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Lattice dynamics of nickel
Physica 34 257-271
H a u t e c l e r , S. Van Dingenen, W.
1967
LATTICE D Y N A M I C S OF N I C K E L b y S. H A U T E C L E R
a n d W. V A N D I N G E N E N *
Studieeentrum voor Kernenergie, Mol, Belgi6
Synopsis The normal modes of vibration of nuclei in nickel at room temperature have been investigated by means of one-ph0non coherent scattering of neutrons.The incident monochromatic cold neutron beam was obtained by Bragg reflection from a Pb single crystal of a Be-filtered reactor beam, and the energy analysis of the scattered neutrons was performed by the time-of-flight technique.The experimental data were treated to give the scattering curves, in the (011) plane of Ni, corresponding to scattering angles of 70 °, 90 ° and 100°, and to an incident neutron wavelength of 4.094 A. From these scattering curves, points of the phonon dispersion curves v(q) in the symmetry directions [1003, [110] and [111] were deduced. A least squares analysis of the Fourier expansion of v2 for the various branches of the dispersion curves shows that the forces extend up to at least third neighbours, but that nearest neighbour interactions dominate. The results have been compared with calculations based on a model proposed by Krebs for facecentered and body-centered cubic metals. The values of the three most sensitive parameters in the theory were obtained by fitting to measured elastic constants and an appropriate value for the effective density of free electrons was chosen. The calculated dispersion curves and scattering curves show very good agreement with the experimental results.
Introduction. D u r i n g the last t e n years, large i n f o r m a t i o n has b e e n o b t a i n e d on the m o t i o n of a t o m s in c r y s t a l s b y m e a n s of t h e r m a l n e u t r o n inelastic scattering. T h e lattice d y n a m i c s of nickel h a s b e e n i n v e s t i g a t e d first b y the a u t h o r s 1) w i t h the aid of a Be p o l y c r y s t a l l i n e filter a n d slow n e u t r o n c h o p p e r s p e c t r o m e t e r w o r k i n g at the low flux r e a c t o r B R 1 , and, m o r e e x t e n s i v e l y a n d also w i t h b e t t e r a c c u r a c y , b y B i r g e n e a u e.a.~) using a triple-axis c r y s t a l s p e c t r o m e t e r in its c o n s t a n t 0 m o d e of operation. T h e last a u t h o r s h a v e f u r t h e r a n a l y z e d t h e i r d a t a , a c t u a l l y the dispersion curves for the n o r m a l m o d e s of v i b r a t i o n p r o p a g a t i n g along the t h r e e s y m m e t r y directions of the crystal, in t h e f r a m e w o r k of the B o r n - y o n K ~ r m ~ n theory, either with general forces or w i t h a x i a l l y s y m m e t r i c forces 3). A good fit to t h e experim e n t a l d a t a could be o b t a i n e d b y including i n t e r a c t i o n s e x t e n d i n g u p to the f o u r t h - n e a r e s t neighbours, which involves the use of a large n u m b e r of a d j u s t a b l e p a r a m e t e r s . H o w e v e r , t h e i n t e r a t o m i c force s y s t e m is r e l a t i v e l y simple: if i n t e r a c t i o n s b e t w e e n d i s t a n t n e i g h b o u r s are present, t h e y do not influence t h e dispersion curves to a large e x t e n t because t h e i n t e r a t o m i c * Deceased
- - 257 - -
2~8
S. HAUTECLER AND W. VAN DINGENEN
force constants between the first neighbours are at least twenty times higher than the force constants between more distant neighbours. The fact that, with the Born-von K~rm~n model, one needs too m a n y parameters to fit experimental dispersion curves of metals has to be attributed to the presence of the conduction electrons, the influence of which also explains the failure of the Cauchy relations4). Several simple models, that take explicitly into account the influence of the conduction electrons on the lattice vibrations, have recently been proposed by De L a u n a y S ) , B h a t i a 6) and KrebsT). All these models contain a very small number of adjustable parameters, which are chosen so as to give correct values for the elastic constants. De L a u n a y considers a monatomic cubic metal as constituted by an ion point lattice embedded in an electron sea. The ion point lattice is treated in the framework of the Born-von K~rm~n model, central "spring" interactions acting between an ion and its first and second neighbours. The electron sea is supposed to behave as an ordinary gas, i.e. a continuum characterized by one elastic constant. Moreover, when waves propagate through the crystal, the conduction electrons are assumed to follow in phase the motion of the ions. Such a model implies three force constants (one of which is due to the electron gas and the other two to the ion-ion interaction) which can be deduced from the measured values of the three elastic constants of a cubic crystal. Unfortunately, as pointed out by L a x 8), this model, as weli as the one proposed by B h a t i a, violates some symmetry requirements: the degeneracy, which is required by the crystal symmetry at some points of the reciprocal space between some branches of the dispersion relation is not obtained. This stems from the fact that, the electrons being treated as a continuum, the dynamical matrix has to more the periodicity of the reciprocal lattice of the crystal. The symmetry requirements are satisfied in the force model developed by K r e b s , in which the metal is considered as a lattice of spherical ions, of uniform charge distribution, embedded in an electron sea. The interaction potential is divided into two parts; each element of the dynamical matrix is then given by the sum of two terms. The first part, which takes into account the exchange interaction due to the overlapping of the ions and eventually the van der Waals attraction, is described, as in the De L a u n a y model, by central spring interactions between each ion and its first and second neighbours; this involves only two parameters, the force constants ~1 and ~2. The second part of the potential must take into account the Coulombinteraction between the ions as well as the interaction induced through the electrons; these latter two contributions are rather large but, due to the screening effect of the conduction electrons, they tend to cancel each other when the separation between the ions increases. Krebs has thus chosen a screened Coulomb interaction between the ions for the second part of the
LATTICE DYNAMICS OF NICKEL
259
potential. Two parameters enter in the corresponding part of each dynamical matrix element: a parameter A depending on the effective charge of the ions, and the screening parameter ~. The whole set of parameters is related to the elastic constants (Cll, C1% C44) and to the lattice constant a b y the relations o~1 : -
½aC44
(l)
0~2 :
:~a(C11 - - C12 - - C44)
(2)
A = ia(Clz
--
C44) a2A 2
(3)
valid for f.c.c, lattices. The screening parameter is written as = c
(3)
(4)
where ne is the electron density, a0 the Bohr radius (0.529 X), and kF the electron wave vector modulus at the Fermi surface. The value of the constant C appearing in eq. (4) is not well defined; in the Thomas-Fermi theory it takes the value 0.814, while only a value of 0.353 is obtained from the plasma theory of Bohm and Pines which takes into account the electron correlations (more exactly, this theory gives a maximum value of 0.47). K r e b s 7) calculated the dispersion curves in the three symmetry directions for Li, Na and K, b y chosing the Pines C-coefficient, assuming one free electron per atom, determining kF in the free electron approximation, and finally using the measured elastic constants to get the values of the basic set of parameters (al, ~2, A). In the case of Na, for which experimental dispersion relations were available, an excellent fit to the data was obtained. On the same basis, S h u k l a 9) has shown that also for Cu good agreement with the experimental dispersion curves is obtained. Thus it seems that for the s t u d y of the lattice dynamics of simple monovalent metals, the Krebs model with no adjustable parameters is satisfactory. Later, S h u k l a 10) and M a h e s h and Daya111) calculated respectively the heat capacities of noble metals (Cu, Au) and of alkaline metals (Na, K, Rb, Cs) b y means of frequency spectra deduced from the Krebs model. An excellent agreement with the experimental data could be obtained if the proportionality factor C in the expression for the electronic screening parameter was treated as an adjustable parameter. The best-fit values ranged between the Pines and the Thomas-Fermi C-coefficient. Using our time-of-flight spectrometer at the high flux reactor B R 2 , we re-examined the lattice dynamics of Ni a) to see how the experimental data obtained with different techniques compare, and b) to test the Krebs model in the case of a non-monovalent transition metal, and not only for dispersion curves along s y m m e t r y axes but more directly for scattering surfaces in
260
S. H A U T E C L E R A N D W . VAN D I N G E N E N
symmetry planes*). A preliminary account of our work was presented elsewhere 13). E x p e r i m e n t a l method and results. When a beam of monochromatic thermal neutrons is sent onto a single crystal, one generally observes peaks in the energy distribution of the neutrons scattered in a given direction; these peaks correspond to one-phonon coherent scattering processes which are governed 14) by the conservation laws
h2 2 m (k~ - - k'2) = ~: hv 0 ~
k 0 - k' = ~ -
q
(5) (6)
where k0(k') is the incident (scattered) neutron wave vector, m the neutron mass, v the frequency and q the phonon wave vector, Q the momentum transfer vector, and T a reciprocal lattice vector. In the energy conservation equation, the + sign corresponds to the creation of one phonon, t h e - - sign to the absorption of one phonon. From the experimentally determined energy of a peak one obtains, with the help of eq. (5), the frequency of a normal mode of vibration; knowing k0, the scattering angle and the orientation of the specimen one further obtains, b y a graphical representation of eq. (6), the wave vector q of this mode. Each peak in the energy distribution of the neutrons scattered in a given direction thus yields a point of the dispersion relation v = v¢(q) (the index i differentiates the various branches). B y repeated observations of energy spectra either for different incident energies, or for different scattering angles, or for different orientations of the sample, one can thus hope to build up the dispersion relations for the crystal under examination. The experiments were carried out using a neutron time-of-flight spectrometer installed at one of the radial beam holes of the B R 2 material-testing reactor at Mol; the apparatus will be described in detail elsewherelS). Essentially, a collimated beam of neutrons coming from the reactor passes through a nitrogen cooled filter combining 20 cm of polycrystalhne Be and 7 cm of Bi single crystal, and is monochromatized b y Bragg reflection from a Pb single crystal. The monochromator is movable along the reactor beam direction so that, with respect to a fixed sample, the incident neutron energy can be chosen at wiU, in the range below 5 meV when the Be filter is used. Under the conditions of the measurements on Ni, the mean wavelength of the monochromatic beam, entirely free from higher order reflections, was 4.094/~; the full width at half maximum of the energy spectrum of the *) In the m e a n t i m e 12) the Krebs model has been applied with succes to the calculation of dispersion curves of several other f.c.c, and b.c.c, metals: A1, Cr, W. No good fit could, however, be obtained in the case of Pb, Mo and Nb.
LATTICE
DYNAMICS
261
OF NICKEL
neutrons incident on the sample was about 1.8°/o. The analysis of the energy distribution of the scattered neutrons is made using the time-of-flight technique. Before reaching the Ni sample, the monochromatic beam is pulsed by a Fermi chopper and the times the neutrons need to reach the detector (a bank of nine 2" BF3 proportional counters divided into two layers) are recorded in a 1024 channel TMC analyzer. The mean flight path was 4.81 m. Under the conditions of the experiment, the energy resolution of the spectrometer was varying between 6% at 5 meV and 13% at 36 meV; no effort was devoted to obtain a higher resolution. The sample, a cylindrical Ni single crystal, 3 cm in diameter and 7 cm long, was mounted on a goniometer head fixed on a rotating table whose axis is normal to the scattering plane; the Ni crystal was itself oriented with the (110) planes parallel to the scattering plane. We performed, in three months of time, a total of about 90 energy distribution measurements, for different scattering angles (70 °, 90 ° and 100°) and orientations of the sample; for each scattering angle the crystal was rotated by steps of 4 ° for successive runs. A typical time-of-flight spectrum is shown in fig. 1. One clearly sees the one-phonon coherent scattering peaks on top of a socle mainly due to onephonon incoherent scattering. In the high energy region, the rapid rise of the socle must be taken into account in order to locate correctly the mean times-of-flight in the one-phonon peaks. In most of the measurements, the background amounted to less than 10% of the peak heights; this low value is mainly due to the fact that the sample is kept out of the main beam. The horizontal line near the time-of-flight axis in fig. 1 corresponds to a background of 700 counts per hour measured with the bank of detectors.
~ t
Ni
= 90 ° = 50 ° =14h
~•
500
%
%
•
c 250 c
",,.. :
". .,,....,
I 2000
1500
Time
[ 1
I 2 5 0 0 ,us
- of - f l i g h t
I 1,5
1 2
I 2.5 A
Wavelength
Fig. I. A typical time-of-flight distribution of neutrons scattered by a Ni single crystal. ~0is the scattering angle; ~vis the angle between the monochromatic beam and the [200] direction of the (01 i) plane; t is the measuring time. The horizontal line near the time-of-flight axis gives the level of the background.
s. HAUTECLER AND W. VAN DINGENEN
262
~
//
~
o
o? L
'
'°" --"
®o
c o(111: o •
2
-
~.
(022)
~
\
;\
,\\ \
i
I
__
\\
ko
\
\,
/ \\\
/
J/
Fig. 2. Scattering surface, in the (011) plane of Ni, corresponding to a scattering angle 9 of 70° and to an incident neutron wavelength of 4.094 •. The reciprocal lattice points located in the scattering plane are given as crossed open points; the polygons determine the limits of the Brillouin zones built around each reciprocal lattice point. The filled circles correspond to high intensity inelastic peaks of the time-of-flight spectra while the open circles are associated to low intensity peaks; doubtful points are indicated by question-marks. Neighbouring points corresponding to non-resolved peaks in the spectra are indexed by letters a and b. Points surrounded by circles are obtained by reflection of experimental points through the (200) and (022) planes. The error bars indicated in some cases correspond to the full widths at half maximum of the inelastic peaks. The heavy lines passing through, or in the neighbourhood of, the experimental points represent the various branches of the scattering surface as calculated in the framework of the Krebs model; the longitudinal or transversal character of the branches is indicated by the letters L or T. These experimental d a t a are converted, t h r o u g h the conservation equations, into points of the scattering surfaces in the reciprocal space, which are defined as the surfaces described b y the e x t r e m i t y of the vector 0 =-- ko - - k' as the crystal is rotated, the scattering angle being kept constant. Fig. 2 represents the scattering surface corresponding to a scattering angle of 70 °. The reciprocal lattice points located in the s c a t t e r i n g plane [the (01 i) plane of the Ni crystal] are given as crossed open points and characterized b y three n u m b e r s which are the cartesian c o m p o n e n t s , in units 2~/a, of the corresponding reciprocal lattice vectors. The polygons give the boundaries, in the scattering plane, of the first Brillouin zones built a r o u n d each reciprocal
263
LATTICE DYNAMICS OF NICKEL
lattice point. The construction of eq. 6 corresponding to one of the onephonon coherent absorbed peaks is shown as an illustration; the wave vector of the absorbed phonon is the vector joining the extremity of Q to the nearest reciprocal lattice point. On fig. 2, the filled circles correspond to high intensity inelastic peaks, while the open circles correspond to low intensity peaks the location of which is less accurate. Doubtful points are indicated b y question-marks; neighbouring points associated to unresolved peaks are indexed b y letters a and b. Experimentally determined points on the scattering surfaces corresponding to scattering angles of 90 ° and 100 ° are seen on fig. 3 and 4, respectively.
\
\ '\\\
r-
',\
®1 L / \ a o
•
"\\
\
b
T
/ 0
-- ~# /
)
/
/
/
/
// \
~o~ ~~ )
/
/
<,~\,
(ooo) //
/ \,,
/ Fig. 3. S c a t t e r i n g surface, in t h e (01 i) p l a n e of Ni, c o r r e s p o n d i n g t o a s c a t t e r i n g a n g l e 9 of 90 ° a n d to a n i n c i d e n t n e u t r o n w a v e l e n g t h of 4.094 &.
P l a c z e k and V a n H o v e 14) have shown that in general there exist at least three scattering surfaces, one for each branch of the dispersion relation. Obviously, the points on fig. 2, 3 and 4 imply only two extended curves. This arises from the fact that the scattering plane is a symmetry plane of the crystal. Of the three normal modes associated with a q laying in such a plane, one must have a polarization normal to the plane. Since the intensity of a one-phonon coherent peak depends 14) on the scalar product of Q b y the polarization vector, the branch whose polarization is normal to the scattering
264
S. HAUTECLER AND W. VAN DINGENEN
plane cannot contribute to the spectra we measured. The longitudinal or transversal character of the various branches of the scattering surfaces are indicated respectively by letters L and T; this character has been deduced from the intensity variation of the inelastic peaks along the branches. On each figure, the external branch is longitudinal, while the internal one is transversal; the closed curves, around the (l i i) reciprocal lattice point on fig. 3 and 4 and around the (200) point on fig. 4, are also longitudinal. On fig. 2, 3 and 4, we have also indicated, in some cases, the full width at half maximum of the one-phonon peaks by means of error bars in the direction of k'. The uncertainties on lengths and orientations of phonon wave vectors q, obtained by constructing the momentum conservation equation, can be estimated from the divergencies of the incident and scattered neutron beams, from the energy width of the monochromatic beam (BIT}
,~--
L
// /
o
® (222)
a
r }
...... /
L
T b
//
o?
// / b
L •
/
~s
//
,,\\
\ (022) --
.oX,~ i/ /T a~
(0001
\ \'x
"
/) //
/" /
X\ Fig. 4. S c a t t e r i n g surface, in t h e (011) p l a n e of Ni, c o r r e s p o n d i n g to a s c a t t e r i n g a n g l e of 100 ° a n d t o a n i n c i d e n t n e u t r o n w a v e l e n g t h of 4.094 A.
and from a mean width of the inelastic peaks. On the right-hand side of fig. 2, 3 and 4, the dotted lines represent, on an average, the extreme limits of the reciprocal space area seen by the time-oi-flight spectrometer, for two values of the energy of the scattered neutrons; the part of this area enclosed by the heavy line (resolution figure) is estimated to give rise to the observed width at half maximum of the inelastic peak. The location in
LATTICE DYNAMICS OF NICKEL
265
reciprocal space of the end of the vector --k' associated to a neutron peak is, however, generally better defined than b y the area of the resolution figure. We estimate that the error on the center of well-resolved intense peaks does not amount to more than one channel width (i.e. 32 ~s) of the analyzer; the corresponding "uncertainty zones" are given b y the hatched regions. The scattering surfaces have to be symmetrical with respect to the (200) and (02P~) planes; the points surrounded b y circles are obtained b y reflection of the experimental points through these planes. The good agreement between the two sets of points is a check of the good orientation of the sample and, moreover, confirms that, for most of the data, the accuracy on the location of the points is much better than that given b y the resolution figure. From the preceding results we deduced the dispersion curves of the normal modes of vibration along the symmetry axes. From each reciprocal lattice point straight lines were drawn parallel to the symmetry directions; the intersections of these lines with the scattering surfaces (obtained b y drawing a smooth curve through the experimental points) determine the wave vectors and the frequencies which are given in table I; $ is a reduced wave vector defined b y ~ = q/qM where 2qM is the distance between the origin and the nearest reciprocal lattice point in the direction of q. These results are 10
.
Ni
.
.
.
~y
k
O f
i
I
t
I
0.2
0.4
0.6
0.8
[~00]--
1.01.0
0.8 '
Reduced
06i
0.4
0.2
0 0
-£~o3
wave
vector
0.2
0.4
0.6
0.8
1.0
[~3-coordinate
~'
Fig. 5. D i s p e r s i o n c u r v e s of t h e n o r m a l m o d e s p r o p a g a t i n g a l o n g t h e s y m m e t r y a x e s i n r o o m t e m p e r a t u r e Ni. T h e r e d u c e d w a v e v e c t o r ~ is d e f i n e d b y ~ = q[qM w h e r e 2qM is t h e d i s t a n c e b e t w e e n t h e origin a n d t h e n e a r e s t reciprocal l a t t i c e p o i n t i n t h e d i r e c t i o n of q. T h e e x p e r i m e n t a l r e s u l t s are g i v e n b y filled p o i n t s for t r a n s v e r s a l w a v e s a n d b y o p e n p o i n t s for l o n g i t u d i n a l waves. T h e s t r a i g h t lines p a s s i n g t h r o u g h t h e o r i g i n give t h e i n i t i a l slope of e a c h b r a n c h as c a l c u l a t e d f r o m t h e elastic c o n s t a n t s . T h e h e a v y lines are t h e d i s p e r s i o n c u r v e s c a l c u l a t e d o n t h e K r e b s model. T h e b r o k e n a n d d o t t e d lines refer to c a l c u l a t e d d i s p e r s i o n c u r v e s b a s e d r e s p e c t i v e l y o n t h e De L a u n a y a n d B o r n - y o n IZArmAn m o d e l s ; t h e o m i t t e d p o r t i o n s p r a c t i c a l l y coincide w i t h t h e c o r r e s p o n d i n g K r e b s curves.
266
S. H A U T E C L E R
AND W. VAN DINGENEN
TABLE
I
Frequencies of the normal m o d e s propagating along the s y m m e t r y axes in r o o m temperature Ni (in units l0 IS Hz)
[~ o OIL I
[~ ;
o]
g ~ o] T4
v
0.090 0.137 0.163 0.222 0.222 0.266 0.437 0.496 0.651 0.679 0.766 0.942 [~]
g o o] r
1.48 2.02 2.28 3.35 3.28 4.12 6.10 6.55 8.36 7.95 8.85 9.26
I 0,137 0.185 0.202 0.252 0.255 0.297 0.353 0.404 0.867 0.925
1.39 1.89 2.12 2.56 2.49 3.17 3.66 4.14 6.43 6.64
0.061 0.097 0.13l 0.180 0.278 0.291 0.313 0.416 0.666 0.734 0.896 0.910
1,41 2.36 3.19 4.14 6.12 6.29 6.68 7.88 8.14 7.71 6.64 6.70
0.091 0.115 0.192 0.242 0.246 0.274 0.375 0.405 0.521 0.545 0.672 0.781
v 1.25 1.77 2.81 3.29 3.41 4.26 5.14 5.74 6.76 7.07 8.38 8.75
gg] r
L v
0.070 0.126 0.132 0.142 0.162 0.176 0.178 0.194 0.234 0.266 0.272
1.45 1.60 1.98 2.24 2.77 2.56 2.82 2.82 3.38 3.72 3.83
0.298 0.314 0.382 0.398 0.480 0.532 0.716 0.722 0.768 0.874 0.888
4.19 4.98 5.06 5.52 6.57 7.34 8.83 8.83 8.87 9.12 9.21
0.150 0.194 0.214 0.292 0.346 0.362 0.440 0.524 0.696 0.952
1.14 1.54 1.53 2.08 2.36 2.45 2.87 3.51 4.11 4.5l
reported on fig. 5 ; the open circles correspond to the longitudinal branch, the filled circles to the transversal one. In the [110] direction, only one of the two non-degenerate transversal branches has been observed, namely the one (T4) for which the polarization is parallel to the 4-fold axis and thus in the scattering plane; under the conditions of the measurements, the other transversal branch (T2) was unobservable since the polarization, being parallel to a 2-fold axis, is normal to the scattering plane. The straight lines on fig. 5 passing through the origin give the initial slope of each branch of the dispersion curves, as calculated from the values of the elastic constants given in table I116). On fig. 5, one observes very often a good agreement between the coordinates of two neighbouring points of the dispersion curves, originating from very different regions of the scattering surfaces; this is another confirmation that, in general, the accuracy is better than the uncertainty given by the resolution figure of fig. 2, 3 and 4. However, a few points do not lay on the average curve shown; in the analysis, a lower weight must be attributed to these points which originate either from regions where the scattering surface is very oblique with respect to q so that a small error on the position of the
L A T T I C E D Y N A M I C S OF N I C K E L
267
surface gives an amplified error on q, or from regions where the scattering surface, being deduced from unresolved inelastic peaks, is less accurately located. The values of the frequencies found in the present work lay slightly above those obtained by B i r g e n e a u e.a.2), about 5% at the zone boundaries. These discrepancies could not be accounted for by normal experimental errors, so a reasonable explanation is still lacking; tentatively, they could be assigned to an incorrect subtraction of the socle under the high energy one-phonon peaks.
Analysis o/the results. D i s p e r s i o n c u r v e s . Following F o r e m a n and L o m e r 17), the dispersion curves along the s y m m e t r y directions can be used to determine the range of the interatomic forces in cubic metals. The analytical form of such branches is given by co
4~2Mv2 = Z q~n[1 -- cos n ~ ]
(7)
~'t--1
where M is the atomic mass, and q)n, the interplanar force constants, are some linear combinations of the interatomic force constants introduced in the Born-yon K~rm~n model; the combinations appropriated to the various directions and polarizations were given by S q u i r e s l S ) . For the various branches, the ~bn have been determined by a least-squares analysis of the experimental data of table I, and for a number N of terms in the serie (7) going from one to eight. From the general behaviour of the interplanar force constants as a function of N, it could be deduced that the forces extend at least up to the third neighbours, which requires nine interatomic force constants in the Born-yon K~rm~n model. However, the relative importance of the q~n not containing the force constants between the first neighbours is very small, so that in nickel the short-range interaction is by far the most important one. At this point of the analysis it would be possible to deduce, from the interplanar force constants, the values of the m a n y interatomic force constants to be introduced in a Born-von K~rmfin model; we preferred, however, to look if the same good agreement with our experimental results could be obtained by using models containing much less adjustable parameters. In the Born-yon K~rmfin model, general forces between nearest neighbours of f.c.c, metals require three parameters; it is then possible to get their values from the three elastic constants if the force constants between more distant neighbours are not taken into account. The dispersion curves calculated using the values of table II are shown as dotted lines in fig. 5 (the omitted portions practically coincide with the heavy lines). As can be
268
S. HAUTECLER
AND
W.
VAN
DINGENEN
seen, the discrepancies between the results given b y this model and the experimental data are not very important. In the De Launay model, only three force constants enter; the broken lines of fig. 5 refer to dispersion curves calculated with this model, when the parameter values are deduced b y fitting the measured elastic constants of table II. The consequences of the non-fulfilling of the periodicity requirements can be noticed on the figure where the degeneracies at the end points of the [~ 0 01 L- and [~ ~ 0] T4 branches, and of the [~ ~ 01 L- and [~ ~ 0] T2 branches, are not obtained. However, since this violation is not very large, and since moreover the general agreement with the experimental data is not bad, we can hope to improve this agreement b y keeping from the De Launay model the central interactions out to the second neighbours, but treating more suitably the conduction electrons, as for instance in the Krebs model. In the Krebs model, the number of parameters is higher than the number of elastic constants; the latter determine the values of three parameters if the other are treated as adjustable. We performed six sets of calculations of dispersion curves, using either different values of the proportionality factor C, or slightly different values for the elastic constants, or different values of the electron density; in all cases however the conduction electrons have been supposed free so that the electron wave vector ky at the Fermi surface was given b y kF = (3~ 2 ne)L (8) A very good fit to the experimental points was obtained when using the values of table II; the corresponding curves are given b y the heavy lines of fig. 5. The electron density given in table II corresponds to 0.6 conduction TABLE
II
V a l u e s of c o n s t a n t s a n d p a r a m e t e r s used in calculations a = 3.5239 A C l l = 2 . 4 6 10 x2 d y n e / c m 2 C12 = 1.50 l 0 in d y n e / c m 2 C 4 4 = 1 . 2 2 1012 d y n e / e m 2
C
=
n,
= 2.4/a 2
0.353
electrons per atom; this is a reasonable figure since in nickel ten electrons have to be distributed over the 3d and 4s bands, and since the value of the magnetic moment indicates that there are 0.6 holes per atom in the 3d bandlg). Our calculations show that a change of the value of the parameters which determine the screened Coulomb interaction has no large influence on the position of the dispersion curves: the fit to the experimental points remains
L A T T I C E D Y N A M I C S OF N I C K E L
269
reasonably good (indeed, a complete cancellation of the long-range term in the dynamical matrix elements leads to a lowering of the frequencies by about 5%). The fact that the long-range interaction is relatively unimportant in Ni is not surprising. Indeed, the value of the elastic constant C44, which determines for a large part the magnitude of the shortrange interaction, is much higher for Ni than for other f.c.c, metals, whereas the deviation from the Cauchy relation C1~-C44, which affects the magnitude of the long-range interaction, is exceptionally small. S c a t t e r i n g s u r f a c e s . In order to include in the test results obtained in non-symmetry directions, we calculated the intersections of the scattering surfaces with the (01 i) plane in the framework of the Krebs model; we gave to the parameters the values of table II which lead to a good agreement for the dispersion curves. The heavy lines of fig. 2, 3 and 4 represent the calculated scattering curves corresponding respectively to a scattering angle of 70 °, 90 ° and 100 °, and to an incident neutron wavelength of 4.094 A; the longitudinal or transversal character of the various branches is indicated by the letters L of T. The agreement between the theoretical curves and the points resulting from our measurements is generally good. In all cases the calculated scattering surface passes through the areas obtained by drawing the resolution figure around each experimental point; even better, the calculated curve passes very often through the uncertainty zone of the resolution figure. On fig. 4, the agreement seems less good for the external longitudinal branch located in the Brillouin zone centered on the (200) reciprocal lattice point; in this region the accuracy on the position of the experimental points is lower than the average since the inelastic peaks are not resolved in the corresponding spectra. On the three figures, the agreement between the experimental and calculated results is, however, somewhat less satisfactory in the righthand upper part of the zone centered on the (1 i i) reciprocal lattice point. In this region, the transversal branch of the scattering surface is changing very abruptly its orientation and becomes very oblique with respect to the analysis direction of the scattered neutrons; this induces, through the divergency of the scattered beam, very large inelastic peaks which we, indeed, dissociated into two components previously to any analysis. On fig. 2, 3 and 4 one also finds a few experimental points aside the calculated scattering curves. Some of these points are doubtful, and most of the others correspond to low intensity peaks; only two extra points (on fig. 3) are associated to high intensity peaks. These points could originate from multiple scattering processes.
Discussion. It has been seen that the experimental points of the dispersion curves along the s y m m e t r y directions of a Ni single crystal can be fitted very well with curves calculated by means of the Krebs model, when using, for
270
S. H A U T E C L E R A N D W. VAN D I N G E N E N
the three most sensitive parameters of the theory, values derived from the measured elastic constants. In spite of the fact that the contribution of the conduction electrons to the lattice dynamics of Ni is small, this metal is, however, particularly interesting since it allows to control that the analytical form of the short-range part of the interaction potential as chosen by Krebs is adequate; in a recent paper, M a h e s h and Dayal20)23) found necessary to include the third-nearest neighbour interaction in a Krebs treatment of the lattice dynamics of some b.c.c, and f.c.c, metals. On fig. 5, it can be seen that the dispersion curves calculated on a Bornvon KArmAn model with general forces extending only up to the firstnearest neighbours are nearly as good as those given by the Krebs model. P a l and G u p t a 21) have also reported a calculation of the phonon dispersion relations in Ni using the S h a r m a and J o s hi 22) electron gas model. Except for a slight violation of the s y m m e t r y requirements, their results coincide very well with our calculated curves giving the best fit to our experimental data. Therefore, it appears that a still more refined analysis would be necessary to test in detail either models. As an attempt to do so, we compared the experimental points of the scattering sur/aces with theoretical curves derived from the Krebs model; in this way, results obtained in non-symmetry directions also contribute to the test. The agreement is very good except in some regions of reciprocal space where the surfaces are changing very abruptly; we think that these regions are more sensitive to the choice of the parameters so that, from improved experimental data, either their values could be obtained more precisely or the model should be excluded. In conclusion, we can say that the Krebs model, whilst taking into account the effect of the conduction electrons in a rather primitive manner, predicts reasonably well the dynamical properties of Ni. A c k n o w l e d g e m e n t s . We wish to thank Dr. M. N ~ v e de M 6 v e r g n i e s , head of the Neutron Physics Department, for the constant interest shown during our work, and Dr. K. K r e b s for very helpful discussions. We are also indebted to Prof. O. K r i s e m e n t and to Dr. D. M e i n h a r d t for supplying both the Ni single crystal sample and the Pb monochromator. Received 23-9-66
REFERENCES I) H a u t e c l e r , S. and V a n D i n g e n e n , W., J. Physique 25 (1964) 653. 2) B i r g e n e a u , R. J., C o r d e s , J., D o l l i n g , G. and W o o d s , A. D. B., Phys. Rev. 136 (1964) A 1359. 3) L e h m a n , G. W., W o l f r a m , T. and De W a m e s , R. E., Phys. Rev. 1~8 (1962) 1593. 4) F u c h s , K., Proc. Roy. Soc. A 1 5 3 (1935) 622.
LATTICE DYNAMICS OF NICKEL
27 1
5) De L a u n a y , J . , J. t h e m Phys. 21 (1952) 1975; Solid State Physics, Vol. 2, Academic Press, New York (1956) 219. 6) B h a t i a , A. B., Phys. Rev. 97 (1955) 363. 7) K r e b s , K., Phys. Letters 10 (1964) 12; Phys. Rev. 138 (1965) A 143. 8) L a x , M., J. Phys. Chem. Solids, Suppl. 1 (1965) 179. 9) S h u k l a , M. M., Phys. Star. Sol. 7 (1964) K l l . 10) S h u k l a , M. M., Phys. Stat. Sol. 8 (1965) 475. 11) M a h c s h , P. S. and D a y a l , B., Phys. Stat. Sol. 9 (1965) 351. 12) K r e b s , K., private communication (1965). 13) H a u t e c l e r , S. and V a n D i n g e n e n , W., S y m p o s i u m on Inelastic Scattering of Neutrons by Condensed Systems, Brookhaven ,BNL 940 (C-45) (1966) 83. 14) P l a e z e k , G. and V a n H o v e , L., Phys. Rev. 93 (1954) 1207. 15) V a n D i n g e n e n , W. and H a u t e c l e r , S., to be published. 16) de K l e r k , J., Proe. Phys. Soe. 73 (1959) 337. 17) F o r e m a n , A. J. E. and L o m e r , W. M., Prom Phys. Soc. B T 0 (1957) 1143. 18) S q u i r e s , G., Arkiv f. Fysik 25 (1963) 21. 19) M o t t , N. and J o n e s , H., The Theory of the Properties of Metals and Alloys, Oxford University Press, London (1936) 316. 20) M a h e s h , P. S. and D a y a l , B., Phys. Rev. 143 (1966) 443. 21) P a l , S. and G u p t a , R. P., Sol. State Com. 4 (1966) 83. 22) S h a r m a , P. K. and J o s h i , S. K., J. chem. Phys. 3 9 (1963) 2633. 23) S h u k l a , M. M. and D a y a l , B., Phys. Stat. Sol. 16 (1966) 513.
H a u t e c l e r , S. Van Dingenen, W.
1967
LATTICE D Y N A M I C S OF N I C K E L b y S. H A U T E C L E R
a n d W. V A N D I N G E N E N *
Studieeentrum voor Kernenergie, Mol, Belgi6
Synopsis The normal modes of vibration of nuclei in nickel at room temperature have been investigated by means of one-ph0non coherent scattering of neutrons.The incident monochromatic cold neutron beam was obtained by Bragg reflection from a Pb single crystal of a Be-filtered reactor beam, and the energy analysis of the scattered neutrons was performed by the time-of-flight technique.The experimental data were treated to give the scattering curves, in the (011) plane of Ni, corresponding to scattering angles of 70 °, 90 ° and 100°, and to an incident neutron wavelength of 4.094 A. From these scattering curves, points of the phonon dispersion curves v(q) in the symmetry directions [1003, [110] and [111] were deduced. A least squares analysis of the Fourier expansion of v2 for the various branches of the dispersion curves shows that the forces extend up to at least third neighbours, but that nearest neighbour interactions dominate. The results have been compared with calculations based on a model proposed by Krebs for facecentered and body-centered cubic metals. The values of the three most sensitive parameters in the theory were obtained by fitting to measured elastic constants and an appropriate value for the effective density of free electrons was chosen. The calculated dispersion curves and scattering curves show very good agreement with the experimental results.
Introduction. D u r i n g the last t e n years, large i n f o r m a t i o n has b e e n o b t a i n e d on the m o t i o n of a t o m s in c r y s t a l s b y m e a n s of t h e r m a l n e u t r o n inelastic scattering. T h e lattice d y n a m i c s of nickel h a s b e e n i n v e s t i g a t e d first b y the a u t h o r s 1) w i t h the aid of a Be p o l y c r y s t a l l i n e filter a n d slow n e u t r o n c h o p p e r s p e c t r o m e t e r w o r k i n g at the low flux r e a c t o r B R 1 , and, m o r e e x t e n s i v e l y a n d also w i t h b e t t e r a c c u r a c y , b y B i r g e n e a u e.a.~) using a triple-axis c r y s t a l s p e c t r o m e t e r in its c o n s t a n t 0 m o d e of operation. T h e last a u t h o r s h a v e f u r t h e r a n a l y z e d t h e i r d a t a , a c t u a l l y the dispersion curves for the n o r m a l m o d e s of v i b r a t i o n p r o p a g a t i n g along the t h r e e s y m m e t r y directions of the crystal, in t h e f r a m e w o r k of the B o r n - y o n K ~ r m ~ n theory, either with general forces or w i t h a x i a l l y s y m m e t r i c forces 3). A good fit to t h e experim e n t a l d a t a could be o b t a i n e d b y including i n t e r a c t i o n s e x t e n d i n g u p to the f o u r t h - n e a r e s t neighbours, which involves the use of a large n u m b e r of a d j u s t a b l e p a r a m e t e r s . H o w e v e r , t h e i n t e r a t o m i c force s y s t e m is r e l a t i v e l y simple: if i n t e r a c t i o n s b e t w e e n d i s t a n t n e i g h b o u r s are present, t h e y do not influence t h e dispersion curves to a large e x t e n t because t h e i n t e r a t o m i c * Deceased
- - 257 - -
2~8
S. HAUTECLER AND W. VAN DINGENEN
force constants between the first neighbours are at least twenty times higher than the force constants between more distant neighbours. The fact that, with the Born-von K~rm~n model, one needs too m a n y parameters to fit experimental dispersion curves of metals has to be attributed to the presence of the conduction electrons, the influence of which also explains the failure of the Cauchy relations4). Several simple models, that take explicitly into account the influence of the conduction electrons on the lattice vibrations, have recently been proposed by De L a u n a y S ) , B h a t i a 6) and KrebsT). All these models contain a very small number of adjustable parameters, which are chosen so as to give correct values for the elastic constants. De L a u n a y considers a monatomic cubic metal as constituted by an ion point lattice embedded in an electron sea. The ion point lattice is treated in the framework of the Born-von K~rm~n model, central "spring" interactions acting between an ion and its first and second neighbours. The electron sea is supposed to behave as an ordinary gas, i.e. a continuum characterized by one elastic constant. Moreover, when waves propagate through the crystal, the conduction electrons are assumed to follow in phase the motion of the ions. Such a model implies three force constants (one of which is due to the electron gas and the other two to the ion-ion interaction) which can be deduced from the measured values of the three elastic constants of a cubic crystal. Unfortunately, as pointed out by L a x 8), this model, as weli as the one proposed by B h a t i a, violates some symmetry requirements: the degeneracy, which is required by the crystal symmetry at some points of the reciprocal space between some branches of the dispersion relation is not obtained. This stems from the fact that, the electrons being treated as a continuum, the dynamical matrix has to more the periodicity of the reciprocal lattice of the crystal. The symmetry requirements are satisfied in the force model developed by K r e b s , in which the metal is considered as a lattice of spherical ions, of uniform charge distribution, embedded in an electron sea. The interaction potential is divided into two parts; each element of the dynamical matrix is then given by the sum of two terms. The first part, which takes into account the exchange interaction due to the overlapping of the ions and eventually the van der Waals attraction, is described, as in the De L a u n a y model, by central spring interactions between each ion and its first and second neighbours; this involves only two parameters, the force constants ~1 and ~2. The second part of the potential must take into account the Coulombinteraction between the ions as well as the interaction induced through the electrons; these latter two contributions are rather large but, due to the screening effect of the conduction electrons, they tend to cancel each other when the separation between the ions increases. Krebs has thus chosen a screened Coulomb interaction between the ions for the second part of the
LATTICE DYNAMICS OF NICKEL
259
potential. Two parameters enter in the corresponding part of each dynamical matrix element: a parameter A depending on the effective charge of the ions, and the screening parameter ~. The whole set of parameters is related to the elastic constants (Cll, C1% C44) and to the lattice constant a b y the relations o~1 : -
½aC44
(l)
0~2 :
:~a(C11 - - C12 - - C44)
(2)
A = ia(Clz
--
C44) a2A 2
(3)
valid for f.c.c, lattices. The screening parameter is written as = c
(3)
(4)
where ne is the electron density, a0 the Bohr radius (0.529 X), and kF the electron wave vector modulus at the Fermi surface. The value of the constant C appearing in eq. (4) is not well defined; in the Thomas-Fermi theory it takes the value 0.814, while only a value of 0.353 is obtained from the plasma theory of Bohm and Pines which takes into account the electron correlations (more exactly, this theory gives a maximum value of 0.47). K r e b s 7) calculated the dispersion curves in the three symmetry directions for Li, Na and K, b y chosing the Pines C-coefficient, assuming one free electron per atom, determining kF in the free electron approximation, and finally using the measured elastic constants to get the values of the basic set of parameters (al, ~2, A). In the case of Na, for which experimental dispersion relations were available, an excellent fit to the data was obtained. On the same basis, S h u k l a 9) has shown that also for Cu good agreement with the experimental dispersion curves is obtained. Thus it seems that for the s t u d y of the lattice dynamics of simple monovalent metals, the Krebs model with no adjustable parameters is satisfactory. Later, S h u k l a 10) and M a h e s h and Daya111) calculated respectively the heat capacities of noble metals (Cu, Au) and of alkaline metals (Na, K, Rb, Cs) b y means of frequency spectra deduced from the Krebs model. An excellent agreement with the experimental data could be obtained if the proportionality factor C in the expression for the electronic screening parameter was treated as an adjustable parameter. The best-fit values ranged between the Pines and the Thomas-Fermi C-coefficient. Using our time-of-flight spectrometer at the high flux reactor B R 2 , we re-examined the lattice dynamics of Ni a) to see how the experimental data obtained with different techniques compare, and b) to test the Krebs model in the case of a non-monovalent transition metal, and not only for dispersion curves along s y m m e t r y axes but more directly for scattering surfaces in
260
S. H A U T E C L E R A N D W . VAN D I N G E N E N
symmetry planes*). A preliminary account of our work was presented elsewhere 13). E x p e r i m e n t a l method and results. When a beam of monochromatic thermal neutrons is sent onto a single crystal, one generally observes peaks in the energy distribution of the neutrons scattered in a given direction; these peaks correspond to one-phonon coherent scattering processes which are governed 14) by the conservation laws
h2 2 m (k~ - - k'2) = ~: hv 0 ~
k 0 - k' = ~ -
q
(5) (6)
where k0(k') is the incident (scattered) neutron wave vector, m the neutron mass, v the frequency and q the phonon wave vector, Q the momentum transfer vector, and T a reciprocal lattice vector. In the energy conservation equation, the + sign corresponds to the creation of one phonon, t h e - - sign to the absorption of one phonon. From the experimentally determined energy of a peak one obtains, with the help of eq. (5), the frequency of a normal mode of vibration; knowing k0, the scattering angle and the orientation of the specimen one further obtains, b y a graphical representation of eq. (6), the wave vector q of this mode. Each peak in the energy distribution of the neutrons scattered in a given direction thus yields a point of the dispersion relation v = v¢(q) (the index i differentiates the various branches). B y repeated observations of energy spectra either for different incident energies, or for different scattering angles, or for different orientations of the sample, one can thus hope to build up the dispersion relations for the crystal under examination. The experiments were carried out using a neutron time-of-flight spectrometer installed at one of the radial beam holes of the B R 2 material-testing reactor at Mol; the apparatus will be described in detail elsewherelS). Essentially, a collimated beam of neutrons coming from the reactor passes through a nitrogen cooled filter combining 20 cm of polycrystalhne Be and 7 cm of Bi single crystal, and is monochromatized b y Bragg reflection from a Pb single crystal. The monochromator is movable along the reactor beam direction so that, with respect to a fixed sample, the incident neutron energy can be chosen at wiU, in the range below 5 meV when the Be filter is used. Under the conditions of the measurements on Ni, the mean wavelength of the monochromatic beam, entirely free from higher order reflections, was 4.094/~; the full width at half maximum of the energy spectrum of the *) In the m e a n t i m e 12) the Krebs model has been applied with succes to the calculation of dispersion curves of several other f.c.c, and b.c.c, metals: A1, Cr, W. No good fit could, however, be obtained in the case of Pb, Mo and Nb.
LATTICE
DYNAMICS
261
OF NICKEL
neutrons incident on the sample was about 1.8°/o. The analysis of the energy distribution of the scattered neutrons is made using the time-of-flight technique. Before reaching the Ni sample, the monochromatic beam is pulsed by a Fermi chopper and the times the neutrons need to reach the detector (a bank of nine 2" BF3 proportional counters divided into two layers) are recorded in a 1024 channel TMC analyzer. The mean flight path was 4.81 m. Under the conditions of the experiment, the energy resolution of the spectrometer was varying between 6% at 5 meV and 13% at 36 meV; no effort was devoted to obtain a higher resolution. The sample, a cylindrical Ni single crystal, 3 cm in diameter and 7 cm long, was mounted on a goniometer head fixed on a rotating table whose axis is normal to the scattering plane; the Ni crystal was itself oriented with the (110) planes parallel to the scattering plane. We performed, in three months of time, a total of about 90 energy distribution measurements, for different scattering angles (70 °, 90 ° and 100°) and orientations of the sample; for each scattering angle the crystal was rotated by steps of 4 ° for successive runs. A typical time-of-flight spectrum is shown in fig. 1. One clearly sees the one-phonon coherent scattering peaks on top of a socle mainly due to onephonon incoherent scattering. In the high energy region, the rapid rise of the socle must be taken into account in order to locate correctly the mean times-of-flight in the one-phonon peaks. In most of the measurements, the background amounted to less than 10% of the peak heights; this low value is mainly due to the fact that the sample is kept out of the main beam. The horizontal line near the time-of-flight axis in fig. 1 corresponds to a background of 700 counts per hour measured with the bank of detectors.
~ t
Ni
= 90 ° = 50 ° =14h
~•
500
%
%
•
c 250 c
",,.. :
". .,,....,
I 2000
1500
Time
[ 1
I 2 5 0 0 ,us
- of - f l i g h t
I 1,5
1 2
I 2.5 A
Wavelength
Fig. I. A typical time-of-flight distribution of neutrons scattered by a Ni single crystal. ~0is the scattering angle; ~vis the angle between the monochromatic beam and the [200] direction of the (01 i) plane; t is the measuring time. The horizontal line near the time-of-flight axis gives the level of the background.
s. HAUTECLER AND W. VAN DINGENEN
262
~
//
~
o
o? L
'
'°" --"
®o
c o(111: o •
2
-
~.
(022)
~
\
;\
,\\ \
i
I
__
\\
ko
\
\,
/ \\\
/
J/
Fig. 2. Scattering surface, in the (011) plane of Ni, corresponding to a scattering angle 9 of 70° and to an incident neutron wavelength of 4.094 •. The reciprocal lattice points located in the scattering plane are given as crossed open points; the polygons determine the limits of the Brillouin zones built around each reciprocal lattice point. The filled circles correspond to high intensity inelastic peaks of the time-of-flight spectra while the open circles are associated to low intensity peaks; doubtful points are indicated by question-marks. Neighbouring points corresponding to non-resolved peaks in the spectra are indexed by letters a and b. Points surrounded by circles are obtained by reflection of experimental points through the (200) and (022) planes. The error bars indicated in some cases correspond to the full widths at half maximum of the inelastic peaks. The heavy lines passing through, or in the neighbourhood of, the experimental points represent the various branches of the scattering surface as calculated in the framework of the Krebs model; the longitudinal or transversal character of the branches is indicated by the letters L or T. These experimental d a t a are converted, t h r o u g h the conservation equations, into points of the scattering surfaces in the reciprocal space, which are defined as the surfaces described b y the e x t r e m i t y of the vector 0 =-- ko - - k' as the crystal is rotated, the scattering angle being kept constant. Fig. 2 represents the scattering surface corresponding to a scattering angle of 70 °. The reciprocal lattice points located in the s c a t t e r i n g plane [the (01 i) plane of the Ni crystal] are given as crossed open points and characterized b y three n u m b e r s which are the cartesian c o m p o n e n t s , in units 2~/a, of the corresponding reciprocal lattice vectors. The polygons give the boundaries, in the scattering plane, of the first Brillouin zones built a r o u n d each reciprocal
263
LATTICE DYNAMICS OF NICKEL
lattice point. The construction of eq. 6 corresponding to one of the onephonon coherent absorbed peaks is shown as an illustration; the wave vector of the absorbed phonon is the vector joining the extremity of Q to the nearest reciprocal lattice point. On fig. 2, the filled circles correspond to high intensity inelastic peaks, while the open circles correspond to low intensity peaks the location of which is less accurate. Doubtful points are indicated b y question-marks; neighbouring points associated to unresolved peaks are indexed b y letters a and b. Experimentally determined points on the scattering surfaces corresponding to scattering angles of 90 ° and 100 ° are seen on fig. 3 and 4, respectively.
\
\ '\\\
r-
',\
®1 L / \ a o
•
"\\
\
b
T
/ 0
-- ~# /
)
/
/
/
/
// \
~o~ ~~ )
/
/
<,~\,
(ooo) //
/ \,,
/ Fig. 3. S c a t t e r i n g surface, in t h e (01 i) p l a n e of Ni, c o r r e s p o n d i n g t o a s c a t t e r i n g a n g l e 9 of 90 ° a n d to a n i n c i d e n t n e u t r o n w a v e l e n g t h of 4.094 &.
P l a c z e k and V a n H o v e 14) have shown that in general there exist at least three scattering surfaces, one for each branch of the dispersion relation. Obviously, the points on fig. 2, 3 and 4 imply only two extended curves. This arises from the fact that the scattering plane is a symmetry plane of the crystal. Of the three normal modes associated with a q laying in such a plane, one must have a polarization normal to the plane. Since the intensity of a one-phonon coherent peak depends 14) on the scalar product of Q b y the polarization vector, the branch whose polarization is normal to the scattering
264
S. HAUTECLER AND W. VAN DINGENEN
plane cannot contribute to the spectra we measured. The longitudinal or transversal character of the various branches of the scattering surfaces are indicated respectively by letters L and T; this character has been deduced from the intensity variation of the inelastic peaks along the branches. On each figure, the external branch is longitudinal, while the internal one is transversal; the closed curves, around the (l i i) reciprocal lattice point on fig. 3 and 4 and around the (200) point on fig. 4, are also longitudinal. On fig. 2, 3 and 4, we have also indicated, in some cases, the full width at half maximum of the one-phonon peaks by means of error bars in the direction of k'. The uncertainties on lengths and orientations of phonon wave vectors q, obtained by constructing the momentum conservation equation, can be estimated from the divergencies of the incident and scattered neutron beams, from the energy width of the monochromatic beam (BIT}
,~--
L
// /
o
® (222)
a
r }
...... /
L
T b
//
o?
// / b
L •
/
~s
//
,,\\
\ (022) --
.oX,~ i/ /T a~
(0001
\ \'x
"
/) //
/" /
X\ Fig. 4. S c a t t e r i n g surface, in t h e (011) p l a n e of Ni, c o r r e s p o n d i n g to a s c a t t e r i n g a n g l e of 100 ° a n d t o a n i n c i d e n t n e u t r o n w a v e l e n g t h of 4.094 A.
and from a mean width of the inelastic peaks. On the right-hand side of fig. 2, 3 and 4, the dotted lines represent, on an average, the extreme limits of the reciprocal space area seen by the time-oi-flight spectrometer, for two values of the energy of the scattered neutrons; the part of this area enclosed by the heavy line (resolution figure) is estimated to give rise to the observed width at half maximum of the inelastic peak. The location in
LATTICE DYNAMICS OF NICKEL
265
reciprocal space of the end of the vector --k' associated to a neutron peak is, however, generally better defined than b y the area of the resolution figure. We estimate that the error on the center of well-resolved intense peaks does not amount to more than one channel width (i.e. 32 ~s) of the analyzer; the corresponding "uncertainty zones" are given b y the hatched regions. The scattering surfaces have to be symmetrical with respect to the (200) and (02P~) planes; the points surrounded b y circles are obtained b y reflection of the experimental points through these planes. The good agreement between the two sets of points is a check of the good orientation of the sample and, moreover, confirms that, for most of the data, the accuracy on the location of the points is much better than that given b y the resolution figure. From the preceding results we deduced the dispersion curves of the normal modes of vibration along the symmetry axes. From each reciprocal lattice point straight lines were drawn parallel to the symmetry directions; the intersections of these lines with the scattering surfaces (obtained b y drawing a smooth curve through the experimental points) determine the wave vectors and the frequencies which are given in table I; $ is a reduced wave vector defined b y ~ = q/qM where 2qM is the distance between the origin and the nearest reciprocal lattice point in the direction of q. These results are 10
.
Ni
.
.
.
~y
k
O f
i
I
t
I
0.2
0.4
0.6
0.8
[~00]--
1.01.0
0.8 '
Reduced
06i
0.4
0.2
0 0
-£~o3
wave
vector
0.2
0.4
0.6
0.8
1.0
[~3-coordinate
~'
Fig. 5. D i s p e r s i o n c u r v e s of t h e n o r m a l m o d e s p r o p a g a t i n g a l o n g t h e s y m m e t r y a x e s i n r o o m t e m p e r a t u r e Ni. T h e r e d u c e d w a v e v e c t o r ~ is d e f i n e d b y ~ = q[qM w h e r e 2qM is t h e d i s t a n c e b e t w e e n t h e origin a n d t h e n e a r e s t reciprocal l a t t i c e p o i n t i n t h e d i r e c t i o n of q. T h e e x p e r i m e n t a l r e s u l t s are g i v e n b y filled p o i n t s for t r a n s v e r s a l w a v e s a n d b y o p e n p o i n t s for l o n g i t u d i n a l waves. T h e s t r a i g h t lines p a s s i n g t h r o u g h t h e o r i g i n give t h e i n i t i a l slope of e a c h b r a n c h as c a l c u l a t e d f r o m t h e elastic c o n s t a n t s . T h e h e a v y lines are t h e d i s p e r s i o n c u r v e s c a l c u l a t e d o n t h e K r e b s model. T h e b r o k e n a n d d o t t e d lines refer to c a l c u l a t e d d i s p e r s i o n c u r v e s b a s e d r e s p e c t i v e l y o n t h e De L a u n a y a n d B o r n - y o n IZArmAn m o d e l s ; t h e o m i t t e d p o r t i o n s p r a c t i c a l l y coincide w i t h t h e c o r r e s p o n d i n g K r e b s curves.
266
S. H A U T E C L E R
AND W. VAN DINGENEN
TABLE
I
Frequencies of the normal m o d e s propagating along the s y m m e t r y axes in r o o m temperature Ni (in units l0 IS Hz)
[~ o OIL I
[~ ;
o]
g ~ o] T4
v
0.090 0.137 0.163 0.222 0.222 0.266 0.437 0.496 0.651 0.679 0.766 0.942 [~]
g o o] r
1.48 2.02 2.28 3.35 3.28 4.12 6.10 6.55 8.36 7.95 8.85 9.26
I 0,137 0.185 0.202 0.252 0.255 0.297 0.353 0.404 0.867 0.925
1.39 1.89 2.12 2.56 2.49 3.17 3.66 4.14 6.43 6.64
0.061 0.097 0.13l 0.180 0.278 0.291 0.313 0.416 0.666 0.734 0.896 0.910
1,41 2.36 3.19 4.14 6.12 6.29 6.68 7.88 8.14 7.71 6.64 6.70
0.091 0.115 0.192 0.242 0.246 0.274 0.375 0.405 0.521 0.545 0.672 0.781
v 1.25 1.77 2.81 3.29 3.41 4.26 5.14 5.74 6.76 7.07 8.38 8.75
gg] r
L v
0.070 0.126 0.132 0.142 0.162 0.176 0.178 0.194 0.234 0.266 0.272
1.45 1.60 1.98 2.24 2.77 2.56 2.82 2.82 3.38 3.72 3.83
0.298 0.314 0.382 0.398 0.480 0.532 0.716 0.722 0.768 0.874 0.888
4.19 4.98 5.06 5.52 6.57 7.34 8.83 8.83 8.87 9.12 9.21
0.150 0.194 0.214 0.292 0.346 0.362 0.440 0.524 0.696 0.952
1.14 1.54 1.53 2.08 2.36 2.45 2.87 3.51 4.11 4.5l
reported on fig. 5 ; the open circles correspond to the longitudinal branch, the filled circles to the transversal one. In the [110] direction, only one of the two non-degenerate transversal branches has been observed, namely the one (T4) for which the polarization is parallel to the 4-fold axis and thus in the scattering plane; under the conditions of the measurements, the other transversal branch (T2) was unobservable since the polarization, being parallel to a 2-fold axis, is normal to the scattering plane. The straight lines on fig. 5 passing through the origin give the initial slope of each branch of the dispersion curves, as calculated from the values of the elastic constants given in table I116). On fig. 5, one observes very often a good agreement between the coordinates of two neighbouring points of the dispersion curves, originating from very different regions of the scattering surfaces; this is another confirmation that, in general, the accuracy is better than the uncertainty given by the resolution figure of fig. 2, 3 and 4. However, a few points do not lay on the average curve shown; in the analysis, a lower weight must be attributed to these points which originate either from regions where the scattering surface is very oblique with respect to q so that a small error on the position of the
L A T T I C E D Y N A M I C S OF N I C K E L
267
surface gives an amplified error on q, or from regions where the scattering surface, being deduced from unresolved inelastic peaks, is less accurately located. The values of the frequencies found in the present work lay slightly above those obtained by B i r g e n e a u e.a.2), about 5% at the zone boundaries. These discrepancies could not be accounted for by normal experimental errors, so a reasonable explanation is still lacking; tentatively, they could be assigned to an incorrect subtraction of the socle under the high energy one-phonon peaks.
Analysis o/the results. D i s p e r s i o n c u r v e s . Following F o r e m a n and L o m e r 17), the dispersion curves along the s y m m e t r y directions can be used to determine the range of the interatomic forces in cubic metals. The analytical form of such branches is given by co
4~2Mv2 = Z q~n[1 -- cos n ~ ]
(7)
~'t--1
where M is the atomic mass, and q)n, the interplanar force constants, are some linear combinations of the interatomic force constants introduced in the Born-yon K~rm~n model; the combinations appropriated to the various directions and polarizations were given by S q u i r e s l S ) . For the various branches, the ~bn have been determined by a least-squares analysis of the experimental data of table I, and for a number N of terms in the serie (7) going from one to eight. From the general behaviour of the interplanar force constants as a function of N, it could be deduced that the forces extend at least up to the third neighbours, which requires nine interatomic force constants in the Born-yon K~rm~n model. However, the relative importance of the q~n not containing the force constants between the first neighbours is very small, so that in nickel the short-range interaction is by far the most important one. At this point of the analysis it would be possible to deduce, from the interplanar force constants, the values of the m a n y interatomic force constants to be introduced in a Born-von K~rmfin model; we preferred, however, to look if the same good agreement with our experimental results could be obtained by using models containing much less adjustable parameters. In the Born-yon K~rmfin model, general forces between nearest neighbours of f.c.c, metals require three parameters; it is then possible to get their values from the three elastic constants if the force constants between more distant neighbours are not taken into account. The dispersion curves calculated using the values of table II are shown as dotted lines in fig. 5 (the omitted portions practically coincide with the heavy lines). As can be
268
S. HAUTECLER
AND
W.
VAN
DINGENEN
seen, the discrepancies between the results given b y this model and the experimental data are not very important. In the De Launay model, only three force constants enter; the broken lines of fig. 5 refer to dispersion curves calculated with this model, when the parameter values are deduced b y fitting the measured elastic constants of table II. The consequences of the non-fulfilling of the periodicity requirements can be noticed on the figure where the degeneracies at the end points of the [~ 0 01 L- and [~ ~ 0] T4 branches, and of the [~ ~ 01 L- and [~ ~ 0] T2 branches, are not obtained. However, since this violation is not very large, and since moreover the general agreement with the experimental data is not bad, we can hope to improve this agreement b y keeping from the De Launay model the central interactions out to the second neighbours, but treating more suitably the conduction electrons, as for instance in the Krebs model. In the Krebs model, the number of parameters is higher than the number of elastic constants; the latter determine the values of three parameters if the other are treated as adjustable. We performed six sets of calculations of dispersion curves, using either different values of the proportionality factor C, or slightly different values for the elastic constants, or different values of the electron density; in all cases however the conduction electrons have been supposed free so that the electron wave vector ky at the Fermi surface was given b y kF = (3~ 2 ne)L (8) A very good fit to the experimental points was obtained when using the values of table II; the corresponding curves are given b y the heavy lines of fig. 5. The electron density given in table II corresponds to 0.6 conduction TABLE
II
V a l u e s of c o n s t a n t s a n d p a r a m e t e r s used in calculations a = 3.5239 A C l l = 2 . 4 6 10 x2 d y n e / c m 2 C12 = 1.50 l 0 in d y n e / c m 2 C 4 4 = 1 . 2 2 1012 d y n e / e m 2
C
=
n,
= 2.4/a 2
0.353
electrons per atom; this is a reasonable figure since in nickel ten electrons have to be distributed over the 3d and 4s bands, and since the value of the magnetic moment indicates that there are 0.6 holes per atom in the 3d bandlg). Our calculations show that a change of the value of the parameters which determine the screened Coulomb interaction has no large influence on the position of the dispersion curves: the fit to the experimental points remains
L A T T I C E D Y N A M I C S OF N I C K E L
269
reasonably good (indeed, a complete cancellation of the long-range term in the dynamical matrix elements leads to a lowering of the frequencies by about 5%). The fact that the long-range interaction is relatively unimportant in Ni is not surprising. Indeed, the value of the elastic constant C44, which determines for a large part the magnitude of the shortrange interaction, is much higher for Ni than for other f.c.c, metals, whereas the deviation from the Cauchy relation C1~-C44, which affects the magnitude of the long-range interaction, is exceptionally small. S c a t t e r i n g s u r f a c e s . In order to include in the test results obtained in non-symmetry directions, we calculated the intersections of the scattering surfaces with the (01 i) plane in the framework of the Krebs model; we gave to the parameters the values of table II which lead to a good agreement for the dispersion curves. The heavy lines of fig. 2, 3 and 4 represent the calculated scattering curves corresponding respectively to a scattering angle of 70 °, 90 ° and 100 °, and to an incident neutron wavelength of 4.094 A; the longitudinal or transversal character of the various branches is indicated by the letters L of T. The agreement between the theoretical curves and the points resulting from our measurements is generally good. In all cases the calculated scattering surface passes through the areas obtained by drawing the resolution figure around each experimental point; even better, the calculated curve passes very often through the uncertainty zone of the resolution figure. On fig. 4, the agreement seems less good for the external longitudinal branch located in the Brillouin zone centered on the (200) reciprocal lattice point; in this region the accuracy on the position of the experimental points is lower than the average since the inelastic peaks are not resolved in the corresponding spectra. On the three figures, the agreement between the experimental and calculated results is, however, somewhat less satisfactory in the righthand upper part of the zone centered on the (1 i i) reciprocal lattice point. In this region, the transversal branch of the scattering surface is changing very abruptly its orientation and becomes very oblique with respect to the analysis direction of the scattered neutrons; this induces, through the divergency of the scattered beam, very large inelastic peaks which we, indeed, dissociated into two components previously to any analysis. On fig. 2, 3 and 4 one also finds a few experimental points aside the calculated scattering curves. Some of these points are doubtful, and most of the others correspond to low intensity peaks; only two extra points (on fig. 3) are associated to high intensity peaks. These points could originate from multiple scattering processes.
Discussion. It has been seen that the experimental points of the dispersion curves along the s y m m e t r y directions of a Ni single crystal can be fitted very well with curves calculated by means of the Krebs model, when using, for
270
S. H A U T E C L E R A N D W. VAN D I N G E N E N
the three most sensitive parameters of the theory, values derived from the measured elastic constants. In spite of the fact that the contribution of the conduction electrons to the lattice dynamics of Ni is small, this metal is, however, particularly interesting since it allows to control that the analytical form of the short-range part of the interaction potential as chosen by Krebs is adequate; in a recent paper, M a h e s h and Dayal20)23) found necessary to include the third-nearest neighbour interaction in a Krebs treatment of the lattice dynamics of some b.c.c, and f.c.c, metals. On fig. 5, it can be seen that the dispersion curves calculated on a Bornvon KArmAn model with general forces extending only up to the firstnearest neighbours are nearly as good as those given by the Krebs model. P a l and G u p t a 21) have also reported a calculation of the phonon dispersion relations in Ni using the S h a r m a and J o s hi 22) electron gas model. Except for a slight violation of the s y m m e t r y requirements, their results coincide very well with our calculated curves giving the best fit to our experimental data. Therefore, it appears that a still more refined analysis would be necessary to test in detail either models. As an attempt to do so, we compared the experimental points of the scattering sur/aces with theoretical curves derived from the Krebs model; in this way, results obtained in non-symmetry directions also contribute to the test. The agreement is very good except in some regions of reciprocal space where the surfaces are changing very abruptly; we think that these regions are more sensitive to the choice of the parameters so that, from improved experimental data, either their values could be obtained more precisely or the model should be excluded. In conclusion, we can say that the Krebs model, whilst taking into account the effect of the conduction electrons in a rather primitive manner, predicts reasonably well the dynamical properties of Ni. A c k n o w l e d g e m e n t s . We wish to thank Dr. M. N ~ v e de M 6 v e r g n i e s , head of the Neutron Physics Department, for the constant interest shown during our work, and Dr. K. K r e b s for very helpful discussions. We are also indebted to Prof. O. K r i s e m e n t and to Dr. D. M e i n h a r d t for supplying both the Ni single crystal sample and the Pb monochromator. Received 23-9-66
REFERENCES I) H a u t e c l e r , S. and V a n D i n g e n e n , W., J. Physique 25 (1964) 653. 2) B i r g e n e a u , R. J., C o r d e s , J., D o l l i n g , G. and W o o d s , A. D. B., Phys. Rev. 136 (1964) A 1359. 3) L e h m a n , G. W., W o l f r a m , T. and De W a m e s , R. E., Phys. Rev. 1~8 (1962) 1593. 4) F u c h s , K., Proc. Roy. Soc. A 1 5 3 (1935) 622.
LATTICE DYNAMICS OF NICKEL
27 1
5) De L a u n a y , J . , J. t h e m Phys. 21 (1952) 1975; Solid State Physics, Vol. 2, Academic Press, New York (1956) 219. 6) B h a t i a , A. B., Phys. Rev. 97 (1955) 363. 7) K r e b s , K., Phys. Letters 10 (1964) 12; Phys. Rev. 138 (1965) A 143. 8) L a x , M., J. Phys. Chem. Solids, Suppl. 1 (1965) 179. 9) S h u k l a , M. M., Phys. Star. Sol. 7 (1964) K l l . 10) S h u k l a , M. M., Phys. Stat. Sol. 8 (1965) 475. 11) M a h c s h , P. S. and D a y a l , B., Phys. Stat. Sol. 9 (1965) 351. 12) K r e b s , K., private communication (1965). 13) H a u t e c l e r , S. and V a n D i n g e n e n , W., S y m p o s i u m on Inelastic Scattering of Neutrons by Condensed Systems, Brookhaven ,BNL 940 (C-45) (1966) 83. 14) P l a e z e k , G. and V a n H o v e , L., Phys. Rev. 93 (1954) 1207. 15) V a n D i n g e n e n , W. and H a u t e c l e r , S., to be published. 16) de K l e r k , J., Proe. Phys. Soe. 73 (1959) 337. 17) F o r e m a n , A. J. E. and L o m e r , W. M., Prom Phys. Soc. B T 0 (1957) 1143. 18) S q u i r e s , G., Arkiv f. Fysik 25 (1963) 21. 19) M o t t , N. and J o n e s , H., The Theory of the Properties of Metals and Alloys, Oxford University Press, London (1936) 316. 20) M a h e s h , P. S. and D a y a l , B., Phys. Rev. 143 (1966) 443. 21) P a l , S. and G u p t a , R. P., Sol. State Com. 4 (1966) 83. 22) S h a r m a , P. K. and J o s h i , S. K., J. chem. Phys. 3 9 (1963) 2633. 23) S h u k l a , M. M. and D a y a l , B., Phys. Stat. Sol. 16 (1966) 513.