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Temperature and wave number dependence of a central peak
Solid State Communications, Vol. 37, pp. 807—811.
0038—1098/81/100807—05$02.00/0
Pergamon Press Ltd. 1981. Printed in Great Britain. TEMPERATURE AND WAVE NUMBER DEPENDENCE OF A CENTRAL PEAK N. Ohata College of Science and Engineering, Aoyama Gakuin University, Chitosedai, Setagaya-ku, Tokyo 157, Japan (Received 22 September 1980 by Y. Toyozawa) The first six moments of the dynamical structure factor is calculated in the spirit of the functional integral for equilibrium quantities. The half-width of the central peak is evaluated with these moments. It is found to be narrower than that obtained by Krumhansl and Schrieffer. 1. SOME YEARS AGO Krumhansl and Schrieffer [1] studied a simple model for structural phase transition which is a linear chain of atoms with one-site doublewell potentials and nearest-neighbor couplings. They calculated the thermodynamic properties of the system exactly by functional integral methods, and showed that the low-temperature thermodynamic behavior can be explained in terms of phonon and domain wall excitations. They also attributed, on the basis of a phenomenological discussion, the central peak in the dynamical structure factor to the motion of domain walls. The central peak and the domain wall motion have been subjects of absorbing interest to many authors. The molecular dynamics computer simulations by Koehler, Bishop, Krumhansl and Schrieffer [2] revealed that the motion rather looks like Brownian motion. In fact the diffusion constant of domain walls was evaluated by Wada and Schrieffer [3]. Aubry [4] exactly calculated the first six moments of the dynamical structure factor by solving the eigenvalue equation for the transfer operator numerically, Using Mon’s method of continued fractions, he outlined the spectrum on the basis of the knowledge of these moments. The spectrum was also obtained by molecular dynamics computations by Aubry [5] and by Schneider and StoIl [6], and the possibility that the domain walls participate in developing the central peak was pointed out. Recently Sahni and Mazenko [7] have obtained, on a rather phenomenological basis, the results which is in qualitative agreement with the molecular dynamics results, There are another group of papers [8, 9] in which the effects of thermal disturbances on the atoms are taken into account by the use of stochastic models. 2. It seems to be useful, at this stage, to calculate ab initio the first few moments of the dynamical structure factor “in the spirit of the functional integral for equilibrium quantities [1]”, to show the temperature-dependence and the wave-number dependence of the moments explicitly, and if possible, to
derive some definite information about the central peak from the moments. This is the purpose of the present paper. If we want to know the detailed shape of a certain peak, the central peak or a phonon side-peak, we have to calculate the moments of the intensity distribution associated with this particular mode. The main weakness of the moment method is that an important contribution to the value of a moment comes from the wings of the curve which are not actually observed. The contributions of other parts of the spectrum have to be carefully excluded from the calculated moments of the peak concerned. In magnetic resonance problems, the way to get rid of them is to separate the perturbing Hamiltonian, responsible for line broadening, into the secular part and the non-secular part and to retain the secular part only [10, 11]. In the present problem, however, since it is not known how to separate the Hamiltonian of the system into the domain wall mode, the phonon mode and the interaction between them, we will be contented with calculating the moments of the whole spectrum and trying to draw from them some definite conclusions about the spectrum in the following special cases: Case I. It is assumed that only the central peak is observed for small wave numbers. Case II. It is assumed that the spectrum is composed of three Lorentzian curves, one for the central peak and two for the phonon side-peaks. The results to be obtained in these cases may be compared with those of Sahni and Mazenko [71and with those of Krumhansl and Schrieffer [1], respectively. .
3. The model Hamiltonian conventionally reads N 1 N IA B q,2 + q~ H = P,2 + —
‘°
N
+
q,
,~o2 807
)2 -
(1)
808
TEMPERATURE AND WAVE NUMBER DEPENDENCE OF A CENTRAL PEAK Vol. 37, No. 10
where q, is the displacement of the lth atom and the periodic boundary conditions q~v+1= q0 and q -1 = ~N
I A V(q)
are imposed. The nth moment of the dynamical structure factor 100
S(k, w)
=
N
dt ~ ~,(t) exp (ikal-— iwt),
27r ~
i=o
(2)
where 0q,(t))
(3)
and a is the lattice constant, is defined by ~ s(k. w) w’~dw,,/f
M~(k) =
2
q +
— ~-
B ~ —
B > 0).
(JO)
q
and DI q I n) is the matrix element of q between the lowest eigenstate and the nth eigenstate. 4. The potential V(q) is a double-well potential with
2 and depth ~A2/B. Let
us denoteat the minima q = eigenvalues ±q = ±(A/B)” and the eigenfunctions of a harmonic oscillator with mass m* in a potential with the
(q
=
=
A
2
S(k, w) dw.
(4)
Since /.,(t) is an even function oft, all the moments for odd n vanish. The moments for even n (= 2p) are clearly given by the formulae
and the latter far larger than the thermal energy at tern-
1 —~Ø,(t)I dt
1d2P
~
()P
the case in which the following conditions are satisfied: 2/B ~- (2A/m*)h/2, (i) ~A (ii) (2A/m*)~2~‘ k~T. That is, the height of the potential hump is far larger than the level distance of the harmonic oscillator.
N
M2~(k)=
curvature equal to that of the potential wells by E~and ~ji~ respectively: E,~ = (n -1- ~)(2A/m*)~ 2 In order to objain definite results, we consider here
exp (ikal) perature T. If we introduce the dimensionless parameters
x ~N ~,(0) exp (ikal).
(5)
2 *\ 12 —rn ~ = q (A2B /
(II)
l=0
and it is lengthy but straightforward to obtain formal
expressions for the moments by functional integral methods with the aid of the equations of motion for q,,
= C/A~ (1 2) the above assumptions (i) and (ii) are equivalent to
with the results (i’)~/4~1
-1
M 2(k)
=
i~~n(k)l(o~qIn)l2l ,
(6)
(11’)
and 2~1
m~n
ii
M 4(k) =
rn
1 -1
a~(k)I (0 I q In) 12 n
(12)
(ii’)(y/2)~
I
respectively. The physical meanings of ~ and ‘y are as
j
follows: The number of harmonic oscillator levels in
2
2
x ~ a~k) A + 4C sin
ka~
~)((o
+ B(A + 4Csin2
ka\ 2jI 1(0 q In) 12
each of the potential wells is ~/4, and kB T is2.smaller The above restrictions may, in effect, relaxed a little more, howthan the level distance by abe factor of (y/2)” ever, since i~ and y appear in the arguments of exponential functions.
—
Iq In)(n 1q310)
+ (0 I q3 In) (n 1q10)) + B2 1(0 I q3 In) 12].
(7)
Here the coefficients a~(k)are defined by e2n~o)
= e2P
in the limit iV
—
—~ eo,
approximation leads to the results
i
n~~o)—2 e~n~0)
cos
0
ka + i
L
~
j3 being kBT. The energy e~,is the nth
4(2A/m*)~/2(1 T ~
I 13)
= ‘I’o) ~,
I
—
(l/~/2)[~ 0(q—q)±~0(q +~)]
(14)
1
+ V(q) ‘1’~(q)= 2Cc~’l’~(q), and the potential with the effective mass rn* = ~ ~
~
(8)
eigenvalue of the effective Schrodinger equation I
Under assumption (i) a tunnelling approximation is valid l~rthe lowest harmonic oscillator state, and this
—~
(9)
Krurnhansl for the lowest andpair Schrieffer of states of may the system, as shown by Under assumption (ii) [11. we approximate equation (8) by
Vol. 37, No. 10 TEMPERATURE AND WAVE NUMBER DEPENDENCE OF A CENTRAL PEAK 2
M
2
‘7
/
1.0
.,,......._~
(/
+
(17)
i5~2])
+ 4y sin
+ ~~‘(5
12
809
If the tunnelling approximation is also valid for the first excited state of the harmonic oscillator, it is
_-.—
,/‘\ ‘78
straightforward to proceed to a higher-order approximation, in which [a 2(k) 1] and [a3(k) 1] are 2M retained as nonzero. In Figs. 1 and 2 (A/m)’M2(k) and (A/m) 4(k) are respectively plotted as functions of ka for ‘y = 1/8 and = 8 and 12. Though these values of and y may seem to violate conditions (i’) and (ii’), the corrections, due to the higher-order approximation, have been found to be less than 5%. In1e’7[ particular, for k 0 we have (18) 1 + 2ye2’7(ka)2] (A/m)-’M2(k) ~i~ (A/m)2M 2], (19) 4(k) ~i~~[1+ ~‘yr~(ka) and for k = ir/a —
—
05
I
-
=
0 25
/
ir
i~
I
10
0
20
3.0
ka
Fig. 1. k-dependence of the second moment. Here definedis by which lower equation for ~ = (11), 12 than is related for r~ to= the 8; -ytemperature, is defined by equation (12). i~,
i~
M
(A/m)IM2(ir/a)~4(y/2)i/Z,
4
(20)
2M 6.0—
‘7—
40
(A/m)~ 4(ir/a)~4{(1 + 2y)2+ ~(5 +4y)ri~}. (21) 5.Casel In the following we will try to guess a plausible
I2
‘7=
8 y
2.0
shape of the central peak for small wave numbers with the knowledge of the second and the fourth moments, under the assumption that only the central peak is observed. For this purpose the usual method [10] of
_
0.125
-
IT
I
0
10
I
20
30
ka
Fig. 2. k-dependence of the fourth moment, =
for n ~ 2
1
(15)
and as a result of this approximation we obtain, for instance, 2= (OIq2IO) ~~(k)I(0IqIn)j
studying the ratio 2/M r(k) = 3[M2(k)] 4(k) (22) may be applied to the present case. This method has been applied not only to magnetic resonance problems [10, 11], but also the neutron scattering problems [12] and electrical conduction problems [13], and has been proved to be useful. According to the general theory of line shapes, it is a good approximation to assume a Gaussian shape if r(k) and a Lorentzian shape if r(k)~l. The actual values of r(k) are plotted against ka in Fig. 3.
+ [a 1(k)
—
1] 1(0 1q11 I2.
2 I 0), etc. are easily The matrixbyelements q I 1), (0 functions Iq calculated the use (0 of Ithe wave (14). Thus we arrive at the final expressions forM 2 and M4: 2[cii(k)+~fl_iJl, (16) M2(k) = (A/m)i7(y/2)~~’ M 2 2[a,(k)+~~~]1 4(k) = (A/rn) x
+ 4yi~sin2 ~)ai(k)+ ~ [(i+ 2y ~~ri2 ~)2
From this figure we see at once that for small wave numnormalized dynamical factor reprebers (ka ~ 0.06 for =structure 8, ka ~ 0.01 foris well = 12) the sented by the cut-off Lorentzian i~
S(k,w)
= irw2+~(k)2 1 &(k) =
0
i~
IwIw~(k),
(23)
where the half-width ~(k)and the cut-off value w~(k)
810
TEMPERATURE AND WAVE NUMBER DEPENDENCE OF A CENTRAL PEAK Vol. 37, No. 10
plotted against ka in Fig. 4. The equations are also
R
solved for wi,, and for the small wave numbers the larger than those of 6. Thus the cut-off is far away in the wings, and our assumption of a cut-off Lorentzian WE~
I 0
-
y
= 0 125
is justified. Especially for k 0 we obtain values of 2’7[ w~turn out to be several orders of magnitude 1 + 4ye2’7(ka)2], (25) r(k) ~e 6(k)~(A/rn)”2 ~ 3’ye2’7(ka)2].
01
1 2~/3
~I2 05
(26)
6. Case II The dynamical structure factor is assumed to be of the form P ö~ QI 62
flie
S(k, w) ______________________________________ I
I
0
I
0
I
20
3.0
=
—
+
—
—
~
+
62 ____________________
ko
(w +
Fig. 3. r, defined by equation (22), plotted vs k. =
w 2+ 0) (P, Q >0) 0
I wI
(27)
IwI>cs,
where w 0 is the phonon frequency and a is the cut.off. The coefficients P and Q and the half-widths 6, and 8
0.1
are to be determined so that equation (27) gives the exact moments. Neglecting quantities of order a/6~and order (a ± w0)/62, we obtain for the second and the fourth moments 62
y
=
0 125
2a
M2
r
—(P61+2Q62)/(P+2Q),
=
M4
=
1712
0.5-
(28)
IT
2a
(P61 + 2Q62)/(P + 2Q).
(29)
IT
0
0.1
If equation (28) is identified with equation (16) and equation (29) with equation (17), 6, and 62 are related to 5 by 6
=
(P61 + 2Q62)/(P + 2Q),
(30)
and a is equal to2, w~. Since w~is approximately it has been found that a w given by (2A/rn)” 0 6~ for the small wave numbers, so that the assumed form —
~-‘
0 I
.0
2.0 I
T
3.0
I
Fig. 4. Half-width plotted vs k.
c5, ~ 62. Thus, if both the central peak and the phonon
are related to the moments by the relations M2(k)
=
2w~(k)6(k)/ir,
M4(k)
equation is justified. From(27) equation (30) we have 6 ~ 6, according as
=
36(k)/x. (24)
~w~(k)
These equations are solved for 6, and (A/m)~”26is
peaks are represented by I_orentzian curves, 6 determined by (24) gives the upper limit or the lower limit values which the half.width of the peak to canthe take, according as the central peak is central narrower than the phonon peaks or vice versa. In particular, if we put 6, = 62, S(k, w) given by equation (27) has the
Vol. 37, No. 10 TEMPERATURE AND WAVE NUMBER DEPENDENCE OF A CENTRAL PEAK same functional form as that obtained by Krumhansl and Schrieffer [1], as far as the w-dependence is concerned. The half-widths of the three Lorentzians are all givenby6. In conclusion, we obtain the half-width 6 determined by equation (24) for the small wave numbers, ka ~ 0.06 for = 8 and ka ~ 0.01 for r~= 12, whether we assume a single peak at w = 0, or three peaks which, however, are all Lorentzians with the same half-width. The half-width is narrower than that of Krumhansl and Schrieffer.
2. 3. 4. 5. 6.
i~
Acknowledgements — I would like to thank Professor R. Kubo for enlightening discussions of the moment method. Case II discussed in the text is due to the suggestion by Professor Y. Wada. I would like to express my special thanks to him for useful suggestions and discussions. I also appreciate discussions with Dr Y. Ono and Mr M. Imada. REFERENCES 1.
J.A. Krunthansl & J.R. Schrieffer,Phys. Rev. Bli, 3535 (1975).
7.
8. 9. 10. 11. 12. 13.
811
T.R. Koehler, A.R. Bishop, J.A. Krumhansl & J.R. Schrieffer, Solid State Commun. 17, 1 515 (1975). Y. Wada & J.R. Schrieffer, Phys. Rev. B18, 3897 (1978). S.Aubry,J. Chem.Phys. 62, 3217(1975). S. Aubry, J. Chern. Phys. 64, 3392 (1976). T. Schneider & E. Stoll, Phys. Rev. Lett. 35, 296 (1975). P.S. Sahni & G.F. Mazenko, Phys. Rev. 820, 4674 (1979). M. Imada,J. Phys. Soc. Japan 47, 699 (1979). R.A. Guyer, Preprint. A. Abragam, The Principles of Nuclear Magnetism. Oxford (1961). R. Kubo & K. Tomita, J. Phys. Soc. Japan 9, 888 (1954). PG. de Gennes,J. Phys. Chem. Solids 4, 223 (1958). N. Ohata & R. Kubo, J. Phys. Soc. Japan 28, 1402 (1970).
0038—1098/81/100807—05$02.00/0
Pergamon Press Ltd. 1981. Printed in Great Britain. TEMPERATURE AND WAVE NUMBER DEPENDENCE OF A CENTRAL PEAK N. Ohata College of Science and Engineering, Aoyama Gakuin University, Chitosedai, Setagaya-ku, Tokyo 157, Japan (Received 22 September 1980 by Y. Toyozawa) The first six moments of the dynamical structure factor is calculated in the spirit of the functional integral for equilibrium quantities. The half-width of the central peak is evaluated with these moments. It is found to be narrower than that obtained by Krumhansl and Schrieffer. 1. SOME YEARS AGO Krumhansl and Schrieffer [1] studied a simple model for structural phase transition which is a linear chain of atoms with one-site doublewell potentials and nearest-neighbor couplings. They calculated the thermodynamic properties of the system exactly by functional integral methods, and showed that the low-temperature thermodynamic behavior can be explained in terms of phonon and domain wall excitations. They also attributed, on the basis of a phenomenological discussion, the central peak in the dynamical structure factor to the motion of domain walls. The central peak and the domain wall motion have been subjects of absorbing interest to many authors. The molecular dynamics computer simulations by Koehler, Bishop, Krumhansl and Schrieffer [2] revealed that the motion rather looks like Brownian motion. In fact the diffusion constant of domain walls was evaluated by Wada and Schrieffer [3]. Aubry [4] exactly calculated the first six moments of the dynamical structure factor by solving the eigenvalue equation for the transfer operator numerically, Using Mon’s method of continued fractions, he outlined the spectrum on the basis of the knowledge of these moments. The spectrum was also obtained by molecular dynamics computations by Aubry [5] and by Schneider and StoIl [6], and the possibility that the domain walls participate in developing the central peak was pointed out. Recently Sahni and Mazenko [7] have obtained, on a rather phenomenological basis, the results which is in qualitative agreement with the molecular dynamics results, There are another group of papers [8, 9] in which the effects of thermal disturbances on the atoms are taken into account by the use of stochastic models. 2. It seems to be useful, at this stage, to calculate ab initio the first few moments of the dynamical structure factor “in the spirit of the functional integral for equilibrium quantities [1]”, to show the temperature-dependence and the wave-number dependence of the moments explicitly, and if possible, to
derive some definite information about the central peak from the moments. This is the purpose of the present paper. If we want to know the detailed shape of a certain peak, the central peak or a phonon side-peak, we have to calculate the moments of the intensity distribution associated with this particular mode. The main weakness of the moment method is that an important contribution to the value of a moment comes from the wings of the curve which are not actually observed. The contributions of other parts of the spectrum have to be carefully excluded from the calculated moments of the peak concerned. In magnetic resonance problems, the way to get rid of them is to separate the perturbing Hamiltonian, responsible for line broadening, into the secular part and the non-secular part and to retain the secular part only [10, 11]. In the present problem, however, since it is not known how to separate the Hamiltonian of the system into the domain wall mode, the phonon mode and the interaction between them, we will be contented with calculating the moments of the whole spectrum and trying to draw from them some definite conclusions about the spectrum in the following special cases: Case I. It is assumed that only the central peak is observed for small wave numbers. Case II. It is assumed that the spectrum is composed of three Lorentzian curves, one for the central peak and two for the phonon side-peaks. The results to be obtained in these cases may be compared with those of Sahni and Mazenko [71and with those of Krumhansl and Schrieffer [1], respectively. .
3. The model Hamiltonian conventionally reads N 1 N IA B q,2 + q~ H = P,2 + —
‘°
N
+
q,
,~o2 807
)2 -
(1)
808
TEMPERATURE AND WAVE NUMBER DEPENDENCE OF A CENTRAL PEAK Vol. 37, No. 10
where q, is the displacement of the lth atom and the periodic boundary conditions q~v+1= q0 and q -1 = ~N
I A V(q)
are imposed. The nth moment of the dynamical structure factor 100
S(k, w)
=
N
dt ~ ~,(t) exp (ikal-— iwt),
27r ~
i=o
(2)
where 0q,(t))
(3)
and a is the lattice constant, is defined by ~ s(k. w) w’~dw,,/f
M~(k) =
2
q +
— ~-
B ~ —
B > 0).
(JO)
q
and DI q I n) is the matrix element of q between the lowest eigenstate and the nth eigenstate. 4. The potential V(q) is a double-well potential with
2 and depth ~A2/B. Let
us denoteat the minima q = eigenvalues ±q = ±(A/B)” and the eigenfunctions of a harmonic oscillator with mass m* in a potential with the
(q
=
=
A
2
S(k, w) dw.
(4)
Since /.,(t) is an even function oft, all the moments for odd n vanish. The moments for even n (= 2p) are clearly given by the formulae
and the latter far larger than the thermal energy at tern-
1 —~Ø,(t)I dt
1d2P
~
()P
the case in which the following conditions are satisfied: 2/B ~- (2A/m*)h/2, (i) ~A (ii) (2A/m*)~2~‘ k~T. That is, the height of the potential hump is far larger than the level distance of the harmonic oscillator.
N
M2~(k)=
curvature equal to that of the potential wells by E~and ~ji~ respectively: E,~ = (n -1- ~)(2A/m*)~ 2 In order to objain definite results, we consider here
exp (ikal) perature T. If we introduce the dimensionless parameters
x ~N ~,(0) exp (ikal).
(5)
2 *\ 12 —rn ~ = q (A2B /
(II)
l=0
and it is lengthy but straightforward to obtain formal
expressions for the moments by functional integral methods with the aid of the equations of motion for q,,
= C/A~ (1 2) the above assumptions (i) and (ii) are equivalent to
with the results (i’)~/4~1
-1
M 2(k)
=
i~~n(k)l(o~qIn)l2l ,
(6)
(11’)
and 2~1
m~n
ii
M 4(k) =
rn
1 -1
a~(k)I (0 I q In) 12 n
(12)
(ii’)(y/2)~
I
respectively. The physical meanings of ~ and ‘y are as
j
follows: The number of harmonic oscillator levels in
2
2
x ~ a~k) A + 4C sin
ka~
~)((o
+ B(A + 4Csin2
ka\ 2jI 1(0 q In) 12
each of the potential wells is ~/4, and kB T is2.smaller The above restrictions may, in effect, relaxed a little more, howthan the level distance by abe factor of (y/2)” ever, since i~ and y appear in the arguments of exponential functions.
—
Iq In)(n 1q310)
+ (0 I q3 In) (n 1q10)) + B2 1(0 I q3 In) 12].
(7)
Here the coefficients a~(k)are defined by e2n~o)
= e2P
in the limit iV
—
—~ eo,
approximation leads to the results
i
n~~o)—2 e~n~0)
cos
0
ka + i
L
~
j3 being kBT. The energy e~,is the nth
4(2A/m*)~/2(1 T ~
I 13)
= ‘I’o) ~,
I
—
(l/~/2)[~ 0(q—q)±~0(q +~)]
(14)
1
+ V(q) ‘1’~(q)= 2Cc~’l’~(q), and the potential with the effective mass rn* = ~ ~
~
(8)
eigenvalue of the effective Schrodinger equation I
Under assumption (i) a tunnelling approximation is valid l~rthe lowest harmonic oscillator state, and this
—~
(9)
Krurnhansl for the lowest andpair Schrieffer of states of may the system, as shown by Under assumption (ii) [11. we approximate equation (8) by
Vol. 37, No. 10 TEMPERATURE AND WAVE NUMBER DEPENDENCE OF A CENTRAL PEAK 2
M
2
‘7
/
1.0
.,,......._~
(/
+
(17)
i5~2])
+ 4y sin
+ ~~‘(5
12
809
If the tunnelling approximation is also valid for the first excited state of the harmonic oscillator, it is
_-.—
,/‘\ ‘78
straightforward to proceed to a higher-order approximation, in which [a 2(k) 1] and [a3(k) 1] are 2M retained as nonzero. In Figs. 1 and 2 (A/m)’M2(k) and (A/m) 4(k) are respectively plotted as functions of ka for ‘y = 1/8 and = 8 and 12. Though these values of and y may seem to violate conditions (i’) and (ii’), the corrections, due to the higher-order approximation, have been found to be less than 5%. In1e’7[ particular, for k 0 we have (18) 1 + 2ye2’7(ka)2] (A/m)-’M2(k) ~i~ (A/m)2M 2], (19) 4(k) ~i~~[1+ ~‘yr~(ka) and for k = ir/a —
—
05
I
-
=
0 25
/
ir
i~
I
10
0
20
3.0
ka
Fig. 1. k-dependence of the second moment. Here definedis by which lower equation for ~ = (11), 12 than is related for r~ to= the 8; -ytemperature, is defined by equation (12). i~,
i~
M
(A/m)IM2(ir/a)~4(y/2)i/Z,
4
(20)
2M 6.0—
‘7—
40
(A/m)~ 4(ir/a)~4{(1 + 2y)2+ ~(5 +4y)ri~}. (21) 5.Casel In the following we will try to guess a plausible
I2
‘7=
8 y
2.0
shape of the central peak for small wave numbers with the knowledge of the second and the fourth moments, under the assumption that only the central peak is observed. For this purpose the usual method [10] of
_
0.125
-
IT
I
0
10
I
20
30
ka
Fig. 2. k-dependence of the fourth moment, =
for n ~ 2
1
(15)
and as a result of this approximation we obtain, for instance, 2= (OIq2IO) ~~(k)I(0IqIn)j
studying the ratio 2/M r(k) = 3[M2(k)] 4(k) (22) may be applied to the present case. This method has been applied not only to magnetic resonance problems [10, 11], but also the neutron scattering problems [12] and electrical conduction problems [13], and has been proved to be useful. According to the general theory of line shapes, it is a good approximation to assume a Gaussian shape if r(k) and a Lorentzian shape if r(k)~l. The actual values of r(k) are plotted against ka in Fig. 3.
+ [a 1(k)
—
1] 1(0 1q11 I2.
2 I 0), etc. are easily The matrixbyelements q I 1), (0 functions Iq calculated the use (0 of Ithe wave (14). Thus we arrive at the final expressions forM 2 and M4: 2[cii(k)+~fl_iJl, (16) M2(k) = (A/m)i7(y/2)~~’ M 2 2[a,(k)+~~~]1 4(k) = (A/rn) x
+ 4yi~sin2 ~)ai(k)+ ~ [(i+ 2y ~~ri2 ~)2
From this figure we see at once that for small wave numnormalized dynamical factor reprebers (ka ~ 0.06 for =structure 8, ka ~ 0.01 foris well = 12) the sented by the cut-off Lorentzian i~
S(k,w)
= irw2+~(k)2 1 &(k) =
0
i~
IwI
(23)
where the half-width ~(k)and the cut-off value w~(k)
810
TEMPERATURE AND WAVE NUMBER DEPENDENCE OF A CENTRAL PEAK Vol. 37, No. 10
plotted against ka in Fig. 4. The equations are also
R
solved for wi,, and for the small wave numbers the larger than those of 6. Thus the cut-off is far away in the wings, and our assumption of a cut-off Lorentzian WE~
I 0
-
y
= 0 125
is justified. Especially for k 0 we obtain values of 2’7[ w~turn out to be several orders of magnitude 1 + 4ye2’7(ka)2], (25) r(k) ~e 6(k)~(A/rn)”2 ~ 3’ye2’7(ka)2].
01
1 2~/3
~I2 05
(26)
6. Case II The dynamical structure factor is assumed to be of the form P ö~ QI 62
flie
S(k, w) ______________________________________ I
I
0
I
0
I
20
3.0
=
—
+
—
—
~
+
62 ____________________
ko
(w +
Fig. 3. r, defined by equation (22), plotted vs k. =
w 2+ 0) (P, Q >0) 0
I wI
(27)
IwI>cs,
where w 0 is the phonon frequency and a is the cut.off. The coefficients P and Q and the half-widths 6, and 8
0.1
are to be determined so that equation (27) gives the exact moments. Neglecting quantities of order a/6~and order (a ± w0)/62, we obtain for the second and the fourth moments 62
y
=
0 125
2a
M2
r
—(P61+2Q62)/(P+2Q),
=
M4
=
1712
0.5-
(28)
IT
2a
(P61 + 2Q62)/(P + 2Q).
(29)
IT
0
0.1
If equation (28) is identified with equation (16) and equation (29) with equation (17), 6, and 62 are related to 5 by 6
=
(P61 + 2Q62)/(P + 2Q),
(30)
and a is equal to2, w~. Since w~is approximately it has been found that a w given by (2A/rn)” 0 6~ for the small wave numbers, so that the assumed form —
~-‘
0 I
.0
2.0 I
T
3.0
I
Fig. 4. Half-width plotted vs k.
c5, ~ 62. Thus, if both the central peak and the phonon
are related to the moments by the relations M2(k)
=
2w~(k)6(k)/ir,
M4(k)
equation is justified. From(27) equation (30) we have 6 ~ 6, according as
=
36(k)/x. (24)
~w~(k)
These equations are solved for 6, and (A/m)~”26is
peaks are represented by I_orentzian curves, 6 determined by (24) gives the upper limit or the lower limit values which the half.width of the peak to canthe take, according as the central peak is central narrower than the phonon peaks or vice versa. In particular, if we put 6, = 62, S(k, w) given by equation (27) has the
Vol. 37, No. 10 TEMPERATURE AND WAVE NUMBER DEPENDENCE OF A CENTRAL PEAK same functional form as that obtained by Krumhansl and Schrieffer [1], as far as the w-dependence is concerned. The half-widths of the three Lorentzians are all givenby6. In conclusion, we obtain the half-width 6 determined by equation (24) for the small wave numbers, ka ~ 0.06 for = 8 and ka ~ 0.01 for r~= 12, whether we assume a single peak at w = 0, or three peaks which, however, are all Lorentzians with the same half-width. The half-width is narrower than that of Krumhansl and Schrieffer.
2. 3. 4. 5. 6.
i~
Acknowledgements — I would like to thank Professor R. Kubo for enlightening discussions of the moment method. Case II discussed in the text is due to the suggestion by Professor Y. Wada. I would like to express my special thanks to him for useful suggestions and discussions. I also appreciate discussions with Dr Y. Ono and Mr M. Imada. REFERENCES 1.
J.A. Krunthansl & J.R. Schrieffer,Phys. Rev. Bli, 3535 (1975).
7.
8. 9. 10. 11. 12. 13.
811
T.R. Koehler, A.R. Bishop, J.A. Krumhansl & J.R. Schrieffer, Solid State Commun. 17, 1 515 (1975). Y. Wada & J.R. Schrieffer, Phys. Rev. B18, 3897 (1978). S.Aubry,J. Chem.Phys. 62, 3217(1975). S. Aubry, J. Chern. Phys. 64, 3392 (1976). T. Schneider & E. Stoll, Phys. Rev. Lett. 35, 296 (1975). P.S. Sahni & G.F. Mazenko, Phys. Rev. 820, 4674 (1979). M. Imada,J. Phys. Soc. Japan 47, 699 (1979). R.A. Guyer, Preprint. A. Abragam, The Principles of Nuclear Magnetism. Oxford (1961). R. Kubo & K. Tomita, J. Phys. Soc. Japan 9, 888 (1954). PG. de Gennes,J. Phys. Chem. Solids 4, 223 (1958). N. Ohata & R. Kubo, J. Phys. Soc. Japan 28, 1402 (1970).