- Email: [email protected]
A study of the transition hcp 4He to fcc 4He in the vicinity of the triple point
Volume
63. number
CHEMICAL
I
A STUDY OF THE TRANSITION
PHYSICS
-RS
hcp 4He TO fee 4He IN THE VICINITY
1 Bhy 1979
OF THE TRIPLE POINT
JP. FRANCK * Depar~menr of P@icz. Receired
Wkirersify
of Alberta. Edmonton. Alberta T6G 111, Canada
24 J3nuriry 1979
The tmnsition hcp to fcc?He wzs studied by therm3IatxxIysis from the triple point (14.992 K) to 15.9 K. ?he slope of the transition line. &‘fdT. increases continually from 510 bar/K 3t 15.9 Ii to possibIy infinity 3: the triple point. 7he transition entropy is constant zt 0.0183 f 0.002 J/mote K owr rbis tempcmture range. whereas the moIar %otume chrtnge %aries from 0.36 mm3/mole to possibly zero at the triple point_ For densities within 1% of melting, the trsnsition shows complicated irrerers!dle bebavicur.
In a previous report [ I] it was shown that the transition between the two close-packed phases of solid “He exhibits temperature-hysteresis, and is generalIy of a nature to suggest a marten&k character_ The size of the temperature hysteresis_ i-e_ the difference between the onset temperature of the transition in heating and in cooling, R as shown to increase with temperature (the maximum hysteresis obsened at present is 1 K at a transition pressure of X67 kbar)_ On approaching the triple point between the two soIid phases and the fluid_ on the other hand, the temperature hysteresis becomes quite small (a vaIue as Icw as 8 nrK was observed); preliminary results showed. however. that the hysteresis might increase again before actualIy reaching the tripte point_ It was decided therefore to study the transition as close to the tripIe point as possibIe_ It was found possibte to observe the transition as close as 3 mK below melting; and in two cases the transition W;IS observed actuaIly on the melting Iine, i-e_ with both soIid and fluid present in the sample ceil_ The measurements were performed by the method of thermal analysis, detltiIs of which are described in ref_ [I ] _ The “transition temperature in heating”. Th, is defined 35 the Iowest temperature at which there is an indication for the transition when heating the sampie_ The transition is then found to progress over a fi-
* Pxsent address: Department of Physics. Sharp Laboratory. Unirrrsity
100
of Dehuxe.
Newuk.
Delaware 19711.
USA.
nite temperature interval, which is referred to as the width of the transition_ A simiIar definition hoIds for the “transition temperature in cooling”, T,, i.e. this is the highest temperature at which the transition starts to proceed when cooiing the sample. The widths of the transition in heating and cooling are not necessarily the same. The temperature hysteresis is defined as TI,-Tc_ In aI cases it was found that Th > T,, the transition ranges in heating and cooiing therefore neber overlap- In addition to the transition temperatures, the latent heat rsH of the transition was also measured for each transition in heating, from this the transition entropy was obtained as 4S= M/T,,, where T,, = $ (Th + T,)The experiments show that both the equiIibrium properties of the transition, as measured by the transition temperature and the molar volume change. and the kinetic properties, manifest in the temperature hysteresis, transition width and the general progress of the transition, change drastically when approaching the triple point cIoseIy_ This is rather unexpected since 3 triple point in the phase diagram of 3 substance is not usuaiiy connected with singular behaviour. In fig_ 1 the transition is shown in the (P. T) plane_ The transition line in heating shows gradually increasing sIope and goes through infinite slope about 0.2 K below the meIting line. For lower densities, Th increases again. In 3 crystal where the transition occurred 3 mK below the melting point, Th had increased by 20 mK
Volume 63, number 1 17
I
L
lb-
i
20
I
_I
11-4 cm2rmole
.
.A
z
ld-
1
I -i
hcp
. m
0 E
IS -
*-
ISI
4HC2
‘= x
1 May 1979
CHEMICAL PHYSICS LEl-lTRS
_k u-l
.
16-
: E
.
l
l
15 2
I5 4
e
I50
Fig. 2. Transition entropy. of transition temprr.fture.
15 6
I5 8
T (K) Fig_ l_ Phase d&am of4Ht. I&C&Ccircles. transition temperAura in heating. Th:open circles: transition temperatures in cooling. T,-_
above the minimum Th _ A further increxe of Th of about 20 mK was observed for the two crystals where the transition was observed orl the melting line, i.e. with the crystal in equilibrium with the fluid The tnmsition line in cooling shows also grsdu3Ily increasing slope and generally approaches the heating transition line but for the lowest densities starts deviating again. For the two crystals on the melting curve the temper3ture hysteresis has risen 3g3in to about 180 mK. The equilibrium transition line has to be between the heating and cooling transition line and the line shown in fig. I was drawn 3s ,r smooth line in accordance with this principle_ Ciose to the triple point. the larger hysteresis imposes larger uncertainties 3gain in this line. The two dotted lines show other possibilities beyond the one chosen. As in ref. [ I] it was always impossible to trizer the transition in the interval between T, and Th_ The only exception to this w3.s one crystal in equilibrium with the fluid. For this crystal on two occasions the transition could be triggered near 15.0 when heating rapidly (100 mK/min),
ERR04
l
I5 5 T
150
.
Q 12.
fGJ+y
.
1
. 14-
I
ESTlMAiED
tr’
160
K
hcp 411r to fee 4 Me. rls I function The triple point is at Tt, = 14.991 Ii.
although when he3ting with the normal mte used for this crystal (1 mKfmin) the T,, of 1 S.OM K was alwsys observed_ The transition entropies for these crystals rare shown in fig. 1. The estimated uncertainty in these merrsurements is about + 10%. _Afurther uncertainty arises from the observation that close to the triple point the trnnsition is sometimes not complete. In some crystals it happens that the trrmsition proceeds only partirrlly. then stops rind finally runs to completion when the crysta! is starting to melt. It is no doubt due to this effect th3t most transition entropies for Th 5 15.1 K rue appwently low. It ~3s therefore decided to neglect this f3lloff and f.tvour the largest transition entropies observed_ X small correction wrs 3k.o applied because the measurements were made 3t constant volume_ This correction varied between 0.9 and 2.7’%_ Within the experimental uncertainty we hsve therefore 3 constant transition entropy of O.OlS3 t 0.002 J/mole I( between the triple point and 159 K. The molecular dymunics czllculations of Hokm et sl_ [3] give also 0.0183 J/mole I(. in excellent but no doubt somewhat fortuitous agreement with esperiment. The transition is definitely first order over the whole measurement range. Using this value of (ti), r one now ctm cakuiate the change in molar volume. (4 rr)tr. from the Clausius-Clapeyron equation_ We used for this the equilibrium transition line shown in fig. I _ which ~3s drawn with a vertical tangent at the triple point. (4V)tr will therefore go to zero at the triple point for this choice of transition line. Whether this indeed oc101
1 hby 1979
CHEMICAL PHYSICS LETTERS
Volume 63. number 1
Table I Tbermod) namic properties of the transition hcp 4He-fcc 4He -ptr
Ttr
(bar)
(K)
1129 1150 1200 1350 1300 1400 IWO I600 1700 --_---
14992 14-992 15.006 15.049 15.1OL 15.236 15AOl 15.580 15.77;
[Lm31mole) 12.32 12.15 12.07 11.97 11.88 11.69 II-52 11.36 ll.Zl I____~___-_-
c40r (A-%, (mm3/mole) (J/mole) 0 0.047 0.121 O-177 OZ?)lS 0.275 0.317 0.343 0.357
0274 0.269 0.260 0.253 0.248 0.240 0.734 0.230 0.X8
3 A = (Tee) - (bcpx
the tripIe point temperature and volume. There does not appear to exist any theoretic31 explanation for this result_ Based on a constant transition entropy and the transition fine of fig. 1 one can 3Iso calculate the internal energy change at the transition, using:
Fig. 3_ Transition mohr vohmc ckwge. hcp 411e to fee ‘:ie_ Solid Ike:
to teII due to the uncertainties inby the temperature hysteresis_ In any c3se the molar volume change must drop drastically when
troduced
approachkg
the triple point. The results of this caku-
Isticn are shown in fig_ 3. For T>
is
15.9 K, (Av&
constant at a value of
036 mm’/mole, which corresponds to 29 ppm of the molar voIume 3t the triple poinr. The high temperature good agieement
vaiue of (Av),,
with the cakulaticns
is in
of Holian
et a![?I, who predict values of 0.28 and 0.38 mm5/moIe, depending on the interatomic potentia1 used. The large drop towards the tripIe point is, ho\vever, unexpected- It is found that this drop of (Av),, can be described by the foIIowing power law: avJ*,rvo
=
7-7
x
where To = 14992 102
lc+
jcl--
T-o)/T-o]o-3
K and V. = 13.22
?
a&/mole
(1)
are
This quantity, together with other pertinent functions. is given in table I_ The intemd energy change is essentialiy the difference in zero point energy plus the difference in static Iattice potentis1; it is ag3in in exceIIent agreement with the calculations of HoIian et al. PlThe rather complicated kinetic aspects of the transition C;UI be briefly described as follows- For molar volumes beIow about 12.0 cm3/moIe. corresponding to transition pressures above about 1235 bar, the tr3nsition proceeds both in heating 3nd cooling in 3 smooth manner. The trrtnsition widths in heating and cooling sre practically identic31, and increase from about 25 mK to 50 mK with increasing density_ It is 3lw3ys found that increasing temperature hysteresis is connected with increasing transition width. For molar volumes above about l2.0 cm3/moIe, the transitions take on increasing 1~ erratic behaviour_ It is very often found that the transition proceeds in several steps, which are distinctIy separated from each other_ In some cry&Is the tr3nsition will run to completion only after the cryskd has started to melt_ Ever: for crysta!s where the transition
proceeds in one step it is found that the transition r3te is riot constant but tends to increase towards the end of the transition. This speeding up in some cases
Volume 63, number I
CHEMICAL PHYSICS LETTERS
takes an almost explosive character, leading to selfcooling during the heating transition and self-heating during the cooling transition_ In this density range the thermal history of the crystal is also of considerable importance; long annealing times are generally associated with smaller transition width. As the cooling transition tends to deviate from the heating transition again, the cooling transition width becomes larger than the heating transition width. This discrepancy between the cocling and heating transition widths, not observed for transition pressures above about 1235 bar. becomes extreme when the transition is observed in crystals in equilibrium with the fluid. In this case the heating transition is well-behaved again in the sense that it proceeds smoothly in one step and over a narrow temperature range (= 5 mK). The cooling transition, on the other hand, appears to be spread over an enormous temperature range of at least 180 mK_ which makes it almost impossible to observe_ Another peculiarity of crystals in equilibrium with the fluid is that the transition can be triggered in the interval between Tc and Th by using large heating rates (z 100 mK/min and above)_ This effect is never observed away from the melting curve. The complicated behaviour of the transition for molar volumes larger than 12.0 cm3lmole is probably an indication of interference of pre-melting phenomena with the kinetics_ Such pre-melting phenomena are theoretically obtained in molecular dynamics theories of layered discs by Alder et al. ]3], extending over ri density range of 2% previous to melting_ This density range corresponds well to the one where erratic behaviour is observed_ Pre-melting behaviour was found in specific heat measurements both for bee and hcp 4 He by Alder et al_ [4] _ A more detailed analysis obviously has to await a microscopic understanding of the transition kinetics_ Returning to the equilibrium properties, it should be noted that the increase in the transition line slope, and corresponding drop in (Al’),,, starts already at much higher densities than the erratic lrinetic behaviour. At the molar volume of 12.0 cmSilmoIe the slope
1 hlrr)’ 1979
has already increased to 1130 bar/K from the value of 510 bar/K which holds for transition temperatures above 15.9 K. It is therefore unlikely that the proximity of the melting transition should be the cause for this effect. The volume-dependence of the transition temperature is given by: dTtr/dV=
Il/(M)tr]
d(W,r/dp”,
since (M)tr is essentially constant over this density range. An accurate prediction of dTt,/d P requires therefore the knowledge of the volume-derivative of the small quantity (&!I’)~,_ The experimental results req&ire that d(Aff),,/dV becomes zero or at least extremely small at the triple point. Present calculations, as can e.g. be seen from tabIe X of ref. [2], are not accurate enough to decide this question_ This is hardly surprising in view of the small energy differences involved. The interesting question remains about any possible significance of the state variabIes near the observed trip!e point. It shouId be mentioned that the triple point volume, 12.22 cmS/mole_ is extremely close to the molar volume at which a central hump appears in the Wigner-Seitz static cell potential_ This volume can be estimated from the calculation of deWette and Nijboer ,[5] as 12.6 cm3/mole, based on a Lennard-Jones potential (E = 10.22 K. o = 2.556 A)_ This research was supported in parr by grants from the National Research Council of Canada.
References ill J-P. I-nnck, Ph~.s_ Rek. Letters 40 (197s) 1271. [1-I B L. H&an. W.D. Gxkinn, &C. Luntz and B.J. Alder, J. Chem. Phys. 59 (1973) 5444. 131 B J. Alder. W-G_ Hoowr zmd T-E_ WGn~xrri_rht,Phrs- Rer_ Letters 11 (1963) Zfl_ [4] 13J. Alder. W-R. Gardner, J.K. Hoff-x, N.E. Phillips 2nd D.X_ Young,Ph>s_ Reb. Letters 21 (1965) 732. (3 1 F-W. deWette cmd B.R.A. Nijboer, Ph) s. Letters 17 (1965) 1.58;18 0 965) 19.
103
63. number
CHEMICAL
I
A STUDY OF THE TRANSITION
PHYSICS
-RS
hcp 4He TO fee 4He IN THE VICINITY
1 Bhy 1979
OF THE TRIPLE POINT
JP. FRANCK * Depar~menr of P@icz. Receired
Wkirersify
of Alberta. Edmonton. Alberta T6G 111, Canada
24 J3nuriry 1979
The tmnsition hcp to fcc?He wzs studied by therm3IatxxIysis from the triple point (14.992 K) to 15.9 K. ?he slope of the transition line. &‘fdT. increases continually from 510 bar/K 3t 15.9 Ii to possibIy infinity 3: the triple point. 7he transition entropy is constant zt 0.0183 f 0.002 J/mote K owr rbis tempcmture range. whereas the moIar %otume chrtnge %aries from 0.36 mm3/mole to possibly zero at the triple point_ For densities within 1% of melting, the trsnsition shows complicated irrerers!dle bebavicur.
In a previous report [ I] it was shown that the transition between the two close-packed phases of solid “He exhibits temperature-hysteresis, and is generalIy of a nature to suggest a marten&k character_ The size of the temperature hysteresis_ i-e_ the difference between the onset temperature of the transition in heating and in cooling, R as shown to increase with temperature (the maximum hysteresis obsened at present is 1 K at a transition pressure of X67 kbar)_ On approaching the triple point between the two soIid phases and the fluid_ on the other hand, the temperature hysteresis becomes quite small (a vaIue as Icw as 8 nrK was observed); preliminary results showed. however. that the hysteresis might increase again before actualIy reaching the tripte point_ It was decided therefore to study the transition as close to the tripIe point as possibIe_ It was found possibte to observe the transition as close as 3 mK below melting; and in two cases the transition W;IS observed actuaIly on the melting Iine, i-e_ with both soIid and fluid present in the sample ceil_ The measurements were performed by the method of thermal analysis, detltiIs of which are described in ref_ [I ] _ The “transition temperature in heating”. Th, is defined 35 the Iowest temperature at which there is an indication for the transition when heating the sampie_ The transition is then found to progress over a fi-
* Pxsent address: Department of Physics. Sharp Laboratory. Unirrrsity
100
of Dehuxe.
Newuk.
Delaware 19711.
USA.
nite temperature interval, which is referred to as the width of the transition_ A simiIar definition hoIds for the “transition temperature in cooling”, T,, i.e. this is the highest temperature at which the transition starts to proceed when cooiing the sample. The widths of the transition in heating and cooling are not necessarily the same. The temperature hysteresis is defined as TI,-Tc_ In aI cases it was found that Th > T,, the transition ranges in heating and cooiing therefore neber overlap- In addition to the transition temperatures, the latent heat rsH of the transition was also measured for each transition in heating, from this the transition entropy was obtained as 4S= M/T,,, where T,, = $ (Th + T,)The experiments show that both the equiIibrium properties of the transition, as measured by the transition temperature and the molar volume change. and the kinetic properties, manifest in the temperature hysteresis, transition width and the general progress of the transition, change drastically when approaching the triple point cIoseIy_ This is rather unexpected since 3 triple point in the phase diagram of 3 substance is not usuaiiy connected with singular behaviour. In fig_ 1 the transition is shown in the (P. T) plane_ The transition line in heating shows gradually increasing sIope and goes through infinite slope about 0.2 K below the meIting line. For lower densities, Th increases again. In 3 crystal where the transition occurred 3 mK below the melting point, Th had increased by 20 mK
Volume 63, number 1 17
I
L
lb-
i
20
I
_I
11-4 cm2rmole
.
.A
z
ld-
1
I -i
hcp
. m
0 E
IS -
*-
ISI
4HC2
‘= x
1 May 1979
CHEMICAL PHYSICS LEl-lTRS
_k u-l
.
16-
: E
.
l
l
15 2
I5 4
e
I50
Fig. 2. Transition entropy. of transition temprr.fture.
15 6
I5 8
T (K) Fig_ l_ Phase d&am of4Ht. I&C&Ccircles. transition temperAura in heating. Th:open circles: transition temperatures in cooling. T,-_
above the minimum Th _ A further increxe of Th of about 20 mK was observed for the two crystals where the transition was observed orl the melting line, i.e. with the crystal in equilibrium with the fluid The tnmsition line in cooling shows also grsdu3Ily increasing slope and generally approaches the heating transition line but for the lowest densities starts deviating again. For the two crystals on the melting curve the temper3ture hysteresis has risen 3g3in to about 180 mK. The equilibrium transition line has to be between the heating and cooling transition line and the line shown in fig. I was drawn 3s ,r smooth line in accordance with this principle_ Ciose to the triple point. the larger hysteresis imposes larger uncertainties 3gain in this line. The two dotted lines show other possibilities beyond the one chosen. As in ref. [ I] it was always impossible to trizer the transition in the interval between T, and Th_ The only exception to this w3.s one crystal in equilibrium with the fluid. For this crystal on two occasions the transition could be triggered near 15.0 when heating rapidly (100 mK/min),
ERR04
l
I5 5 T
150
.
Q 12.
fGJ+y
.
1
. 14-
I
ESTlMAiED
tr’
160
K
hcp 411r to fee 4 Me. rls I function The triple point is at Tt, = 14.991 Ii.
although when he3ting with the normal mte used for this crystal (1 mKfmin) the T,, of 1 S.OM K was alwsys observed_ The transition entropies for these crystals rare shown in fig. 1. The estimated uncertainty in these merrsurements is about + 10%. _Afurther uncertainty arises from the observation that close to the triple point the trnnsition is sometimes not complete. In some crystals it happens that the trrmsition proceeds only partirrlly. then stops rind finally runs to completion when the crysta! is starting to melt. It is no doubt due to this effect th3t most transition entropies for Th 5 15.1 K rue appwently low. It ~3s therefore decided to neglect this f3lloff and f.tvour the largest transition entropies observed_ X small correction wrs 3k.o applied because the measurements were made 3t constant volume_ This correction varied between 0.9 and 2.7’%_ Within the experimental uncertainty we hsve therefore 3 constant transition entropy of O.OlS3 t 0.002 J/mole I( between the triple point and 159 K. The molecular dymunics czllculations of Hokm et sl_ [3] give also 0.0183 J/mole I(. in excellent but no doubt somewhat fortuitous agreement with esperiment. The transition is definitely first order over the whole measurement range. Using this value of (ti), r one now ctm cakuiate the change in molar volume. (4 rr)tr. from the Clausius-Clapeyron equation_ We used for this the equilibrium transition line shown in fig. I _ which ~3s drawn with a vertical tangent at the triple point. (4V)tr will therefore go to zero at the triple point for this choice of transition line. Whether this indeed oc101
1 hby 1979
CHEMICAL PHYSICS LETTERS
Volume 63. number 1
Table I Tbermod) namic properties of the transition hcp 4He-fcc 4He -ptr
Ttr
(bar)
(K)
1129 1150 1200 1350 1300 1400 IWO I600 1700 --_---
14992 14-992 15.006 15.049 15.1OL 15.236 15AOl 15.580 15.77;
[Lm31mole) 12.32 12.15 12.07 11.97 11.88 11.69 II-52 11.36 ll.Zl I____~___-_-
c40r (A-%, (mm3/mole) (J/mole) 0 0.047 0.121 O-177 OZ?)lS 0.275 0.317 0.343 0.357
0274 0.269 0.260 0.253 0.248 0.240 0.734 0.230 0.X8
3 A = (Tee) - (bcpx
the tripIe point temperature and volume. There does not appear to exist any theoretic31 explanation for this result_ Based on a constant transition entropy and the transition fine of fig. 1 one can 3Iso calculate the internal energy change at the transition, using:
Fig. 3_ Transition mohr vohmc ckwge. hcp 411e to fee ‘:ie_ Solid Ike:
to teII due to the uncertainties inby the temperature hysteresis_ In any c3se the molar volume change must drop drastically when
troduced
approachkg
the triple point. The results of this caku-
Isticn are shown in fig_ 3. For T>
is
15.9 K, (Av&
constant at a value of
036 mm’/mole, which corresponds to 29 ppm of the molar voIume 3t the triple poinr. The high temperature good agieement
vaiue of (Av),,
with the cakulaticns
is in
of Holian
et a![?I, who predict values of 0.28 and 0.38 mm5/moIe, depending on the interatomic potentia1 used. The large drop towards the tripIe point is, ho\vever, unexpected- It is found that this drop of (Av),, can be described by the foIIowing power law: avJ*,rvo
=
7-7
x
where To = 14992 102
lc+
jcl--
T-o)/T-o]o-3
K and V. = 13.22
?
a&/mole
(1)
are
This quantity, together with other pertinent functions. is given in table I_ The intemd energy change is essentialiy the difference in zero point energy plus the difference in static Iattice potentis1; it is ag3in in exceIIent agreement with the calculations of HoIian et al. PlThe rather complicated kinetic aspects of the transition C;UI be briefly described as follows- For molar volumes beIow about 12.0 cm3/moIe. corresponding to transition pressures above about 1235 bar, the tr3nsition proceeds both in heating 3nd cooling in 3 smooth manner. The trrtnsition widths in heating and cooling sre practically identic31, and increase from about 25 mK to 50 mK with increasing density_ It is 3lw3ys found that increasing temperature hysteresis is connected with increasing transition width. For molar volumes above about l2.0 cm3/moIe, the transitions take on increasing 1~ erratic behaviour_ It is very often found that the transition proceeds in several steps, which are distinctIy separated from each other_ In some cry&Is the tr3nsition will run to completion only after the cryskd has started to melt_ Ever: for crysta!s where the transition
proceeds in one step it is found that the transition r3te is riot constant but tends to increase towards the end of the transition. This speeding up in some cases
Volume 63, number I
CHEMICAL PHYSICS LETTERS
takes an almost explosive character, leading to selfcooling during the heating transition and self-heating during the cooling transition_ In this density range the thermal history of the crystal is also of considerable importance; long annealing times are generally associated with smaller transition width. As the cooling transition tends to deviate from the heating transition again, the cooling transition width becomes larger than the heating transition width. This discrepancy between the cocling and heating transition widths, not observed for transition pressures above about 1235 bar. becomes extreme when the transition is observed in crystals in equilibrium with the fluid. In this case the heating transition is well-behaved again in the sense that it proceeds smoothly in one step and over a narrow temperature range (= 5 mK). The cooling transition, on the other hand, appears to be spread over an enormous temperature range of at least 180 mK_ which makes it almost impossible to observe_ Another peculiarity of crystals in equilibrium with the fluid is that the transition can be triggered in the interval between Tc and Th by using large heating rates (z 100 mK/min and above)_ This effect is never observed away from the melting curve. The complicated behaviour of the transition for molar volumes larger than 12.0 cm3lmole is probably an indication of interference of pre-melting phenomena with the kinetics_ Such pre-melting phenomena are theoretically obtained in molecular dynamics theories of layered discs by Alder et al. ]3], extending over ri density range of 2% previous to melting_ This density range corresponds well to the one where erratic behaviour is observed_ Pre-melting behaviour was found in specific heat measurements both for bee and hcp 4 He by Alder et al_ [4] _ A more detailed analysis obviously has to await a microscopic understanding of the transition kinetics_ Returning to the equilibrium properties, it should be noted that the increase in the transition line slope, and corresponding drop in (Al’),,, starts already at much higher densities than the erratic lrinetic behaviour. At the molar volume of 12.0 cmSilmoIe the slope
1 hlrr)’ 1979
has already increased to 1130 bar/K from the value of 510 bar/K which holds for transition temperatures above 15.9 K. It is therefore unlikely that the proximity of the melting transition should be the cause for this effect. The volume-dependence of the transition temperature is given by: dTtr/dV=
Il/(M)tr]
d(W,r/dp”,
since (M)tr is essentially constant over this density range. An accurate prediction of dTt,/d P requires therefore the knowledge of the volume-derivative of the small quantity (&!I’)~,_ The experimental results req&ire that d(Aff),,/dV becomes zero or at least extremely small at the triple point. Present calculations, as can e.g. be seen from tabIe X of ref. [2], are not accurate enough to decide this question_ This is hardly surprising in view of the small energy differences involved. The interesting question remains about any possible significance of the state variabIes near the observed trip!e point. It shouId be mentioned that the triple point volume, 12.22 cmS/mole_ is extremely close to the molar volume at which a central hump appears in the Wigner-Seitz static cell potential_ This volume can be estimated from the calculation of deWette and Nijboer ,[5] as 12.6 cm3/mole, based on a Lennard-Jones potential (E = 10.22 K. o = 2.556 A)_ This research was supported in parr by grants from the National Research Council of Canada.
References ill J-P. I-nnck, Ph~.s_ Rek. Letters 40 (197s) 1271. [1-I B L. H&an. W.D. Gxkinn, &C. Luntz and B.J. Alder, J. Chem. Phys. 59 (1973) 5444. 131 B J. Alder. W-G_ Hoowr zmd T-E_ WGn~xrri_rht,Phrs- Rer_ Letters 11 (1963) Zfl_ [4] 13J. Alder. W-R. Gardner, J.K. Hoff-x, N.E. Phillips 2nd D.X_ Young,Ph>s_ Reb. Letters 21 (1965) 732. (3 1 F-W. deWette cmd B.R.A. Nijboer, Ph) s. Letters 17 (1965) 1.58;18 0 965) 19.
103