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On the convulution approximation in the theory of coherent neutron scattering from liquids
Volume 27A. number 2
PHYSICS LETTER S
scribes a spin density wave state [3] of wavelength 2r/Q. In the limit of small Q “, the energy expression corresponding to this wave-function is easily expanded to give (L = 1), 1Q2.{l _vr/v}(4) 0(Q) = EFer1o(M)+(417)_ where E~1’~’0(M) is the energy of the ferromagnetic state with magnetization M. Expression (4) shows that a ferromagnetic state goes over into a long wave-length spin density wave state at the same critical coupling constant as when the spin wave gets unstable. Secondly, in the three-dimensional case the magnon dispersion is such that
E~!
3 June 1968
Both conditions are plotted in fig. lb. One sees that in this case the Stoner criterion is the more restrictive one, in opposition to the one-dlimensional case. This means that when considering the stability of a ferromagnetic state one has to consider both conditions. Just as in the one-dimensional case one shows that the spin density wave state with N+ particles in the A-branch and N_ particles in the B-branch and of long wavelength will be the new HF-state. The author wishes to thank Prof. I. Prigogine for extending to him the hospitality of Universitd Libre de Bruxelles.
D 3
=
‘~i-[’ --~~{(1+M)~- (1 _M)~}]=
=
~
[1
~v~’/v]
for a given magnetization M. Stability then requires V> V~. The Stoner condition on the other hand gives 3 - ~l ~ — 2 M ~ J . — 317 J(~ ÷M~ * A full variation of Q will of course lead to the result given in ref. 4. However, here we are just interested in the meaning of the magnon instability.
References 1. A.Katsuki and E.P.Wohlfarth, Proc. Roy. Soc. 295 182. and M.M.Antonoff, Physica 30 (1964) 2. (1966) F.Englert 429. 3. A.W.Overhauser, Phys. Rev. Letters 4 (1960) 415, 463. 4. V. Celli, G. Fano and G. Morandi, Nuovo Cimento 43 (1966) 42.
ON THE CONVOLUTION APPROXIMATION COHERENT NEUTRON SCATTERING
IN THE THEORY OF FROM LIQUIDS
C. D. ANDRIESSE Reactor Instituut, Delft, The Netherlands Received 3 May 1968
A simple extension of the convolution approximation is proposed which gives a proper fit to quasi-elastic
neutron scattering from liquid argon.
In order to relate the coherent neutron scattering cross section of liquids to the incoherent one Vineyard proposed the convolution approximation [1]. However, it does not satisfy the mo-
an additional convolution: ~ d”~” ‘
F
—
rs/,-~
—
I
~
-
11
/ dt~F
Fi’~
V
5~,~,,t-t~
ment relations nor does it give a correct hydrodynamic limit. Though Singwi discussed a model [2] in which a solution of this problem is given, we propose a different approach by introducing
I1~
,~,
,
(1)
0
where Fd(Q, t) and F5(Q, t) are the distinct and self intermediate scattering functions, S(Q) is the structure factor and the function f(Q, t) form93
Volume 27A, number 2
PHYSICS
ally accounts for the limitations of Vineyard’s model. Physically f(Q, t) has to do with the time development of coherent motions in liquids. From the relations~for the zero moment and the second moment it can be found that the Fourier transfrom of f(Q, t), J(Q, w), should satisfy
f dwS5(Q, w)~Q,w) 2S I dw w 5(Q, w)7(Q, w)
=
1
ITDQ2
b(Q)
~ ~(~±x2) 217 1 -
I
.i.
I
I
I
2
2.25
3
2 Ox10 .125 2
=
0
1
,
,
exp(x~)erfc(x 2) erfc (x) exp (x 0) ~
I 01
1.25
1 5
I 1.75 _______
(~
Fig. 1. Full width at half height of the quasi-elastic peak from liquid argon at 84°K; the curves are calcu~< i0~2sand = 1.7 x iO-5 cm2/s, the dots lated for threeD different values of T with ~o = are 1 X measured points [51. quasi-elastic peak according to Dasannacharya and Rao [~1 (fig. 1). Reasonable agreement is found for T 1.5 X 10~s, ~ = 1 X i012s and 2/s. D =The 1.7 results x iO-5cm obtained are remarkable in view of the simplicity of this treatment. The curves in fig. 1 may be compared directly to the ones calculated by Singwi. The pronounced oscillatory behaviour is due to the long tail of f(Q, t). The high value for T implies that the decay time of locally ordered motions in liquids is much longer than the time of diffusive steps. The magnitude
(4)
of the latter is comparable to
(5)
0, which gives build the time up. scale in which coherent motions are The author thanks C.Bruin, A. Compagner,
2T and = ~DQ2r where x = ~DQ 0. Both a(Q) and and b(Q) are positive, which means that initially f(Q, t) is increasing and than decreasing, provided T0
94
_,
3 June 1968
(2)
the incoherent scattering law being S5(Q, w). The first moment is zero if S5(Q, w)7(Q, w), and thus ~(Q,~‘) are even. According to eqs. (2) ~(Q,w) should be in part negative, which means that f(Q,t) is not a monotonically decreasing function. A simple even function that satisfies these conditions is 2 + (3) f(Q,t) = a(Q) exp -(t/T) ± b(Q)(t/r 2 exp -(t/r 2 0) 0) Recently Glass and Rice [3] proposed a similar but slightly more complicated function which they multiply with F lations for some of the 5(Q,t) coefficients rather than which fold. they Reintroduced are found from a careful examination of the atomic motion at short times. Since in our treatment we consider a simple diffusion model with the self diffusion coefficient D as the only parameter, we are able to make exact calculations of the coefficients a and b. Eqs. (2) are satisfied if a(Q)
LETTERS
T
A.Hasman and Prof.Dr. J.J.Van Loef for their valuable comments on and interest in this work. A full account will be published elsewhere.
References ~ G.H.Vineyard, Phys. Rev. 110 (1958) 999. 2. K. S. Singwi, Physica 31 (1965) 1257. 3. L.Glass and S.A.Rice, Phys. Rev. 165 (1968) 186. ~ D.G.Henshaw, Phys. Rev. 105 (1957) 976. 5. B.A.Dasannacharya and K.R.Rao, Phys. Rev. 137 (1965) A417.
PHYSICS LETTER S
scribes a spin density wave state [3] of wavelength 2r/Q. In the limit of small Q “, the energy expression corresponding to this wave-function is easily expanded to give (L = 1), 1Q2.{l _vr/v}(4) 0(Q) = EFer1o(M)+(417)_ where E~1’~’0(M) is the energy of the ferromagnetic state with magnetization M. Expression (4) shows that a ferromagnetic state goes over into a long wave-length spin density wave state at the same critical coupling constant as when the spin wave gets unstable. Secondly, in the three-dimensional case the magnon dispersion is such that
E~!
3 June 1968
Both conditions are plotted in fig. lb. One sees that in this case the Stoner criterion is the more restrictive one, in opposition to the one-dlimensional case. This means that when considering the stability of a ferromagnetic state one has to consider both conditions. Just as in the one-dimensional case one shows that the spin density wave state with N+ particles in the A-branch and N_ particles in the B-branch and of long wavelength will be the new HF-state. The author wishes to thank Prof. I. Prigogine for extending to him the hospitality of Universitd Libre de Bruxelles.
D 3
=
‘~i-[’ --~~{(1+M)~- (1 _M)~}]=
=
~
[1
~v~’/v]
for a given magnetization M. Stability then requires V> V~. The Stoner condition on the other hand gives 3 - ~l ~ — 2 M ~ J . — 317 J(~ ÷M~ * A full variation of Q will of course lead to the result given in ref. 4. However, here we are just interested in the meaning of the magnon instability.
References 1. A.Katsuki and E.P.Wohlfarth, Proc. Roy. Soc. 295 182. and M.M.Antonoff, Physica 30 (1964) 2. (1966) F.Englert 429. 3. A.W.Overhauser, Phys. Rev. Letters 4 (1960) 415, 463. 4. V. Celli, G. Fano and G. Morandi, Nuovo Cimento 43 (1966) 42.
ON THE CONVOLUTION APPROXIMATION COHERENT NEUTRON SCATTERING
IN THE THEORY OF FROM LIQUIDS
C. D. ANDRIESSE Reactor Instituut, Delft, The Netherlands Received 3 May 1968
A simple extension of the convolution approximation is proposed which gives a proper fit to quasi-elastic
neutron scattering from liquid argon.
In order to relate the coherent neutron scattering cross section of liquids to the incoherent one Vineyard proposed the convolution approximation [1]. However, it does not satisfy the mo-
an additional convolution: ~ d”~” ‘
F
—
rs/,-~
—
I
~
-
11
/ dt~F
Fi’~
V
5~,~,,t-t~
ment relations nor does it give a correct hydrodynamic limit. Though Singwi discussed a model [2] in which a solution of this problem is given, we propose a different approach by introducing
I1~
,~,
,
(1)
0
where Fd(Q, t) and F5(Q, t) are the distinct and self intermediate scattering functions, S(Q) is the structure factor and the function f(Q, t) form93
Volume 27A, number 2
PHYSICS
ally accounts for the limitations of Vineyard’s model. Physically f(Q, t) has to do with the time development of coherent motions in liquids. From the relations~for the zero moment and the second moment it can be found that the Fourier transfrom of f(Q, t), J(Q, w), should satisfy
f dwS5(Q, w)~Q,w) 2S I dw w 5(Q, w)7(Q, w)
=
1
ITDQ2
b(Q)
~ ~(~±x2) 217 1 -
I
.i.
I
I
I
2
2.25
3
2 Ox10 .125 2
=
0
1
,
,
exp(x~)erfc(x 2) erfc (x) exp (x 0) ~
I 01
1.25
1 5
I 1.75 _______
(~
Fig. 1. Full width at half height of the quasi-elastic peak from liquid argon at 84°K; the curves are calcu~< i0~2sand = 1.7 x iO-5 cm2/s, the dots lated for threeD different values of T with ~o = are 1 X measured points [51. quasi-elastic peak according to Dasannacharya and Rao [~1 (fig. 1). Reasonable agreement is found for T 1.5 X 10~s, ~ = 1 X i012s and 2/s. D =The 1.7 results x iO-5cm obtained are remarkable in view of the simplicity of this treatment. The curves in fig. 1 may be compared directly to the ones calculated by Singwi. The pronounced oscillatory behaviour is due to the long tail of f(Q, t). The high value for T implies that the decay time of locally ordered motions in liquids is much longer than the time of diffusive steps. The magnitude
(4)
of the latter is comparable to
(5)
0, which gives build the time up. scale in which coherent motions are The author thanks C.Bruin, A. Compagner,
2T and = ~DQ2r where x = ~DQ 0. Both a(Q) and and b(Q) are positive, which means that initially f(Q, t) is increasing and than decreasing, provided T0
94
_,
3 June 1968
(2)
the incoherent scattering law being S5(Q, w). The first moment is zero if S5(Q, w)7(Q, w), and thus ~(Q,~‘) are even. According to eqs. (2) ~(Q,w) should be in part negative, which means that f(Q,t) is not a monotonically decreasing function. A simple even function that satisfies these conditions is 2 + (3) f(Q,t) = a(Q) exp -(t/T) ± b(Q)(t/r 2 exp -(t/r 2 0) 0) Recently Glass and Rice [3] proposed a similar but slightly more complicated function which they multiply with F lations for some of the 5(Q,t) coefficients rather than which fold. they Reintroduced are found from a careful examination of the atomic motion at short times. Since in our treatment we consider a simple diffusion model with the self diffusion coefficient D as the only parameter, we are able to make exact calculations of the coefficients a and b. Eqs. (2) are satisfied if a(Q)
LETTERS
T
A.Hasman and Prof.Dr. J.J.Van Loef for their valuable comments on and interest in this work. A full account will be published elsewhere.
References ~ G.H.Vineyard, Phys. Rev. 110 (1958) 999. 2. K. S. Singwi, Physica 31 (1965) 1257. 3. L.Glass and S.A.Rice, Phys. Rev. 165 (1968) 186. ~ D.G.Henshaw, Phys. Rev. 105 (1957) 976. 5. B.A.Dasannacharya and K.R.Rao, Phys. Rev. 137 (1965) A417.