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Neutron scattering in solid 4He at large momentum transfer
Solid State Communications,
Vol. 11, pp. 1307—1310, 1972. Pergamon Press.
Printed in Great Britain
NEUTRON SCATTERING IN SOLID 4He AT LARGE MOMENTUM TRANSFER V.F. Sears Atomic Energy of Canada Limited, Chalk River Nuclear Laboratories, Chalk River, Ontario, Canada (Received 4 August 1972 by M.F. Collins)
Recent neutron inelastic scattering experiments on solid 4He at large momentum transfer are used to determine the moments of the velocity spectrum of the crystal, f(~). The results obtained suggest that, as a consequence of anharmonic effects, f(w) differs significantly from the phonon-frequency spectrum and, in particular, has a long high-frequency tail.
SLOW neutron scattering experiments on solid 4He have recently been reported’ for wave vector transfers, K, in the range 2.4—5.4 A~ In this range the Debye—Waller factor is small compared with unity (<0.1) and the scattering is dominated by multiphonon processes or, equivalently, by single-particle excitations. The single-particle region has not previously been studied by neutron inelastic scattering in crystals although it has 2~liquid neon,7 been in liquid helium, liquidinvestigated rubidium, and hydrogen gas. In the present ~—14 article a theory which has been developed in connection with liquids and gases is applied in the interpretation of the solid helium experiments.
1
f C4
0.269.~ ±
—
—
.
For sufficiently large values of K the scattering function S(K,w), in which the frequency c~is the energy transfer in units of h, consists essentially of a single, broad peak which can be characterized by the frequency of the maximum we,,, the mean frequency w~= j(Cü÷+ &~),and the full width at half maximum & = ~t. where c~are the frequencies at which S(K,c&) falls to one half its maximum value. At T = 0 the large -K expansions of the above quantities’4 take the form —
=
— ~.
2f~
÷-!. CI..tr
.L~_—
8f~2
+ t...
12f~~ 4f~
±Q(c~ç~),
(1)
1307
0 0255
.~
—
0 0346
—0 0254
+
0(z)
f4
(2) 1 =
4 ~~Jlh2“(‘U 1
0.192
—
4
,f —0.125 ~—0.162 ~1
÷! 0.0363
12
1
—
0.173
4
÷0.439f~
fi
f4
1’
f5
1
1
1
1
—
0.286
!~.
+
f6 j 1
0(aç~)j. j
(3)
Here a,. = hK2/2m is the recoil frequency, m being the mass of an atom, andu = (1iJ /4mY’~ is, apart from a factor 6~’2 the root mean square velocity of an atom. The quantity j~= is the nth moment of the velocity spectrum: 4 f(c~)= .—~—tanh (hw/2kT) Recosc~itdt, ,
3TTh&.
f
(4) in which v (0 is the velocity of an atom in the Heisenberg picture and the remaining quantities have their usual meanings. In each of the expansions (1), (2) and (3) the leading term gives the result for the impulse
NEUTRON SCATTERING IN SOLID 4He
1308
Vol. 11, No. 10
approximation, in which the scattering atom is assumed to recoil freely, and the remaining terms SOLID HELIUM • 20.8 crn3/mole
give the corrections due to final-state interactions as an asymptotic expansion in inverse powers of eu,.. The above expressions apply to incoherent scattering and are based on the gaussian approximation15 but are otherwise exact.
‘I,
In the present context we note first that 4He is a coherent scatterer so that with increasing K the quantities ~ ~ and A~Jwill oscillate” about their corresponding incoherent values. However these oscillations are generally of relatively small amplitude, and in fact have not been resolved in the present experiments, so that the incoherent approximation is appropriate. Secondly, for a harmonic crystal the gaussian approximation is exact and f(w) is the normalized phononfrequency spectrum. Although solid helium is highly anharmonic in the Born—Von Kdrm~nsense 16
a self-consistent harmonic theory is in reasonable agreement with the observed’7 phonon dispersion curves for b.c.c. 4He. Consequently one would expect that non-gaussian effects are small in solid helium and hence that the expansions (1), (2) and (3) are applicable. This assumption is consistent with recent theoretical calculations
13
=
15
=
20.5,
15
-
-
~
0
= =
6.82, 65.3.
/
\ OBSERVED
/‘
-
-
/ 5-
-
, /
2
/ 0
I
I
I
2
I 3
I
4
5
6
7
(A)
K
1. The full width at half maximum of the scattering solid 4He. The line is the function impulse of approximation and dashed the solid curves labelled 1 and 2 show the result of inFIG.
cluding respectively the first and second cotrections for final-state interactions. The observed curve is from reference 1.~
_____________________________ I I I I I/I SOLID HELIUM 20
-
15
-
/
hcp polycrystol obcc 1100) • bcc ether
/
Q
/
f
2 14
-
/
/•
V • 20.8 cm3/mote
With ~ in units of iOta sec1 the moments of the phonon-frequency spectrum of b.c.c. 4He at 20.9 cm3/mole are taken to be 2.45,
-
/
which indicate that non-gaussian effects are negligible in the Debye—Waller factor.
f~=
ao
(5)
216.
0
10
-
/
/
/
-
I. I. / °
S -
2
/
°
-
*
/
/
0
I’
5
-
/
These values have been obtained scaling 21 ofbyb.c.c. ~Hethe at calculated frequency spectrumprocedure is based 21.46 cm3 /mole. The scaling on the observation that the ratios of the calculated phonon frequencies’6’2’ at equivalent points in the first Brillouin zone are constant to within 3 per cent, Substituting the values (5) into (3) and (1) we obtain the curves shown in Figs. 1 and 2 respectively. In each figure the dashed line shows the impulse approximation and the solid curves labelled 1 and 2 show the result of including
/
Oo—
I
/ /
I
I
2
3
4
5
6
7
(45 FIG. 2. The frequency of the maximum of the scattering function of solid 4He. The dashed curve is the impulse approximation and the solid curves labelled 1 and 2 show the result of including respectively the first and second corrections for final-state interactions. The data are from reference 1. K
NEUTRON SCATTERING IN SOLID 4He
Vol. 11, No. 10
respectively the first and second corrections for final-state interactions. The calculated width in Fig. 1 is seen to be in reasonable agreement with the width obtained from the observed’ half-height contours. This is, however, to some extent fortuitous since the observed width has not been corrected for resolution effects. The observed values of ~ in Fig. 2 are significantly less than the calculated values. These discrepancies are, to some extent, due to anharmonic effects which result in a velocity spectrum which is substantially different from the phonon-frequency spectrum. To the extent that non-gaussian effects are negligible the data can be used to determine empirical values for the moments of the velocity spectrum. If K > 2.5 A” we can take = a,. —~/2f,, —
respectively. The former value gives 12 = 16 ±7 which is considerably larger~than value 6.82 in (5) for the second moment of the phonon-frequency spectrum. It would therefore appear that the velocity spectrum of solid helium differs significantly from the phonon-frequency spectrum and, in particular, has a long high-frequency tail. Such tails are characteristic of velocity spectra in liquids.23 Finally we note from (6) that, quite apart from the question of the precise value of f 2 If,, w0 is less than Wm at large K so that S(k,w) is not symmetrical about ~ but is skewed toward high frequencies such that the high-frequency wing is enhanced and the low-frequency wing depressed. has the been noted pre4 This and property agrees with observed viously” scattering’ in solid ~.
0.269 f 2 If
=
1309
4 ‘t/(ln2)
,
(6)
KU,
Acknowledgements Discussions with Dr. A.D.B. Woods and Prof. R.A. Cowley are gratefully acknowledged. —
where u = (‘hi1 /4m . The data for &ti give 1, = 2.1 which is close to the calculated value f, = 2.45 in (5). The data for w0 and OJm give values of j~‘ft equal to 7.6 ±3.3 and 10.8 “~
REFERENCES 1.
KITCHENS T.A., SHIRANE G., MINKIEWICZ V.J. and OSGOOD E.B.,Phys. Rev. Lett. 29, 552 (1972).
2.
COWLEY R.A. and WOODS A.D.B., Phys. Rev. Lett. 21, 787 (1968).
3.
WOODS A.D.B. and COWLEY R.A., Neutron Inelastic Scattering, Vol. I, p. 609. International Atomic Energy Agency, Vienna (1968). HARLING O.K., Phys. Rev. Lett. 24, 1046 (1970). HARLING O.K., Phys. Rev. A3, 1073 (1971).
4. 5. 6. 7. 8.
COWLEY R.A. and WOODS A.D.B., Can. J. Phys. 49, 177 (1971). BUYERS W.J.L., SEARS V.F., LONNGI P.A. and LONNGI D.A., mt. Atomic Energy Agency Symp. on Neutron inelastic Scattering, Grenoble, France, March 6—10 (1972). EGELSTAFF P.A., PAGE D.I. and DUFFILL C., to be published.
10.
LEFEVRE Y., CHEN S.H. and YIP S., mt. Atomic Energy Agency Symp. on Neutron Inelastic Scattering, Grenoble, France, March 6—10 (1972). SEARS V.F., Phys. Rev. 185, 200 (1969).
11. 12. 13.
SEARS V.F., Phys. Rev. Al, 1699 (1970). SEARS V.F., Ber. Bunsenges 75, 376 (1971). SEARS V.F., Phys. Rev. A5, 452 (1972).
14.
SEARS V.F., to be published.
9.
1310
NEUTRON SCATTERING IN SOLID ‘He
Vol. 11, No. 10
15.
RAHMAN A., SINGWI K.S. and SJOLANDER A., Phys. Rev. 126, 986 (1962).
16.
GLYDE HR. and KHANNA F.C., Can. J. Phys. 50, 1152 (1972).
17.
OSGOOD E.B., MINKIEWICZ V.J., KITCHENS T.A. and SHIRANE G., Phys. Rev. A5, 1537 (1972).
18..
SEARS V.F. and KHANNA F.C., Phys. Rev. Lett. 29, 549 (1972).
19.
McMAHAN A.K, and GUYER R.A., Phys. Rev. Lett. 29, 556 (1972).
20.
HORNER H., to be published.
21. 22.
GLYDE HR. and KHANNA F.C., Can. J. Phys. 49, 2997 (1971). We estimate that this value may, however, be reduced by as much as 30 per cent when allowance is made for the variation of the resolution with analyser angle.
23.
EGELSTAFF P.A., An Introduction to the Liquid State, p. 142. Academic Press, New York (1967).
Des experiences recentes de lagrandes diffusion neutrons de aux valeurs de inélastique transfers dedes la quantité dans 1’ 4He solide mouvement sont utilisées pour determiner les moments du spectre de la vitesse du cristal, f(a.t). Les résultats obtenus suggérent que f(ci~)dévie appreciablement du spectre des fréquences phononiques, a cause des effets anharmoniques, et en particulier que f(~)a une longue queue aux hautes frdquences.
Vol. 11, pp. 1307—1310, 1972. Pergamon Press.
Printed in Great Britain
NEUTRON SCATTERING IN SOLID 4He AT LARGE MOMENTUM TRANSFER V.F. Sears Atomic Energy of Canada Limited, Chalk River Nuclear Laboratories, Chalk River, Ontario, Canada (Received 4 August 1972 by M.F. Collins)
Recent neutron inelastic scattering experiments on solid 4He at large momentum transfer are used to determine the moments of the velocity spectrum of the crystal, f(~). The results obtained suggest that, as a consequence of anharmonic effects, f(w) differs significantly from the phonon-frequency spectrum and, in particular, has a long high-frequency tail.
SLOW neutron scattering experiments on solid 4He have recently been reported’ for wave vector transfers, K, in the range 2.4—5.4 A~ In this range the Debye—Waller factor is small compared with unity (<0.1) and the scattering is dominated by multiphonon processes or, equivalently, by single-particle excitations. The single-particle region has not previously been studied by neutron inelastic scattering in crystals although it has 2~liquid neon,7 been in liquid helium, liquidinvestigated rubidium, and hydrogen gas. In the present ~—14 article a theory which has been developed in connection with liquids and gases is applied in the interpretation of the solid helium experiments.
1
f C4
0.269.~ ±
—
—
.
For sufficiently large values of K the scattering function S(K,w), in which the frequency c~is the energy transfer in units of h, consists essentially of a single, broad peak which can be characterized by the frequency of the maximum we,,, the mean frequency w~= j(Cü÷+ &~),and the full width at half maximum & = ~t. where c~are the frequencies at which S(K,c&) falls to one half its maximum value. At T = 0 the large -K expansions of the above quantities’4 take the form —
=
— ~.
2f~
÷-!. CI..tr
.L~_—
8f~2
+ t...
12f~~ 4f~
±Q(c~ç~),
(1)
1307
0 0255
.~
—
0 0346
—0 0254
+
0(z)
f4
(2) 1 =
4 ~~Jlh2“(‘U 1
0.192
—
4
,f —0.125 ~—0.162 ~1
÷! 0.0363
12
1
—
0.173
4
÷0.439f~
fi
f4
1’
f5
1
1
1
1
—
0.286
!~.
+
f6 j 1
0(aç~)j. j
(3)
Here a,. = hK2/2m is the recoil frequency, m being the mass of an atom, andu = (1iJ /4mY’~ is, apart from a factor 6~’2 the root mean square velocity of an atom. The quantity j~= is the nth moment of the velocity spectrum: 4 f(c~)= .—~—tanh (hw/2kT) Re
3TTh&.
f
(4) in which v (0 is the velocity of an atom in the Heisenberg picture and the remaining quantities have their usual meanings. In each of the expansions (1), (2) and (3) the leading term gives the result for the impulse
NEUTRON SCATTERING IN SOLID 4He
1308
Vol. 11, No. 10
approximation, in which the scattering atom is assumed to recoil freely, and the remaining terms SOLID HELIUM • 20.8 crn3/mole
give the corrections due to final-state interactions as an asymptotic expansion in inverse powers of eu,.. The above expressions apply to incoherent scattering and are based on the gaussian approximation15 but are otherwise exact.
‘I,
In the present context we note first that 4He is a coherent scatterer so that with increasing K the quantities ~ ~ and A~Jwill oscillate” about their corresponding incoherent values. However these oscillations are generally of relatively small amplitude, and in fact have not been resolved in the present experiments, so that the incoherent approximation is appropriate. Secondly, for a harmonic crystal the gaussian approximation is exact and f(w) is the normalized phononfrequency spectrum. Although solid helium is highly anharmonic in the Born—Von Kdrm~nsense 16
a self-consistent harmonic theory is in reasonable agreement with the observed’7 phonon dispersion curves for b.c.c. 4He. Consequently one would expect that non-gaussian effects are small in solid helium and hence that the expansions (1), (2) and (3) are applicable. This assumption is consistent with recent theoretical calculations
13
=
15
=
20.5,
15
-
-
~
0
= =
6.82, 65.3.
/
\ OBSERVED
/‘
-
-
/ 5-
-
, /
2
/ 0
I
I
I
2
I 3
I
4
5
6
7
(A)
K
1. The full width at half maximum of the scattering solid 4He. The line is the function impulse of approximation and dashed the solid curves labelled 1 and 2 show the result of inFIG.
cluding respectively the first and second cotrections for final-state interactions. The observed curve is from reference 1.~
_____________________________ I I I I I/I SOLID HELIUM 20
-
15
-
/
hcp polycrystol obcc 1100) • bcc ether
/
Q
/
f
2 14
-
/
/•
V • 20.8 cm3/mote
With ~ in units of iOta sec1 the moments of the phonon-frequency spectrum of b.c.c. 4He at 20.9 cm3/mole are taken to be 2.45,
-
/
which indicate that non-gaussian effects are negligible in the Debye—Waller factor.
f~=
ao
(5)
216.
0
10
-
/
/
/
-
I. I. / °
S -
2
/
°
-
*
/
/
0
I’
5
-
/
These values have been obtained scaling 21 ofbyb.c.c. ~Hethe at calculated frequency spectrumprocedure is based 21.46 cm3 /mole. The scaling on the observation that the ratios of the calculated phonon frequencies’6’2’ at equivalent points in the first Brillouin zone are constant to within 3 per cent, Substituting the values (5) into (3) and (1) we obtain the curves shown in Figs. 1 and 2 respectively. In each figure the dashed line shows the impulse approximation and the solid curves labelled 1 and 2 show the result of including
/
Oo—
I
/ /
I
I
2
3
4
5
6
7
(45 FIG. 2. The frequency of the maximum of the scattering function of solid 4He. The dashed curve is the impulse approximation and the solid curves labelled 1 and 2 show the result of including respectively the first and second corrections for final-state interactions. The data are from reference 1. K
NEUTRON SCATTERING IN SOLID 4He
Vol. 11, No. 10
respectively the first and second corrections for final-state interactions. The calculated width in Fig. 1 is seen to be in reasonable agreement with the width obtained from the observed’ half-height contours. This is, however, to some extent fortuitous since the observed width has not been corrected for resolution effects. The observed values of ~ in Fig. 2 are significantly less than the calculated values. These discrepancies are, to some extent, due to anharmonic effects which result in a velocity spectrum which is substantially different from the phonon-frequency spectrum. To the extent that non-gaussian effects are negligible the data can be used to determine empirical values for the moments of the velocity spectrum. If K > 2.5 A” we can take = a,. —~/2f,, —
respectively. The former value gives 12 = 16 ±7 which is considerably larger~than value 6.82 in (5) for the second moment of the phonon-frequency spectrum. It would therefore appear that the velocity spectrum of solid helium differs significantly from the phonon-frequency spectrum and, in particular, has a long high-frequency tail. Such tails are characteristic of velocity spectra in liquids.23 Finally we note from (6) that, quite apart from the question of the precise value of f 2 If,, w0 is less than Wm at large K so that S(k,w) is not symmetrical about ~ but is skewed toward high frequencies such that the high-frequency wing is enhanced and the low-frequency wing depressed. has the been noted pre4 This and property agrees with observed viously” scattering’ in solid ~.
0.269 f 2 If
=
1309
4 ‘t/(ln2)
,
(6)
KU,
Acknowledgements Discussions with Dr. A.D.B. Woods and Prof. R.A. Cowley are gratefully acknowledged. —
where u = (‘hi1 /4m . The data for &ti give 1, = 2.1 which is close to the calculated value f, = 2.45 in (5). The data for w0 and OJm give values of j~‘ft equal to 7.6 ±3.3 and 10.8 “~
REFERENCES 1.
KITCHENS T.A., SHIRANE G., MINKIEWICZ V.J. and OSGOOD E.B.,Phys. Rev. Lett. 29, 552 (1972).
2.
COWLEY R.A. and WOODS A.D.B., Phys. Rev. Lett. 21, 787 (1968).
3.
WOODS A.D.B. and COWLEY R.A., Neutron Inelastic Scattering, Vol. I, p. 609. International Atomic Energy Agency, Vienna (1968). HARLING O.K., Phys. Rev. Lett. 24, 1046 (1970). HARLING O.K., Phys. Rev. A3, 1073 (1971).
4. 5. 6. 7. 8.
COWLEY R.A. and WOODS A.D.B., Can. J. Phys. 49, 177 (1971). BUYERS W.J.L., SEARS V.F., LONNGI P.A. and LONNGI D.A., mt. Atomic Energy Agency Symp. on Neutron inelastic Scattering, Grenoble, France, March 6—10 (1972). EGELSTAFF P.A., PAGE D.I. and DUFFILL C., to be published.
10.
LEFEVRE Y., CHEN S.H. and YIP S., mt. Atomic Energy Agency Symp. on Neutron Inelastic Scattering, Grenoble, France, March 6—10 (1972). SEARS V.F., Phys. Rev. 185, 200 (1969).
11. 12. 13.
SEARS V.F., Phys. Rev. Al, 1699 (1970). SEARS V.F., Ber. Bunsenges 75, 376 (1971). SEARS V.F., Phys. Rev. A5, 452 (1972).
14.
SEARS V.F., to be published.
9.
1310
NEUTRON SCATTERING IN SOLID ‘He
Vol. 11, No. 10
15.
RAHMAN A., SINGWI K.S. and SJOLANDER A., Phys. Rev. 126, 986 (1962).
16.
GLYDE HR. and KHANNA F.C., Can. J. Phys. 50, 1152 (1972).
17.
OSGOOD E.B., MINKIEWICZ V.J., KITCHENS T.A. and SHIRANE G., Phys. Rev. A5, 1537 (1972).
18..
SEARS V.F. and KHANNA F.C., Phys. Rev. Lett. 29, 549 (1972).
19.
McMAHAN A.K, and GUYER R.A., Phys. Rev. Lett. 29, 556 (1972).
20.
HORNER H., to be published.
21. 22.
GLYDE HR. and KHANNA F.C., Can. J. Phys. 49, 2997 (1971). We estimate that this value may, however, be reduced by as much as 30 per cent when allowance is made for the variation of the resolution with analyser angle.
23.
EGELSTAFF P.A., An Introduction to the Liquid State, p. 142. Academic Press, New York (1967).
Des experiences recentes de lagrandes diffusion neutrons de aux valeurs de inélastique transfers dedes la quantité dans 1’ 4He solide mouvement sont utilisées pour determiner les moments du spectre de la vitesse du cristal, f(a.t). Les résultats obtenus suggérent que f(ci~)dévie appreciablement du spectre des fréquences phononiques, a cause des effets anharmoniques, et en particulier que f(~)a une longue queue aux hautes frdquences.