POLARISED NEUTRON STUDIES OF THE
PARAMAGNETIC STATE OF CHROMIUM
J. Booth, K. Ziebeck, H. Capellmann, P. Brown
To cite this version:
J. Booth, K. Ziebeck, H. Capellmann, P. Brown. POLARISED NEUTRON STUDIES OF
THE PARAMAGNETIC STATE OF CHROMIUM. Journal de Physique Colloques, 1982, 43
(C7), pp.C7-301-C7-304. <10.1051/jphyscol:1982743>. <jpa-00222350>
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https://hal.archives-ouvertes.fr/jpa-00222350
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JOURNAL DE PHYSIQUE
Colloque
C7, supplément
au n°12,
Tome 43, dêcembve
1982
page
C7-301
POLARISED NEUTRON STUDIES OF THE PARAMAGNETIC STATE OF CHROMIUM
J.G. Booth*, K.R.A. Ziebeck* , H. Capellmann + and P.J. Brown**
*Physies Department,
Salford
University,
U.K.
**I.L.L.
Grenoble,
France
+
Inst.
Theor. Physik,
T.H. Aachen,
F.R.G.
Résumé - La diffusion paramagnétique d'un monocristal de chrome, spectroscopiquement pur, a été mesurée en utilisant des neutrons polarisés et la technique
d'analyse de polarisation. La diffusion présente un pic très marqué au voisinage de (100). L'intensité du pic, mesurée à 3 températures, décroît de 90 %
entre 1.18 TJJ et 2.23 Tfj ; dans le même domaine de températures la longueur
d'onde caractéristique de l'ordre à courte distance associée décroît de 57 Â à
26 Â. La diffusion observée est isotrope autour de (100) et décroît jusqu'à
un niveau très faible lorsque le vecteur d'onde mesuré à partir de (100) est
supérieur à 0.2 TT A .L'intégration de la diffusion observée dans la 1ère zone
de Brillouin conduit à une valeur moyenne du moment, par atome de chrome, de
0.1 ± 0.05 y]}, indépendamment de la température.
Abstract - Polarised neutrons and polarisation analysis havebeen used to investigate the paramagnetic scattering of neutrons from a single crystal of
spectrographically pure chromium. The scattering was found to be strongly
peaked around (100). For the three temperatures at which measurements were
made, 1.18, 1.53 and 2.23 T N the peak intensity was found to decrease by 90 %
and the characteristic wavelength for short range magnetic order from.57 A to
26 A. The scattering was found to be isotropically disposed about 100 and to
fall to a low background level for wavevectors greater than .2 n A . Integration of the observed scattering throughout the Brillouin zone indicated the
average moment per chromium atom to be 0.1 ± 0.05 yjj independent of temperature.
1. Introduction - It is well known that the properties of the magnetic transition
metals present problems of interpretation for both localised and itinerant theories.
On the one hand it is difficult using a localised model to explain the occurence of
non integral moments and their magnitude while on the other observations of CurieWeiss behaviour and the magnitude of the Curie temperatures cannot readily be explained in terms of a band model in which the band splitting goes to zero at the
ordering temperature as a result of excitations from the majority to the minority
band.
Many of the properties of the antiferromagnetic metal chromium have however been
explained satisfactorily in terms of an itinerant model in which the average amplitude of the local spin density vanishes at TJJ. The latter assumption conforms to
earlier observations (1,2) that no significant paramagnetic scattering exists in
the paramagnetic phase. But recent inelastic neutron scattering measurements (18,14)
have suggested that some part of the band remains split at temperatures beyond
M . 5 Tfq and the present paper gives results of an attempt using a single crystal and
polarised neutrons with polarisation analysis to determine uniquely the magnetic
contribution to the scattering in the paramagnetic regime . The results have some
features in common with measurements made recently by the authors and co-workers
ref. T23] on other ferromagnetic and antiferromagnetic systems such as Fe and
a-Mn and which have been interpreted in terms of short range magnetic order and
the retention of the band splitting into the paramagnetic regime. These results
are highly relevant to current theories of the existence of magnetism in the
transition metals (3,4,5).
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1982743
C7-302
JOURNAL DE PHYSIQUE
2. P r o p e r t i e s of pure chromium - Below t h e NGel temperature (311 K) chromium o r d e r s
a n t i f e r r o m a g n e t i c a l l y with a modulated s i n u s o i d a l
s t r u c t u r e which i s t r a v e r s e l y
p o l a r i s e d t o t h e propagation v e c t o r 5 confined t o a (100) d i r e c t i o n (TSDW). On cool i n g below 120 K t h e d i r e c t i o n of p o l a r i s a t i o n [61 changes t o become p a r a l l e l t o
fi [7] (LSDW). A t low temperatures t h e maximum moment i s 0.59 VB and t h e magnitude
of t h e wavevector i s found t o vary slowly w i t h temperature. Careful s i n g l e c r y s t a l
measurements c l o s e t o TN suggest t h a t t h e antiferromagnetic-paramagnetic t r a n s i t i o n
i s of f i r s t o r d e r [81. The small anomaly i n t h e h e a t c a p a c i t y a t TN showing t h a t t h e
change i n entropy i s only 0.0044 191 cal/deg, h a s been i n t e r p r e t e d a s i n d i c a t i n g a
c o l l e c t i v e e l e c t r o n mechanism f o r t h e antiferromagnetic t r a n s i t i o n . The absence of
s i g n i f i c a n t paramagnetic neutron s c a t t e r i n g a t 2.27 TN seemed t o add weight t o t h i s
i n t e r p r e t a t i o n [21. According t o a two band model developed by Overhauser [ l o ] and
Lomer [ l l l t h e antiferromagnetic s t a t e a r i s e s from t h e s t r o n g c o r r e l a t i o n between
t h e e l e c t r o n s i n a band a t t h e
p o i n t and t h e holes i n a band l o c a t e d a t t h e H
p o i n t . As a r e s u l t of exchange i n t e r a c t i o n s between t h e s e , t h e e l e c t r o n s of one band
move coherently with t h e h o l e s i n t h e o t h e r t o form a condensed s t a t e with long
range o r d e r . The thermal and magnetic p r o p e r t i e s a r e e x p r e s s i b l e i n terms of an
2a
energy gap 2g, between the two bands and a s p i n d e n s i t y wavevector
ll(lgl = -(I
6,
0,O) 6 = 0.04) which i s t h e s m a l l e s t wavevector connecting t h e two i e c e s o? Fermi
s u r f a c e . Using t h i s model t h e changes i n magnetic moment, TN and
which a r e observed when chromium i s alloyed can be explained i n terms of the geometrical feat u r e s of t h e Fermi s u r f a c e s [ I l l .
-
*
Recent i n e l a s t i c coherent neutron s c a t t e r i n g experiments (12,13,14,15) have suggested t h a t a t l e a s t some magnetic c o n t r i b u t i o n remains a t temperatures w e l l above TN.
In t h e paramagnetic region a broad Gaussian c o n t r i b u t i o n i s observed centred on t h e
magnetic (1,0,0) p o s i t i o n with very l i t t l e a t t e n u a t i o n below 11 THz.
3. Experimental technique - The m e t a l l u r g i c a l h i s t o r y and p u r i t y of t h e high
p u r i t y chromium c r y s t a l have been described previously by Ziebeck and Booth 1131.
The c y l i n d e r a x i s was c l o s e t o a
d i r e c t i o n and t h e c r y s t a l of diameter 18 mrn
and l e n g t h 25 mm was mounted with t h i s a x i s v e r t i c a l . The D5 spectrometer a t I.L.L.
was used t o determine t h e paramagnetic s c a t t e r i n g a t t h r e e temperatures up t o 2.2 TN.
The d e t a i l s of t h e experimental arrangement have been given by Ziebeck e t a l . [16].
DOO]
The paramagnetic s c a t t e r i n g was placed on an absolute s c a l e w i t h r e s p e c t t o t h e nuc l e a r incoherent s c a t t e r i n g using t h e i s o t o p i c non-spin-flip (b;(NSF) = 1.38 f
13
barns) and s p i n f l i p (bi(SF) = 0.38 f
08) incoherent s c a t t e r i n g cross-sections
reported by Cywinski [17]. The c o r r e c t e d r a t i o of t h e SF count t o the NSF count was
found t o be 1.73 i n good agreement w i t h the value 1.64 found by Cywinski e t a l .
1171.
.
.
The paramagnetic response was measured a t each temperature from scans made a t a
s e r i e s of p o s i t i o n s throughout t h e f i r s t B r i l l o u i n zone. S i g n i f i c a n t s c a t t e r i n g was
observed only near t h e (100) p o s i t i o n . This s c a t t e r i n g appeared t o be d i s t r i b u t e d
i s o t r o p i c a l l y about (1,0,0). The paramagnetic s c a t t e r i n g was obtained a s described
by Ziebeck and Brown [161. An effectivemoment M(0) can then be defined a s
Here f(Q) i s t h e form f a c t o r of t h e magnetic c a r r i e r s and t h e i n t e g r a t i o n i s over
t h e i n c i d e n t neutron energy E; of t h e instrument. From t h e f l u c t u a t i o n - d i s s i p a t i o n
theorem
x
.
I f t h e energy r e s o l u t i o n i s s u f f i c i e n t t o include a l l f l u c t u a t i o n s .
i s t h e d.c.
s u s c e p t i b i l i t y . The s c a t t e r i n g f u n c t i o n of equation (1) f o r a p a r t i c u l a r momentum
t r a n s f e r Q i s r e l a t e d t o t h e magnetisation d e n s i t y - magnetisation d e n s i t y correl a t i o n f u n c t i o n by t h e equation
o,
sNp(e..)=
J-_--
<M~(Q,L)
M'(-Q,O))~U
Ziebeck e t a l . (22) have shown t h a t t h e q u a s i - e l a s t i c cross-section measured i n t h e
.
p r e s e n t experiment i s given by
d d ~ ~ ~ (j A 9
' < ~ *~
(q,4
k)
SJC
*
M~(-Q,O))
2
"
where A t = 2 ~ r / A w
and i s thus t h e time average over an i n t e r v a l
A t of t h e c o r r e l a t i o n function.
4. R e s u l t s - The paramagnetic response c l o s e l y resembled t h a t measured i n previous
unpolarised experiments with s u p e r i o r energy r e s o l u t i o n (14,18). A t each temperat u r e the e l a s t i c s c a t t e r i n g was s t r o n g l y peaked i n t h e v i c i n i t y of (1,0,0). The
q-width of t h i s r e s onse increased with i n c r e a s i n g temperature from (FWHM) 0.11 2-I
a t 367 K t o 0.24 g- a t 687 K. The q-dependence could not be f i t t e d t o any simple
f u n c t i o n a l form. Constant q scans a t s i x wave-vectors i n t h e range between (0.9,0,0)
and (1,0,0) showed t h a t t h e s c a t t e r i n g f e l l predominantly w i t h i n t h e energy resolut i o n of t h e instrument f o r w = 0 , namely 12 THz. Although no e f f e c t of temperature
on t h e energy width a t (1,0,0) was observed a small i n c r e a s e w i t h i n c r e a s i n g temp e r a t u r e was found i n constant q-scans a t (0.9,0,0), (0.92,0,0) and (0.94,0,0) : f o r
t h e e l a s t i c condition t h e v a r i a t i o n of t h e i n t e n s i t y i n t h e peak i s i l l u s t r a t e d i n
Fig. 1. I n order t o o b t a i n an e s t i m a t e of t h e magnetic moment p i n t h e Wigner-Seitz
II
c e l l the equivalent expression
P
was used where t h e i n t e g r a t i o n i s
over t h e e n t i r e B r i l l o u i n zone.
Using our r e s u l t s t h i s y i e l d s
= 0.1 c 0.05 YB independently
2
of temperature s i n c e t h e f a c t o r Q
h e a v i l y weights c o n t r i b u t i o n s
away from t h e zone c e n t r e .
Figure 1. Paramagnetic s c a t t e r i n g i n chromium
a s a f u n c t i o n of reduced wavevector i n t h e
(hOO) d i r e c t i o n a t t h r e e d i f f e r e n t temperatures.
.
Previous paramagnetic s c a t t e r i n g
measurements on chromium by S h u l l
[ I ] and more r e c e n t l y by Wilkinson
e t a l . 121 have been c a r r i e d out
on p o l y c r y s t a l l i n e m a t e r i a l s using
two a x i s spectrometry and unpolar i s e d neutrons. I n n e i t h e r case
was any s i g n i f i c a n t paramagnetic
s c a t t e r i n g observed. The f a i l u r e
t o observe paramagnetic s c a t t e r i n g
i n t h e s e previous experiments does
n o t c o n f l i c t w i t h our p o s i t i v e res u l t here. The observation of weak
paramagnetic s c a t t e r i n g w i t h a
narrow Q-width i s very d i f f i c u l t
using a p o l y c r y s t a l l i n e sample f o r
which t h e response i s spread out
over a r i n g and t h e background
l e v e l from m u l t i p l e s c a t t e r i n g and
thermal d i f f u s e processes i s high.
For a s i n g l e c r y s t e l sample t h e
background l e v e l i s lower and the
response i s concentrated i n a
small a r e a of s c a t t e r i n g space.
Furthermore p o l a r i s a t i o n a n a l y s i s
allows t h e paramagnetic s c a t t e r i n g t o be obtained by subtract i o n of cross-sections obtained
f o r e x a c t l y t h e same s t a t e
C7-304
JOURNAL DE PHYSIQUE
of t h e sample whereas t h e s u b t r a c t i o n of c r o s s - s e c t i o n s o b t a i n e d above and below t h e
Nee1 temperature i s a more q u e s t i o n a b l e procedure and t h e a s s o c i a t e d u n c e r t a i n t i e s
may mask a small s i g n a l .
5. Conclusions - The width of t h e (100) peak s u g g e s t s t h a t t h e s p a t i a l c o r r e l a t i o n s
extend over 30-60 1 depending on temperature and t h e s p i n d e n s i t y would appear t o b e
v e r y s t r o n g l y c o r r e l a t e d over t h e s e d i s t a n c e s . The s m a l l amplitude of t h e observed
moment compared w i t h t h a t observed i n t h e o r d e r e d phase may r e f l e c t t h e importance
of l o n g i t u d i n a l f l u c t u a t i o n s . A l t e r n a t i v e l y t h e l i m i t e d time r e s o l u t i o n of t h e neut r o n experiment (Eq. 3) averages t h e c o r r e l a t i o n o v e r a d i s t a n c e which i n m e t a l s may
be o v e r s e v e r a l u n i t c e l l s . I f t h i s i s much g r e a t e r t h a n t h e c o r r e l a t i o n l e n g t h t h e
measured moment w i l l b e c o n s i d e r a b l y l e s s t h a n t h e mean moment w i t h i n t h e c o r r e l a t e d
r e g i o n s . The o b s e r v a t i o n of a moment i n t h e paramagnetic phase is supported by rec e n t t h e o r e t i c a l and e x p e r i m e n t a l evidence. Holden and Heine [I91 from a reexaminat i o n of t h e t h e o r y of t h e magneto volume e f f e c t , conclude t h a t t h e atomic moments
a r e reduced by o n l y a few p e r c e n t a t most from t h e ground s t a t e v a l u e s . Shimizu [20]
i n f o r m u l a t i n g t h e thermal average over l o n g i t u d i n a l and t r a n s v e r s e f l u c t u a t i o n s of
t h e magnetic f r e e energy a s power s e r i e s of t h e m a g n e t i s a t i o n d e n s i t y , f i n d s a sig n i f i c a n t and temperature dependent moment above TN f o r chromium. From h i s model
Shimizu c a l c u l a t e d a moment p e r C r atom of 0.2 VB a t 700 K , i n r e a s o n a b l e agreement
w i t h t h e p r e s e n t experiment. F i n a l l y Kotani and Masuda [21] have measured n u c l e a r
s p i n r e l a x a t i o n r a t e s f o r 5 3 ~ ir n C r metal and Cr - 2 At.%V b o t h below and above TN.
For b o t h m a t e r i a l s t h e temperature dependent r e l a x a t i o n r a t e above TN i s i n d i c a t i v e
of s p i n f l u c t u a t i o n s which can b e i n t e r p r e t e d q u a l i t a t i v e l y . The v a l u e of t h e prop o r t i o n a l i t y c o n s t a n t f o r chromium 1s almost a n o r d e r of magnitude l a r g e r t h a n t h a t
f o r Cr-2%V and comparable t o t h e v a l u e s observed f o r a n t i f e r r o m a g n e t i c B manganese
a l l o y s , above TN.
The importance of s p a t i a l c o r r e l a t i o n s i n chromium i s c l e a r and t h e p r e s e n t measurements i n d i c a t e t h a t t h e magnetic moment p e r s i s t s beyond 2.2 TN. T h i s i s a t var i a n c e w i t h models u s u a l l y proposed f o r chromium which assume t h i s t o c o l l a p s e c l o s e
t o 311 K.
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