Journal of Magnetism and Magnetic Materials 28 (1982) 97- 108

North-Holland Publishing Company 91

THE MAGNETIC ORDERING AND INCOMMENSURATE PHASE TRANSITION OF Tb,Ge, - A NEUTRON DIFFRACTION STUDY *

Penelope SCHOBINGER-PAPAMANTELLOS

Insritut fiir Krisiallographie und Petrographic der ETH Ziirich, Switzerland

Received 1 December 1981; in revised form 5 March 1982

Neutron powder diffraction studies have shown that upon cooling Tb,Ge, (hexagonal, Mn,Si, type) orders antiferromagnet-

ically at TN =85 K, and undergoes an incommensurate magnetic transition in the range of 75-50 K. The moment alignment

for the two end members of the transition is within each chemical cell collinear and ferromagnetic, changing the direction of

magnetization with the temperature as a spiral along the hexagonal axis. At 77 K the spiral angle measures 180” and the

structure can be described by the magnetic space group C,Z/c. The moment distribution for both Tb positions is sine

modulated and the moment angles with the hexagonal axis are 67.7’ for the 4(d) and 60.2” for the 6(g) atoms. The 4.2 K

structure is described as a flat spira! with a helical angle of 166.14’ between second nearest neighbours, nearest neighbours

being paired. The ordered saturation moment for both Tb positions measures 8.9(1)~,.

1. Introduction

Tb,Ge, is a Nowotny phase [l] and crystallizes at room temperature in the hexagonal D8,

Mn,Si,-type structure, with space group P6,/mcm. The lattice parameters are a = 8.495, c = 6.351 A,

c/u = 0.75 [2], and the structure contains two for-

mula units per unit cell. The 10 Tb atoms are located in two symmetry positions: 4(d) and 6(g)

with x = 0.25, and the Ge atoms are at 6(g) with x = 0.61. The Tb atoms in 4(d) form linear chains

along the c-axis with an interspace of c/2(3.176 A) which is about 10% shorter than twice the metallic radius of Tb. To,Ge, orders antiferromagnetically below TN = 85 K [2]. The positive paramagnetic Curie temperature (0 = 93 K) indicates a strong ferromagnetic interaction, that according to

Buschow [2] may be connected to the short inter-

atomic distance of the Tb atoms in 4(d). In the present paper we report on the magnetic structure

and temperature dependent magnetic phase transi- tion of Tb,Ge, deduced from neutron powder diffraction data.

* Dedicated to Prof. Dr. H. Nowotny for his 70th birthday

2. Experimental

The powder sample, synthesized by arc melting of the elements under an atmosphere of argon gas

[2], was found to be a single phase (X-ray Guinier focusing camera, CuKa).

The neutron diffraction data were collected at the temperatures 293, 77 and 4.2 K with the two

axis spectrometer at the reactor Diorit in Wtire- nlingen. The profiles shown in fig. 1 were corrected for absorption. The data taken at 293 and 77 K

were refined using line profile analysis [3], whereas

integrated intensities were calculated for the data

taken at 4.2 K. A peak at 70’ 28 due to aluminium in the temperature variable cryostat was observed in the profile at 77 K.

Certain parts of the diagram, fig. 2, were also measured for the whole temperature range 77-

4.2 K in order to follow the magnetic phase transi- tion.

3. The nuclear structure

The nuclear intensities at 293 K were refined using the starting parameters given in ref. [2], with

0304-8853/82/0000-0000/$02.75 Q 1982 North-Holland

98 P. Schobinger-Papamantellos / Neutron diffraction study of TbsGe ~

!nX 10 -3

15

T b 5 G e 3

a) 293 K

= 2,319

hex. cell

a,a,c

o oo~ o~ oo

c) 4t2 K

hex. cell

10 20 30

K I I

i +

+ o l i+i + i ~1 + ~++ot+ + 1 i + r

II I ! II I III I I III I I I I I I1 I

40 50 60 70 80 90 2@ (o)

I I I I I I I

Fig. 1. The absorption corrected neutron diffraction patterns of TbsGe 3 at the temperatures (a) 293 K, (b) 77 K and (c) at 4.2 K. The indexing refers to the hexagonal, P2¢ orthohexagonal and to the incommensurate cell expressed as 00q ~ satellites of the hexagonal nuclear reflections, respectively.

I I I I I I I I I

I l II I 11 II II II I i I I l I i II I II 11 1

P. Schobinger-'Papamantellos / Neutron diffraction study of Tb~Ge ~ 99

ZT~

" ° I I : , n ~ l ;_

..- •" T : .? " "'.1 I :

• L " " . . - . ~ l . : : . : . . . , . . , ' . k - . . - . . , " , ~ .: x~,~" . ' ~ \ : "-: ~..._/",.-

" " " ~ , 5 2 K " " ~x-_f: " " ; " ' ' " : .,p__.-. ; :%..., . . . .

:-. ~, 6 0 x : ' . " :". • • • . r .

:~ 6 e x "~, r : ' . . " . .,,. . . =. .

• f 7 O K . " #.

~' ' I 7 4 K " :. I - " I ;~-.,~ I I ',,=o,, : : ."..', ~ _: " 2 }v'".. . . . , • ~ . , ~ - . . . . . . . ~ " ~ J " , _ . - _ . ' N . .

41o i 51o 1 a'o ~ e ~ I

Fig. 2. The 20 shift of the magnetic reflections with tempera- ture. The indices refer to the P2~ cell at 77 K. The nuclear peaks remain unmoved.

-l.n~ 10"3 2 0 (o) d (A)

. , , ~ , 0 0 1 ) -10.51 12.66 - ~ ~ - 1 0 . 3 7 1 2 . 8 3

~'~ - 1 0 . 2 2 13 .01

1 4 - ~ ' ~ - 1 0 . 0 8 13 .20 \ - 9 , 94 13 .39

1 2 - - 9 .79 13.58 - 9 . 65 13 .79

1 0 - I~ - 9 .51 13 .98

° \

" T b s G e 3 T r

2 ' 0 gO ' "- 160 T( K)-

Fig. 3. The temperature variation of the magnetic intensity and 2e position of the (001) peak and the corresponding c value in A.

the scattering lengths brb = 0.76, b~e = 0.819F and an overall temperature factor of 0.75 ~2. The re- fined parameters are x = 0.240(1) for Tb at 6(g), x =0.606(1) for Ge at 6(g) and lattice constants a = 8.483(5), c = 6.384(2) A. The small difference compared with the X-ray parameters [2] may be related to the sample preparation. The obtained reliability factors, R n and weighted profile Rwp are

0.08 and 0.13, respectively. The fitting of the calculated to the observed data is satisfactory (ta- ble 1).

3.1. The magnetic ordering

Below the ordering temperature T N = 85 K ad- ditional diffraction lines of magnetic nature ap-

Table I

The integrated observed and calculated neutron intensities of TbsGe 3 at 293 K including the multiplicity and the Lorenz factor

h k 1 Ic~ lob s h k I /talc lob s

0 0 2 10488 9261 0 0 4 11003 10972 1 0 2 6795 4524 4 0 2 9281 8950 2 1 0 13705 11093 1 0 4 222 18 1 1 2 41893 43342 4 1 1 676 195 2 1 1 54277 56 125 2 2 3 4060 5204 3 0 0 22146 22612 1 ! 4 83 74 2 0 2 5155 4563 3 2 2 6409 5730 3 I 0 3392 3712 3 1 3 6113 7611 2 2 1 6494 6709 5 0 0 6450 8016 3 0 2 484 505 2 0 4 144 153 3 i 1 9356 9620 4 1 2 3390 4612 1 1 3 346 209 3 3 0 344 0 2 2 2 13 829 15 912 4 2 0 9541 8854 3 1 2 513 132 2 1 4 8640 8042 2 I 3 25525 24980 3 3 1 10187 9342 3 2 0 12 0 4 2 1 5772 3777 3 2 1 7607 6185 3 0 4 17799 17595 4 I 0 6634 8302 5 0 2 28566 28471

100 P. Schobinger-Papamantellos / Neutron diffraction study of TbsGe 3

Table 2 77K

The splitting of the two Tb sites through the symmetry lowering P63/mcm ~ Cm. The (xyz) coordinates and the cos and sine values of the structure factor F =cos 2~rk. r--i sin 2~rk,r, of the 001 and 003 reflections correspond to the orthohexagonal double cell

P63/mcm /Lj within Cm (001 ) (003) Tb site plane(00z )

Tb site x y z cos sin cos sin

~ / ~ / - - 4(b) 0 ~ 0 4 0 4 0

-- 2(a) I 4 0 ~s 6/v~ 6/~/2 - 6 / ( 2 - 6/~f2 4(d) 4(b) ~ ~ ~

6(g) 4(b) 0 ~ 41 0 4 0 4

2(a) ] 0 - - 4 (b) ~ s , ~' - 6 / v ~ - - 6 / ~ / 2 6 / ( 2 6 / ( 2

pear. In the temperature range 85-75 K the mag- netic lines may be indexed with a c-doubled hexagonal cell. Below 75 K the magnetic lattice becomes incommensurate , its period is changing cont inuously with the temperature to about 50 K (figs. 2 and 3). N o similar changement was ob- served for the nuclear lines.

3.2. The P2c magnetic cell at 77 K

The magnetic reflections of the 77 K neutron diffraction pat tern can be indexed in the P2,- hexagonal lattice ( k = 0 0 ½ ) using the notat ion given in ref. [4]. However, for the calculation of the magnetic intensities the or thohexagonal cell O~

[5] (aorth=ahex, borth=ahexV/3, Corth= 2Chex) was used together with the space group Cm for the atomic positions, and the formula of Halpern and Johnson [6] for all equivalent reflections (table 5, figs. lb, 4a). In fig. lb only the indices of the allowed magnetic reflections h + k = 2n and 1 = 2n + 1 are given.

The symmetry lowering of the hexagonal axis can be assumed from the existence of a moment componen t perpendicular to it, which gives rise to the 001 magnetic peak. Most likely the atoms within each (00z) plane have their moments paral- lel. The structure factors of the 001 and 003 reflec- tions (table 2) indicate that both Tb positions (4(d) and 6(g)) contr ibute to the 001 intensity, otherwise 003 would have non-zero intensity. The choice of the monoclinic space group Cm has certain para-

metrization advantages. In the TbsGe 3 structure, the Tb magnetic atoms are distributed on 4 equidistant layers perpendicular to the c axis (ta- ble 2). Using the Cm description one can assign a moment to each layer and refine the angle between them. In table 2 the expected splitting of the Tb positions from two to six due to a symmetry lowering from P 6 3 / m c m to Cm(77 K) is shown. Cm is an isotranslational subgroup of P63 /mcm of index twelve. The calculations of the magnetic intensities refer to the chemical cell. The antitrans- lation operat ion is incorporated into the calcula- tion as a constant factor of 2:

E # s e : ' i * ' " ( 1 - e~i') = 2 E / t j e2=i*'", l = 2n + 1. J J

r/is the positional vector in the chemical cell and k a vector in reciprocal space.

3.2.1. The magnetic space group of the collinear modulated structure

As shown in table 4, two different models were refined: a collinear and a canted one. The relia- bility factors, however, do not differ essentially. According to group theory [4] there are two possi- ble magnetic space groups of Cm(Ccm and C~c) associated with the wave vector k = 00½. The C~m space group allows only one E~Fy(+ + - - ) mode for the rnv position (2(a)), while Ccc allows two modes FxE~(+ + - - ) and F z ~ ( + + - - ) , see ta- ble3. The refinement has shown that Ccc is the most probable magnetic space group. Because of

P. Schobinger-Papamantellos / Neutron diffraction study of TbsGe 3 101

Table 3 The magnetic modes of position 4(b) in the lower half of the magnetic space group Ccc. The rest is obtained by adding the c antitranslation

Cm Ccc Cem

4(b) position F~, Ay F z A x Fy A z

x y z + + + + + + x )7 z + -- + -- + --

x + ' 2 y - -½ z + -- + -- + -- x + ' 2 y + ½ z + + + + + +

the magnetic domain distribution the neutron powder data of uniaxial systems [8-10] do not provide any information about the moment direc- tion in the plane perpendicular to the unique axis. In our case, however, the description in the Ccc magnetic space group suggests that the moment component in the plane (00z) points along the ahe x : aorth axis (fig. 4a). The strong negative cor- relations between the moment components per layer during the refinement indicated an interde- pendence among them. The structure factor of the extinguished reflection 003 provides for any collin- ear model two linear equations, because both its

real and imaginary parts must be zero (table 2):

2pt 1 - - 3 / ~ ( ~ 2 - - P , 4 ) = 0 , 003, q = 1,

2]/,3 - - 3 / / ~ - ( ~ 2 + p, 4 ) : 0 .

Assuming the same average moment value per layer for the 6(g) atoms at z = ~ and 3, l l a21 = I i a 4 l, the correlation between the moment results in two non-distinguishable structures, differing by a c/2 origin shift:

1) ~1 = 0 a n d ~ 3 = ( 3 / ~ - ) / t 2 = ( 3 / ~ - ) / ~ 4 ,

2) ~3 = 0 and #1 = (3/f2-)/t 2 = ( - 3 / ~ - ) ~ 4 .

Table 4 The refined parameters from the 77 K neutron data of ThsGe3 for the collinear and the canted models. The estimated standard deviations are given in parentheses and correspond to the last digit . / , is the ordered moment of Th and ~xyz its xyz components, ex, • , are the moment angles with the x, z axis, respectively. Rn, Rm, R,% are the agreement values for the nuclear, magnetic and weighted profile intensities. Rex p is the expected value related to the statistical accuracy of the data

T = 7 7 K Tb t Tb 2 Tb 3 Tb 4 R , __1 __3 at z = 0 at z =18 at z - z at z - ~ R m

R w o

Rexp

Collinear ~x(#e) 0 2.97(3) 6.35(5) #z(#n) 0 1.7 (3) 2.6 (7) #T(P'n) 0 3.4 (2) 6.8 (3) ~z(deg) 60.2 (2) 67.7 (4)

Canted lax(#B) 5.1(8) 2.4 (4) - #y(#n) - 1.2 (1.0) 5.1 (8) #z(#B) 0.6(1.1) 2.5 (7) 0.6 (1.1) #T(#B) 5.2(7) 3.7 (2) 5.2 (7) • ,,(deg) 0 26.8(27.4) 90

2 .97(3) 0.06 1.7 ( 3 ) 0.04 3.4 ( 2 ) 0.10

60.2 ( 2 ) 0.03

--2.4 1.2 2.5 3.7

153

4) 0.06 1.0) 0.02 7) 0.10 2) 0.03

(27)

aorta = 8.493(4), b = av~- = 14.71 (1), c = ! 2.733(4) (.~), overall temperature factor = 0.75 (,~2).

102 P. Schobinger-Papamantellos / Neutron diffraction study of TbsGe x

p 2 w

• °

~ t ' . . ~ °

/ ' '/ ' " I t i

, Tb I Z=O,114 ~ i j

oTb 2 • 1#8 ~,.,,,~..~0 t # ' • Tb 2 ~ 3/8 a ! 0 Fig. 4. T h e co l l inear m o d u l a t e d s t ruc tu re o f T b s G e 3. (a) the p ro j ec t i on o f the lower ha l f of the o r t h o h e x a g o n a l cell. T h e m o m e n t s in

the p l a n e p o i n t a l o n g ah~x = aorth a n d c h a n g e s ign in the next cell; b) the p ro j ec t i on o f the re f ined ave rage l ayer m o m e n t s o n the p l a n e

Oxz. T h e sine m o d u l a t e d c h a r a c t e r a n d p a r a m e t e r co r r e l a t i ons are exp l a ined b y t w o sine w a v e s / t o fo r the 4(d) a t o m s a n d / ~ o for the

6(g) a t o m s wi th the s ame or igin .

T a b l e 5

T h e i n t eg ra t ed o b s e r v e d a n d c a l c u l a t e d n e u t r o n in tens i t ies o f T b s G e 3 a t 77 K

re fe r to the o r t h o r h o m i c cell a n d to the co l l inear m o d e l

i n c l u d i n g the mul t ip l i c i ty a n d the L o r e n z fac tor ; hkl

h k / Inu c lmag lob s h k 1 I . . c Imag lob s

0 0 1 -- 136535 140295 1 5 0 3231 - 3156

0 0 3 -- 0 335 0 4 3 74 72

0 1 3 -- 0 0 3 1 i 656 696

2 0 1 -- 78 78 3 1 1 -- 8 8

0 1 -- 2956 2976 2 4 1 40 42

1 3 1 -- 8846 8976 2 4 1 -- 5242 5592

3 1 -- 146 150 1 5 1 -- 62 66

1 1 3 -- 1754 2198 1 5 1 6492 6952

1 3 -- 4168 5226 3 1 2 13052 - 13766

0 2 3 -- 3600 4526 2 0 4 12092 -- 12560

2 2 1 -- 920 940 2 4 2 13954 -- 14491

2 1 -- 2278 2328 1 3 4 2 2 3 7 4 - 2 3 2 2 8

0 4 1 -- 3750 3850 0 0 5 0 0

0 0 4 8007 6804 1 5 2 13253 - 13757

2 0 3 -- 159 155 3 3 0 13275 -- 12201 0 3 -- 6728 6558 0 6 0 6140 5642

1 3 3 -- 9160 8858 3 3 1 1012 114

T 3 3 -- 1492 1452 3 3 1 1904 2096

1 1 4 2870 2125 2 2 4 2747 -- 3027

0 2 4 1350 999 1 1 5 843 920 3 I 0 2817 2842 1 1 5 1446 1594

2 2 3 -- 38 38 0 2 5 1236 1364

2 3 -- 46 46 0 4 4 1605 -- 1770

2 4 0 3768 3723

P. Schobinger.-Paparnantellos / Neutron diffraction study of TbsGe 3 103

The nodes at z = 0 or ¼ (#1 or # 3 = 0 ) of the collinear structures can be considered as a sine or cosine modulation of the magnetic moment along the c-axis with the modulated amplitude along a (fig. 4b). Insertion of the above parameter correla- tions in the least square refinement reduces the number of free parameters from eight to two. The profile calculated on the base of this model fits well the observed data (table5). The reliability factors are R n = 0.06, R m = 0.04 and R% = 0.10. The Tb 3+ magnetic form factor used is from ref. [11]. The refined average moment values at 77 K, refer to moment per layer divided by the number of atoms in it and are:

#1 = 0, #2 = #4 = 3.4(2) and #3 = 6-8(3)#B,

as expected, lower than the free ion value for Tb 3 + (9.7/~B). The modulated character of the structure along a is better schematized in fig. 4b which presents the refined #x and #~ values at the 4Tb layers together with the sign change due to the c antitranslation. The moment angle with the c-axis is 67.7°(4) for the 4(d) atoms and 60.2°(2) for the 6(g) atoms.

The parameter correlation between the #x val- ues of the two Tb positions can be understood as a phase relation between the atoms at ~r/2 and ~r/4 on two sine waves with amplitude relation (#0

3 0 = ~ # 2 ) and going through the same origin. A simple explanation of this amplitude relation would be to assume that one 6(g) atom per layer has no resulting ordered moment as for example MnsSi 3 [15]. The /~x value of the remaining 6(g) atoms would then be (3)# 2 = #1/vc2 -. It is obvious that the parameter correlation introduced by the ex- tinguished 003 reflection may also render other collinear modulated structures possible, i.e.

[#1[ = 1#3[ and # 4 = 0 , #2 = (2¢ r~/3)#l.

We limited our model to modulation in the linear chain due to its higher symmetry. The symmetry of the collinear structure is finally higher than Ccc and can be described in the magnetic space group Cc2/c, with the moment values and directions given in table4. The magnetic space group C¢2/c Sh~ has a higher symmetry than the earlier sug- gested [71 S h ~ B 2 ' / m = C 2 ' / m or Sh6~ B 2 / m ' =

C2/m' . Cc2/c results from C 2 / m ' by adding the c antitranslation.

3.2.2. The canted model In the canted model the moments of the 4(d)

atoms turn by ~r/2 in the (00z) plane by going from #1 to # 3 like a 41 screw axis having the same absolute value. For the 6(g) atoms we assumed, however, the same moment value 1#21 = 1#41 and an angle (between them) of 180 ° - 2 ~ x where dP x is the angle between #2 and #1. The large error on the resultant angle ~x = (26.8 -~ 27.4) ° does not make this model preferable to the collinear one.

3.3. The temperature dependent magnetic phase transition

The temperature dependence (fig. 3) of the in- tegrated magnetic intensity of the 001 reflection is in fair agreement with the magnetic data [2], both methods result in the same Nrel temperature T N = 85 K.

Below 85 K the magnetic intensity increases, as expected, with decreasing temperature; the satura- tion moment value is reached at 20 K. At the same time a continuous shift at the 20 position, of the same peak, towards lower values could be ob- served in the temperature interval 75-50 K. Simi- lar displacements have been observed for most of the magnetic peaks (fig. 2). For the same tempera- ture interval the nuclear peaks remained unmoved (compare the peaks 002 and 210 in fig. 2). This fact proves that the phase transition causes a con- tinuous shift of only the magnetic peaks, and is therefore an incommensurate [14] phase transition of the magnetic structure. The remarkable 20 shift of (1 w 0.15) ° of the first magnetic satellite 00q w- (q - -½ at 77K) indicates that the propagation vector of the modulation has a component in the c* direction. At 4.2 K q was 0.4616(2)c* or 0.0725 .~-1. With this q value inserted in the formula:

d .2 ---- 4 sin20//~ 2

= (h 2 + hk + k2)la*] 2 + (l]c* I ~ q)2,

all magnetic peaks could be indexed as 00qW- satellites of the allowed [12] nuclear peaks. This holds for the whole temperature range of the phase

104 P. Schobinger-Papamantellos / Neutron diffraction study of TbsGe J

h k I ooq ~' d" P2c (~)-i

i15 !12 + 0,458

213 211' 9,430

210~ O.3Eg 211 211-

113 111" 0.334 112-

201 200t 0.202

103 102- 0272

110~ 0.246 =~ iii Iii-

001 000~ 0078

20(~)

50.81 ~

=¢~57,59

~5Q,53 ~

>~45,44 " ~

=~--33,31

~" 36,78

=~i 33.45

2-1045_._____

o.s 0,~9 0,q~ 2 51 25

t ooq ± 2 8 ( )

112 ~ 3

211* 59,29

3(I0 -+ 57.4

I02 + 56,72

211- 50,74

2]C ~ q'], 3t4

IlU 48.09

111" 44.81

2f10-* 38,07

102- 31.54

111- 33.74 110 -+ 33.18

000-* 9,81

0,47 0,46 q . c

17 13 SPIRAL PERIOD IN C

74 ~5 60 52 50 T I K I

Fig. 5. The splitting of the fundamental hexagonal peaks into satellites h k l -,- as a function of q and 7". The commensurate n c

values of the spiral periods are noted. The left column refers to the orthohexagonal indexing P2c for q = ~2 with corresponding d* and 20 values. The right column to the 4.2 K data.

transition. In fig. 5 we derive systematically the d* and 20 shifts of the magnetic peaks as a function of q. In the abscissa we note also for five q values the corresponding temperatures for which the neu-

0 0 2 T~ol: .1 . , . , - - . . , - . . . - . . . ,

ooo

Fig. 6. The magnetic reflections (crosses) as satellites of the allowed nuclear reflections at 4.2 K, q =0.4616(2) in e* units.

tron patterns are indexed by this q value. Although the phase transition is in its nature

incommensurate there are intermediary q values like 0.46, 0.47, 0.48 for which the period of the

u ~ OOq I

o.5o,-2 . J 4 18o

/ - 2 . 0 8 172 .8

2 . 1 6 6 - , 0 . 4 6 1 6 5 , 6

T [K 0 . 4 2 I I I I

0 2 0 4 0 6 0 8 0

Fig. 7. The left-hand ordinate gives the variation of the wave vector q and of the spiral periodicity as a function of tempera- ture for TbsGe 3. The right-hand ordinate gives the variation of the spiral angle vs. temperature.

modulation is a rational multiple of c. The corre- sponding magnetic cell is commensurate and has a rather large c-period, 13xc, 17xc, 25xc, etc.

Fig. 6 shows the magnetic reflections (crosses) occurring at 4.2 K as satellites (q = 0.4616) of the allowed nuclear reflections in reciprocal space. The phase transition in reciprocal space is explained as a temperature dependent shift of all satellites from the original value of q = ½ at 77 K to q = 0.4616(2) below 50 K. How this shift takes place as a func- tion of the temperature is illustrated in fig. 7 which refers to the behaviour of the zero point satellite. The change of the propagation vector describes a smooth convex parabolic curve with increasing temperature. The period of the incommensurate cell varies between 2c and 2.16c.

3.3.1. The helical structure at 4.2 K Modulated structure described by only one wave

vector could correspond to one of the following types [ 12-15]: sine modulated, conical spiral, flat spiral or antiphase domain.

The antiphase domain type is ruled out due to the absence of higher order satellites nq. Similarly from the absence of ferromagnetic contribution to the nuclear reflections the conical spiral can be excluded.

The choice between a sinusoidal or a flat spiral structure is based on the comparison of the free ion with the experimental magnetic moment value

P. Schobinger-Papamantellos / Neutron diffraction study of TbsGe ~ 105

for each model derived from the zero point satel- lite.

The absolute value in #8/cel l of the magnetic structure amplitude of the zero point satellite at 4.2 K is proportional to the square root of the measured integrated neutron intensity in barns [15].

&,ooo+ = ,0c t2mc2)2] J2

M X L P X f m 2 ~ e 'y 2

= 63.25/~B/cell,

M = 2 is the multiplicity, LP the Lorenzfactor, fm the Tb 3+ form factor, c the scale factor and (e'y2/2mc2)=0.2695 × 10 -12 cm is a conversion factor from barns to/~ B.

For the flat spiral the calculated structure am- plitude factor [13] of 000 ~ satellite is given as:

~ 1 + COS2tO F~'°°°+ = 4 ~ttpf~( q) e x p ( - i ~ ) ,

P

ff l +cos260 _ 1

4

~o = 0, being the angle between the spiral cone axis and the scattering vector e. /~ is the magnetic moment of the Vth atom and ¢~ its phase angle in the chemical cell.

The corresponding formula for the sine mod- ulated structure is

sin F~'°°°+- 2 ~'ll°f~(q) exp( - - iO, ) ,

P

½sin 0~ = ½,

~o = 90 ° is the angle between the scattering vector e and z, a unit vector in the direction of the varying spin component and / to the amplitude of the sinusoidal variation of the moment of the Pth atom. The two formulas differ essentially by a multiplicity factor.

Since the relative phases (I)~ of the 10 Tb atoms, refering to the origin atom at z = 0 of the chemical cell are not known a priori, we first compare the resulting minimum moment values for the two models on the assumption that all atoms scatter in phase within a cell.

For the sine modulation the average minimum moment value, # = 63.25 × 2 / 2 p e x p ( - i ~ v ) = 12.65ga which is unrealistic and leaves the spiral model with ~ttmi n = 12.65/vr2 - = 8.9gB as the most probable model.

Since the average moment value of the spiral model is close to the saturation value of the free ion Tb 3+ (9.7~a) the relative phases of the Tb atoms within the chemical cell are near zero.

The spiral angle within the limits of the experi- mental error remains unchanged in the temper- ature interval 50-4.2 K and is ~v = 2~r X 0.4616 = 166.14 ° from cell to cell. It appears that the q for the modulation has to a good approximation the value 6//13. This means that 6 helix periods add up to 13c lattice spacings i.e. 82.74 ,~, or, that after 13 of the original unit cells the magnetic moment takes its initial orientation.

b

Fig. 8. The magnetic spiral structure of TbsGe 3 at 4.2 K show- ing two unit cells. For clarity the scale of the c axis is doubled. The full circles correspond to Tb at 6(g) while the open circles with arrows to the Tb 4(d) position. The open circles (without arrows) correspond to the Ge 6(g) atoms. The spiral angle of 166.14 ° is between second nearest neighbours.

106 P. Schobinger-Papamantellos / Neutron diffraction study of TbsGe 3

In table 6 the observed and the calculated mag- netic intensities at 4.2 K are compared for the flat spiral model. The moment value per Tb atom is 8.9(1)# a (as derived from the 000 ~ satellite abso- lute intensity). The good agreement between Icalc and lob s supports the choice of this model. The scale factor and the overall temperature factor are the same as for the 77 K refinement. The corre- sponding magnetic structure is shown in fig. 8 for two chemical cells at 4.2 K. The main feature of this structure is that within each cell the magnetic moments are parallel and confined to the hexago- nal plane, the direction of the magnetization trac- ing a helical spiral with angle 166.14 ° as shown in fig. 9. The symmetry of the spiral axis can be described as 136 , i.e. after 13 cells and six turns the direction of magnetization returns to its original position.

Both Tb positions have a negative exchange between second nearest neighbours in the c direc- tion. Since the two 6(g) Tb layers are related by the screw rotation operation of the 63 axis and the moments are parallel, the spiral along the 63 axis

has a double coiling. The same is true for the linear chains of the 4(d) atoms. The 4(d) atoms behave like conjugate because they change their moment direction in the plane in pairs, presuma- bly this is connected to the covalent character of the bond.

4. Conclusions

The TbsGe 3 neutron data confirm the X-ray parameter values as well as the antiferromagnetic properties and the ordering temperature T N = 85 K

[21. The temperature dependent magnetic phase

transition has been explained by a one-dimen- sional and continuous change of the wave vector of the modulation showing that the transition is incommensurate in its nature and the magnetic structure aperiodic. Effectively the angle ~. be- tween the moment directions in two successive cells changes. The exchange interaction within a chemical cell is ferromagnetic for the whole tem-

Table 6 The integrated observed and calculated magnetic neutron intensities of TbsGe 3 at 4.2 K including the multiplicity and the Lorenz factor. The indexing refers to the hexagonal chemical cell and the reflections hkl + are given as 00q ~ satellites (q = 0.4616) of the nuclear hkl reflection. 20 c deg is the calculated peak position (bt = 8.9(1)#a)

h k I 2 0c(deg ) Imag lmag lob ~

0 0 0 ~- 9.61 976 895 976 895 972 301 0 0 2 - 32.58 3157 7 1 1 0 ~ 33.18 9861 ~ 52053 52383 1 1 1 33.74 39035

1 0 2 - 37.54 48531 ~ 76962 76967 2 0 0 w 38.07 28 431 J 1 1 1 ÷ 44.81 27427 "l 1 1 2 46.09 58 988 ~ 86 415 82 942

2 0 2 - 49.92 29422 1

2 1 0 + 50.34 29 289 I 59 748 64 459 2 1 1 -- 50.74 1037 1 0 2 + 56.72 20297 ] 3 0 0 ~ 57.4 24 212 j* 44 509 48 500

2 1 1 + 59.29 24 874 24 874 26 977 1 1 2 + 63.30 31 975 31 975 27 887 1 1 3 - 64.99 13441 1 3 0 2 - 65.71 2281 2 0 2 + 66.44 0 17549 17985 2 2 0 + 67.06 1827

P. Schobinger-Papamantellos / Neutron diffraction study of TbsGe 3 107

perature range. This assumption agrees with the high positive paramagnetic Curie temperature (8p = 93 K). We have derived here the magnetic struc- ture of the end members of the phase transition.

' / / /

,,/S / , 0 /

6

9

10 ,.~121 5

7

Fig. 9. The spiral variation of the magnetization direction within the magnetic cell of period 13c, at 4.2 K.

The collinear antiferromagnetic model (~v = rr, P2¢ as magnetic cell) used to explain the 77 K data results in a modulated structure and correlations

between the magnetic moment values of Tb 3+ at the 6(g) and 4(d) positions. The simplest assump- tion arising from the parameter correlations is that one of the 6(g) atoms per layer has a zero moment value. A similar result was found in MnsSi 3 [15] from single crystal data, and is interpreted as a moment modulation in the plane.

From the possible models fitting the data the one having the nodes of the modulation in the linear chains of the 4(d) atoms is chosen because of it's higher symmetry. The corresponding mag- netic space group is C c 2 /c and the magnetic struc- ture is described by two ferromagnetic compo- nents in the chemical cell. Within this description the moment component in the hexagonal plane is parallel to the a axis. The antiferromagnetic com- pensation is realized via the c-antitranslation oper- ation. The magnetic modes for all Tb positions are Vx x,

The refined parameters for the canted model are also given for comparison. It is hoped that one-domain single crystal data will be helpful in choosing a unique model of this structure [8].

In the spiral structure at 4.2 K the antiferro- magnetic arrangement between second nearest neighbours deviates by 14 ° from that of the P2¢ phase ( ~ = 180°). The splitting of the magnetic reflections occurs along only one reciprocal space direction and the wave vector value of 0.4616(2)c* gives an unique interpretation of the observed reflection positions (table6, fig. 1). Because the experimentally defined moment value (from the 000 T- satellite) of 8.9(1)/~ B is close to the free ion value it is assumed that the moments are confined to the hexagonal plane and that the crystal field effects do not affect the moment value.

In contrast to the P2¢ structure t h e / ~ compo- nents are zero. Besides the temperature factor, which was t aken from the 77 K data, this spiral structure is defined by only two parameters. The hard axis of magnetization is parallel to the hexag- onal axis c and probably the magnetic moment behaves isotropically within the hexagonal plane.

The spiral angle of 166.14" indicates that the structure is not very different from the collinear antiferromagnetic model. Another characteristic is that the 4(d) atoms, supposedly connected by co- valent bonds, change their angle in pairs forming a

108 P. Schobinger-Paparnantellos / Neutron diffraction study of TbsGe 3

doub le spiral a long c. The doub le spiral is also

fo rmed by the layers of the 6(g) a toms re la ted by

the 63 screw rota t ion . To our knowledge, in most of the cases of

k n o w n spirals [14-17] the spiral angle refers to neares t neighbours , while here to second neares t neighbours . I t is the d i rec t ion of the magne t i za t ion tha t traces a hel ical spiral . The absence of a criti- cal field value when p lac ing T b s G e 3 in a pulsed magnet ic field also agrees with the spiral config-

u ra t ion [2].

Acknowledgements

I would like to express my gra t i tude to Prof. Dr. A. Niggl i at E T H Z , Ins t i tu t for Kr is ta l logra- phie und Pet rographie , for the suppor t of this work, to Dr. P. Fischer , E T H Z , Ins t i tu t for Re-

ak tor technik , for useful discussions and to Mr. R. Haefel i , Ab te i lung Meta l lurg ie EIR, for the sam- p le p repara t ion .

[2] K.H.J. Buschow and J.F. Fast, Phys. Stat. Sol. 21 (1967) 593.

[3] H.M. Rietveld, RCN Rep. 104, Petten, The Netherlands (1970). W. von Wartburg, AF-SSP-46 Wiirenlingen (1970).

[4] W. Opechowski and R. Guccione, In: Treatise of Mag- netism, vol. IIA, eds. H. Suhl and G. Rado (Academic Press, New York, 1965) p. 105.

[5] International Tables for X-ray Crystallography, vol. I (The Kynoch Press, Birmingham, England, 1969) p. 19.

[6] O. Halpern and M.H. Johnson, J. Phys. Rev. 55 (1939) 898.

[7] P. Schobinger-Papamantellos, Z. f. Krist. 147 (1978) 330. [8] G. Shirane, Acta Cryst. 12 (1959) 282. [9] C. Wilkinson and J. Lisher, Acta Cryst. A 29 (1973) 453.

[10] C. Wilkinson, Acta Cryst. A 31 (1975) 856. [I1] A.J. Freeman and J.P. Desclaux, J. Magn. Magn. Mat. 12

(1979) 11. [12] W.C. Kohler, Acta Cryst. 14 (1961) 535. [13] B. van Laar, RCN-92, Petten, The Netherlands (1968). [14] J. Przystawa, IAEA-SMR-46/115, vol. 2 (1980) p. 213. [15] G.E. Bacon, Neutron Diffraction (Clarendon Press, Ox-

ford, 1975). [16] G.H. Lander, P.J. Brown and J.P. Forsyth, Proc. Phys.

Soc. 91 (1967) 332. [17] A. Oles, F. Kajzar, M. Kucab and W. Sikora, Magnetic

Structures Determined by Neutron Diffraction PWN, Warsaw (1976).

References

[1] K. Schubert, Kristallstrukturen zweikomponentiger Pha- sen (Springer, Verlag, Berlin, 1964) p. 307.