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Temperature dependence of S(Q) for liquid 4He
TE 5
Physica 108B (1981) 1317-1318 North-ttolland Publishing Company
TEMPERATURE
DEPENDENCE OF S(Q) FOR LIQUID
4He
E.C. Svensson and A.F. Murray*
Atomic Energy of Canada Limited Chalk River Nuclear Laboratories Chalk River, Ontario, Canada KOJ iJO High accuracy (typically ±0.4%) results for the total neutron scattering by liquid ~He have been obtained for Q = 2.04 ~-i, the.position of the principal peak of S(Q), at 27 temperatures in the range 1.04 < T < 4.23 K with particular emphasis on the region near T%. Less complete results have also been obtained for Q = 1.87 and 2.30 ~-l. The principal peak is very clearly seen to sharpen and increase in height as one approaches T% from either side. There is also a surprising discontinuity of 1½% in peak height on crossing T~. Two recent predictions for the temperature variation of S(Q) for superfluid ~H$ are found to give good agreement with our new results.
There has recently been considerable interest [1-5] in the anomalous temperature variations of the static structure factor S(Q) and the pair correlation function g(r) of liquid ~He. The amplitudes of the oscillations of S(Q) and g(r), which are a measure of the atomic correlations, exhibit [1,3] the normal increase with decreasing temperature above T%, but then there is a reversal and the amplitudes decrease [1,3] with decreasing temperature below T%. At present there are two competing explanations for this anomalous behavior which is unique to superfluid ~He. Hyland et al. [6] originally attributed it to the changing occupation of the zero-momentum condensate state. More recently, Reatto and co-workers [4] have attributed it to the changing thermal population of rotons and have given results which agree [1,4] with experiment to within the relatively large uncertainties. More accurate and extensive experimental results are clearly needed to provide more stringent tests, but to determine the complete S(Q) and g(r) with high accuracy for a great many temperatures is prohibitively time consuming. We have therefore, in this paper, taken a new approach and determined with high accuracy only the most crucial information on S(Q), namely how the height and width of the principal peak vary with temperature. The principal peak of S(Q) is shown in Fig.l. The position, Qm' of the maximum
i
i
1.4 1.2
0.8 0.6 1.5
[
I
2.0
2.5
30
Old") Fig. i. The principal peak of S(Q) from ref. i.
0378-4363/81/0000-0000/$02.50
© North-Holland Publishing Company
o- I is [i] 2.04 A , essentially independent of temperature. In the present study we have measured the fast-neutron background and the total scattering by the empty specimen containe~ and by the container filled with liquid ~Hepfor scattering angles of 14.7 ° , 16.1 ° and 18.2 ° corresponding to Q values of 1.87, 2.04 and 2.30 ~-I (see Fig. i). The pressure above the 4He bath, of which the specimen was a part, was controlled to ±0.01 mm Hg (and hence the temperature to ±0.0001 K near T%). The experimental conditions and the data-correction procedures were identical to those of ref. i. The net scattering, C*, by the helium is shown in Fig. 2. All results have been normalized to a constant count in the incident-beam monitor. Between 4.23 K and T~ = 2.172 K, where the density increases by 17.3%~ C*(14.7 °) and C*(18.2 °) increase by 18.0% and 19.6%, respectively, while C*(16.1 °) increases by 28.0%. On further cooling C*(16.1 °) reverses and decreases by 4.7% in the superfluid phase (where the density decreases by only 0.7%) exhibiting a typical order-parameter behavior. In contrast, C*(18.2 °) exhibits no significant variation for T < Tl while C*(14.7 °) varies by at most half as much ~s C*(16.1°). We thus see very clearly that the principal peak of S(Q) both sharpens and increases in height as one approaches T% from either side. There is a surprising discontinuity of 1½% in C*(16.10) on crossing T% which we tentatively attribute to a decrease in density Caused by the violent boiling that occurs just above Tl. Unfortunately, the results for C*(14.7 °) and C*(18.2 °) are not accurate enough to either establish or rule out a similar discontinuity at these positions. The C*(16.1 °) values, after correction for the remaining shielding effect, multiple scattering, and density changes, give structure factors S*(2.04) which are shown in Fig. 3. The normalization constant [i] was fixed by requiring that S*(2.04) for 1.04 and 1.38 K agree on average with S(Qm ) for 1.00 and 1.38 K from ref. i. The errors on the solid circles (±0.4% on average)
1317
t318
146
'i
56
+
+ ,
~ :,812O
52
: 2 3o~,-'
~
62
I
+ 1.42
x b
-w(.0
+
o
i
{ .44
o cJ
66 64
I
x
+
6~ o
'
.', ~,+
54
5O
o
{
i i
+
(~
•
o"
I
1.40
= 16. l °
TX
W,
I I
Q : 2.04~. -I {.38
6O
]
1.6
,
I
18
~
l
,
2.0
TEMPERATURE
It 22
(K)
58 56 54 ~
+*"
;++
+ Fig. 3. The temperature dependence of S*(2.04). Crosses indicate the S(O ) values of Svensson et al. (ref. I). Solid and ~ashed curves indicate the predictions of eq. (I) and of Gaglione et al. (ref. 4), respectively. Caution: the open circles are not S(Q) values (see text).
{
=i4T"
i I
= "871-J
t
52
Tx 1
5O
+
i
48
,
,
I
i
,
2
I
,
3
TEMPERATURE
I
4
(K)
F i g . 2. The t e m p e r a t u r e d e p e n d e n c e scattering from liquid 4He.
of
the net
are correct as regards relative values of S(Q), but do not allow for a possible systematic error arising from the normalization. The open circles for T > T% have been included only to show more clearly the discontinuity on crossing T%. They must not be interpreted as S(Q) values slnce, in the presence of boiling, we do not know what density values to use to convert from C* to S*. To obtain reliable S(Q) values in this temperature region, one must carry out measurements at large Q where S(Q) = 1 in order to fix the normalization. Except at 1.77 K where the reported value of 1.36 is now known [7] to be incorrect, the S(Qm) values of ref. 1 (crosses) give a temperature variation in good agreement with the present results. It has recently been proposed
[5] that
S(Q) = naSa(Q) + nnSn(Q)
(i)
where n = p /p and n = l - n are the superf luld . and s normal-flui~ s fractions and S (Q) and S (Q) are the values of S(Q) at T = 0 ~nd T = T~ n A respectively. The variation predicted by (i) is shown as a solid curve in Fig. 3. It agrees with experiment to within the uncertainty allowed by the errors. The dashed curve in Fig. 3 shows the recent prediction of Gaglione at al. [4] which also gives a good description of experiment except possibly very close to T~. Gaglione at al. have ignored density changes,-and they also find that Qm is temperature dependent, so the dashed
curve, which is their prediction for the maximum change in S(Q), is not strictly the same as our S*(2.04). Nevertheless, we feel that the two results probably agree to within the combined uncertainties. This agreement lends support to the contention [4] that the anomalous behavior of S(Q) is attributable to the changing thermal population of rotons. Unfortunately, our new results do not provide a direct test of the alternative hypothesis of Hyland et al. [6] since their explicit predictions pertain only to the variations of g(r). We wish to acknowledge numerous discussions with Dr. V.F. Sears and the invaluable technical assistance of J.C. Evans, H.F. Nieman, M.M. Potter, and especially D.C. Tennant. * Present address, Physics Department, University of Edinburgh, Edinburgh EH9 3JZ, U.K. [i] Svensson, E.C., Sears, V.F., Woods, A.D.B. and Martel, P., Phys. Rev. B21 (1980) 3638. [2] Sears, V.F. and Svensson, E.C., Phys. Rev. Lett. 43 (1979) 2009; Int. J. Quant. Chem. 14 (1980) 715. [3] Robkoff, H.N., Ewen, D.A. and Hallock, R.B., Phys. Rev. Lett. 43 (1979) 2006. [4] De Michelis, C., Masserini, G.L. and Reatto, L., Phys. Lett. 66A (1978) 484; Gaglione,G., Masserini, G.L. and Reatto, L., Phys. Rev. B23 (1981) 1129. [5] Svensson, E.C., Sears, V.F. and Griffin, A., Phys. Rev. B (in press). [6] Hyland, G.J., Rowlands, G. and Cummings, F.W., Phys. Lett. 31A (1970) 465. [7] See footnote i0 in ref. 5.
Physica 108B (1981) 1317-1318 North-ttolland Publishing Company
TEMPERATURE
DEPENDENCE OF S(Q) FOR LIQUID
4He
E.C. Svensson and A.F. Murray*
Atomic Energy of Canada Limited Chalk River Nuclear Laboratories Chalk River, Ontario, Canada KOJ iJO High accuracy (typically ±0.4%) results for the total neutron scattering by liquid ~He have been obtained for Q = 2.04 ~-i, the.position of the principal peak of S(Q), at 27 temperatures in the range 1.04 < T < 4.23 K with particular emphasis on the region near T%. Less complete results have also been obtained for Q = 1.87 and 2.30 ~-l. The principal peak is very clearly seen to sharpen and increase in height as one approaches T% from either side. There is also a surprising discontinuity of 1½% in peak height on crossing T~. Two recent predictions for the temperature variation of S(Q) for superfluid ~H$ are found to give good agreement with our new results.
There has recently been considerable interest [1-5] in the anomalous temperature variations of the static structure factor S(Q) and the pair correlation function g(r) of liquid ~He. The amplitudes of the oscillations of S(Q) and g(r), which are a measure of the atomic correlations, exhibit [1,3] the normal increase with decreasing temperature above T%, but then there is a reversal and the amplitudes decrease [1,3] with decreasing temperature below T%. At present there are two competing explanations for this anomalous behavior which is unique to superfluid ~He. Hyland et al. [6] originally attributed it to the changing occupation of the zero-momentum condensate state. More recently, Reatto and co-workers [4] have attributed it to the changing thermal population of rotons and have given results which agree [1,4] with experiment to within the relatively large uncertainties. More accurate and extensive experimental results are clearly needed to provide more stringent tests, but to determine the complete S(Q) and g(r) with high accuracy for a great many temperatures is prohibitively time consuming. We have therefore, in this paper, taken a new approach and determined with high accuracy only the most crucial information on S(Q), namely how the height and width of the principal peak vary with temperature. The principal peak of S(Q) is shown in Fig.l. The position, Qm' of the maximum
i
i
1.4 1.2
0.8 0.6 1.5
[
I
2.0
2.5
30
Old") Fig. i. The principal peak of S(Q) from ref. i.
0378-4363/81/0000-0000/$02.50
© North-Holland Publishing Company
o- I is [i] 2.04 A , essentially independent of temperature. In the present study we have measured the fast-neutron background and the total scattering by the empty specimen containe~ and by the container filled with liquid ~Hepfor scattering angles of 14.7 ° , 16.1 ° and 18.2 ° corresponding to Q values of 1.87, 2.04 and 2.30 ~-I (see Fig. i). The pressure above the 4He bath, of which the specimen was a part, was controlled to ±0.01 mm Hg (and hence the temperature to ±0.0001 K near T%). The experimental conditions and the data-correction procedures were identical to those of ref. i. The net scattering, C*, by the helium is shown in Fig. 2. All results have been normalized to a constant count in the incident-beam monitor. Between 4.23 K and T~ = 2.172 K, where the density increases by 17.3%~ C*(14.7 °) and C*(18.2 °) increase by 18.0% and 19.6%, respectively, while C*(16.1 °) increases by 28.0%. On further cooling C*(16.1 °) reverses and decreases by 4.7% in the superfluid phase (where the density decreases by only 0.7%) exhibiting a typical order-parameter behavior. In contrast, C*(18.2 °) exhibits no significant variation for T < Tl while C*(14.7 °) varies by at most half as much ~s C*(16.1°). We thus see very clearly that the principal peak of S(Q) both sharpens and increases in height as one approaches T% from either side. There is a surprising discontinuity of 1½% in C*(16.10) on crossing T% which we tentatively attribute to a decrease in density Caused by the violent boiling that occurs just above Tl. Unfortunately, the results for C*(14.7 °) and C*(18.2 °) are not accurate enough to either establish or rule out a similar discontinuity at these positions. The C*(16.1 °) values, after correction for the remaining shielding effect, multiple scattering, and density changes, give structure factors S*(2.04) which are shown in Fig. 3. The normalization constant [i] was fixed by requiring that S*(2.04) for 1.04 and 1.38 K agree on average with S(Qm ) for 1.00 and 1.38 K from ref. i. The errors on the solid circles (±0.4% on average)
1317
t318
146
'i
56
+
+ ,
~ :,812O
52
: 2 3o~,-'
~
62
I
+ 1.42
x b
-w(.0
+
o
i
{ .44
o cJ
66 64
I
x
+
6~ o
'
.', ~,+
54
5O
o
{
i i
+
(~
•
o"
I
1.40
= 16. l °
TX
W,
I I
Q : 2.04~. -I {.38
6O
]
1.6
,
I
18
~
l
,
2.0
TEMPERATURE
It 22
(K)
58 56 54 ~
+*"
;++
+ Fig. 3. The temperature dependence of S*(2.04). Crosses indicate the S(O ) values of Svensson et al. (ref. I). Solid and ~ashed curves indicate the predictions of eq. (I) and of Gaglione et al. (ref. 4), respectively. Caution: the open circles are not S(Q) values (see text).
{
=i4T"
i I
= "871-J
t
52
Tx 1
5O
+
i
48
,
,
I
i
,
2
I
,
3
TEMPERATURE
I
4
(K)
F i g . 2. The t e m p e r a t u r e d e p e n d e n c e scattering from liquid 4He.
of
the net
are correct as regards relative values of S(Q), but do not allow for a possible systematic error arising from the normalization. The open circles for T > T% have been included only to show more clearly the discontinuity on crossing T%. They must not be interpreted as S(Q) values slnce, in the presence of boiling, we do not know what density values to use to convert from C* to S*. To obtain reliable S(Q) values in this temperature region, one must carry out measurements at large Q where S(Q) = 1 in order to fix the normalization. Except at 1.77 K where the reported value of 1.36 is now known [7] to be incorrect, the S(Qm) values of ref. 1 (crosses) give a temperature variation in good agreement with the present results. It has recently been proposed
[5] that
S(Q) = naSa(Q) + nnSn(Q)
(i)
where n = p /p and n = l - n are the superf luld . and s normal-flui~ s fractions and S (Q) and S (Q) are the values of S(Q) at T = 0 ~nd T = T~ n A respectively. The variation predicted by (i) is shown as a solid curve in Fig. 3. It agrees with experiment to within the uncertainty allowed by the errors. The dashed curve in Fig. 3 shows the recent prediction of Gaglione at al. [4] which also gives a good description of experiment except possibly very close to T~. Gaglione at al. have ignored density changes,-and they also find that Qm is temperature dependent, so the dashed
curve, which is their prediction for the maximum change in S(Q), is not strictly the same as our S*(2.04). Nevertheless, we feel that the two results probably agree to within the combined uncertainties. This agreement lends support to the contention [4] that the anomalous behavior of S(Q) is attributable to the changing thermal population of rotons. Unfortunately, our new results do not provide a direct test of the alternative hypothesis of Hyland et al. [6] since their explicit predictions pertain only to the variations of g(r). We wish to acknowledge numerous discussions with Dr. V.F. Sears and the invaluable technical assistance of J.C. Evans, H.F. Nieman, M.M. Potter, and especially D.C. Tennant. * Present address, Physics Department, University of Edinburgh, Edinburgh EH9 3JZ, U.K. [i] Svensson, E.C., Sears, V.F., Woods, A.D.B. and Martel, P., Phys. Rev. B21 (1980) 3638. [2] Sears, V.F. and Svensson, E.C., Phys. Rev. Lett. 43 (1979) 2009; Int. J. Quant. Chem. 14 (1980) 715. [3] Robkoff, H.N., Ewen, D.A. and Hallock, R.B., Phys. Rev. Lett. 43 (1979) 2006. [4] De Michelis, C., Masserini, G.L. and Reatto, L., Phys. Lett. 66A (1978) 484; Gaglione,G., Masserini, G.L. and Reatto, L., Phys. Rev. B23 (1981) 1129. [5] Svensson, E.C., Sears, V.F. and Griffin, A., Phys. Rev. B (in press). [6] Hyland, G.J., Rowlands, G. and Cummings, F.W., Phys. Lett. 31A (1970) 465. [7] See footnote i0 in ref. 5.