode ( H < HC): 2ec2H2 = F -

(F 2 - G)~

(1)

HF mode ( H < H c ) : 2ot2H02= F + (/72 - G)~

(2)

OR mode (H = HC):

H 0 = HC,

(3)

SF mode ( H > HC):

H02 = H 2 + a/-/2,

(4)

if the static magnetic field is parallel to the easy axis. In these formulae F = (1 + a2)/5/2 + a 2 ( H 2 + H 2 ) and G = 4a 2 (~¢/1~ 2 - H~)(cff-/d 2 2 - H~). 2 The behavlour • of the resonance fields as a function of temperature is determined by the quantities a, Hw, H B and H C. In Fig. 4 a characteristic behaviour is sketched for the case Hto < H C. The resonance modes can be observed in a limited temperature range only. The temperatures T1, T2, T3, and T4 (see fig. 4) can be found from the relations: (1-a)H2=H

2,

ot/2=H 2,

d - / 2 = H 2,

and

respectively. At high microwave frequencies ( H w ~>/-/C)

122 N.J. Zimmerman and A. J. Van Duyneveldt/Electron paramagnetic and antiferromagnetic resonance in LiCuCI3.2H20 in which

S

~

~

e =H2/~H 2,

r,

f = [(g~H0 cos/3)2 + (gnH0 sin/3)2]/~g~H 2, and

I

tg 2q~ = sin 2/3/(cos 2/3 - a f). / HF 0

T

T2 T3

Fig. 4. The temperature dependence of the resonance fields of an orthorhombic antiferromagnet for H~ < Hc(T = 0) and the magnetic field parallel to the easy axis. the temperatures T1 and T4 are given by another relation, but this situation is of no interest for our present study. The paramagnetic resonance mode in fig. 4 has been marked by PAR. (b) H//r~-~ plane. Only one resonance mode occurs in the case where the static magnetic field is directed in the plane perpendicular to the easy axis. The components of the resonance fields in the direction of r7 and ~', Hn and H~., obey the relation:

U (g,/gO 2 n (gdgO 2 +

- 1.

(S)

The resonance field H 0 can be derived from these equations by calculating the solutions for q~, then f from q~ and finally H 0 fromf. For Hto < H C and small angles/3 two resonance modes occur at temperatures below T2. For/3 ~ 0 one of these modes changes into the LF mode and the other into the OR or the SF mode, depending on the actual temperature. (d) H//~-f plane. When the magnetic field is oriented in this plane, the spin-flopping that occurs for 0 ~ 0 must also be considered. The spin-flop fields are given by the so-called critical hyperbola [3]

Hc(O ) = HC/[cos 2 0 + ((gJg~ )2 /(1 - H2/H2)} sin 2 0 ]~

(7) This expression also applies to the resonance fields of the OR mode in this plane. For H(O ) > Hc(O) the resonance condition of the SF mode is Hg cos 2 0

It appears that for H/h7 no resonance occurs below the temperature T2. For H//f no resonance occurs below T3. (c) H//~-~7 plane. When the static magnetic field lies in the G-r/plane and/3 is the angle between this field and the easy direction, the resonance condition is e 2 -- e [f(&2 cos 2 xIt + 1) + t9 + cos 2 ( ~ --/3) _ 2sin 2 ( ~ _/3)] + f 2 a 2 cos 2

- f [ a cos 2 • cos 2(@ -/3) + a sin/3 cos q~ sin ('II -/3) + cos/3 sin @ sin (@ -/3) + p(a

COS 2 @ - -

+ [p - sin 2 (@ -/3)] cos 2(@ - fl) = 0,

sin 2 @)] (6)

Ho2(g~/g~)2 sin 2 0 = 1.

(8)

F o r H ( 0 ) < HC(O) only one resonance mode exists. The resonance condition of this mode is given by eq. (6) if the indices B and C in this equation are interchanged. The actual resonance frequencies can be derived directly from the expressions ( 1 ) - ( 8 ) for the resonance conditions. In fig. 5 these frequencies have been indicated for T = 0 K, where a is chosen to be 1. The modification of these frequency versus field diagrams at increasing temperature can be found from ref. 3 sub 2. In the above summary we restricted ourselves to the cases where the microwave field Hhf / H. The orthorhombic antiferromagnet as described above differs from the uniaxial antiferromagnet in having

N. J. Zirnmerman and A. J. Van Duyneveldt/Electron paramagnetic and antiferromagnetic resonance in LiCuCI3.2H20

123

/

//

/ / I I

/ I/'/

I I

. . . . . . i

:/

tO

I

I~/ . . . . .

q

I

/ /

/

/ /

/

/

He(O)

i

["

/

0 HII~

/

HII~.

H//~

Fig. 5. Frequency-field diagrams for an orthorhombic antiferromagnet at T = 0 with easy axis ~, second-easy axis r/and hard axis ~'. non-degenerate zero-field resonance modes. This nondegeneracy causes the existence of a temperature region from T2 to T3 in which neither the LF nor the HF resonance mode exists. 3. Experimental results Measurements on LiCuC13.2H20 have been performed with a superheterodyne ESR spectrometer. The resonance frequency amounted to 9.60 GHz. Three samples (I, II and III) were cut from larger crystals that were cleaved in slices parallel to the a - c plane. The samples were about 2 × 0.5 X 0.5 m m and were mounted against the wall o f the cavity, so that the microwave magnetic field was always parallel to the a - c plane. The paramagnetic resonance spectrum o f LiCuC13. 2 H 2 0 consists of one narrow absorption line, o f which b o t h the position and the shape are independent o f the orientation o f the microwave field. The g values were measured at 20 K. Using the procedure (GEWA) as described in [16], the g values in the directions o f the magnetic x, y and z axes appeared to be gx = 2.050(2), gy = 2.136(2) and gz = 2.224(2). The magnetic y axis coincides with the crystallographic b axis, which can be expected on account of the monoclinic structure o f the crystal. The x and z directions lie within the a - c plane. The angle between the z and a axes as well as between the z and c axes is 54.5 °. The paramagnetic resonance line has a lorentzian shape. T h e line width at 77K is 186(4), 234(5) and 308(6) G for H//x, b and z, respectively. The antiferromagnetic ordering direction, the easy axis e, has been determined from the angular dependence o f the antiferromagnetic resonance lines, as de-

scribed below. This axis lies in the a - c plane at an angle of 70.5 ° to the a axis and 38.5 ° to the c axis. This means that the easy axis coincides with the crystaUographic (101) direction within the precision of mounting the sample. The AFM resonance spectra for H//e at three different temperatures are shown in fig. 6. At 3.092 K the spectrum consisted of two lines, which appeared to be the LF and OR lines. The OR line was observed below 4.23 K. A third resonance line ( ' N I ' in I

I

J

I

@

LF

]

Li CuCI3.2H20 H//ce T=3.092 K

©

LF

OR/

T=1,628K

©

LF I OR~ I

O

H

I

I

I

,4-

6

8

I

T=1.369K

I

I

10

12

kG

Fig. 6. The antiferromagnetic resonance spectrum of LiCuC13. 2H20 at three temperatures for the magnetic field H parallel to the easy axis e.

124 N.J. Zirnrnerman and A. J. Van Duyneveldt/Electron paramagnetic and antiferromagnetic resonance in LiCuCI 3. 2H20 fig. 6) was measured at 2.4 K and at lower temperatures, while two additional lines ('N2' and 'N3') were detected below 1.5 K. The LF line showed a clearly visible structure at the lowest temperatures. At 1.260 K as many as four components can be distinguished, as illustrated in fig. 7. The temperature dependence of the position of the antiferromagnetic resonance lines and the paramagnetic resonanc~ line (PAR) are given in fig. 8 for the cases H//x, b and e. The dashed lines in this figure indicate those fields at which the absorption reaches half its maximum value. The start of a strong decrease o f the PAR line position occurs at T = 4.440 -+ 0.005 K for each of the three orientations o f the magnetic field. The open circles in fig. 8 represent measurements on crystal II, the black circles those on crystal III. The resonance lines ' N I ' , 'N2' and 'N3' have been observed on crystal III only, simply because these measurements were performed in a smaller cryostat, so that a stronger magnetic field could be applied. The angular dependence o f the resonance fields of the A F M R lines is shown in fig. 9, for orientations of the external magnetic field within the a - c and the e - b plane at temperatures of about 1.6 and 1.2 K. For H / / e - b plane, the resonance lines ' N I ' , 'N2' and 'N3' existed in the narrow region of only 1° around the e axis. The temperature dependence of the full width at half intensity o f the PAR line and the antiferromagnetic LF line is shown for H//e in fig. 10. A considerable line broadening occurs near the N6el temperature. A pure Lorentzian shape of the PAR line is observed down to 30 mK above TN. Closer to 7~ the PAR line becomes asymmetrical, the width at fields

....

'

'''

''

kO

.....

. l ~,~ ~r..=?+o-::::!

_

. . . . .

.

HN 2 ~ .

i

1

E,

~

,

5K

~

,- . . . . . . . .

7 ._._.~o-°'°-, /, / _ * - ? - * ~ * ~

o.

8

. . . . .

,

4

.,, 12 7- 7 T . . . . ~"~. ,_.+•_~.__°rl~,

H,,I,llb,,

"/o/..:,",, "'""'-.~*L~'-,

LF

4

" ',%', < 7 z

H/

H F ~i~:,"

PAR

i J

o

T

2

1

3

4-

5

K

Fig. 8. The temperature dependence of the resonance field of the paramagnetic resonance line (PAR) and the AFMR-lines HF, LF, OR, 'NI', 'N2' and 'N3' in LiCuCI3.2H20 for H//easy axis e. The insets in the upper right corner show the resonance field as a function of temperature for H//x and H//b.

'

kG

'

i

~-

~7/-~

~

,NI' I ;

I

'

JJk 2Ji

~,i

; ,, ~LjN 3"

11 /./'"

i/

Ic

r°%

OR

•

,,"o,#f'~ii\ "\

~",.,

"\

N

1.62 K

\

,," i

(. °~°~" ~.

L F"

Ho

,'"

kG \

-~

t o

~

'

l

',/

\ \\o /

'.

"~ /

-,.e

~_//, 6°

-e

-~

t o

'

l

'

'//

lr

12

",

/

i p//

e

-~

OR 1.21 K

,,

,

3

'

,''

~x ,''

/

t\

6°

,,

' I

I

/I t

. t, ~, \ , N 2 , ~\ .,._,.

g

'I ,'/ //

~

• :

LICuCs2H20

•,,,o~_..,,.."

I

X

~°!"--,',n/);

Ht le

-

T

' .....

T=I.£60K

OR

OR

1.62 K

1.18 K

I( \

9

-30 H

8.5

KG

/

",.

.ot 8.0

\

i

LF

,

\

../

, I i,

I i r

-15

0

\

,

"', ,_L~ 15

30 °

LF

/ I . . . .

l

-30

0

-15

/ I , I i i I 15

30 °

9.O

Fig. 7. The shape of the antiferromagnetic LF-line of LiCuC13. 2H20 at 1.260 K and H//e.

Fig. 9. The angular dependence of the resonance fields of the AFMR-lines in LiCuC13.2H20 for H//e-b plane and H//a-c plane at about 1.6 and 1.2 K.

N. J. Zimmerman and A. J. Van Duyneveldt/Electron paramagnetic and antiferromagnetic resonance in LiCuCI3.2H20 125 4.2. The EPR and AFMR spectra I

I

i, 2

! L~

H//~

1pAR

5

~;

~5

K

2o

Fig. 10. The temperature dependence of AH, the full linewidth at half height of the PAR-line and the antiferromagnetic LFline in LiCuC13.2H20 for HUe. below the resonance field being larger than above. This asymmetry continues through the N6el temperature where the PAR line changes into the HF branch of the antiferromagnetic resonance spectrum. For a correct interpretation of fig. 10 it should be remembered that the LF line does not occur above the temperature T2 (see figs. 4 and 8).

4. Discussion

4.1. g values The g values of LiCuC13.2H20 as determined at 20 K, gx = 2.050(2), gb = 2.136(2) and gz = 2.224(2), agree nicely with the results of Date and Nagata [11 ] who reported ga = 2.06,g b = 2.14 andgc = 2.22 at temperatures between 77 and 300 K. Our experiments have indicated that the antiferromagnetic ordering direction e is not collinear with any axis of the g tensor, but rather with the crystal (101) direction. This is in striking disagreement with the refs. 8 and 11, where other directions in the a - c plane are suggested to be the easy axis. Susceptibility measurements as well as 7Li nuclear spin resonance experiments support our conclusion that e lies in the a - c plane at an angle of 70.5 ° to the a axis and 38.5 ° to the c axis. The earlier experiments may have suffered from difficulties in recognizing the crystal planes

LiCuC13.2H20 is known as a four-sublattice antiferromagnet with hidden canting, exactly as CuC12. 2H20. The canting angle amounts to 6 degrees. Joenk [17] has calculated that a hidden canting of this magnitude will produce no effect on the resonance modes as derived from the NKK-theory for a two sublattice antiferromagnet. We have identified the OR-, LF- and HF-mode of an orthorhombic antiferromagnet in the LiCuC13. 2H20 resonance spectrum (fig. 8). We will now discuss whether the temperature and angular dependence of these AFMR-lines can be described quantitatively by the NKK-theory. For H//e and T > T1 the resonance field of the ORmode equals the spin-flop field HC. This spin-flop field is known to vary with temperature according to [18]: H c ( T ) = HC(0 ) + D sinh ET. Our measurements of the resonance field of the OR-mode above 1.7 K are fitted to this function, the resulting parameters being HC(0 ) = 10.53 kG, D = 0.1257 kG and E ---0.806 K-1. This result is indicated by a full curve in fig. 8. Static susceptibility measurements of CuC12. 2H20 [14] resulted in an anisotropy parameter a ( T ) which can be described by a ( T ) = 1 - (T/TN) 2"2 within an accuracy of a few percent. Susceptibility measurements by Metselaar and Schumm clearly indicate that the same expression applies to LiCuCI3.2H20 [12]. The field HB(T = 0) can be calculated from the resonance condition of the LF-mode, using the above results for H c ( T ) and a(T), together with H~ = 3.10 kG and H L F ( T = 0) = 8.55 kG. The result is HB(T = O) = 13.51 kG, so that the orthorhombicity parameter p = 1.65, whereas HB(T ) = x / # H c ( T ). The temperature and angular dependence of the AFMR lines is determined completely by this set of parameters. The temperature dependence of the LF-, HF- and SF-modes for H//e is calculated from the formulae in section 2 and indicated by tile full curves in fig. 8. The agreement between the calculations and the experimental data is excellent. From the NKK-theory it follows that the OR-mode does not exist above T4 = 4.250 K; our experiments yielded 4.23 K. The OR-line should change into the SF-line at T1 = 1.51 K, while the resonance field of the SF-line at T = 0 should amount to 11 kG. Experimentally no sharp transition of the OR-line into the SF-line is observed, although

126 N.J. Zimmerman and A. J. Van Duyneveldt/Electron paramagnetic and antiferromagnetic resonance in LiCuCI3.2H20 the slight increase of the resonance field of the OR-line below 1.7 K may be related to this transition. In fig. 11 the temperature dependence of the LFand HF-mode for H//e just~below TN is shown with an enlarged temperature scale. This figure shows that the agreement between the NKK-theory (full curves) and experiment for H//e as well as for H//b is satisfactory. For H//f (~ is the hard axis) the theoretical temperature dependence is shown in the centre graph of fig. 11 by the full curve, while the dashed curve represents the theoretical behaviour of the AFMR-field for H//b. The measurements have been performed in the x direction, which declines 16° from the ~"axis, so a difference between experimental result and the calculated values for H//f has to be expected. However, the coincidence of the measured values with the calculated curve for H//b is surprising. The measured angular dependences of the LF-, SFand OR-resonance fields are compared with the NKKtheory in figs. 12, 13 and 14. The open symbols represent the resonance fields at negative angles/3 and O while the black ones refer to/3, 0 > 0. The squares are

k

if

H//b

I

I

TN~ I

T2

if

H//X

OI

/'//

72

.....

'

-

' T.; r

%

4

H//e 2

Hol o

4-,3

T2

T3

4-.4-

K

Fig. 11. Theoretical and experimental results for the AFMRfields in LiCuC13.2H20just below TN (4.440 K) for H#b, x and e.

related to sample III and the triangles to sample II. The theoretical results are based on the parameters derived above and have been indicated by dashed curves. We have tried some other choices for a and p, as the original set of parameters led only to a qualitative agreement with the experiments. The values of the anisotropy parameter and the orthorhombicity parameter are given in the subscripts of the figs. 12, 13 and 14. The theoretical curves belonging to these fits are indicated by full curves. It is striking that these values of c~and p agree with those reported by De Jong [21], but differ by a factor of about 1/2 and 2 respectively from the values which have been used successfully for the description of the temperature dependence of the axial resonance fields. Above the spin-flop field, 'new' AFMR-lines have been discovered [19], which cannot be explained simply by the NKK-theory. The temperature dependence of the resonance fields of these lines for H//e is given in fig. 8. The AFMR-line marked by ' N I ' has not been observed above 2.5 K and is dividing into two lines 'N2' and 'N3' below 1.5 K. The angular dependence of the position of the new AFMR-lines is shown in figs. 9 and 15. In the e - b plane the resonance lines are found in the vicinity of the e axis only. In fact we were able to measure this angular dependence in the e - b plane because of the accurate adjustment possibilities of the magnet and the sample holder. Assuming that LiCuC13.2H20 contains two weakly coupled orthorhombic antiferromagnetic two-sublattice systems having slightly different exchange, the AFMR-line ' N I ' can be identified as the orientation resonance line OR' of the (dashed) second system. The temperature dependence of the resonance field of the OR'-line and also the angular dependence in the a - c plane (critical hyperbola) are pointing to the occurrence of a second spin-flop transition in LiCuC13.2H20, 0.06 kG above the spin-flop transition of the first system. The presence of the two spin-flop transitions has been affirmed by susceptibility measurements of Metselaar [12]. The critical hyperbola of the OR'-line, which is represented by a full curve in fig. 15b, conforms to the formula given in section 2 with p = 3.4. As to the angular dependence, the AFMR-line 'N3' seems to be the continuation of the OR'4ine below 1.5 K. At 1.175 K the angular dependence of this line obeys the NKK-theory [eq. (7)] for p = 2.15. In our model the AFMR-line 'N2' can be identified as the

N. J. Zimmerman and A. J. Van Duyneveldt/Electron paramagnetic and an tiferromagnetic resonance in LiCuCI3.2H20 I

I

I

127

I

@ 11

/JJ~

\\

\

\

\ \ \ \

_

o X~\\

1

\

/i/I/

/ /

I

/

J J

/

/

• /

/

// //////

/

H

I

O~

~ 2

4

I

6

8°

0

p

I 2

I 4

I 6

I °

8

Fig. 12. T h e o r e t i c a l f i t t i n g o f t h e a n g u l a r d e p e n d e n c e in t h e e - b - p l a n e o f t h e r e s o n a n c e field o f t h e A F M R - l i n e s in L i C u C 1 3 . 2 H 2 0 at 1.2 K (a) a n d 1 . 8 0 7 K (b). (a) H C = 1 0 . 7 0 k G : - ~ = 0 . 4 8 8 a n d p = 3.4, - . . . . ~ = 0 . 5 5 a n d p = 2.8, - - - a = 0.95 a n d p = 1.65. (b) H C = 1 0 . 8 5 k G : - a = 0 . 4 6 8 a n d p = 3.4, - - - a --- 0 . 8 6 a n d p = 1.65.

kG

I

I

12

g H 0

9.0

I

r

kG_ (~

,,9" 10

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Fig. 13. T h e o r e t i c a l f i t t i n g o f t h e a n g u l a r d e p e n d e n c e in t h e a - c p l a n e o f t h e r e s o n a n c e field o f t h e a n t i f e r r o m a g n e t i c L F - l i n e in L i C u C 1 3 . 2 H 2 0 a t 1.17 K (a) a n d 1.69 K (b). (a) H C = 1 0 . 7 0 k G : - a = 0 . 5 5 a n d p = 2.8, - - - a = 0.95 a n d p = 1.65. (b) H C = 1 0 . 8 5 k G : - ~ = 0 . 4 6 6 a n d p = 3.4, - - - a = 0 . 9 0 a n d p = 1.65.

128 N.J. Zimrnerman and A. J. Van Duyneveldt/Electron paramagnetic and antiferromagnetic resonance in LiCuCI3.2H20 I

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Fig. 14. Theoretical fitting of the angular dependence in the a-c plane of the resonance field of the OR-line in LiCuC13.2H20 at 1.619 K (a) and of the SF-line at 1.17 K (b). (a) HC = 10.86 kG: - a = 0.452 and p = 3.4, - - - - a = 0.90 and p = 1.65. (b) HC = 10.70 kG: - a = 0.95 and p = 2.6, - - - - a = 0.95 and p = 1.65. spin-flop mode SF o f the dashed system. The angular dependence of the SF'-line in the a - c plane obeys the NKK-theory with values/9 = 1.65 and a = 0.95 at 1.175 K (dash-dotted curve in fig. 15a). Presumably the structure which has been observed in the LF-line is indicative of the existence o f more than one antiferromagnetic two-sublattice system in LiCuC13.2H20. However, the angular dependence o f the new AFMRlines cannot be explained by the proposed model. It should be remarked that the behaviour o f the 'Nl'-line in the e - b plane shows strong resemblance to that of a 'new' not yet explained AFMR-line which has been observed by Galkin and Kovner [20] in CuC12.2H20.

4.3. Linebroadening in the vicinity o f T N LiCuC13.2H20 shows critical broadening o f the paramagnetic resonance line above TN, as is usual in antiferromagnetic compounds. The reduced paramagnetic line width A H ( T ) / A H (77 K) is plotted as a function of the reduced temperature e = ( T I T N - 1) in fig. 16. For these plots TN is chosen to be 4.440 K. The straight lines in fig. 16 indicate that the line width obeys the relation A H ( T ) / A H (77) = ce-P within a region of three orders o f magnitude in e, in each of the

three directions of the external magnetic field. The parameters corresponding to the lines in fig. 16 are c = 1.01,p = 0 . 3 5 ; c = 1.13,p = 0 . 2 9 a n d c = 0 . 9 4 , p = 0.45 forH//e,H//b and H//x, respectively. A similar analysis of the line widths in CuC12.2H20 yields a (critical) exponent p = 0.58 independent o f the direction of the external magnetic field, but over a temperature interval of less than 2 orders o f magnitude in e. The actual line width A H of the antiferromagnetic resonance lines depends on the frequency co(= getaBHU~), because co is not proportional to the resonance field H (see fig. 5). Therefore, the experimentally determined line width A H has to be multiplied by the factor ( d H U d H 0 ) in order to obtain a corrected line width AHto, which is related to the frequency. The factor ] d H U d H o ] i s shown in fig. 17 as it was calculated from eqs. (1) and (2) in order to perform the correction. Gerritsen [ 15 ] described the antiferromagnetic LF line width of CuC12.2H20 by the relation AHto(T ) = 9 exp (T/1.05). We plotted AHto of the LF line width o f LiCuC13.2H20 as a function of temperature in fig. 18a in order to perform a similar analysis. The experiments are well described b y AHto ( T ) = 17 exp (T/0.94) which corresponds to the straight line in fig. 18a. This fit shows a strong resem-

N. J. Zimmerman and A. J. Van Duyneveldt/Electron paramagnetic and antiferromagnetic resonance in LiCuCI3.2H20 [

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Fig. 15. Theoretical fitting of the angular dependence in the a-c plane of the resonance fields of the AFMR-lines 'N2' (SF') and 'N3' (OR') in LiCuC13.2H20 at 1.175 K (a) and the AFMR-line ' N I ' (OR') at 1.619 K (b). (a) - H C = 11.11 kG, ~ = 0.95 and p = 2.15, - - - H C = 11.11 kG, a = 0.95 and p = 1.65, - . . . . H C = 11.66 kG, c~ = 0.95 and p = 1.65, (b) - HC = 1 1 . 4 9 k G , c~ = 0 . 4 6 5

and p = 3.4, ----

H C = 1 1 . 4 9 k G , c~ = 0 . 9 0 a n d p = 1 . 6 5 .

blance to the analysis in [21 ], also numerically. Further we performed an analysis of AHto from a double logarithmic plot (fig. 18b). It now appears that AH~ can be described by AHto oc(1 - T/TN)-Pwithp = 1.4 for T < 3.7 K andp = 0.55 for T > 3.7 K. Finally the AMFR line width might also be described by A H w a T P . Curve fitting resulted in two values for p, p = 2.2 for 1.5 K < T < 3.4 K and p = 6 for T > 3.4 K. We enumerated the various descriptions for A H ( T ) in order to show that great care is necessary in assigning a temperature relationship to line width data.

12

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Fig. 17. The temperature dependence of IdHto/dHol for the resonance field H 0 of the antiferromagnetic LF- and HF-lines.

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Fig. 16. The reduced paramagnetic line width AH(T)/AH (77 K) of LiCuC13.2H20 as a function of the reduced temperature e = TITN - 1 for H//e, b and x, The triangular s y m b o l s refer to sample I and the circles to sample II.

/

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i i

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,

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Fig. 18. The temperature dependence of the line width AHto of the antiferromagnetic LF-line in the LiCuC13.2H20 crystals I (triangles) and II (circles) on a semi-logarithmic scale (a) and a double-logarithmic scale (b).

130 N.J. Zimmerman and A. J. Van Duyneveldt/Electron paramagnetic and antiferromagnetic resonance in LiCuCI3.2H20 Acknowledgements The authors wish to express their thanks to J. D. Bastmeijer, W. S c h u m m , A. T. M. Scholten and J. van der Schee for their help with the m e a s u r e m e n t s and the calculations and to W. F . Elbers for technical assistance. The authors are pleased to acknowledge their indebtedness to Prof. dr. J. van den Handel for his stimulating interest.

References [1] H. Forstat and D. R. McNeely, J. Chem. Phys. 35 (1961) 1594. [2] C. Kittel, Phys. Rev. 82 (1951) 565. [3] T. Nagamiya, Progr. theor. Phys. 6 (1951) 342; 11 (1954) 309; 15 (1956) 306. [4] F. Keffer and C. Kittel, Phys. Rev. 85 (1952) 329. [5 ] T. Nagamiya, K. Yosida and R. Kubo, Antiferromagnetism, Adv. in Phys. 4 (1955) 1. [6] F. H. Getman, J. Phys. Chem. 26 (1922) 377. [7] P. H. Vossos, D. R. Fitzwater and R. E. Rundle, Acta Cryst. 16 (1963) 1037.

[8] S. C. Abrahams and H. J. Williams, J. chem. Phys. 39 (1963) 2923. [9] P. H. Vossos, L. D. Jermings and R. E. Rundle, J. chem. Phys. 32 (1960) 1590. [10] T. O. Klaassen, private communication. [11] M. Date and K. Nagata, J. appl. Phys. 34 (1963) 1038. [12] J. W. Metselaar, Physica 63 (1973) 499. [13] F. Keffer, Spin Waves, Handbuch der Physik Bd XVIII/2 (Springer, Berlin, 1968) p. 1. E. A. Turov, Physical Properties of Magnetically Ordered Crystals (Acad. Press, New York, 1965). M. S. S. Brooks, Phys. Rev. B2 (1970) 1346. [14] J. Ubbink, Thesis, Leiden (1953). [15] H. J. Gerritsen, Thesis, Leiden (1955). [16] N. J. Zimmerman, Thesis, Leiden (1974). [17] R. J. Joenk, Phys. Rev. 126 (1962) 565; 127 (1962) 2287. [18] G. J Butterworth, V. S. Zidell and J. A. Woollam, Phys. Letters 32A (1970) 24. [19] N. J. Zimmerman, J. D. Bastmeijer and J. van den Handel, Phys. Letters 40A (1972) 259. [20] A. A. Galkin and S. N. Kovner, Soviet Physics-JETP Letters 9 (1969) 275. [21 ] W. M. de Jong, W. L. C. Rutten and J. C. Verstelle, Physica 82B (1976) 303 (Commun. Kamerlingh Onnes Lab., Leiden No. 421b).