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Coherent α-α′ phase transition of hydrogen in niobium
COHERENT a-x'PHASE TRANSITION OF HYDROGEN IN NIOBIUM and H. PEISL
H. ZABELt
Sektion Physikdcr Ludwig-Maximilians Univcrsitlit Miinchcn. X Miinchen 22, Germany (Receicrud19 Jarluory 1979; irr revisedform
23 October 1979)
Abstract-The a-a’ phase transition of hydrogen in niobium closely resembles a gas-liquid transition of a real gas and is attributed to the elastic interaction via the lattice distortions due to the hydrogen atoms. Because of the range of the interaction, the critical fluctuations at the critical point of the a-a’ transition are drastically suppressedin a coherent lattice. X-ray scattering studies of Nb crystals. loaded in situ with hydrogen, show that, below the critical point, coherent inhomogeneous hydrogen density fluctuations exist. They are due to a smooth density variation over the whole sample size and depend sensitively on its shape through the fulfilment of the elastic boundary condition. R&urn&-On attribue la transition a-a’ de I’hydroghne dans le niobium, qui ressemble g la transition gaz-liquide d’un gaz r&L B lfnteraction dlastique produite par les atomes d’hydrogkne par I’intermtdiaire des distorsions du rbeau. Du fait de la port&e des interactions, les fluctuations critiques de la transition CY-CC’ sont considhabkment aflaiblies dans un r&au cohtrent. Des etudes par diffraction de rayons X de cristaux de niobium chargts in situ avec de I’hydrog&e montrent qu’il existe au-dessous du point critique des fluctuations cohtrentes h&&&es de la densit d’hydrogtne. Elles proviennent d’une kg4re variation de densit sur tout l%chantillon et elles dependent beaucoup de sa forme par l’interm& diaire des conditions aux limites Clastiques. Zwammenfassung-Der a-a’ Phaseniibergang von Wasserstoff in Niob kann in Analogie zum Phaseniibergang gastirmig-fliissig freier realer Gase betrachted werden. Er geht auf die elastische Wechselwirkung zuriick. die tiber die Gitterverzerrungen durch den Wasserstoff hervorgerufen wird. Auf Grund der Reichweite der Wechselwirkung werden in einem kohiienten Gitter kritische Fluktuationen am kriti-
schen Punkt des a-a’ Phasentibergangs stark unterdriickt. RSntgen-Streuexperimente an Nb Kristallen. die in situ mit Wasserstoff beladen wurden, zeigen, dass unterhalb des kritischen Punktes kohirente inhomogene Wasserstoffdichtefluktuationen existieren. Sie werden von einer
langsam variierenden
Dichtevertcilung des Wasserstoffs fiber die gesamte Probendimension Dichtefluktuationen ab.
verursacht. Die makroskopischen hilngen iiber die elastische Randbedingung empfindlich von der Probengeometiie
As this phase transition takes place in an elastically and plastically deformable crystal, it is necessary to differentiate between coherent and incoherent a-a’ transitions. The coherent transition is characterized by a continuous variation of lattice parameter through regions of continually varying hydrogen concentration, while the incoherent transition shows a discontinuous change of lattice parameter, and thus the lattice disregistry at the interface of the a-a’ phases. A part of the incoherent phase diagram of H in Nb is shown in Fig, 1. as discussed in detail elsewhere [IO, 241. Here we will concentrate on the coher-
1. lNTRODI_JClION In recent years much research effort has been spent on the study of metal-hydrogen alloys. The discussion of thermodynamics and phase transitions of hydrogen in metals was especially stimulated. by the models of elastic interaction amdng interstitially dissolved hydrogen atoms [I] and of macroscopic hydrogen density modes[Z]. In a series of publications[3-8], these concepts have been further extended and clarified. In niobium, hydrogen occupies tetrahedral interstitial sites and expands the lattice proportional to the dissolved hydrogen concentration [9, IO]. The high mobility (jump rates are in the order of 10” s-l) guarantees a quick attainment of thermal equilibrium conditions. The LX-Z’transition of H in Nb corresponds to the gas-liquid transition of a real gas. In this transition the Nb lattia retains its cubic symmetry and the phases are only distinguishable by their different hydrogen concentrations and thus by their different lattice expansions.
ent phase transition. A preliminary summary of some of our results on sampleshape dependent macroscopic modes, to be described below, has been published separately [ 111. In binary alloys the coherent spinodal decomposition by microscopic density modes of about 8A is established theoretically [12,13], as well as experimentally [14, IS]. In the NbH system the microscopic modes also have been observed [ 16,171.The data pre-
sented in the following, describe the first observation
t Present address: Department of Physics, University of Illinois at Urbana-Champaign, Urbana, IL 61801 U.S.A. 589
of H in Nb by macroscopic density modes. The wavelength of these macroscopic modes is on the order of the sample size
of coherent spinodal decomposition
590
ZABEL
ANI)
PEISL:
PHASE TRANSITION
OF HYDROGEN
IN NIOBIUM
180 - T,= 17PC -
C(H/Nb)-
Fig I. Part of the incoherent phase diagram of H in Nb [IO]. d. Ther$ore, they depend on the sample shape through the fulfilment of the elastic boundary condition. In contrast to these, the microscopic modes have a wavelength J 4 d, and the boundary conditions are irrelevant in a physical sense.
elastic dipole moments, leading to the same lattice distortion as the defects itself[25]. For a given p(r). the strain field q,(r)must be calculated by solving the elastic equilibrium condition a&r)
= 0
r in V
(3)
and the boundary condition 2 THEORETICAL
BACKGROUND
np&)
The theory of elastic interaction and phase transitions in coherent metal-hydrogen alloys will be outlined briefly. For more details we refer to the original papers of Wagner and Horner [Z] and Horner and Wagner [3] and a recent review article by Wagner
Cl0 In the continuum version the elastic energy of the hydrogen atoms in an isotropic medium is given by If,, = - f
I
&)%,(r,
(p(r))) dV
(I)
Here, eij is the strain tensor Q(r) = $CGj(r) +
api(r)l.
(2)
8, is a short form for the partial differentiation in respect to Cartesian coordinates Zi = a/&,. c,, is a function of the position r and the hydrogen density distribution p(r) of the hydrogen in the lattice. u describes ttie deviations of the metal atoms from their ideal positions in a hydrogen-free lattice. The socalled double force tensor flj replaces the action of point defects on the host lattice by a distribution of
r on S
(4)
where Q)
= &s,(r)
rcprcscntntion
(5)
- P(r)Pij
is the stress tensor and C,, are the elastic constants. The unit normal vector n is directed outward on the surface S of the volume V. Besides the conditions (3) and (4) the compatibility relation must be taken into account: 4a,~rr + v,~,,
- a,+,
- aik,
= 0
(6)
This relation assures, .that q,(r)follows from a continuously differentiable displacement field u(r). If the condition (6) is fulfilled, we may speak of a coherent state of the crystal; otherwise we have an incoherent state. The terms ‘coherent’ and ‘incoherent* state are closely related to the stresses involved. A trivial solution of the equilibrium condition (3) and boundary condition (4) is simply a&) = 0, and from (5) it follows for the strain field: c,,(r) =
P(rPijkr%.
(7)
In general, this strain field will not fulfil the compatibility condition (6) for an arbitrary density distribu(b)
(a)
Fig. 2. Simplilisd
= 0
of a coherent (a) nnd an incohcrcnt
S~IIL‘ (h) of a crystal.
ZABEL
ASI> f’EiSL:
PHASE
TRANS1710N
OF HYDROGEN
IN NIOBIUM
591
tion o(r). Thus the stress-freestain hclongs IO an incohcrcnt crystal. Howcvcr. in two casts. that is when it(r) is either a constant or a linear function of r. +?&) = 0 satislics the condition (6). All deviations from these two density distributions will crcatc so called ‘coherency stresses’in a coherent lattice. The cohcrcncy stressesfor a given p(r) might be so high.
I;, is the Boltzmann constant. ML = E,_ - p2/c,, is the attractive part of the elastic interaction after substracting all short ranged repulsive electronic and elastic contributions, and.[&) ‘is the second derivative of a fret energy with respect lo the density. where
that they exceed the critical shear stress for the onset of plastical deformation. Then dislocations will be created in order to diminish the internal stressesand the crystal transforms to an incoherent state (Fig. 2).
Because MO = MI for reasons explained above, the spinodal temperatures Tz** for the homogeneous and the constant density gradient mode are equal and have the highest critical temperature. which is identical with the T, for an incoherent phase transition. In
It turns out that the calculation of the elastic energy for an arbitrary density fluctuation p(r), by solving the proper boundary value problem, leads to an eigenvalue problem, in which the fluctuations can
be analyzed in terms of eigenmodes elastic energy :
Y,(r) of the
dr) = C pf.Ydr) L
and the total energy is represented by a summation over all eigenmodes
Yt(r) is an orthonormal set of eigenfunctions-called “density eigenmodes’-to the eigenvalues Efi The energy spectrum of the eigenmodes is discrete and divided into those connected with macroscopic density modes and those connected with microscopic modes. The macroscopic modes have a macroscopic wavelength comparable to the sample size and the spacing of their energy levels is volume-independent In isotropic media the energy levels of microscopic modes are degenerate and situated well below the lowest macroscopic energy level. Formally one finds the energy eigenvalues of microscopic modes by replacing condition (4) with a periodic boundary condition. Plane waves, which are solutions of the periodic boundary condition, are restricted to wavelengths I 6 d (d = sample size). in order to be insensitive to the real boundary condition. The density modes with L = 0 and L = 1, describing a homogeneous density mode and a constant density gradient respectively, have the lowest elastic energy, because they are not accompanied by coherency stresses. They do not depend on the boundary condition and occur in all saxnptes regardiess &f their shape. All modes with L ;r 2 describe density ffuctuations, which depend strongly on the sample shape and create coherency stresses resulting in a higher elastic energy. For suitable temperatures and concentrations, the elastic interaction leads to phase transitions involving a decomposition of the hydrogen in the metal lattice. In the molecular field approximation the critical spinodal temperature Tf; where the system becomes unstable, is given by ML
bT,L= fi(pf
only short range interactions contribute. and which can be approximated by an ideal mixing entropy.
the case of a coherent lattice, equation (IO) shows that each density mode becomes unstable at a different temperature, i.e. the spinodals associated with differ-
ent modes are well separated. Consequently the number of modes exhibiting critical behaviour in the vicinity of T, is strongly suppressed when compared to a normal liquid, where the spectrum of critical fluctuations is continuous. Between the temperature of the highest ‘incoherent’ T and the spinodal temperature of the boundary insensitive microscopic modes there exists an entire range of spinodals, each connected with a macroscopic mode.
The dominant fluctuation wavelength of the microscopic modes, once they are excited, is selected by the fastest growing wave[12]. In contrast to this the macroscopic wavelengths are defined by the related spinodals and grow one after the other by coofing the system [SJ. In practice only a few modes of the higher spinodals become unstable and grow. At lower temperatures the increasing coherency stresses will destroy the coherent lattice. Macroscopic density modes exist also for the decomposition of binary alloys, but only as far as long-range elastic interactions play an important role in the interaction Hamiltonian (usually, short range forces dominate). The smali diffusion constant of the constituent species on the one hand, and the macroscopic diffusion distances required on the other, make those systems further impractica1 from experimental point of view. In contrast to this. the diffusion constant of hydrogen in some metals is extraordinarily high (D x IO-’ cm’/s for H in Nb at room temperature [l9]), and makes these systems most favourable for investigating coherent phase separation via macroscopic modes. 3. EXPERIME~AL
PROCEDURE
The coherent spinodal decomposition has been investigated here by means of X-ray scattering. The principal of the method and the experimental set-up are discussed elsewhere [lo]. Only some special aspects are outlined below. In order to investigate coherent macroscopic hydrogen density fluctuations, the starting condition of the material has to be coherent. Therefore all Nb samples were loaded with the critical hydrogen con~nt~tion from the gaseous phase in situ in a special high temperature furnace
592
ZABEL
PEISL:
AND
PHASE
TRANSITION
above TP In this way incoherent phase transitions, which destroy the lattice coherency before beginning the actual experiments. are avoided. However, even though a careful loading, the crystal will not be ideally coherent. Lattice defects always interrupt the coherency locally. Nevertheless, it can be shown [20] that, with regard to coherent density modes, it is not the virginal (grown-in) dislocation which determine the state of the crystal, but rather those which are newly created by the coherency stresses of the macroscopic modes themselves, Only the latter transform the crystal from a coherent to an incoherent state. Thus by in situ loading of the samples with the critical hydrogen concentration we expect to find one sharp Bragg reflection, shifted to lower scattering angles, due to an expanded lattice and indicating a homogeneous hydrogen distribution in the crystal above T, Below T, we expect a broadened reflection, due to the onset of macroscopic density modes expanding the lattice inhomogeneously. The transition from a coherent state to an incoherent a-a’ phase separation should be observable in a splitting of the broadened peak into (eventuallj) two w reflections, in agreement with the homogeneous concentrations c, and c,. of the coexisting phases a and a’. The Nb samples (single crystals, foils, wires) were purchased from MRC with a stated purity of 99.98%. Single crystals were spark cut, orientated, ground and polished. Polycrystals didn’t attain a special treatment. All samples were degassed at.22OO”C for several hours in a vacuum of lo-” torr, and then mounted in an X-ray oven, which had a longtime stability and homogeneity of better than 0.1 degree. For the different sample geometries, the sample holders in the oven were constructed to allow the samples to relax freely under the action of an inhomogeneous hydrogen density distribution. In the following only lattice parameters but not concentrations will be given. It has been proven that the relationship between lattice parameter and concentration is linear, at least in the concentration range of the a-a’ part of the phase diagram [lo], with
OF HYDROGEN
well
Aa - &(T - 293) ; a > E = 17.24 (H/Nb) is the concentration coefficient [9]. ir = 10.8 x 10-6K-’ the corrected thermal expansion coefficient for NbH [lo], here for the critical hydrogen concentration. Thus the phase diagram can be presented equally in concentration or lattice parameter. However. the latter has the advantage of +I = E -
being directly comparable
with the present data, inde-
which are difficult to estimate. especially for inhomogeneous density distributions. pendent of temperature
corrections.
For the Iatticc parameter measurement usuul scuttcring geometry. gcncrator
(Sci%rt).
MO-tube
(AEG)
(Sicmcns). Associotcd electronics by Wcnzl and. Bicron.
we used the
with a commercial
X-ray
and goniometer
were manufactured
IN NIOBIUM
RESULTS
4.
The &et of spinodul decomposition by coherent hydrogen density tluctuations was investigated in two different sample geometries: a disk-shaped single crystal and a polycristalline wire. These yielded different information concerning the spatial resolution of the density fluctuations. In the case of the single crystal (thickness 0.5 mm, diameter 12.5 mm), the X-ray beam penetrated about 20% from the surface, thus integrating over all fluctuations in this region. The grain size in the wire (length 13 mm, diameter 0.76 mm) was about SOpm. Because of their different orientations, the grains played the role of little probes coming into position of reflection one after the other when the wire turned around and thus displaying the averaged hydrogen concentration in a grain via the lattice parameter measurement. In both experiments described here the samples were cooled below the critical point (7’, = 17l”C, C, = 0.31 H/Nb [lo]) slowly, in the order of hours. The isothermal development of the density inhomogeneities was studied, step by step, over long periods of time. Afterwards the samples were reheated above T, in order to prove the reversibility of these density inhomogeneities. It has been shown [21] that an incoherent phase transition below T, always leads to a time independent density inhomogeneity above T,, stabilized by internal stresses. On the other hand a homogeneous density distribution reflects a coherent lattice. Thus, whether or not the Bragg-peak position and its shape are reversible above T, can be regarded as a test of a coherent vs incoherent phase transition. 4.1 Coherent densityjluctuations in a disk-shaped single crystal In a single crystal with a [llO] orientation, the (440) Bragg reflection was investigated using MO& radiation. After in situ loading, the average hydrogen concentration in the crystal was 0.32 H/Nb. Figure 3 shows the particular steps of the experiment. On the left the lattice parameter measurements are outlined. The solid lines always indicate the incoherent phase boundary [IO]. Some examples of Bragg reflections are given on the right, corresponding to those lattice parameters connected with the Bragg reflection by an arrow. Whenever the isothermal (temporal) progress of the lattice constants was measured, the last measurement in the sequence is recorded in Fig. 3. The different runs in detail are: Run No. /: coolirty jkotn 600 to 160°C (Fig. 3a). At 182°C. or about IO degrees above T,. a reference measurement was made for later comparison. The crystal then was cooled to 169°C in 250 min. At 167. 165, 162 and 160°C the isothermal shift of the Bragg reflection was studied. A slight increase of the lattice parameter connected with on increase in full width at half maximum
(FWHM) Intticc
of the Bragg peak could
obscrvcd.
All
paxmctcrs
arc situated
miscibility
gap and do not lit with the incohcrcnt
bc
in the x-z
ZAHEL
AND
PEISL:
PHASE
TRANSITION
180
i
y
160
t
to
gap. now tend
phase. Thus. it runs did
140b 332
334
3.36 338
593
IX
more
to the
concluded,
any density
dislocations.
As
low density
ting
::fter 4 h; after
fluctuations
before
rcturncd
to
its
a further
origiinal
C(
that the foregoing the
annealing at 182°C Icd to :I cancellation
2i?/Grad-
a/A-
C;III
not stahilizc
gcncr;lling
3.40
(down
144°C). The I:ltticc parameters, still within the mis-
cibility
s
a
IN NIOBIUM
time as well as with the dccrcasing temperature
__
t
OF HYDROGEN
of the split-
IO h, the
position.
by
isothermal reflection
The
FWHM
incrcascd by about 30%. The isothermal
relaxation
the Bragg reflection is shown in Fig. 4.
The data are
of
fitted by an exponential function with a time constant of T = 150min.
b.
332
334
336 338
3.40
72
a/B, -
73
Run No. 4: coohgfiorn 182 fo 125°C (Fig. 3~). The crystal was cooled down slowly and held at 144°C.
74
2t’?/Gmd-
Fig. 5 demonstrates Bragg
peak
intensity
the isothermal and
shape.
variation The
of the
broadening
still more with decreasing temperature and the reflection split at 133°C. At 125°C. first an enormous asymmetry of the peak was observed, which vanished at the same time as its position shifted in the direction of the incoherent a-phase boundary. After 6.5 h finally a symmetric reflection was measured. It then did not change position and shape with further time. Because of the stability of the symmetrical, sharp and time-independent Bragg reflection it is most likely that, at this temperature, a transition from the coherent to the incoherent state has occurred, even if the lattice parameter deviates from the incoherent phase boundary. This may be caused by a misalignment of the sample which appears during its relaxation under the effect of an inhomogeneous hydrogen distribution. Due to a macroscopic density fluctuation, a thin disk-shaped crystal bends like a section of a spherical shell, in such a manner that the @w density a-phase is located on the inner compressed surface and the higher density a’ phase on the outer expanded surface. This will be discussed in more detail in 4.3. Thus it is reasonable to find here only one phase (a) with the a’ phase situated on the opposite surface, where the X-rays were not scattered. The reversibility test at 182°C indeed confirms the observations at 125°C. The Bragg reflection no longer moved back to the original position but remained shifted even after an isothermal anneal for 25 h increased
,/ C.
@-yy-+yq,
3.32 334
3.36 330
3.40
72
73
74
2@Grad -)
a/A-
Fig. 3. Coherent phase transition of hydrogen in a thin disk-shaped single crystal. For explanations see text. phase boundary. Starting from 160°C the sample was reheated to 182°C. At this temperature the lattice parameter returned to its original value to within 0.025% and its former FWHM after 17 h. This last measure ment is given in Fig. 3b as the first point of the second run.
Run No. 2: cooling from 182 to 150°C (Fig. 36). After to 160°C the same state as in the first run was reached in 7 h. At 157°C the reflection became asymmetric and more broadened. After a further 20 h the reflection split and a considerable de crease in intensity was observed. The splitting is really rather more a broad intensity distribution with maxima on both sides than an overlap of two otherwise symmetric reflections. Thus, here, and in the following, the points in the T vs a diagram indicate the
slow cooling
lattice parameters according to intensity maxima
and
full bar in between characterizes the intensity distribution. At 155°C the splitting of the maxima increased slightly and fluctuated in the 29 position the
down to 150°C. The wide distribution ameters within reflects
an
the incoherent
inhomogeneous
of lattice par-
phase boundary density
fluctuation
still of
hydrogen in the lattice. The broad intensity distribution completely
vanished after re-heating
again to 182°C and annealing
the sample
isothermally
for 11 h.
The Bragg reflection moved to the first position and its shape matched the original.
Run No. 3: coolingfrom 182 to 144°C (graphically not outlined). At 151°C again a strong decrease in the intensity and a splitting of the peak was observed at 148°C. The amount
of the splitting
fluctuated
with
Fig. 4. Isothermal relaxation at 182°C of the lattice constant after re-heating from 144°C. Data are fitted with an exponential function (full line). T = I50 min.
594
ZABEL
73.4
AND
PE!SL:
PHASE
736
735
TRANSITION
t3.7
scatt6fing
angle
OF HYDROGEN
IN NlORlUM
736 28
Fig. 5. Isothermal fluctuation of the (440) Bragg reflection taken at 144°C. This is due to an irreversible density distribution above T, after a preceding incoherent phase transition has taken place below T, [213. 4.2 Coherent densityjluctuationsin a wire Four small grains along the wire axis were investigated using both the (600) and (433) reflections with MO& radiation. The wire was in situ loaded with Deuterium and had an averaged concentration of 0.34 D/Nb. The different steps of the experiment are recorded in Fig. 6, The solid line indicates the into-
a.
(201 / 200
c.
\ 3.34
336
1
336
x40
3.36
340
1
334
336 o/A-
Fig. 6. Coherent phase transition of Deuterium in a Nb wire. Larfiee pxitmster R~e~~silrernent during cooling (0) am‘ heat&g ( x ) of ditTorent gr:tins in the wire. For further exphmations see test.
herent e-a’ phase boundary which is almost the same for hy&ogen and deuterium [IO]. In the first cooling run from 600% to 165°C (Fig. 6a) the spatial hydrogen distribution is homogeneous,, i.e. all grains yield the same lattice constant within the experimental error, down to the incoherent phase boundary. With further temperature decrease the density distribution becomes in~ea~ngly inhom~eneou~ For each temperature we find more*than two lattice parameters spread over the range between the incoherent phase boundaries The hydrogen density fluctuation completely disappears when reheating the sample above the phase boundary. In the second cooling run from 200 to 161°C (Fig, 6b) the density distribution becomes even more inhomogeneous in the range from 165 to 161°C. Moreover the local density in a grain fluctuates with time. In this case again the density dis~bution reverts to its original state by re-heating the sample above T, In the third cooling run from 184 to 145°C (Fig. SC), first spatial and temporal fluctuations of the density were observed to 150°C. Keeping the temperature constant at ISOT, the lattice constants jumped discontinuously to the outer phase boundaries after about an hour. Hen~forth the lattice constants changed only with temperature according to the phase diagram but not with time when keeping the temperature constant. The temporal sequence of the last cooling run is outlined in Fig. 7. Heating from 14.5to 18S°C shows clearly that the density distribution now stays inhomogeneous independent of time. As in the experiment on the disk-shaped crystal discussed before. the irreversibility above T, is an indication of a foregoing incoherent phase transition well below ‘&. On the other hnnd the direrent. temporally nuctu~ting lattice parameters along the wire axis with values lying in hctween the a-or’ phase boundary. as well as the reversibility to only one lattice parameter
ZABEL
ANIJ PEISL:
PHASE
TR~~NSlTlON
OF
HYDROGEN
IN NIOBIUM
595
going spinodal decomposition by macroscopic density modes without regard to nucfeation centres. 1n the first oxampk of a 0.3 mm thick disk-shaped (dia. 12.5 mm) Nh sin& crystal. the X-rays were scattzrcd ahornately OII both sides of the disk (Fig. 8a). DOW to the incoherent phase boundary they cxhihit the same lattice parameter, which then, below fSo”C moved apart in the direction of the outer phase boundary. On csamination the crystal appeared bent like a section of a spherical she11 (Fig, 8bf. The lattice parameter tending to the lower density EXphase was taken at the inner contracted surface (circles in Fig. ga) while the lattice parameter tending to the higher density 2 phase was at the outer expanded surface (squares in Fig. 8a). It should be
noted that the Bragg reflections taken at both surfaces exhibited an asymmetric shape, indicating a density gradient towards the other phase, but no existence of a second phase on either side of the crystal could be detected. Thus from the X-ray experiment, as well as
Fig. 7. Isothermalfluctuationof Iattice parametersalong the wireaxis.The latticeconstantsof different grains in the wire were measured by the f&owing reSections: 1. (6CQl _U
refiection(x f 2. (600)reflection(0) 3. (433)reflection (0) (433)reflection(A).
4.
above T, show that a ~herent hydrogen density mode has taken place over a t~~rature range of 20 degrees between 170 and 1W’C
OLI
14oc I
334
336
333
Q/)L-
Coherent density fluctuations Mow T, are .connccted with macroscopic variations of the hydrogen concentration in the sample. This will be demonstrated on several sample geometries. Under the action of inhomogeneous density fluctuations, the crystals relax, i.e. they bend in order to reduce internal elastic coherency stresses. In general this is not compktcly possible and the residuat stresses may exceed the critical shear stress for the onset of the coherent-incoherent transition at a certain tempcrature below T, Attendant plastic deformation and generation of cracks which devefop by further cooling then present a frozen picture of the original coherent decomposition. From those pictures together with X-ray examinations we obtain an idea of the spatial arran~ment of the macroscopic hydrogen modes in the coherent state prior ta the onset of incoherence and how it depends on sample geometry. The plastic (bi deformation discussed here has to be clearly distinguished from the electron mi~o~pic investiga- Fig. 8. Ma~~opic hydrogen density mode in a 0.3 mm tion of piastic deformation during hydride precipita- thick disk-shaped Nb single crystal. (a) Lattice parameters taken at both surfaces; scattering geometry is sketched by tion by T. Schober [26]. The latter is in principle an the inset. Full tini: represents the incoherent phase boundincoherent process and precipitation starts at nucIeation centres; the former is the result of a fore-
ary. (b) Photograph of the bent crystal at room temperature.
596
ZABEL
AND
PEISL:
PHASE
TRANSITION
OF HYDROGEN
1N NIOBIUM
piece, was split in the middle of the disk parallel to the surface, and both parts were bent in the same manner as the half thick crystal in the first example
3.34
3.36
33%
o/A69
(b) Fig. 9. Macroscopic hydrogen density mode in a 0.6mm thick disk-shaped Nb single crystal (a) Lattice parameters taken at both surfaces iike in Fig. 7. (b) Splitting and bending of the crystal at room temperature. from the crystal bending, we conclude that a density profile exists, which varies macroscopically from one surface to the other, corresponding to a halfwave-
length fluctuation directed normal to the disk surface. In the second example the disk-shaped crystal was chosen with a double thickness (0.6mm). The X-rays again were scattered from both surfaces and yielded the same lattice ~n~~~ even far below the incolierent phase boundary (Fig 9ak After incoherent deeomposition, the original crystal, consisting of only one
3.34
(Fig. 9b). An X-ray examination of the expanded inner surfaces indeed showed that the 3’ phase is located there as expected. Thus in this case the density profile represents a full wavelength fluctuation normal to the disk surface. Similar experiments on thicker crystals (2 mm up to 1Omm)showed that the low density a phase always is located at the top and bottom of the disks ,or cylinders (Fig lOa).The result is a splitting of small slices at the surfaces, which have similar thickness and bending radius as in the first two examples. A photograph of such a crystal is given in Fig lob. Under the action of density ~h~~en~ti~ the inner parts of thick crystals are not allowed to relax in the same manner as the outer parts by the constrain of the nei~bou~ng layers. Therefore the spatial arrangement of the phases is much more complex and can be demonstrated only by neutron scattering To obtain a rough idea however, a 0.1 mm thick Nb foil was squeezed in the sample holer such that a macroscopic bending was suppressed but other kinds of rel~ation, i.e. ripples in the foil plane, were still possible and have been later observed. The X-ray scattering results are given in Fig 11 and yield a homogeneous hydrogen distribution down to 20 degrees below T, Then a sudden incoherent phase separation occurs whereby the intensities of the Bragg peaks are nearly the same indicating an equal distribution of both phases in the plane of the foil. The next investigations deal with Nb wires. The polycrystalline nature should not be important because the wavelength of the macroscopic modes is so large compared with the grain size, In the first example of a wire with 1.2mm diameter the lattice parameters tend below T, to the lower density side of the phase diagram (Fig 12af. No indication of the a’ phase could be found within a cylindric wall of about 50 m, corresponding to the penetration depth of MoK, radiation. A Debye Scherrer pattern from the same wire confirmed this 6nding[22].
335
a/A-
Fig. IO(a).
338
ZABEL
ANI) PEISL:
PHASE TRANSITION
OF HYDROGEN
IN NlOBIU~
597
Fig. 10. Macroscopic hydrogen density mode in a 2 mm thick disk-shaped Nb single aystal (a) Lattice parameters taken at both surfaas like in Fig. 7. (b) Surface structure of the crystal at room temperature. X-ray examination of the top and bottom of the wire after breaking it in pieces, yielded both coexisting phases with equal scattering intensities. Scanning eleo tron micrographs (Fig 12b) of the top clearly show a macroscopic cylindrically symmetric precipitation structure, whereby the CC’ phase is located in the core, surrounded by the a phase. The macroscopic mode, corresponding to this precipitation structure, represents a full wavelength flu~uation directed normal to the wire axis.
In wires with 0.76 mm diameter we find a cyiindritally symmetric, as well as a half cylinder preclpitation structure (Fig 13). In the latter case, each part of the half cylinder is occupied by one of the coexisting phases 5~and cr’. Wires with a diameter of 0.25 mm exhibit only half cylinder shaped precipitation structure. With regard to the density fluctuation wavelength, the last example of a thin wire corresponds to the first example of a thin disk shaped crystal. In both crystal geometries a half wave density fluctuation becomes unstable below T, whereas its direction depends on
the shorter linear dimension of the sample. In the same manner the 0.6mm thick disk shapccl crystal and the 1.2476 mm wire eorrcsponds. Both exhibit a full wavelength density fluctuation. The wavekngth of the density Ruetuations seems to be nearly constant at about 1 mm. On the other hand, the number of waves depends .on the size of the shorter dimension of the crystal.
I
xi2
336
d-
I, 3400
I
.
0.5
I
1
5. DISCUSSION
1.5
l!!L-..+ Ihlt
Fig. II. Lattice parameters and integrated intensities of Bragg reflections versus temperature taken on a 100~ thick NbHO,:, foil, squeezed in the sample holder. The intensities are normalized at 250°C. The full line represents the incoherent phase boundary.
A quantative comparison of the experiments with the theory of phase transitions in coherent metal hydrogen systems, given by Wagner and Homer
[2,3], cannot be accomplished, as the maeroseopic modes have been calculated explicitly only for a sphere shaped
crystal.
However, the experiments
ZABEL
598
(a)
ASD
PEISL:
PHASE
TRANSITION
1
220 00
160 -
$
t
140
-
100
-
-
ea
0 >o( 000
6ow 3.34
336
a/h
336
4
(W
mode in a Nb wire with 1.2mm diameter. (a) Lattice parameters taken at several grains around the wire. (b) Scanning electron micrograph of the top of a broken part. The a’ phase is located in the core surrounded by the a phase. Fig. 12. Macroscopic hydrogen density
OF HYDROGEN
IN
NIOBIUM
clearly demonstrate the main features of the predictions: 1. Below the critical point the hydrogen density fluctuation becomes rather smoothly inhomogeneous. No phase separation with a steep interface between the two, phases takes place. This can be concluded from the asymmetric and broadened Bragg reflection, taken on the disk-shaped single crystal, described in 4.1. A sharp interface would not contribute to an interference phenomena like a Bragg peak. Even though the peak is broadened, small parts of the crystal are still scattering coherently, i.e. the lattice parameter changes only slowly. 2. The amplitude of the inhomogeneous density distribution increases with lower temperatures, as can be seen in Fig. 6. 3. The macroscopic modes depend on the sample shape. To each geometry only one special fluctuation is connected unequivocally. The direction of the flue tuation wave is determined by the shorter linear dimension of the crystal, and the wavenumber by its extension. 4. The time constant for the re-homogenization above T, is on the order of hours and thus typical for long distance diffusion of the hydrogen in the lattice, in accordance with the estimate made for a sphere [7]. The temperature range below T, where coherent spinodal decomposition by macroscopic modes occurs, is surprisingly large. In an estimate of the shear stresses, a temperature range of about 5 degrees for a single crystal with very low dislocation densities was expected [23]. On the contrary in single crystal slices, temperature ranges of more than 40 degrees were observed, and even in polycrystals a temperature range of about 20 degrees was usual. The occurrence of coherent macroscopic modes in polycrystals is astonishing and shows that they can be
(4
Fig. 13. Scanning electron micrographs of wires with 0.76mm diameter. Two different precipitation structuresareobservcd:(a)cylindricalsymmetricprecipitation. (b) halfcylindrical symmetric precipitation.
ZABEL
AND
PEISL:
PHASE
TRANSITION
OF
HYDROGEN
IN NIOBIUM
599
Tahlc I. Coherent state
Inhomogeneous macroscopic density distribution
Incohcrcnt state Two definite densities. given by the
incoherent phase boundary. 1~ the sin& phases homogeneous density distribution
grows continuously with CT,-‘F]
Phase separation grows continuously with IT,--T]
Macroscopic modes depend on sample shape
sample shape
Crystal is elastically deformed
Crystal is plastically deformed
Reversible change of state
Irreversible change of state
Homogeneous density distribution
Inhomogeneous density distribution
Amplitude
T< r,
Reheating to T> r,
of density fluctuation
superimposed on the already existing dislocation structure. This is discussed in more detail elsewhere c201. The transition from the coherent to the incoherent state is microscopically a very complex procedure. It can be characterized through experimental evidence in the following way: at some temperature below T, a sudden jump in the concentration distribution occurs and only two different concentrations remain, in accordance tith the incoherent phase boundary. After a temperature change, the new equilibrium is attained quickly and no more temporal fluctuations occur. Reheated above the critical temperature a reversible homogeneous hydrogen distribution can no longer be attained and the inhomogeneity becomes temperature dependent and decreases with increasing temperature [21]. The properties of coherent and incoherent NbHo.3 crystals are finally summarized in Table 1, as they have been distinguished ckarly by the experiments. Acknowledgements-We thank the Bundesministerium Rir Forschung und Technologie for financial support of this work. One of us (H.Z) also wishes to thank the U.S. Department of Energy for support during his stay at the University of Houston and Professor S. C. Moss for many helpful’discussions and a critical reading of the manuscript. We would like to thank Professor G. Alefeld for his helpful interest in this work and Professor H. Wagner for many enlightening discussions conccmitig the theoretical background of the experiments. We finally acknowledge with much appreciation the Kristallabor of the Physik-Departmcnt der Technischen Univcrsitit Miinchen for valuable help in sample preparation.
REFERENCES I. G. Alefeld, in Critical Phenomena in Alloys. Magnets
Phase separation is independent of the
and Superconductors (edited by R. E. Mills et al.). McGraw-Hill, New York (1971). 2. H. Wagner and H. Homer Adu. Phys. 23. 587 (1974). 1 H. Horner and H. Wagner, J. Phys. C. 7. 3305 (1974). &: R. Bausch, H. Homer and H. Wagner. J. Phys. C. 8, 2559 (1975). 5. H. K. Janssen, Z. Phys. BU. 245 (1976). 6. H. A. Goldberg. J. Phys. C. IO, 2059 (1977). 7. T. W. Burckhardt, and W. WGger, Z. Phys. B21, 89 (1975). 8. W. Kappus and H. Horner, Z. Phys. B27.215 (1977). 9. H. Pfeiffer and H. Peisl. Phvs. Left. 6OA. 363 119771 10. H. Zabel and H. Peisl, j. Piys. F. 9, 1461 (19j9j ’ Il. H. Zabel and H. Peisl, Phys. Rev. I&L 42, 51 I (1979). 12. J. W. Cahn, Acta Metall. 9, 795 (1961). 13. J. W. Cahn, Acta Metall. 10, 179 (1962). 14. J. E. Woodilla Jr. and B. L. Avervach, Acta Metall. 16, 255 (1968). 15. S. C. Moss and B. L. Averbach, in Small-Angle X-Roy Scattering (edited by H. Brumberger). Gordon and Breach, New York, London, Paris (1967). 16. H. Conrad, G. Bauer, G. Alefeld, T. Springer and W. Schmatz Z. Phys. 266,239 (1974). 17. W. MUnzing and N. Stump, J. appl. Crystullogr. 11,588 (1978). 18. H. Wagner, Elastic Interaction and Phase Transition in Coherent Metal Hydrogen Alloys, in Topics in Applied Physics, Vol 28 (edited by G. Alefeld and J. V6lklj Springer, Berlin, Heidelberg, New York (1978). 19. G. Schaumann. J. V6lkl and G. Alefeld, Phys. Rev. Letr. 21, 891 (1968). 20. H. Zabel and H. Peisl(l979). to be published. 21. H. Zabcl G. Alefeld and H. Peisl, Scripta Met. 9, 1345 (1975). 22. H. Zabel Thesis, Universitat Miinchen, Miinchen (1978). 23. T. W. Burkhardf Z. Phys. 269, 237 (1974). 24. T. Schober and H. Wenzl, in Topics in Applied Physics (edited by G. Alcfeld and J. Viilkl). Vol. 28. Springer, Berlin, Heidelberg, New York (1978). 25. R. Siems, Phys. stat sol. 30, 645 (1968). 26. T. Schober, Phys. stat. sol. (a) 29, 395 (1975). T. Schober, Phys. stat. sol. (a) 30, 107 (1975).
H. ZABELt
Sektion Physikdcr Ludwig-Maximilians Univcrsitlit Miinchcn. X Miinchen 22, Germany (Receicrud19 Jarluory 1979; irr revisedform
23 October 1979)
Abstract-The a-a’ phase transition of hydrogen in niobium closely resembles a gas-liquid transition of a real gas and is attributed to the elastic interaction via the lattice distortions due to the hydrogen atoms. Because of the range of the interaction, the critical fluctuations at the critical point of the a-a’ transition are drastically suppressedin a coherent lattice. X-ray scattering studies of Nb crystals. loaded in situ with hydrogen, show that, below the critical point, coherent inhomogeneous hydrogen density fluctuations exist. They are due to a smooth density variation over the whole sample size and depend sensitively on its shape through the fulfilment of the elastic boundary condition. R&urn&-On attribue la transition a-a’ de I’hydroghne dans le niobium, qui ressemble g la transition gaz-liquide d’un gaz r&L B lfnteraction dlastique produite par les atomes d’hydrogkne par I’intermtdiaire des distorsions du rbeau. Du fait de la port&e des interactions, les fluctuations critiques de la transition CY-CC’ sont considhabkment aflaiblies dans un r&au cohtrent. Des etudes par diffraction de rayons X de cristaux de niobium chargts in situ avec de I’hydrog&e montrent qu’il existe au-dessous du point critique des fluctuations cohtrentes h&&&es de la densit d’hydrogtne. Elles proviennent d’une kg4re variation de densit sur tout l%chantillon et elles dependent beaucoup de sa forme par l’interm& diaire des conditions aux limites Clastiques. Zwammenfassung-Der a-a’ Phaseniibergang von Wasserstoff in Niob kann in Analogie zum Phaseniibergang gastirmig-fliissig freier realer Gase betrachted werden. Er geht auf die elastische Wechselwirkung zuriick. die tiber die Gitterverzerrungen durch den Wasserstoff hervorgerufen wird. Auf Grund der Reichweite der Wechselwirkung werden in einem kohiienten Gitter kritische Fluktuationen am kriti-
schen Punkt des a-a’ Phasentibergangs stark unterdriickt. RSntgen-Streuexperimente an Nb Kristallen. die in situ mit Wasserstoff beladen wurden, zeigen, dass unterhalb des kritischen Punktes kohirente inhomogene Wasserstoffdichtefluktuationen existieren. Sie werden von einer
langsam variierenden
Dichtevertcilung des Wasserstoffs fiber die gesamte Probendimension Dichtefluktuationen ab.
verursacht. Die makroskopischen hilngen iiber die elastische Randbedingung empfindlich von der Probengeometiie
As this phase transition takes place in an elastically and plastically deformable crystal, it is necessary to differentiate between coherent and incoherent a-a’ transitions. The coherent transition is characterized by a continuous variation of lattice parameter through regions of continually varying hydrogen concentration, while the incoherent transition shows a discontinuous change of lattice parameter, and thus the lattice disregistry at the interface of the a-a’ phases. A part of the incoherent phase diagram of H in Nb is shown in Fig, 1. as discussed in detail elsewhere [IO, 241. Here we will concentrate on the coher-
1. lNTRODI_JClION In recent years much research effort has been spent on the study of metal-hydrogen alloys. The discussion of thermodynamics and phase transitions of hydrogen in metals was especially stimulated. by the models of elastic interaction amdng interstitially dissolved hydrogen atoms [I] and of macroscopic hydrogen density modes[Z]. In a series of publications[3-8], these concepts have been further extended and clarified. In niobium, hydrogen occupies tetrahedral interstitial sites and expands the lattice proportional to the dissolved hydrogen concentration [9, IO]. The high mobility (jump rates are in the order of 10” s-l) guarantees a quick attainment of thermal equilibrium conditions. The LX-Z’transition of H in Nb corresponds to the gas-liquid transition of a real gas. In this transition the Nb lattia retains its cubic symmetry and the phases are only distinguishable by their different hydrogen concentrations and thus by their different lattice expansions.
ent phase transition. A preliminary summary of some of our results on sampleshape dependent macroscopic modes, to be described below, has been published separately [ 111. In binary alloys the coherent spinodal decomposition by microscopic density modes of about 8A is established theoretically [12,13], as well as experimentally [14, IS]. In the NbH system the microscopic modes also have been observed [ 16,171.The data pre-
sented in the following, describe the first observation
t Present address: Department of Physics, University of Illinois at Urbana-Champaign, Urbana, IL 61801 U.S.A. 589
of H in Nb by macroscopic density modes. The wavelength of these macroscopic modes is on the order of the sample size
of coherent spinodal decomposition
590
ZABEL
ANI)
PEISL:
PHASE TRANSITION
OF HYDROGEN
IN NIOBIUM
180 - T,= 17PC -
C(H/Nb)-
Fig I. Part of the incoherent phase diagram of H in Nb [IO]. d. Ther$ore, they depend on the sample shape through the fulfilment of the elastic boundary condition. In contrast to these, the microscopic modes have a wavelength J 4 d, and the boundary conditions are irrelevant in a physical sense.
elastic dipole moments, leading to the same lattice distortion as the defects itself[25]. For a given p(r). the strain field q,(r)must be calculated by solving the elastic equilibrium condition a&r)
= 0
r in V
(3)
and the boundary condition 2 THEORETICAL
BACKGROUND
np&)
The theory of elastic interaction and phase transitions in coherent metal-hydrogen alloys will be outlined briefly. For more details we refer to the original papers of Wagner and Horner [Z] and Horner and Wagner [3] and a recent review article by Wagner
Cl0 In the continuum version the elastic energy of the hydrogen atoms in an isotropic medium is given by If,, = - f
I
&)%,(r,
(p(r))) dV
(I)
Here, eij is the strain tensor Q(r) = $CGj(r) +
api(r)l.
(2)
8, is a short form for the partial differentiation in respect to Cartesian coordinates Zi = a/&,. c,, is a function of the position r and the hydrogen density distribution p(r) of the hydrogen in the lattice. u describes ttie deviations of the metal atoms from their ideal positions in a hydrogen-free lattice. The socalled double force tensor flj replaces the action of point defects on the host lattice by a distribution of
r on S
(4)
where Q)
= &s,(r)
rcprcscntntion
(5)
- P(r)Pij
is the stress tensor and C,, are the elastic constants. The unit normal vector n is directed outward on the surface S of the volume V. Besides the conditions (3) and (4) the compatibility relation must be taken into account: 4a,~rr + v,~,,
- a,+,
- aik,
= 0
(6)
This relation assures, .that q,(r)follows from a continuously differentiable displacement field u(r). If the condition (6) is fulfilled, we may speak of a coherent state of the crystal; otherwise we have an incoherent state. The terms ‘coherent’ and ‘incoherent* state are closely related to the stresses involved. A trivial solution of the equilibrium condition (3) and boundary condition (4) is simply a&) = 0, and from (5) it follows for the strain field: c,,(r) =
P(rPijkr%.
(7)
In general, this strain field will not fulfil the compatibility condition (6) for an arbitrary density distribu(b)
(a)
Fig. 2. Simplilisd
= 0
of a coherent (a) nnd an incohcrcnt
S~IIL‘ (h) of a crystal.
ZABEL
ASI> f’EiSL:
PHASE
TRANS1710N
OF HYDROGEN
IN NIOBIUM
591
tion o(r). Thus the stress-freestain hclongs IO an incohcrcnt crystal. Howcvcr. in two casts. that is when it(r) is either a constant or a linear function of r. +?&) = 0 satislics the condition (6). All deviations from these two density distributions will crcatc so called ‘coherency stresses’in a coherent lattice. The cohcrcncy stressesfor a given p(r) might be so high.
I;, is the Boltzmann constant. ML = E,_ - p2/c,, is the attractive part of the elastic interaction after substracting all short ranged repulsive electronic and elastic contributions, and.[&) ‘is the second derivative of a fret energy with respect lo the density. where
that they exceed the critical shear stress for the onset of plastical deformation. Then dislocations will be created in order to diminish the internal stressesand the crystal transforms to an incoherent state (Fig. 2).
Because MO = MI for reasons explained above, the spinodal temperatures Tz** for the homogeneous and the constant density gradient mode are equal and have the highest critical temperature. which is identical with the T, for an incoherent phase transition. In
It turns out that the calculation of the elastic energy for an arbitrary density fluctuation p(r), by solving the proper boundary value problem, leads to an eigenvalue problem, in which the fluctuations can
be analyzed in terms of eigenmodes elastic energy :
Y,(r) of the
dr) = C pf.Ydr) L
and the total energy is represented by a summation over all eigenmodes
Yt(r) is an orthonormal set of eigenfunctions-called “density eigenmodes’-to the eigenvalues Efi The energy spectrum of the eigenmodes is discrete and divided into those connected with macroscopic density modes and those connected with microscopic modes. The macroscopic modes have a macroscopic wavelength comparable to the sample size and the spacing of their energy levels is volume-independent In isotropic media the energy levels of microscopic modes are degenerate and situated well below the lowest macroscopic energy level. Formally one finds the energy eigenvalues of microscopic modes by replacing condition (4) with a periodic boundary condition. Plane waves, which are solutions of the periodic boundary condition, are restricted to wavelengths I 6 d (d = sample size). in order to be insensitive to the real boundary condition. The density modes with L = 0 and L = 1, describing a homogeneous density mode and a constant density gradient respectively, have the lowest elastic energy, because they are not accompanied by coherency stresses. They do not depend on the boundary condition and occur in all saxnptes regardiess &f their shape. All modes with L ;r 2 describe density ffuctuations, which depend strongly on the sample shape and create coherency stresses resulting in a higher elastic energy. For suitable temperatures and concentrations, the elastic interaction leads to phase transitions involving a decomposition of the hydrogen in the metal lattice. In the molecular field approximation the critical spinodal temperature Tf; where the system becomes unstable, is given by ML
bT,L= fi(pf
only short range interactions contribute. and which can be approximated by an ideal mixing entropy.
the case of a coherent lattice, equation (IO) shows that each density mode becomes unstable at a different temperature, i.e. the spinodals associated with differ-
ent modes are well separated. Consequently the number of modes exhibiting critical behaviour in the vicinity of T, is strongly suppressed when compared to a normal liquid, where the spectrum of critical fluctuations is continuous. Between the temperature of the highest ‘incoherent’ T and the spinodal temperature of the boundary insensitive microscopic modes there exists an entire range of spinodals, each connected with a macroscopic mode.
The dominant fluctuation wavelength of the microscopic modes, once they are excited, is selected by the fastest growing wave[12]. In contrast to this the macroscopic wavelengths are defined by the related spinodals and grow one after the other by coofing the system [SJ. In practice only a few modes of the higher spinodals become unstable and grow. At lower temperatures the increasing coherency stresses will destroy the coherent lattice. Macroscopic density modes exist also for the decomposition of binary alloys, but only as far as long-range elastic interactions play an important role in the interaction Hamiltonian (usually, short range forces dominate). The smali diffusion constant of the constituent species on the one hand, and the macroscopic diffusion distances required on the other, make those systems further impractica1 from experimental point of view. In contrast to this. the diffusion constant of hydrogen in some metals is extraordinarily high (D x IO-’ cm’/s for H in Nb at room temperature [l9]), and makes these systems most favourable for investigating coherent phase separation via macroscopic modes. 3. EXPERIME~AL
PROCEDURE
The coherent spinodal decomposition has been investigated here by means of X-ray scattering. The principal of the method and the experimental set-up are discussed elsewhere [lo]. Only some special aspects are outlined below. In order to investigate coherent macroscopic hydrogen density fluctuations, the starting condition of the material has to be coherent. Therefore all Nb samples were loaded with the critical hydrogen con~nt~tion from the gaseous phase in situ in a special high temperature furnace
592
ZABEL
PEISL:
AND
PHASE
TRANSITION
above TP In this way incoherent phase transitions, which destroy the lattice coherency before beginning the actual experiments. are avoided. However, even though a careful loading, the crystal will not be ideally coherent. Lattice defects always interrupt the coherency locally. Nevertheless, it can be shown [20] that, with regard to coherent density modes, it is not the virginal (grown-in) dislocation which determine the state of the crystal, but rather those which are newly created by the coherency stresses of the macroscopic modes themselves, Only the latter transform the crystal from a coherent to an incoherent state. Thus by in situ loading of the samples with the critical hydrogen concentration we expect to find one sharp Bragg reflection, shifted to lower scattering angles, due to an expanded lattice and indicating a homogeneous hydrogen distribution in the crystal above T, Below T, we expect a broadened reflection, due to the onset of macroscopic density modes expanding the lattice inhomogeneously. The transition from a coherent state to an incoherent a-a’ phase separation should be observable in a splitting of the broadened peak into (eventuallj) two w reflections, in agreement with the homogeneous concentrations c, and c,. of the coexisting phases a and a’. The Nb samples (single crystals, foils, wires) were purchased from MRC with a stated purity of 99.98%. Single crystals were spark cut, orientated, ground and polished. Polycrystals didn’t attain a special treatment. All samples were degassed at.22OO”C for several hours in a vacuum of lo-” torr, and then mounted in an X-ray oven, which had a longtime stability and homogeneity of better than 0.1 degree. For the different sample geometries, the sample holders in the oven were constructed to allow the samples to relax freely under the action of an inhomogeneous hydrogen density distribution. In the following only lattice parameters but not concentrations will be given. It has been proven that the relationship between lattice parameter and concentration is linear, at least in the concentration range of the a-a’ part of the phase diagram [lo], with
OF HYDROGEN
well
Aa - &(T - 293) ; a > E = 17.24 (H/Nb) is the concentration coefficient [9]. ir = 10.8 x 10-6K-’ the corrected thermal expansion coefficient for NbH [lo], here for the critical hydrogen concentration. Thus the phase diagram can be presented equally in concentration or lattice parameter. However. the latter has the advantage of +I = E -
being directly comparable
with the present data, inde-
which are difficult to estimate. especially for inhomogeneous density distributions. pendent of temperature
corrections.
For the Iatticc parameter measurement usuul scuttcring geometry. gcncrator
(Sci%rt).
MO-tube
(AEG)
(Sicmcns). Associotcd electronics by Wcnzl and. Bicron.
we used the
with a commercial
X-ray
and goniometer
were manufactured
IN NIOBIUM
RESULTS
4.
The &et of spinodul decomposition by coherent hydrogen density tluctuations was investigated in two different sample geometries: a disk-shaped single crystal and a polycristalline wire. These yielded different information concerning the spatial resolution of the density fluctuations. In the case of the single crystal (thickness 0.5 mm, diameter 12.5 mm), the X-ray beam penetrated about 20% from the surface, thus integrating over all fluctuations in this region. The grain size in the wire (length 13 mm, diameter 0.76 mm) was about SOpm. Because of their different orientations, the grains played the role of little probes coming into position of reflection one after the other when the wire turned around and thus displaying the averaged hydrogen concentration in a grain via the lattice parameter measurement. In both experiments described here the samples were cooled below the critical point (7’, = 17l”C, C, = 0.31 H/Nb [lo]) slowly, in the order of hours. The isothermal development of the density inhomogeneities was studied, step by step, over long periods of time. Afterwards the samples were reheated above T, in order to prove the reversibility of these density inhomogeneities. It has been shown [21] that an incoherent phase transition below T, always leads to a time independent density inhomogeneity above T,, stabilized by internal stresses. On the other hand a homogeneous density distribution reflects a coherent lattice. Thus, whether or not the Bragg-peak position and its shape are reversible above T, can be regarded as a test of a coherent vs incoherent phase transition. 4.1 Coherent densityjluctuations in a disk-shaped single crystal In a single crystal with a [llO] orientation, the (440) Bragg reflection was investigated using MO& radiation. After in situ loading, the average hydrogen concentration in the crystal was 0.32 H/Nb. Figure 3 shows the particular steps of the experiment. On the left the lattice parameter measurements are outlined. The solid lines always indicate the incoherent phase boundary [IO]. Some examples of Bragg reflections are given on the right, corresponding to those lattice parameters connected with the Bragg reflection by an arrow. Whenever the isothermal (temporal) progress of the lattice constants was measured, the last measurement in the sequence is recorded in Fig. 3. The different runs in detail are: Run No. /: coolirty jkotn 600 to 160°C (Fig. 3a). At 182°C. or about IO degrees above T,. a reference measurement was made for later comparison. The crystal then was cooled to 169°C in 250 min. At 167. 165, 162 and 160°C the isothermal shift of the Bragg reflection was studied. A slight increase of the lattice parameter connected with on increase in full width at half maximum
(FWHM) Intticc
of the Bragg peak could
obscrvcd.
All
paxmctcrs
arc situated
miscibility
gap and do not lit with the incohcrcnt
bc
in the x-z
ZAHEL
AND
PEISL:
PHASE
TRANSITION
180
i
y
160
t
to
gap. now tend
phase. Thus. it runs did
140b 332
334
3.36 338
593
IX
more
to the
concluded,
any density
dislocations.
As
low density
ting
::fter 4 h; after
fluctuations
before
rcturncd
to
its
a further
origiinal
C(
that the foregoing the
annealing at 182°C Icd to :I cancellation
2i?/Grad-
a/A-
C;III
not stahilizc
gcncr;lling
3.40
(down
144°C). The I:ltticc parameters, still within the mis-
cibility
s
a
IN NIOBIUM
time as well as with the dccrcasing temperature
__
t
OF HYDROGEN
of the split-
IO h, the
position.
by
isothermal reflection
The
FWHM
incrcascd by about 30%. The isothermal
relaxation
the Bragg reflection is shown in Fig. 4.
The data are
of
fitted by an exponential function with a time constant of T = 150min.
b.
332
334
336 338
3.40
72
a/B, -
73
Run No. 4: coohgfiorn 182 fo 125°C (Fig. 3~). The crystal was cooled down slowly and held at 144°C.
74
2t’?/Gmd-
Fig. 5 demonstrates Bragg
peak
intensity
the isothermal and
shape.
variation The
of the
broadening
still more with decreasing temperature and the reflection split at 133°C. At 125°C. first an enormous asymmetry of the peak was observed, which vanished at the same time as its position shifted in the direction of the incoherent a-phase boundary. After 6.5 h finally a symmetric reflection was measured. It then did not change position and shape with further time. Because of the stability of the symmetrical, sharp and time-independent Bragg reflection it is most likely that, at this temperature, a transition from the coherent to the incoherent state has occurred, even if the lattice parameter deviates from the incoherent phase boundary. This may be caused by a misalignment of the sample which appears during its relaxation under the effect of an inhomogeneous hydrogen distribution. Due to a macroscopic density fluctuation, a thin disk-shaped crystal bends like a section of a spherical shell, in such a manner that the @w density a-phase is located on the inner compressed surface and the higher density a’ phase on the outer expanded surface. This will be discussed in more detail in 4.3. Thus it is reasonable to find here only one phase (a) with the a’ phase situated on the opposite surface, where the X-rays were not scattered. The reversibility test at 182°C indeed confirms the observations at 125°C. The Bragg reflection no longer moved back to the original position but remained shifted even after an isothermal anneal for 25 h increased
,/ C.
@-yy-+yq,
3.32 334
3.36 330
3.40
72
73
74
2@Grad -)
a/A-
Fig. 3. Coherent phase transition of hydrogen in a thin disk-shaped single crystal. For explanations see text. phase boundary. Starting from 160°C the sample was reheated to 182°C. At this temperature the lattice parameter returned to its original value to within 0.025% and its former FWHM after 17 h. This last measure ment is given in Fig. 3b as the first point of the second run.
Run No. 2: cooling from 182 to 150°C (Fig. 36). After to 160°C the same state as in the first run was reached in 7 h. At 157°C the reflection became asymmetric and more broadened. After a further 20 h the reflection split and a considerable de crease in intensity was observed. The splitting is really rather more a broad intensity distribution with maxima on both sides than an overlap of two otherwise symmetric reflections. Thus, here, and in the following, the points in the T vs a diagram indicate the
slow cooling
lattice parameters according to intensity maxima
and
full bar in between characterizes the intensity distribution. At 155°C the splitting of the maxima increased slightly and fluctuated in the 29 position the
down to 150°C. The wide distribution ameters within reflects
an
the incoherent
inhomogeneous
of lattice par-
phase boundary density
fluctuation
still of
hydrogen in the lattice. The broad intensity distribution completely
vanished after re-heating
again to 182°C and annealing
the sample
isothermally
for 11 h.
The Bragg reflection moved to the first position and its shape matched the original.
Run No. 3: coolingfrom 182 to 144°C (graphically not outlined). At 151°C again a strong decrease in the intensity and a splitting of the peak was observed at 148°C. The amount
of the splitting
fluctuated
with
Fig. 4. Isothermal relaxation at 182°C of the lattice constant after re-heating from 144°C. Data are fitted with an exponential function (full line). T = I50 min.
594
ZABEL
73.4
AND
PE!SL:
PHASE
736
735
TRANSITION
t3.7
scatt6fing
angle
OF HYDROGEN
IN NlORlUM
736 28
Fig. 5. Isothermal fluctuation of the (440) Bragg reflection taken at 144°C. This is due to an irreversible density distribution above T, after a preceding incoherent phase transition has taken place below T, [213. 4.2 Coherent densityjluctuationsin a wire Four small grains along the wire axis were investigated using both the (600) and (433) reflections with MO& radiation. The wire was in situ loaded with Deuterium and had an averaged concentration of 0.34 D/Nb. The different steps of the experiment are recorded in Fig. 6, The solid line indicates the into-
a.
(201 / 200
c.
\ 3.34
336
1
336
x40
3.36
340
1
334
336 o/A-
Fig. 6. Coherent phase transition of Deuterium in a Nb wire. Larfiee pxitmster R~e~~silrernent during cooling (0) am‘ heat&g ( x ) of ditTorent gr:tins in the wire. For further exphmations see test.
herent e-a’ phase boundary which is almost the same for hy&ogen and deuterium [IO]. In the first cooling run from 600% to 165°C (Fig. 6a) the spatial hydrogen distribution is homogeneous,, i.e. all grains yield the same lattice constant within the experimental error, down to the incoherent phase boundary. With further temperature decrease the density distribution becomes in~ea~ngly inhom~eneou~ For each temperature we find more*than two lattice parameters spread over the range between the incoherent phase boundaries The hydrogen density fluctuation completely disappears when reheating the sample above the phase boundary. In the second cooling run from 200 to 161°C (Fig, 6b) the density distribution becomes even more inhomogeneous in the range from 165 to 161°C. Moreover the local density in a grain fluctuates with time. In this case again the density dis~bution reverts to its original state by re-heating the sample above T, In the third cooling run from 184 to 145°C (Fig. SC), first spatial and temporal fluctuations of the density were observed to 150°C. Keeping the temperature constant at ISOT, the lattice constants jumped discontinuously to the outer phase boundaries after about an hour. Hen~forth the lattice constants changed only with temperature according to the phase diagram but not with time when keeping the temperature constant. The temporal sequence of the last cooling run is outlined in Fig. 7. Heating from 14.5to 18S°C shows clearly that the density distribution now stays inhomogeneous independent of time. As in the experiment on the disk-shaped crystal discussed before. the irreversibility above T, is an indication of a foregoing incoherent phase transition well below ‘&. On the other hnnd the direrent. temporally nuctu~ting lattice parameters along the wire axis with values lying in hctween the a-or’ phase boundary. as well as the reversibility to only one lattice parameter
ZABEL
ANIJ PEISL:
PHASE
TR~~NSlTlON
OF
HYDROGEN
IN NIOBIUM
595
going spinodal decomposition by macroscopic density modes without regard to nucfeation centres. 1n the first oxampk of a 0.3 mm thick disk-shaped (dia. 12.5 mm) Nh sin& crystal. the X-rays were scattzrcd ahornately OII both sides of the disk (Fig. 8a). DOW to the incoherent phase boundary they cxhihit the same lattice parameter, which then, below fSo”C moved apart in the direction of the outer phase boundary. On csamination the crystal appeared bent like a section of a spherical she11 (Fig, 8bf. The lattice parameter tending to the lower density EXphase was taken at the inner contracted surface (circles in Fig. ga) while the lattice parameter tending to the higher density 2 phase was at the outer expanded surface (squares in Fig. 8a). It should be
noted that the Bragg reflections taken at both surfaces exhibited an asymmetric shape, indicating a density gradient towards the other phase, but no existence of a second phase on either side of the crystal could be detected. Thus from the X-ray experiment, as well as
Fig. 7. Isothermalfluctuationof Iattice parametersalong the wireaxis.The latticeconstantsof different grains in the wire were measured by the f&owing reSections: 1. (6CQl _U
refiection(x f 2. (600)reflection(0) 3. (433)reflection (0) (433)reflection(A).
4.
above T, show that a ~herent hydrogen density mode has taken place over a t~~rature range of 20 degrees between 170 and 1W’C
OLI
14oc I
334
336
333
Q/)L-
Coherent density fluctuations Mow T, are .connccted with macroscopic variations of the hydrogen concentration in the sample. This will be demonstrated on several sample geometries. Under the action of inhomogeneous density fluctuations, the crystals relax, i.e. they bend in order to reduce internal elastic coherency stresses. In general this is not compktcly possible and the residuat stresses may exceed the critical shear stress for the onset of the coherent-incoherent transition at a certain tempcrature below T, Attendant plastic deformation and generation of cracks which devefop by further cooling then present a frozen picture of the original coherent decomposition. From those pictures together with X-ray examinations we obtain an idea of the spatial arran~ment of the macroscopic hydrogen modes in the coherent state prior ta the onset of incoherence and how it depends on sample geometry. The plastic (bi deformation discussed here has to be clearly distinguished from the electron mi~o~pic investiga- Fig. 8. Ma~~opic hydrogen density mode in a 0.3 mm tion of piastic deformation during hydride precipita- thick disk-shaped Nb single crystal. (a) Lattice parameters taken at both surfaces; scattering geometry is sketched by tion by T. Schober [26]. The latter is in principle an the inset. Full tini: represents the incoherent phase boundincoherent process and precipitation starts at nucIeation centres; the former is the result of a fore-
ary. (b) Photograph of the bent crystal at room temperature.
596
ZABEL
AND
PEISL:
PHASE
TRANSITION
OF HYDROGEN
1N NIOBIUM
piece, was split in the middle of the disk parallel to the surface, and both parts were bent in the same manner as the half thick crystal in the first example
3.34
3.36
33%
o/A69
(b) Fig. 9. Macroscopic hydrogen density mode in a 0.6mm thick disk-shaped Nb single crystal (a) Lattice parameters taken at both surfaces iike in Fig. 7. (b) Splitting and bending of the crystal at room temperature. from the crystal bending, we conclude that a density profile exists, which varies macroscopically from one surface to the other, corresponding to a halfwave-
length fluctuation directed normal to the disk surface. In the second example the disk-shaped crystal was chosen with a double thickness (0.6mm). The X-rays again were scattered from both surfaces and yielded the same lattice ~n~~~ even far below the incolierent phase boundary (Fig 9ak After incoherent deeomposition, the original crystal, consisting of only one
3.34
(Fig. 9b). An X-ray examination of the expanded inner surfaces indeed showed that the 3’ phase is located there as expected. Thus in this case the density profile represents a full wavelength fluctuation normal to the disk surface. Similar experiments on thicker crystals (2 mm up to 1Omm)showed that the low density a phase always is located at the top and bottom of the disks ,or cylinders (Fig lOa).The result is a splitting of small slices at the surfaces, which have similar thickness and bending radius as in the first two examples. A photograph of such a crystal is given in Fig lob. Under the action of density ~h~~en~ti~ the inner parts of thick crystals are not allowed to relax in the same manner as the outer parts by the constrain of the nei~bou~ng layers. Therefore the spatial arrangement of the phases is much more complex and can be demonstrated only by neutron scattering To obtain a rough idea however, a 0.1 mm thick Nb foil was squeezed in the sample holer such that a macroscopic bending was suppressed but other kinds of rel~ation, i.e. ripples in the foil plane, were still possible and have been later observed. The X-ray scattering results are given in Fig 11 and yield a homogeneous hydrogen distribution down to 20 degrees below T, Then a sudden incoherent phase separation occurs whereby the intensities of the Bragg peaks are nearly the same indicating an equal distribution of both phases in the plane of the foil. The next investigations deal with Nb wires. The polycrystalline nature should not be important because the wavelength of the macroscopic modes is so large compared with the grain size, In the first example of a wire with 1.2mm diameter the lattice parameters tend below T, to the lower density side of the phase diagram (Fig 12af. No indication of the a’ phase could be found within a cylindric wall of about 50 m, corresponding to the penetration depth of MoK, radiation. A Debye Scherrer pattern from the same wire confirmed this 6nding[22].
335
a/A-
Fig. IO(a).
338
ZABEL
ANI) PEISL:
PHASE TRANSITION
OF HYDROGEN
IN NlOBIU~
597
Fig. 10. Macroscopic hydrogen density mode in a 2 mm thick disk-shaped Nb single aystal (a) Lattice parameters taken at both surfaas like in Fig. 7. (b) Surface structure of the crystal at room temperature. X-ray examination of the top and bottom of the wire after breaking it in pieces, yielded both coexisting phases with equal scattering intensities. Scanning eleo tron micrographs (Fig 12b) of the top clearly show a macroscopic cylindrically symmetric precipitation structure, whereby the CC’ phase is located in the core, surrounded by the a phase. The macroscopic mode, corresponding to this precipitation structure, represents a full wavelength flu~uation directed normal to the wire axis.
In wires with 0.76 mm diameter we find a cyiindritally symmetric, as well as a half cylinder preclpitation structure (Fig 13). In the latter case, each part of the half cylinder is occupied by one of the coexisting phases 5~and cr’. Wires with a diameter of 0.25 mm exhibit only half cylinder shaped precipitation structure. With regard to the density fluctuation wavelength, the last example of a thin wire corresponds to the first example of a thin disk shaped crystal. In both crystal geometries a half wave density fluctuation becomes unstable below T, whereas its direction depends on
the shorter linear dimension of the sample. In the same manner the 0.6mm thick disk shapccl crystal and the 1.2476 mm wire eorrcsponds. Both exhibit a full wavelength density fluctuation. The wavekngth of the density Ruetuations seems to be nearly constant at about 1 mm. On the other hand, the number of waves depends .on the size of the shorter dimension of the crystal.
I
xi2
336
d-
I, 3400
I
.
0.5
I
1
5. DISCUSSION
1.5
l!!L-..+ Ihlt
Fig. II. Lattice parameters and integrated intensities of Bragg reflections versus temperature taken on a 100~ thick NbHO,:, foil, squeezed in the sample holder. The intensities are normalized at 250°C. The full line represents the incoherent phase boundary.
A quantative comparison of the experiments with the theory of phase transitions in coherent metal hydrogen systems, given by Wagner and Homer
[2,3], cannot be accomplished, as the maeroseopic modes have been calculated explicitly only for a sphere shaped
crystal.
However, the experiments
ZABEL
598
(a)
ASD
PEISL:
PHASE
TRANSITION
1
220 00
160 -
$
t
140
-
100
-
-
ea
0 >o( 000
6ow 3.34
336
a/h
336
4
(W
mode in a Nb wire with 1.2mm diameter. (a) Lattice parameters taken at several grains around the wire. (b) Scanning electron micrograph of the top of a broken part. The a’ phase is located in the core surrounded by the a phase. Fig. 12. Macroscopic hydrogen density
OF HYDROGEN
IN
NIOBIUM
clearly demonstrate the main features of the predictions: 1. Below the critical point the hydrogen density fluctuation becomes rather smoothly inhomogeneous. No phase separation with a steep interface between the two, phases takes place. This can be concluded from the asymmetric and broadened Bragg reflection, taken on the disk-shaped single crystal, described in 4.1. A sharp interface would not contribute to an interference phenomena like a Bragg peak. Even though the peak is broadened, small parts of the crystal are still scattering coherently, i.e. the lattice parameter changes only slowly. 2. The amplitude of the inhomogeneous density distribution increases with lower temperatures, as can be seen in Fig. 6. 3. The macroscopic modes depend on the sample shape. To each geometry only one special fluctuation is connected unequivocally. The direction of the flue tuation wave is determined by the shorter linear dimension of the crystal, and the wavenumber by its extension. 4. The time constant for the re-homogenization above T, is on the order of hours and thus typical for long distance diffusion of the hydrogen in the lattice, in accordance with the estimate made for a sphere [7]. The temperature range below T, where coherent spinodal decomposition by macroscopic modes occurs, is surprisingly large. In an estimate of the shear stresses, a temperature range of about 5 degrees for a single crystal with very low dislocation densities was expected [23]. On the contrary in single crystal slices, temperature ranges of more than 40 degrees were observed, and even in polycrystals a temperature range of about 20 degrees was usual. The occurrence of coherent macroscopic modes in polycrystals is astonishing and shows that they can be
(4
Fig. 13. Scanning electron micrographs of wires with 0.76mm diameter. Two different precipitation structuresareobservcd:(a)cylindricalsymmetricprecipitation. (b) halfcylindrical symmetric precipitation.
ZABEL
AND
PEISL:
PHASE
TRANSITION
OF
HYDROGEN
IN NIOBIUM
599
Tahlc I. Coherent state
Inhomogeneous macroscopic density distribution
Incohcrcnt state Two definite densities. given by the
incoherent phase boundary. 1~ the sin& phases homogeneous density distribution
grows continuously with CT,-‘F]
Phase separation grows continuously with IT,--T]
Macroscopic modes depend on sample shape
sample shape
Crystal is elastically deformed
Crystal is plastically deformed
Reversible change of state
Irreversible change of state
Homogeneous density distribution
Inhomogeneous density distribution
Amplitude
T< r,
Reheating to T> r,
of density fluctuation
superimposed on the already existing dislocation structure. This is discussed in more detail elsewhere c201. The transition from the coherent to the incoherent state is microscopically a very complex procedure. It can be characterized through experimental evidence in the following way: at some temperature below T, a sudden jump in the concentration distribution occurs and only two different concentrations remain, in accordance tith the incoherent phase boundary. After a temperature change, the new equilibrium is attained quickly and no more temporal fluctuations occur. Reheated above the critical temperature a reversible homogeneous hydrogen distribution can no longer be attained and the inhomogeneity becomes temperature dependent and decreases with increasing temperature [21]. The properties of coherent and incoherent NbHo.3 crystals are finally summarized in Table 1, as they have been distinguished ckarly by the experiments. Acknowledgements-We thank the Bundesministerium Rir Forschung und Technologie for financial support of this work. One of us (H.Z) also wishes to thank the U.S. Department of Energy for support during his stay at the University of Houston and Professor S. C. Moss for many helpful’discussions and a critical reading of the manuscript. We would like to thank Professor G. Alefeld for his helpful interest in this work and Professor H. Wagner for many enlightening discussions conccmitig the theoretical background of the experiments. We finally acknowledge with much appreciation the Kristallabor of the Physik-Departmcnt der Technischen Univcrsitit Miinchen for valuable help in sample preparation.
REFERENCES I. G. Alefeld, in Critical Phenomena in Alloys. Magnets
Phase separation is independent of the
and Superconductors (edited by R. E. Mills et al.). McGraw-Hill, New York (1971). 2. H. Wagner and H. Homer Adu. Phys. 23. 587 (1974). 1 H. Horner and H. Wagner, J. Phys. C. 7. 3305 (1974). &: R. Bausch, H. Homer and H. Wagner. J. Phys. C. 8, 2559 (1975). 5. H. K. Janssen, Z. Phys. BU. 245 (1976). 6. H. A. Goldberg. J. Phys. C. IO, 2059 (1977). 7. T. W. Burckhardt, and W. WGger, Z. Phys. B21, 89 (1975). 8. W. Kappus and H. Horner, Z. Phys. B27.215 (1977). 9. H. Pfeiffer and H. Peisl. Phvs. Left. 6OA. 363 119771 10. H. Zabel and H. Peisl, j. Piys. F. 9, 1461 (19j9j ’ Il. H. Zabel and H. Peisl, Phys. Rev. I&L 42, 51 I (1979). 12. J. W. Cahn, Acta Metall. 9, 795 (1961). 13. J. W. Cahn, Acta Metall. 10, 179 (1962). 14. J. E. Woodilla Jr. and B. L. Avervach, Acta Metall. 16, 255 (1968). 15. S. C. Moss and B. L. Averbach, in Small-Angle X-Roy Scattering (edited by H. Brumberger). Gordon and Breach, New York, London, Paris (1967). 16. H. Conrad, G. Bauer, G. Alefeld, T. Springer and W. Schmatz Z. Phys. 266,239 (1974). 17. W. MUnzing and N. Stump, J. appl. Crystullogr. 11,588 (1978). 18. H. Wagner, Elastic Interaction and Phase Transition in Coherent Metal Hydrogen Alloys, in Topics in Applied Physics, Vol 28 (edited by G. Alefeld and J. V6lklj Springer, Berlin, Heidelberg, New York (1978). 19. G. Schaumann. J. V6lkl and G. Alefeld, Phys. Rev. Letr. 21, 891 (1968). 20. H. Zabel and H. Peisl(l979). to be published. 21. H. Zabcl G. Alefeld and H. Peisl, Scripta Met. 9, 1345 (1975). 22. H. Zabel Thesis, Universitat Miinchen, Miinchen (1978). 23. T. W. Burkhardf Z. Phys. 269, 237 (1974). 24. T. Schober and H. Wenzl, in Topics in Applied Physics (edited by G. Alcfeld and J. Viilkl). Vol. 28. Springer, Berlin, Heidelberg, New York (1978). 25. R. Siems, Phys. stat sol. 30, 645 (1968). 26. T. Schober, Phys. stat. sol. (a) 29, 395 (1975). T. Schober, Phys. stat. sol. (a) 30, 107 (1975).