studied by neutron inelastic scattering
TRANSCRIPT
PHYSICAL REVIEW A VOLUME 25, NUMBER 6 JUNE 1982
Dynamics of liquid N2 studied by neutron inelastic scattering
Karen Schou Pedersen~Department of Physics, University of Edinburgh, Edinburgh, Scotland
Kim CarneiroPhysics Laboratory I, Universitetsparken 5, DK 2100-Kttbenhavn (tt and Ristt National Laboratory,
DK-4000 Roskilde, Denmark
Flemming Yssing HansenFysisk Kemi-sk Institut, The Technical University of Denmark, DK 2800 L-yngby, Denmark
(Received 10 August 1981)
Neutron inelastic-scattering data from liquid N2 at wave-vector transfers a between
0.18 and 2.1 A. ' and temperatures ranging from T =65 —77 K are presented. The dataare corrected for the contribution from multiple scattering and incoherent scattering. Theresulting dynamic structure factor S(~,c0) is compared with $(~,co) determined by amolecular-dynamics calculation and with that calculated from a linearized Vlasov equa-tion. For a. &0.3 L ' the experimental S(tc, tv) is compared with the values predicted byhydrodynamic theory.
I. INTRODUCTION
Being a homonuclear diatomic liquid, the con-densed phase of N2 is one of the simplest liquidspresenting rotational degrees of freedom. As such,considerable interest has been devoted to thisliquid. Studies of the molecular structure havebeen made by Dore et al. ' and by Pedersen et al.using neutron diffraction. The microscopicdynamics has been investigated by Carneiro et al.using neutron inelastic scattering, as well as byWeis et al. performing a molecular-dynamics(MD) simulation. The dynamic structure factor,S(v,co), agrees fairly well in these two studies atlarger wave vectors (~) 1.7 A '), but at smallerwave vectors there are larger discrepancies between
the neutron-scattering results and the MD results.These discrepancies were, in Ref. 4, inferred tostem from lack of correction for multiple scatter-ing. In order to derive more accurate values forS(~,co) of liquid N2 at smaller wave vectors, we
have performed neutron inelastic-scattering experi-ments at wave-vector transfers ~ between 0.18 and2.1 A ' and temperatures ranging from T =65 to77 K. The data have been corrected for multiplescattering using the method of Copley and a com-parison is made between this method and that ofSears. We also show that in order to derive accu-
rate values for S(~,co) at the smaller wave vectors,one must subtract the incoherent scattering fromthe measured intensities. The measurements havepreferably been performed at wave vectors andtemperatures such that the MD simulations couldbe compared directly with our results.
II. BASIC FORMULAS
The sample dynamics is related to the neutron-scattering properties via the double differentialscattering cross section
[o„hS(tr,to)+o;„eS;„,(tr, to)],d0 dto 4tr
where N is the number of scattering nuclei, ko andk are the incoming and outgoing wave vectors ofthe neutron, which are related to the wave vector~, and the energy fico, transferred to the system bya neutron through the relations ~ = ko —k andAco=A (ko —k )/2m„, where m„ is the neutronmass. 0„~ and 0.;„,are the coherent and in-coherent neutron-scattering cross sections. Since inisotropic systems such as liquids only the length ofthe scattering vector is of importance, we use ~
25 3335 1982 The American Physical Society
3336 SCHOU PEDERSEN, CARNEIRO, AND HANSEN
here. The two dynamic structure factors, S(~,r0)
and S;„,(~,~), both obey the principle of detailedbalance
where kz is the Boltzmann constant. It is con-venient to represent the experimental data in theform of the symmetrized dynamic structure factor
—Are/2k~ TS(~,a)) =e S(pc,cg) .
%hen comparing to classical theories such asmolecular dynamics or hydrodynamics, S(~,co) isused since it corrects for quantum effects to firstorder fi. The static structure factor, S(a), equalsthe zeroth moment of the coherent dynamic struc-ture factor
f S(~,ro)dc0=S(s),
while the zeroth moment of the incoherent scatter-ing function equals unity:
f S;„,(v, co)da) = I .
The expression for the first energy moment, firstobtained by Placzek, takes the form
00 Kf LES (K,co) dco =
~here M is the effective mass of the scatterer. Forthe second energy moment de Gennes derived thefollowing approximate result for a diatomichomonuclear liquid:
ATf co'S{s.,co) dro= [—,+~ (g)],
where d is the molecular bond length, g=sd,
l3 [2gcosg+(g —2) slug],
and M, the atomic mass.
III. EXPERIMENTAI.
The experiments were performed as constant ascans using triple-axis spectrometers at Ris@ Na-
tional Laboratory. A summary of the experimentsis given in Table I. Experiment IV (a=2.3 A.
and 4 A ') was only made to get data for themultiple-scattering correction and will not beanalyzed any further. All the other experiments
were performed at the cold source using the triple-
axis spectrometer TAS I. Pyrolytic graphite (PG)crystals were used as monochromators and
analyzers. Higher-order reflections were removed
TABLE I. Summary of neutron inelastic-scattering experiments on liquid N2.
ExperimentNo.
Wave-vectortransfer ~
{A-')
Energy transferfun (meV}
5-meV incident 14-meV incidentenergy energy
Temperature{K}
0.180.220.260.30
—2.1 —+1.05—1.5—+1.3—1.8—+1.5—2.1 —+1.6
—1.9~1.8—2.4~2.2—2.9~2.6—3.3—+3.0
65,74,76
0.300.450.751.051.401.90
—2.1~1.6—3.0~2.0—3.0—+2.0—3.0~2.0—3.0~2.0—3.0~2.0
—3.3~3.0—5.0~4.4—9.2—+6.8
—10.0~7.0—10.0—+7.0—10.0~7.0
1.701.902.10
—3.0~2.0—3.0—+2.0—3.0~2.0
—10.0~7.0—10.0~7.0—10.0~7.0
IV 2.304.00
—3.0~0.0 —10.0~7.0—10.0~0.0
72
DYNAMICS OF LIQUID Nt STUDIED BY NEUTRON. . . 3337
by a cooled Be crystal filter in case of 5-meV in-
cident energy and a PG filter in case of 14-meV
incident energy. The sample chamber consisted ofan aluminium cylinder with an inner radius of 0.9cm and a wall thickness of 1 mm. To reduce themultiple scattering it was divided into smallerchambers by parallel horizontal plates at intervals
of 0.45 cm. The energy resolution width (FWHM)was 0.17 meV for the 5-meV incident energy and
0.93 meV for the 14-meV incident energy. As an
example the measured intensity data for ~=1.9' and T =66 K, T =74 and 77 K are shown in
Fig. 1 together with the data for the empty Alcylinder.
IV. DATA CORRECTIONS
A. Instrumental corrections
The data of each of the scans in Table I werecorrected for the scattering from the emptycylinder and for instrumental sensitivity. ' The in-
strumental energy resolution width is lower for the5-meV incident energy data, while on the otherhand the higher energy transfers are only reached
by the 14-meV incident energy neutrons. Since theresolution is most important for the co values closeto zero where S(a,co) may vary considerably with
co, the 5-meV incident energy data are used for
I%co
I& 1 —2 meV, the exact value depending on
the wave-vector transfer, and the 14-meV data forhigher energy transfers. Both coherent and in-
coherent as well as multiple scattering fulfill theprinciple of detailed balance as given by Eq. (2).Therefore, when I (~,co) represents the backgroundand sensitivity corrected scattering intensity, thefunction
—4u/2k~ TI(]c,co) =e I(x',co)
should be symmetric in co. Figure 2 comparesI(~,co) for positive and negative energy transfers at«=1.9 A. ' and T =72 K. The data for ro&0 are
generally slightly higher than those for co &0, but
the difference is considered small enough to be
neglected and the average values of I(~,co) for po-sitive and negative energy transfers are used.
In principle our data should also be corrected forthe finite instrumental ~ and co resolution, and this
correction was made in the study of liquid Ar bySkold et al." We have approached this problem
differently by using sufficient high resolution when
necessary. Since all measured energy widths arebroader than 0.9 meV (FWHM), i.e., much largerthan the instrumental resolution of 0.17 meV for5-meV neutrons, our S(~,co) is not significantly af-fected by energy resolution. On the other hand,the wave-vector resolution plays a role very closeto ~=0, and this probably influences our S(~,co) atthe smallest wave vectors (experiment II). Atlarger ~'s the effect of wave-vector resolution is
negligible.
N 10000
0IU
5000lLJ
K
0-4.0
(b)
5 200000O
1-tf)X~ 10000X
0 x=19 A
P
2.0
000
oo
g o
-2,0 0.0
20000 6
0ti)
C30tJ
I-
w 10000IJJ
Z
E,= 5meV
0
0
II
I
I
I X=19AI
I
I
I
I
I
I
E,=14, meV
0 h(JI+0U he&0
-8.0 -4.0ENERGY TRANSFER hu)(meV)
00
I0900RROQ@~~yen
2 3 4 5h(v (meVj
FIG. 1. Measured intensity data from the empty cell
(5) and from the cell filled with liquid N2 for ~=1.9' and T =65 K (0), T =74 K (~), and T =77 K
(0). (a) 5-rneV incident energy. (b) 13.7-meV incident
energy.
FIG. 2. Symmetrized scattering law for liquid Nq at«.= 1.9 A ', to test the intensity corrections for thetriple-axis spectrometer. The dashed line indicates the
energy transfer below which the Eo——5-meV data areused and above which the Eo ——14-meV data are used.
3338 SCHOU PEDERSEN, CARNEIRO, AND HANSEN
B. Multiple-scattering correction
The available computer programs for multiple-scattering correction requires some knowledge ofthe dynamic structure factor, S(x,~). This prob-lem is bypassed if one assumes. S(a,co) from theoryor MD, "' but this is not fully satisfactory if onewants to derive a truly experimental dynamicstructure factor. %e have derived S(~,co) withoutany assumptions about the shape in the wave-vector region 0.30 A ' &x (4.0 A ', namely, byperforming the multiple-scattering calculation as a
self-consistent correction procedure using theMonte Carlo simulation program of Copley. Inthe mentioned interval the data of experiment I,III, and IV [corrected as described above and nor-malized to the tabulated values of S(~)' at the ap-propriate temperature] were used as a first estimateof S(x,e). For ~ and co values between thosecovered by measurements, S(x,co) is found by in-
terpolation in (x,m}, while it is set to zero outsidethe measured ~ interval. For a &0.3 A ' a hydro-dynamic model is used':
y —1 2DT& 1 I a IxS(~,~)= S(&)
2 2 2+— 22 22+ 22y a) +(DTx ) y (~+c,x ) +(I x ) (co —c,a ) +(I x )
where y is equal to the ratio between the specificheat at constant pressure Cz
——4.07' 10 J/kg Kand the specific heat at constant volume
C„=2.23X10 J/kg K. The thermal diffusivity
BT——A, /pC„y, where A, =0.149 J/ms K is the ther-mal conductivity and p=804 kg/m is the density.The sound attenuation coefficient
A, (y —11 4/3il +g
pCy'
p
where q =2.07& 10 kg/ms is the shear viscosityand (=2.47X10 kg/ms the bulk viscosity. Thevelocity of sound e, =914 m/s. The above valuesfor C&, C„, A, , il, g, and c, are from Ref. 15 and atT =72 K. The value of the static structure factor,S(a), may be calculated independently in the so-called compressibility limit:
S(v)=pk&T+ when a —+0,
where p is the number density and X=M/(pe, ) isthe isothermal compressibility.
For v p 4 A ', a free-particle model is used':' 1/2
S(z,co)=
'2I iri scXexp — Stcam
The effective mass of a N2 molecule is set to16.8m„.' ' After each run of the multiple-scat-tering program, the input data for S(x,~) were
I
corrected accordingly, and the runs were continueduntil the corrected data and the latest input datawere identical.
The second and following runs of the multiple-scattering computer program only cause minorchanges in the dynamic structure factors but forv &0.75 A ' a perfect consistency is not obtaineduntil after 4—5 complete runs. The final multi-
ple-scattering corrected data were obtained by per-forming two additional runs using a slightly dif-ferent normalization procedure. The intensity datafor i'd=1. 9 A. ' corrected for the multiple-
scattering contribution as described above were ateach step of this procedure renormalized to the ta-bulated value of S(~). The preliminary correcteddata for the other wave-vector transfers were thennormalized using the normalization constant foundfor &=1.9 A '. The idea behind this is that atir= 1.9 A ', $(a') is well determined and the rela-
tive contribution to the scattering intensity frommultiple and incoherent scattering is small, whichmeans that the normalization constant can bedetermined with high accuracy. This S(~,~) wasused in a comparison between the multiple scatter-ing calculated by the Monte Carlo method and bythe Sears method. The results are shown in TableII. Since the Sears method is much more expen-sive in computer time than the Monte Carlomethod, we have only made a comparison betweenthe two methods for 5-meV neutrons. From TableII is seen that one generally finds a highermultiple-scattering contribution by the Searsmethod than by the Monte Carlo method. Forwave-vector transfers where S(a) is small, minor
DYNAMICS OF LIQUID N, STUDIED BY NEUTRON. . . 3339
TABLE II. Summary of the multiple-scattering corrections of the neutron inelastic-scattering data on liquid Nz atT=72 K.
I(x)(A ') X10'
I(x)-Mult. Scat.&10
Eo ——5 meV
Mult. Scat. in % ofTotal Scat.
MC AM
I (x)-Mult. Scat.y10(MC)
Eo ——14 meVMult. Scat.
in % of TotalScat.(MC)
0.300.450.751.051.401.90
2.242.932.722.783.39
22.1
0.8531.111.201.762.93
21.5
0.6240.8270.7831.402.73
20.8
3.384.084.715.238.06
42.7
1.721.912.383.396.69
41.1
49534935173.7
'MC represents Monte Carlo.bAM represents Analytical method.
differences in the calculated multiple scatteringhave a pronounced effect on the resulting S(x,m)
as is apparent from Pig. 3, which for x =0.3 Aand an incident energy of 5 meV shows the resultsof both correction procedures. The reason for thediscrepancy is probably the approximations intro-duced in the Sears method but has not been furtheranalyzed. Since the Monte Carlo procedure is ex-act (neglecting numerical and statistical shortcom-ings, of course) and also because the computationis less, me have chosen to base our multiple-scattering correction entirely on the Monte Carlomethod.
The scattering results for 0.18 A ' &a &0.30A ' and 1.7 k '&a&2. 1 A ' and varying Tmerc corrected for multiple scattering by theMonte Carlo method using the values of S(a.,u)
& 1000U
500
FIG. 3. Intensity data at a =0.3 A. ' before (0) and
after multiple-scattering correction by Monte Carlosimulation (8), and using the analytical expression ofSears (C3).
determined for T =72 K. The multiple scatteringconstitutes about 60% of the total scattering in theformer x interval and less than 10% in the latteronc.
C. Corrections for incoherent scattering
Liquid N2 is known to have a very small in-coherent scattering cross section, but the tabulatedvalue' of o;„,=0.3 barn is subject to some uncer-tainty. An incoherent cross section would add ex-tra intensity to I(~,u) centered in m=0 but in theliquid phase it can be difficult to distinguish itfrom the coherent contribution. To investigate thcproblem further, neutron inelastic scattering wasperformed on solid N2 at T=32 K for v values of0.30, 0.45, 0.75, and 1.05 A '. In the solid phasecoherent and incoherent scattering can easily bedistinguished. Any incoherent contribution mill
appear as an elastic line, whereas the coherentscattering for the mentioned wave vectors mill bepurdy inelastic. For all four wave-vector transferssharp dastic peaks mere observed, which demon-strates that N2 docs have a measurable incoherentcross section. The intensity from liquid N2corrected for background, sensitivity, and multiplescattering equals
I(Ic, )=roconst[trqohS (K,co) +0'iegieq(K, cgp) ] .
A simple sdf-diffusion model is used for the sym-metrized incoherent dynamic structure factor
3340 SCHOU PEDERSEN, CARNEIRO, AND HANSEN
I;„,(a,~)= const o;„,
)& J R (a)' —a) )S;„,(II,co') dcO',
(14)
where I;„,(x,co) is the incoherent contribution tothe background subtracted and sensitivity correctedscattering intensity. It is assumed that the resolu-tion function, R, has a Gaussian line shape
R(co)= e2n.b,
(15)
where five=0. 072 meV is the measured instrumen-tal resolution width (corresponding to 0.17 meVFWHM). An independent evaluation of o;„, is ob-tained by varying o;„,and studying the resultingI(~,u) —I;„,(~,co). At least at small x's too highvalues of o.;„,will cause a local minimum in
S(x,co) at co=0, while too small values will add anextra top to S(~,~) at co=0. The self-diffusioncoefficient for liquid N2 at 72 K is set to3.0p 10 cm /s. Figure 4 shows the dynamicstructure factor before and after subtraction of theincoherent contribution for cr;„, values of 0.2, 0.3,and 0.4 barn. The tabulated value of o;„,=0.3barn gives the most reasonable line shape ofS(a,co) and is therefore used here.
I
0,10~ =0.301
III
005 $
ooo ~0.0 1.0
I
X L.A'
II
I
X 40454' ~ =
1
I
:I'I
I
0.754 'I =
I
I
X
1.05A '
! I I I I
1.0 1.0 1.0h(L) ImeV)
FIG. 4. Intensity data corrected for multiple scatter-ing by the Monte Carlo method before (- - -) and aftersubtraction of the incoherent scattering contribution em-
ploying values of cr;„,=0.2 barn (. ), o;„,=0.3 barn( ), and cr;„,=0.4 barn (--- ).
1 DvS;„,(&,~)=—
~ m+(D~)where D is the self-diffusion coefficient. For small~ values S;„,(~,~) is sharply centered around co=0and, as opposed to the above-mentioned treatmentof S(v,m), it is therefore necessary to take into ac-count the instrumental resolution:
V. RESULTS
A. S(a,u) and S(a)
The symmetrized dynamic structure factors ofliquid N2, S(~,co), are tabulated in Tables III—V.Table III gives the structure factors for a &0.30A ' and temperatures between 65 and 77 K, TableIV the structure factors for 0.30 A ' &x & 1.90A ' and T=72 K, and Table V the dynamicstructure factors for 1.70 A '(x (2.10 A ' andtemperatures between 65 and 77 K. Since the reso-lution, in particular the wave-vector resolutioncoming from vertical collimation, becomes moreimportant the smaller the ~ values, and also be-cause the multiple and incoherent scattering consti-tute a 1arge part of the total scattering at small ~values, the data of Table III must be attributed lessaccuracy than those of Table IV and V. A graphi-cal representation of the tabulated data for T =72K is shown in Fig. 5.
Compared with the earlier neutron-scatteringresults of Carneiro and McTague, our values ofS(~,co) represent an appreciable improvement inaccuracy. Firstly, since the multiple-scattering line
shape is in general broader in ~ than S(x,m), thewings of S(K,QP) of Rcf. 3 wcIc overestimated, and,secondly, the incoherent scattering added a quasie-lastic contribution. Both these effects are strongestat low ~'s, and this is indeed where the most re-markable differences between the MD results andthose of Ref. 3 were found.
Relatively little has been done regarding thetemperature dependence of the dynamics of liquids.Even our bare intensity data of Fig. 1 show„how-ever, that there is an appreciable variation in
S(x,~) of liquid N2 with temperature. This tem-perature dependence of the dynamic structure fac-tor is, however, confined to wave vectors aroundthe peak in S(~), i.e., x =2 A ', where for instanceS(a.,0) varies with 30%%uo when T is raised from 64to 76 K. Our results are shown in Fig. 6. S(~)varies 23% in this temperature region. As onewould expect from the temperature dependence ofthe hydrodynamic parameters in (9), S(a', m) forsmall wave vectors (Table III) varies only a little.Figure 7 shows the static structure factor, S(~), ofliquid N2 at T =72 K obtained as
m8xJ „S(N.,co)denmin
from the experimental dynamic structure factorsand from the molecular-dynamics calculation com-pared with S(v) of Ref. 13. Also shown in Fig. 7
DYNAMICS OP LIQUID Ng STUDIED BY NEUTRON. . .
c0
I
0Vl
V)
I
Qo
0
V
c5
efabQC
c4V
V~ t+&I
V~ &
Q
CP
Q
aV
zQ'
0
V
V~ tel
6CO
Qk
oo ~II
000
Qooo
Ok0
Ok0
00
000
Ok0
g k
g k
0
0
Ck0
0
00
m
0
QO
00
Ok
0
cVQk
0
0
0
Ok0
Qk0
000
00
oo
OkQk
0
00
CD0
Ok0 0 0 CD 0 Ok 0 0 0 0 0 0 0 0 00 0
000 00 Ok0
QO
00 Ok Ok0
0
0 0
Ch W 0 oo ~ m m oo eq oo ~ m ~88888888888888O 0 Q Q O 0 O 0 0 0 0 0 0 O
Ch
0
0Ok
~ ~ 0 W 0 Ch
8888880000 0 0
Ok0 Ok0 00 Qk0 00 0
W~Qt M~~~oo~888888888800kQQQOOQQQ
H H H H H H 8 8 8 8 H 8 8 8 8 8 8 8 80QQQQQOQQ
QO
Qk0 Ok0 00 0 0 8 80 0 0 0 0
Ok0
Qo
CD0 00
Qo
0Qr mooooooWWW~W
0 OOQQOO
0 Ok0 Ok0
00
0
0
QO0Ok0 88880000
Ok0
Qo
Qk0 Qk 00000 0 Ok Qk
00 0
CD0
0oo
Ok Ok0
QO
Ok0
0
0
85
o m H H H m 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8
0 0
Ck0 0 0 0 QO0 0 t oo CA 0 ~ ~ ~ ~ M ~ I Qo Ch
3342 SCHOU PEDERSEN, CARNEIRO, AND HANSEN 25
TABLE IV. Dynamic structure factors, S(v,co) of liquid Nz at T =72 K determined fromneutron inelastic-scattering data for 0.30 L ' &s &1.90 A
hE(meV
K—1) 0.30 0.45 0.75 1.05 1.40 1.90
0.00.1
0.20.30.40.50.60.70.80.91.01.11.21.31.41.51.61.71.82.02.22.42.62.83.03.23.43.63.84.04.24.44.64.85.05.2545.65.86.06.26.46.66.87.07.27.47.67.88.0
0.0524
0.0431
0.03460.03090.0253
0.0198
0.0178
0.0131
0.0120
0.0114
0.01090.008600.008480.006630.005550.003930.002810.001040.00040
0.03540.03350.03520.03190.02890.02490.02190.01910.01630.01450.0136
0.0131
0.0117
0.0107
0.009920.008480.009160.008810.008350.008240.007340.008030.007170.005880.005030.004780.002480.001500.001900.000960.00062
0.04120.04000.03960.03180.02830.02330.02050.01970.01710.01610.01490.01410.0132
0.0114
0.00951
0.008760.008690.007790.007090.007950.007160.007040.006980.007910.006770.007190.005960.006290.006470.006470.006500.006450.007340.006510.005830.004870.005610.003740.004130.003490.003110.003530.003450.003030.003050.0021S
0.00200
0.05190.04920.04700.04450.04220.03980.02910.02780.02480.02370.02250.02170.01900.01900.01690.01700.01370.01560.01350.01450.01370.01360.01130.01100.01100.009720.01070.009920.01050.01010.008360.008650.009100.009300.006620.008360.006840.006970.006730.005100.006490.005370.006030.007070.006570.004730.004870.004910.004070.00310
0.1580.1510.1430.1280.1200.1070.07850.07290.06490.06190.06620.06410.05840.05920.0478
0.0411
0.03830.03650.03070.03020.02590.02230.02120.02090.02010.01670.01690.01550.01370.01050.01150.01070.008950.01090.01000.009820.008680.007960.006480.006170.007280.007320.006350.005020.004770.003830.004560.00429
2.492.542.141.671.220.9220.7210.6020.5220.4120.3550.3080.2690.2320.2130.2010.1650.1610.1290.1090.07640.06160.05260.04620.03840.03320.03010.02770.02380.02070.01650.01800.01610.01600.01360.01290.01210.01090.01240.01010.01110.009800.008760.008320.007340.006460.006020.005410.006140.00429
DYNAMICS OF LIQUID N2 STUDIED BY NEUTRON. . . 3343
TABLE &V. (Cononued. )
0.30 0.45 0.75 1.05 1.40 1.90
8.28.48.68.89.09.29.49.69.810.0
0.002420.001740.000720.000830.000630.00053
0.003030.003040.002660.002700.001780.002110.001380.000750.000210.00015
0.004330.002730.002560.002280.002490.001800.001590.001500.001080.00098
0.004640.003240.003780.003210.003240.002630.002970.003330.002590.00175
TABLE V. Dynamic structure factor, S(x,m) of liquid N2 determined from neutron inelastic-scattering data for 1.70(s.(2.1 A at different temperatures.
v= 1.70 A «=1.90 A(meV) T=66 K T=74 K T=77 K T=66 K T=74 K T=77 K
«=2. 10 LT66 K T74 K T77 K
0.00.1
0.20.30.40.50.60.70.80.91.01.11.21.31.41.51.61.82.02.22.42.62.83.03.23.43.63.84.04.244
0.8020.8130.7880.6540.5130.4100.3440.2810.2480.2120.1930.1760.1630.1470.131
0.1010.08050.06530.05820.04940.04180.03680.03170.02700.02490.02230.02090.01710.01700.0142
0.9501.021.000.8570.7170.5980.4950.4250.3740.3240.2780.2540.2180.1850.1630.1530.1210.09640.07860.06450.05350.04660.04140.03230.03200.02830.02400.02260.01900.01650.0165
1.011.071.060.9450.7990.6590.5840.5190.4610.4050.3480.3080.2690.2320.1960.1770.1570.1210.09590.08040.06320.05480.04570.03820.03300.02820.02530.02310.01930.01750.0166
2.82
2.571.991.400.9650.6910.5330.4390.3560.3100.2740.2590.243
0.199
0.1400.1060.09030.07150.05960.04810.04350.03890.03220.02760.02490.02170.02040.01810.0165
2.232.232.051.721.331.090.8630.7250.6250.5120.4360.3720.3280.2860.2500.2170.1740.1420.1120.08260.06570.05410.04680.03930.03700.03350.02710.02260.02070.01810.0166
2.032.081.911.661.341.090.9110.7420.6910.603O.S140.4400.3660.3260.2870.2410.1960.1590.1150.09590.07570.0658O.OS600.046S0.03990.03480.02960.02660.02340.02140.0193
1.000.9490.9030.7540.6130.4170.3910.3420.2710.2430.208
0.200
0.1130.09280.08650.07570.06370.05880.04990.04670.04220.03710.03200.03010.02550.02580.0198
0.588
0.5710.5570.5100.4550.4160.3880.3660.306
0.206
0.157
0.1200.1020.09390.08450.06960.06430.04930.05480.04600.04030.03540.03050.03020.02570.0206
0.6110.5670.5670.5330.4890.4440.4190.3930.37S
0.2580.207
0.1920.1680.1500.1380.1240.1090.09630.08350.07220.06080.05760.05430.04670.03900.04670.03370.03020.0267
TABLE V. (Continued. )
i6co «=1.70 A «=1.90 A «=2 10 A
(meV) T=66 K T=74 K T=77 K T=66 K T=74 K T=77 K T=66 K T=74 K T=77 K
4.64.85.05.25.45.65.86.06.26.46.66.87.07.27A7.67.88.08.28.48.68.89.09.29.49.69.8
10.0
0.01580.01350.01310.01230.01170.01120.008930.008850.008930.009110.007110.066680.006710.067339.006910.006190.605690.004700.904620.004870.064120.004540.081070.003390.003200.003680.001929.60231
0.01500.01670.01330.01250.91220.01190.909430.01060.61630.609120.008020.008426.067340.006560.005360.005410.005290.005580.004150.004620.603830.004330.004360.903950.002720.002400.602720.06237
0.01610.01480.01410.01270.01220.01040.01166.009490.009060.009150.007990.007480.008240.007000.006590.006540.005710.005050.005110.004930.004270.603840.664090.603646.602960.003150.602679.60235
0.01620.01560.61410.01190.01170.01040.01100.009640.008750.008170.006920.007970.607480.005100.667110.096310.004680.003990.003740.004930.004310.004170.004670.003056.903640.002970.002760.00192
0.01730.01460.01480.0116-0.01190.01106.909990.01930.068430.009520.068340.008560.005700.007510.006880.006340.006050.004780.90426O.OQ468
0.094950.004190,003650.604060.602450.003330.002969.00239
6.01819.01680.01590.01480.01350.01220.01160.01090.609970.009020.008610.008190.007790.007610.007100.096826.006190.005560.005290.004980.064670.004370.604380.004390.003760.003116.062640.06217
0.01960.01620.01770.01460.01440.01110.01390.009800.067940.009230.906810.607260.008120.007220.006090.096630.00634O.OQ443
0.095380.004310.004726.004496.004530.003619.003286.002366.002526.06169
6.02020.62100.01590.01536.01460.01466.01150.01250.009420.609800.009360.007820.907640.008360.006760.007690.006010.005350.005220.005260.604930.00456Q QOAAA
0.004196.004270.002980.003029.00250
6.02510.02349.02060.01779.91670.01560.01440.01310.01220.01120.01000.008810.008410.008000.008010.098020.007420.006810.006236.005650.005330.005000.604600.004190.903976.003740.002740.06174
- 25—LIQUID Np
g= K
2.0
is the center-of-mass structure factor S, ~ (x) cal-
culated from the x-ray data for SIx):
S(«)=f~(«)+fq(«)[S, («)—1],
f)(«)=1+Kd
FIG. 5. Perspective view of S(a,m).
where d is the molecular bond length. Belo~
S~ ~ (K) is of importance in our calculation ofS(x,m) using the modified Vlasov equation.
B. The moments of S(x,ap)
It is of course of interest to compute the mo-
ments of the experimental S(x,u) to see how they
25 DYNAMICS OF LIQUID N2 STUDIED BY NEUTRON. . . 3345
1.0T =65.9 K
0.5
0.0
3.Q
x=1.9A '
T =65.3 K2.0
E
31.0
I I I(
I/
r
x=1.9A 'I J I
( I [I
(I
x=1.9 A-'
T=77.0 K
0.0
1Q
I/
I ! I [I ! I
x-214 ' x=2.1 tc,'
T =74.2 K
(' ( '
I'
I
x=2.1 A'
T =770 K
0.5
0.00.0 2.0 4.0 0.0 2.0 4.0 0.0 2.0 4.0
W& (meV)
FIG. 6. S(x,co) of liquid N2 determined from neutron inelastic-scattering data (experiment III). The dotted linesrepresent S(x,co) determined from a linearized Vlasov equation.
3.0—
2.0—
1.0—
Liquid N2
T=72 K
IIIIIIIIIII
agree with (4), (6), and (7). The zeroth moment (4)relates the dynamic structure factor to the staticstructure factor S(~), and in Fig. 7 we demonstratethat our data are in excellent agreement with S(~)determined by x-ray diffraction. It is worth men-
tioning that for small wave vectors S(~) assumesthe compressibility value of 0.08.
The first frequency moment of S(~,co), or thePlaczek sum rule, is demonstrated in Fig. 8, where
maxf frcoS(v, cu) dfuomin
0.0 1.0x (g-I)
2.0
FIG. 7. The static structure factor S(x) of liquid N2.The line represents the data of Ref. 13 at T =72 K.The circles are the values of this work (T =72 K). Thenormalization point is indicated by a full circle. The tri-angles are values found by a molecular-dynamics simu-
lation at T =71 K. The dashed line represents thecenter-of-mass structure factor S, (a) calculated fromthe data of Ref. 13.
is shown versus ~ . As expected from Eq. (6) we
get a straight line, but it should be noticed that theslope corresponds to the mass M =23.9m„, as ob-tained from a best fit to the data, i.e., close to themolecular mass of N2. This result is not in con-tradiction to the earlier results, which indicatedthat S(~,co) was shifted in co corresponding to therecoil of free nitrogen particles. ' The apparentdiscrepancy may occur from either the fact that wehave taken the first moment of S(~,co), whereas
3346 SCHOU PEDERSEN, CARNEIRO, AND HANSEN
5.0—1.0o 3.0-20
X1.0
I I I
0.0 1.0 2.0 3.0 4.0x' (A ')
FIG. 8. The 1st-moment sum rule of S(x,m) forT=66 K (o), T=72 K (), T=74 K (V), and T=77K (6).
the earlier work considered the average frequency,or from the fact that our results are taken at lowerx values. In any case our result is in accordance
6
with the expectation, that for a & 2 A ', S(a',u) re-
flects primarily the center of mass dynamics ofliquid N2.
The second frequency moment (7) of the syrn-
metrized dynamic structure factor, S(~,r0), multi-
plied by S(ir)/[ —,+m(g)] is shown versus a in
Fig. 9. Here the best-fitted slope corresponds to anominal mass of 14.4m„, i.e., close to the atomicmass. It is to our knowledge the first time the "deGennes sum rule" (7) has been demonstrated exper-imentally for a diatomic liquid.
VI. DISCUSSION AND CONCLUSIONS
The experimentally determined dynamic struc-ture factor, S(x,co), covers the wave vectors fromthe collective (or small-x) region to the single parti-cle (or large-~) region. As mentioned above, in theregion 0.3 A '&a &1.9 A. ', S(ir, co) was derived
by computer simulation and we have chosen wavevectors close to those of this study in order to fa-cilitate comparison. Apart from such a compar-ison, wc discuss below our results in thc light oftwo relevant theories, covering opposite v regions.
The linear hydrodynamic theory of fluids ac-counts in general very well for experiInental results
at small wave vectors, i.e., when wavelengths arelarge compal cd to intcrmolccular distances. Abovewe used the hydrodynamic results (9) for S(~,co) inconnection with the multiple-scattering corrections.These corrections do not depend on the details ofthe structure factor, so that deviations between themeasured S(~,m) and (9) do not influence our finalresult. It is therefore relevant to compare S(x,~)to (9) to investigate how well the hydrodynamiccontinuum theory applies at wavelengths as shortas 20 A, i.e., quite close to the intermolecular dis-tances in the liquid. In Fig. 10 we show this com-parison for wave vectors from 0.18 A ' to 0.3A '. Figure 10 demonstrates that hydrodynamictheory accounts relatively well for the experimentalstructure factor, although it shows less broadeningof both the central Rayleigh line and of the Bril-loum lj.nes. In order to Invest&gate this more quan-titively, we have fitted the measured S(x,~) to thehydrodynamic line shape
A0 I0 A) I )S(K,OJ)=
2 2 +m +I'0 ~ (m —Q)) +I )
Comparing (19) to (9) and (10), hydrodynamictheory predicts the following relations:
30+RA ) ——kg T/(Meg),
1 0 DTa——
0)——cga,
I ) ——1 x
Figure 11 shows the results of the fit comparedto the results of the theory when using the num-
006 ) t f j j
„ix=0.18 A'
I I I
1Q Q — — J(s) 5 {'K,b) I d4)
(P+m(g)l
E 5.0-lX
G.GA
E
3.~ 0.02
i~
0.000,0 1.0 2.0 1.0 2.0 1.0 2.0
t)u) ImeVI
1.0 2.0 3,0
I I I
0.0 1.0 2.0 3.0 4,.0x'(A 'j
FIG. 9. The 2nd-moment sum rule of S(a,~) forT =66 K (o), T =72 K (), T =74 K (V), and T =77K (6).
FIG. 10. Dynamic structure factors S(x,co) of liquidN2 determined from neutron inelastic-scattering data(experiment II) at T =74 K. The dashed lines representthe S(I(,„co) obtained from hydrodynamic theory.
DYNAMICS OF LIQUID N2 STUDIED BY NEUTRON. . .
joE 1—
0 ~g
3—
L lQUlo Nq
o T =65K
0.10— —1.0
Ao+ 2A 2A, IAO
I I
q -0.5
FIG. 11. Line-shape analysis of S(x,~) according tohydrodynamic theory. The dashed lines are extrapola-tions of the theory vnth parameters taken from trans-port measurements (at a.=O).
bcfs foI' cg» I, and BT glvcn above. It shows thathydrodynamic theory accounts quantitatively forthe observed Brillouin line parameters cq and I, aswell as for the integrated intensity $(a). The ther-
mal diffusivity DT is not well reproduced. Thiscould bc fclated to thc problem of finite K fcsolu-
tion, which may have a broadening effect on theexperimental S(x,ra). The odd behavior of I o vs xpoints towards the same deficiencies in the data.Finally» thc spcclflc heat 1'atlo Cp/CI)
=28~ /Ho+1 is derived to be quite small, but the
significance of this observation is not quite clear inthe light of the problems with the central part ofour S(x',co). The composite evidence is that hydro-dynRInlc theory cxplalns ouf measured S(K,Q)) welleven at the very short wavelengths considered. Inparticular, Figs. 10 and 11 indicate that at a =0.30
A the agreement is quantitative.Secondly, it is interesting to compare our results
with those of the molecular-dynamics calculationby Nels Rnd Levesque ln the wave-vectof region0.3 A '&x &1.9 A '. This is done in Fig. 12.However, the comparison is complicated by thefact that S(x) in the calculation is appreciably lessthan the mell-established experimental value exceptwhen ~= 1.9 A '. %c have therefore chosen tocompare our S(x',u) both to the absolute valuesgiven in Ref. 4, as well as to the relative values byadjusting S(s,0). Apart from when s = 1.9 Athere are important differences between the calcu-lated S(K,Q)) Rnd thc IIlcasurcd onc ln Flg. 12. At0a =0.30 A ', where the neutron results agree qual-itatively with the hydrodynamic theory, moleculardynamics shows a mell-resolved side peak, indicat-ing a sound velocity of almost twice the hydro-dynamic value. However, the widths of the linesagree rather well with the neutron results as well aswith hydrodynamics. This is true both for thecentral line, where the width is determined by ther-mal diffusivity BT, and for the Brillouin linewhere the width stems from the finite lifetimes of
0.05
0,04E
0.033
0.02
u 0.01
3A'0.04
gp
E o.o3—3g 002
o Q01
t I
x=045A '
0.04,
—003E
30.02
~ 0.01
0.0 1.0 2.0 3.0 &0 5Q
htu (meVj
0.0 1.0 2.Q 3.Q go 50hut ImeVj
0.0 10,0
0.05
0.04E= 0033
0.02
,~001
0.0
a 0.15E
3. 0.10
o 005IV)
0.0 5.Q
htu ImeVj
2.0E
1.53
FIQ. 12. Dynamic stfuctufc factofs S(K'»QP) of liquid N2 determined from ncutfon inelastic-scattering data at. T =72K (experiment I) and by molecular dynamics simulation at T =71 K. The MD data are shown normalized to the sameS(g,O) as the experimentally determined S(x,O) (---) and applying the S(~) value determined by the MD simulation(- ~ - ~). The line (. ~ .) represents the S(a,e) determined from a linearized Vlasov equation (T =72 K).
3348 SCHOU PEDERSEN, CARNEIRO, AND HANSEN 25
the sound waves, as given by I . As ~ increasesthere is in both the experimental and the computedS(K,co) an increasing smearing and merging of thetwo lines, but the trend is the same as when v=0. 30A '. The results are in reasonable agreement as
regards the central peak, but the computed line
shape extends far out in co. Only at 1.9 A ' dothe two results coalesce at all frequencies. The
comparison therefore indicates that the cornpressi-
bility or inverse sound velocity of the computerliquid is underestimated but that relaxationprocesses are well accounted for in the calculation.It should of course be noted that an underestimat-
ed compressibility in the computer liquid decreasesS(~) in the compressibility region (10). This is
indeed observable in Fig. 7 where the static struc-ture factors are shown. Here the computed S(~)approaches a value which is about one-third of the
correct value. Literally we find from both S{~,co)
and from S(K) that molecular dynamics implies asound velocity of 1.7c~, a strong indication thatour consistent analysis of the deficiencies in thecomputer liquid is correct.
Figure 11 also compares our results with amodification of the theory presented by Singwiet al. for the collective motions. It is based on alinearized Vlasov equation and claimed to bevalid at short wavelengths and high frequencies.The dynamic structure factor is predicted from thestatic structure factor, S(~). We have applied thistheory to liquid N2. Because N2 consists of dia-tomic molecules, the calculations turned out to bemore complicated than in the atomic case. To getthrough to an expression for S(a,co) we had to re-strict the calculations to center-of-mass motions.We then obtained the following expression:
1 CO
exp 2~7TQjp COp
S, (a)1 —S, (v) 2~
exp —2 J exp(t )dt
S m (v) p cop
Nexp 2
COp
where
(2kB T 2)1/2
M(22)
For ~ values below 1.7 A ' the agreement withboth experimental data and MD data is very bad.At «=1.05 A. ' and «=1.40 A ' the calculateddynamic structure factors have a well-defined side
peak at Ace=3 meV. It is in agreement with whatis found by Nelkin et al. applying a linearizedVlasov equation to a classical liquid, but as is ap-parent from Fig. 10 it is not confirmed by the ex-periments. For a) 1.7 A ' and Ace) 0.5 meVthere is a reasonable agreement between thedynamic structure factors predicted by the theoryand those determined from the experimental data.Furthermore it is noteworthy that the theory
(21)I
predicts the general trends of the effect of varyingthe temperature, i.e., whether the heights andwidths of the central peaks are increased or de-creased with temperature. This is demonstrated inFig. 6, where we compare the temperature depen-dence of S(~,co) at ~=1.9 A ' to that of Eq. (21)for these temperatures. Thus the noticeable tem-perature dependence in liquid nitrogen at wave vec-tors around the peak in S(a ) can be understood inthe basis of the modified Vlasov equation.
ACKNOWLEDGMENT
We wish to thank the Danish Natural ScienceResearch Council for sponsoring the computer cal-culations.
'Present address: Instituttet for Kemiteknik, TheTechnical University of Denmark, DK-2800 Lyngby.
'J. C. Dore, G. Walford, and D. I. Page, Mol. Phys.29, 565 (1975).
2K. S. Pedersen, F. Y. Hansen, and K. Carneiro, J.Chem. Phys. 70, 1051 (1979).
K. Carneiro and J. P. McTague, Phys. Rev. A 11, 1744(1975).
DYNAMICS OF LIQUID Nt STUDIED BY NEUTRON. . .
4J. J. %eis and D. Levesque, Phys. Rev. A 13, 450(1976).
5J. R. D. Copley, Comput. Phys. Commun. 7„289(1974).
6V. F. Sears, Adv. Phys. 24, 1 (1975).7P. Schofield, Phys. Rev. Lett. 4, 239 (1960).SG. Placzek, Phys. Rev. 86, 377 (19S2).9P. G. de Gennes, Physica 25, 825 (1959).~OG. Dolling and V. F. Sears, Nucl. Instrum. Methods
106, 419 (1973)."K.Skold, J. M. Rowe, G. Ostrowski, and P. D. Ran-
dolph, Phys. Rev. A 6, 1107 (1972).'2J. R. D. Copley and J. M. Rowe, Phys. Rev. A 9,
1656 (1974).'3A. %'. Furumuto and C. %. Shaw, Phys. Fluids 7,
1026 (1974); tabulated by P. %'. Schmidt and C. %.Tompson in Ref. 1S.
'~J. P. Hansen and I. R. McDonald, Theory of SimpleI.iquids (Academic, London, 1976).
'5H. L. Frisch and Z. %. Salsburg, Simp/e Dense F/uids
(Academic, New York, 1968),' See, for example, %. Marshall and S. %. I ovesey,
Theory of Thermal Neutron Scattering (OxfordUniversity Press, Oxford, 1971).
'7R. G. Sachs and E. Teller, Phys. Rev. 60, 18 (1941).18J. G. Powles Adv. Phys. 22 1 (1973)'98. T. M. Willis, Chemical Application of Thermal
Neutron Scattering (Clarendon, Oxford, 1972), Appen-dix I.
2oJ. Barojas, D. Levesqne, and B. Qnentrec, Phys. Rev.A 7, 1092 (1973).
2'R. N. Sinclair, J. H. Clarke, and J. C. Dore, J. Phys.C 8, L41 (1975).
22K. S. Singwi, K. Skold, and M. P. Tosi, Phys. Rev. A1, 454 (1970).
23A. A. Vlasov, Many Particle Theory and its App/ica-tion to P/asmas (Gordon and Breach, New York,1961).
24M. Nelkin and S. Ranganathan, Phys. Rev. 164, 222(1967).