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Neutron scattering from itinerant magnets in the paramagnetic region
Volume 112A, number 6,7
PHYSICS LETTERS
4 November 1985
NEUTRON SCATI'ERING FROM ITINERANT MAGNETS IN THE PARAMAGNETIC REGION Joseph C A L L A W A Y Department of Physics, Louisiana State University, Baton Rouge, 1.,.4 70803- 4001, USA
Received 30 May 1985; revised manuscript received 12 August 1985; accepted for publication 21 August 1985
Neutron scattering from itinerant magnets in the paramagnetic region is considered on the basis of a simple approach based on band theory. It is possible to account for the variation with q of the energy width, and to obtain a maximum in the scattering cross section for fixed q and o~> 0 without invoking the existence of spin waves.
I have proposed, in a recent paper, that measurements of neutron scattering from itinerant ferromagnets in the paramagnetic region above T c might be interpreted in terms of an expression for the wave vector and frequency dependent magnetic susceptibility derived from a simple band model based on local spin density functional theory [1]. In this note, two problems which have been discussed in the recent literature are analyzed from the point of view of ref. [ 1]. The problems a r e : ( 1 ) t h e q dependence of the energy width of a constant q scan, and (2) the report. ed appearance of damped spin waves above T c. I will first review briefly the theoretical results contained in ref. [1]. The double differential cross section (in energy and angle) in the paramagnetic region is, according to ref. [1], proportional to a function I shall simply denote by o which is given by o = (1 - e - a ~ / K T ) -1 Im[x(q, 6o, T)] ,
(1)
in which X is the wave vector and frequency depen. dent susceptibility at temperature T. Here q is the momentum transfer and ti6o is the energy transfer. Hereafter, I shall suppress factors o f ~ and K. An exact expression for × is not known, but I shall use an expression which results either from local spin density functional theory [2], or from the random phase approximation to the Hubbard model [3] (although the meaning of the interaction constant U is different in the two cases): 0.375-9601/85/$ 03.30 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
X = XO(q, 60, 79/[ 1 - U XO(q, 6o, 791 ,
(2)
in which X0 is the non-self.consistent susceptibility. If 6O/uFq is small, but q is not too large (compared, say, with k F or more appropriately with the BriUouin zone dimensions) the following result should hold: o = ( 1 - e - ~ / T ) -1
×ix) (q, 6o, 79 [1 - U X [ R) (q, 6o, 79]2 + U2X~I)2 (q, to, T) = (U2p) -1 [60/(1 - e - w / T ) ]
X
q 6o2 + (q2/U2p2) {C + U[q2S + (6o2/q2)R]}2 ' (3)
in which P, R, and S are weakly temperature depen. dent quantities, here simply regarded as adjustable constants, and C is C = 1 - U X(0, 0, 79.
(4)
The quantity C vanishes at Te, and is positive for T > T c.
In the case of a constant q scan, consider the value of the energy transfer for which the cross section has fallen to ½ the value at q = 0. Denote this by 6Ol/2 (it will simply be referred to as the line width). Then 337
Volume 112A, number 6,7
PHYSICS LETTERS
[for the present ignoring the slowly varying factor
~/(1
3 IO
- e-~/r)l,
COl/2 = (q/p) (C/U+ Sq2) .
I
I
I
I
I
I
I
I
I
Fe3Pt
/
[
I
I
elf ,{'
2.0
(5)
Just at T c, C vanishes, and the expression, which is a mean field result, gives e I/2 ~ q3, which is at variance with and presumably inferior to the scaling result o~1/2 ~ q5/2 [4]. However somewhat above T c, eq. (5) may be reasonably satisfactory. The approach described briefly above has been considered by other authors; in particular Izuyama et al. [3] and Lowde and Windsor [5]. The present note applies this model to some recent experimental results. B6ni et al. [6] have compared the scattering from paramagnetic 3' (fcc) and ~ (bcc) iron. They fit the scattering to a product of lorentzians in energy and q and were able to extract a width. They assume that Wl/2 ~ q8 ; and find that 6 ~ 1.3 for 7 iron whereas for a iron a previous measurement had found 6 to be in the range from 2 to 2.5 [7]. In the latter case 6 showed some tendency to increase withq, being close to 2 for small q and nearer to 2.5 for large q; in agreement with ref. [4]. However, the small value of 6 for 7 iron could not be understood. I offer the following argument. Empirically, eq. (5) is roughly compatible with a range of values of 8 from 1 to 3. What is observed depends essentially on q and on C. In 7 iron C probably is considerably larger (7 iron is not observed to be ferromagnetic) than for iron at similar temperatures. Hence one expects that Wl/2 may be close to linear for q not too large. In c~ iron a higher power dependence is expected. These considerations are further supported by recent measurements concerning the ordered invar alloy Fe3Pt (T c ~ 504 K) [8]. An extensive series of line width measurements are reported at several different temperatures (508, 518, 602, and 701 K). The measurements do not show much difference between the 508 K and 518 K results, and I consider here only the 7 0 1 , 6 0 2 , and 518 K data. I have made least squares fits to this data, using eq. (5). The results are shown in fig. 1. The fits are quite satisfactory. The parameters are given in table 1. The quantity C/UP decreases by a factor of 4 from 701 to 518 K, while S/Pis much more slowly varying. In particular this approach is able to explain the departure of the results from a simple power law (q5/2) for small q at T = 701 and 602 K. 338
4 November 1985
/~
~._ 0 5 @ u_l
_z --J
02
T //)} // /
OI
x 602K u 701 K --
K
._. . ._.
602K701K
' 0i2
'
zi/~
005
: 518
4~ ' ''J'' 005 O.I
q
'
0'.5
)
Fig. 1. Line widths as functions o f q for Fe3Pt. The curves
are least squares fits using eq. (5). The data is from ref. [7].
It is not possible at this time to calculate the parameters from band theory for Fe3Pt. The approximately linear temperature dependence of C/UP is consistent with the model, but this quantity should vanish at T c = 504 K whereas a linear extrapolation indicates instead T c = 456 K. Probably, the mean field theory on which this calculation is based is losing validity at 518 K. However the fits are reasonably good. The quantity SIP would be expected according to a (very rough) parabolic band model to be a slowly increasing function of temperature (see eq. (13) of ref. [ 1 ]). A numerical estimate based on this model (Fe3Pt is cubic with a = 3.75 A) indicates the value of SIP obtained in the fit is reasonable.
Table 1 Parameters in a least squares fit of eq. (5) for Fe3Pt. T(K)
C/UP(meVA)
S/P(meV A3)
518 602 701
0.378 0.931 1.50
94.6 105. 127.
Volume 112A, number 6,7
PHYSICS LETTERS
I will now consider the question o f spin waves above T c. This topic has a long and very controversial history, which I will not attempt to review here (see ref. [8] and the references listed in that paper). It suffices to say here that recently reported high resolution measurements o f magnetic scattering above T c in nickel in a constant q procedure show cross sections which decrease with energy monotonically for small q, but have a weak maximum not at q = 0 for larger q [8]. These maxima are interpreted by the authors o f ref. [8] as implying the existence o f spin waves at large q above T c. The point I wish to make here is that the cross section as given b y eq. (3) will have a maximum for fixed q and co > 0 which is not related to spin waves. The function co/(1 - e - c ° / T ) is an increasing function o f co, roughlylinear in co for small co/T. The remainder of the function in eq. (2) varies with co for small co as A - Bco 2 where A and B depend on temperature and q. This function has, therefore, a
>rr cr Fco cr
g
E(meV) Fig. 2. Observed and fit neutron scattering cross sections as functions of energy transfer E (points are experimental neutron counts from ref. [8]) for q = (0.04, 0.04, 0.04) Ao I for nickel at T = 1.06Tc. The fit is made using eq. (2). The data is normalized to theory at E = 4 meV.
4 November 1985
120
b 9O
o
4 E(meV)
Fig. 3. Same as fig. 2 except that q = (0.14, 0.14, 0.14) Ao I . The data is normalized to theory at E = 0. Note that the zero of o is off the scale.
maximum for co > 0. For small q the maximum is very weak and so close to co = 0 so that it is not seen in the experiments, but as q increases, the maximum becomes noticeable but it is not large. This behavior characterizes the data o f ref. [8]. I show in figs. 2 and 3, fits to the data o f ref. [8] for small q and larger q. A maximum does appear in the second case. It must be emphasized that this theoretical maximum is not in any way connected with spin waves. I suggest that the question o f spin waves above T c can be studied more effectively i f the data in constant q scans are multiplied by [(1 - e - c ° / T ) / w ] , which will remove the factor which causes the maxima in the theoretical curve. If the data, after this multiplication, still show a maximum, then one may suspect resonant behavior coming from the denominator (or its modification in a more correct theory), which would then indicate the presence o f spin waves. Finally, it seems from the curves in ref. [8], that in order to obtain the observed variation in the general shape o f the curves: steep for small q, much flatter for large q; the interaction quantity U in eq. (1) must be allowed to be a decreasing function o f q . A good fit is not possible with a constant U. This work was supported in part b y the Division o f Materials Research o f the US National Science Foundation. 339
Volume 112A, number 6,7
PHYSICS LETTERS
References [1 ] J. CaUaway, Phys. Lett. 104A (1984) 487. [2] J. Callaway and A.K. Chatterjee, J. Phys. F8 (1978) 2569. [3] T. Izuyama, D.J. Kim and R. Kubo, J. Phys. Soc. Japan 18 (1963) 1025. [4] P. R6sibois and C. Piette, Phys. Rev. Lett. 24 (1970) 514.
340
4 November 1985
[5] R.D. Lowde and C.G. Windsor, Adv. Phys. 19 (1970) 813. [6] P. Bbni, G. Shirane, J.P. Wicksted and C. Stassis, Phys. Rev. B31 (1985) 4597. [7] J.P. Wicksted, P. B6ni and G. Shirane, Phys. Rev. B30 (1984) 3655. [8] P. B6ni, B.H. Grief, G. Shirane and Y. Ishikawa, preprint. [9] H.A. Mook and J.W. Lynn, J. Appl. Phys. 57 (1985) 3006.
PHYSICS LETTERS
4 November 1985
NEUTRON SCATI'ERING FROM ITINERANT MAGNETS IN THE PARAMAGNETIC REGION Joseph C A L L A W A Y Department of Physics, Louisiana State University, Baton Rouge, 1.,.4 70803- 4001, USA
Received 30 May 1985; revised manuscript received 12 August 1985; accepted for publication 21 August 1985
Neutron scattering from itinerant magnets in the paramagnetic region is considered on the basis of a simple approach based on band theory. It is possible to account for the variation with q of the energy width, and to obtain a maximum in the scattering cross section for fixed q and o~> 0 without invoking the existence of spin waves.
I have proposed, in a recent paper, that measurements of neutron scattering from itinerant ferromagnets in the paramagnetic region above T c might be interpreted in terms of an expression for the wave vector and frequency dependent magnetic susceptibility derived from a simple band model based on local spin density functional theory [1]. In this note, two problems which have been discussed in the recent literature are analyzed from the point of view of ref. [ 1]. The problems a r e : ( 1 ) t h e q dependence of the energy width of a constant q scan, and (2) the report. ed appearance of damped spin waves above T c. I will first review briefly the theoretical results contained in ref. [1]. The double differential cross section (in energy and angle) in the paramagnetic region is, according to ref. [1], proportional to a function I shall simply denote by o which is given by o = (1 - e - a ~ / K T ) -1 Im[x(q, 6o, T)] ,
(1)
in which X is the wave vector and frequency depen. dent susceptibility at temperature T. Here q is the momentum transfer and ti6o is the energy transfer. Hereafter, I shall suppress factors o f ~ and K. An exact expression for × is not known, but I shall use an expression which results either from local spin density functional theory [2], or from the random phase approximation to the Hubbard model [3] (although the meaning of the interaction constant U is different in the two cases): 0.375-9601/85/$ 03.30 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
X = XO(q, 60, 79/[ 1 - U XO(q, 6o, 791 ,
(2)
in which X0 is the non-self.consistent susceptibility. If 6O/uFq is small, but q is not too large (compared, say, with k F or more appropriately with the BriUouin zone dimensions) the following result should hold: o = ( 1 - e - ~ / T ) -1
×ix) (q, 6o, 79 [1 - U X [ R) (q, 6o, 79]2 + U2X~I)2 (q, to, T) = (U2p) -1 [60/(1 - e - w / T ) ]
X
q 6o2 + (q2/U2p2) {C + U[q2S + (6o2/q2)R]}2 ' (3)
in which P, R, and S are weakly temperature depen. dent quantities, here simply regarded as adjustable constants, and C is C = 1 - U X(0, 0, 79.
(4)
The quantity C vanishes at Te, and is positive for T > T c.
In the case of a constant q scan, consider the value of the energy transfer for which the cross section has fallen to ½ the value at q = 0. Denote this by 6Ol/2 (it will simply be referred to as the line width). Then 337
Volume 112A, number 6,7
PHYSICS LETTERS
[for the present ignoring the slowly varying factor
~/(1
3 IO
- e-~/r)l,
COl/2 = (q/p) (C/U+ Sq2) .
I
I
I
I
I
I
I
I
I
Fe3Pt
/
[
I
I
elf ,{'
2.0
(5)
Just at T c, C vanishes, and the expression, which is a mean field result, gives e I/2 ~ q3, which is at variance with and presumably inferior to the scaling result o~1/2 ~ q5/2 [4]. However somewhat above T c, eq. (5) may be reasonably satisfactory. The approach described briefly above has been considered by other authors; in particular Izuyama et al. [3] and Lowde and Windsor [5]. The present note applies this model to some recent experimental results. B6ni et al. [6] have compared the scattering from paramagnetic 3' (fcc) and ~ (bcc) iron. They fit the scattering to a product of lorentzians in energy and q and were able to extract a width. They assume that Wl/2 ~ q8 ; and find that 6 ~ 1.3 for 7 iron whereas for a iron a previous measurement had found 6 to be in the range from 2 to 2.5 [7]. In the latter case 6 showed some tendency to increase withq, being close to 2 for small q and nearer to 2.5 for large q; in agreement with ref. [4]. However, the small value of 6 for 7 iron could not be understood. I offer the following argument. Empirically, eq. (5) is roughly compatible with a range of values of 8 from 1 to 3. What is observed depends essentially on q and on C. In 7 iron C probably is considerably larger (7 iron is not observed to be ferromagnetic) than for iron at similar temperatures. Hence one expects that Wl/2 may be close to linear for q not too large. In c~ iron a higher power dependence is expected. These considerations are further supported by recent measurements concerning the ordered invar alloy Fe3Pt (T c ~ 504 K) [8]. An extensive series of line width measurements are reported at several different temperatures (508, 518, 602, and 701 K). The measurements do not show much difference between the 508 K and 518 K results, and I consider here only the 7 0 1 , 6 0 2 , and 518 K data. I have made least squares fits to this data, using eq. (5). The results are shown in fig. 1. The fits are quite satisfactory. The parameters are given in table 1. The quantity C/UP decreases by a factor of 4 from 701 to 518 K, while S/Pis much more slowly varying. In particular this approach is able to explain the departure of the results from a simple power law (q5/2) for small q at T = 701 and 602 K. 338
4 November 1985
/~
~._ 0 5 @ u_l
_z --J
02
T //)} // /
OI
x 602K u 701 K --
K
._. . ._.
602K701K
' 0i2
'
zi/~
005
: 518
4~ ' ''J'' 005 O.I
q
'
0'.5
)
Fig. 1. Line widths as functions o f q for Fe3Pt. The curves
are least squares fits using eq. (5). The data is from ref. [7].
It is not possible at this time to calculate the parameters from band theory for Fe3Pt. The approximately linear temperature dependence of C/UP is consistent with the model, but this quantity should vanish at T c = 504 K whereas a linear extrapolation indicates instead T c = 456 K. Probably, the mean field theory on which this calculation is based is losing validity at 518 K. However the fits are reasonably good. The quantity SIP would be expected according to a (very rough) parabolic band model to be a slowly increasing function of temperature (see eq. (13) of ref. [ 1 ]). A numerical estimate based on this model (Fe3Pt is cubic with a = 3.75 A) indicates the value of SIP obtained in the fit is reasonable.
Table 1 Parameters in a least squares fit of eq. (5) for Fe3Pt. T(K)
C/UP(meVA)
S/P(meV A3)
518 602 701
0.378 0.931 1.50
94.6 105. 127.
Volume 112A, number 6,7
PHYSICS LETTERS
I will now consider the question o f spin waves above T c. This topic has a long and very controversial history, which I will not attempt to review here (see ref. [8] and the references listed in that paper). It suffices to say here that recently reported high resolution measurements o f magnetic scattering above T c in nickel in a constant q procedure show cross sections which decrease with energy monotonically for small q, but have a weak maximum not at q = 0 for larger q [8]. These maxima are interpreted by the authors o f ref. [8] as implying the existence o f spin waves at large q above T c. The point I wish to make here is that the cross section as given b y eq. (3) will have a maximum for fixed q and co > 0 which is not related to spin waves. The function co/(1 - e - c ° / T ) is an increasing function o f co, roughlylinear in co for small co/T. The remainder of the function in eq. (2) varies with co for small co as A - Bco 2 where A and B depend on temperature and q. This function has, therefore, a
>rr cr Fco cr
g
E(meV) Fig. 2. Observed and fit neutron scattering cross sections as functions of energy transfer E (points are experimental neutron counts from ref. [8]) for q = (0.04, 0.04, 0.04) Ao I for nickel at T = 1.06Tc. The fit is made using eq. (2). The data is normalized to theory at E = 4 meV.
4 November 1985
120
b 9O
o
4 E(meV)
Fig. 3. Same as fig. 2 except that q = (0.14, 0.14, 0.14) Ao I . The data is normalized to theory at E = 0. Note that the zero of o is off the scale.
maximum for co > 0. For small q the maximum is very weak and so close to co = 0 so that it is not seen in the experiments, but as q increases, the maximum becomes noticeable but it is not large. This behavior characterizes the data o f ref. [8]. I show in figs. 2 and 3, fits to the data o f ref. [8] for small q and larger q. A maximum does appear in the second case. It must be emphasized that this theoretical maximum is not in any way connected with spin waves. I suggest that the question o f spin waves above T c can be studied more effectively i f the data in constant q scans are multiplied by [(1 - e - c ° / T ) / w ] , which will remove the factor which causes the maxima in the theoretical curve. If the data, after this multiplication, still show a maximum, then one may suspect resonant behavior coming from the denominator (or its modification in a more correct theory), which would then indicate the presence o f spin waves. Finally, it seems from the curves in ref. [8], that in order to obtain the observed variation in the general shape o f the curves: steep for small q, much flatter for large q; the interaction quantity U in eq. (1) must be allowed to be a decreasing function o f q . A good fit is not possible with a constant U. This work was supported in part b y the Division o f Materials Research o f the US National Science Foundation. 339
Volume 112A, number 6,7
PHYSICS LETTERS
References [1 ] J. CaUaway, Phys. Lett. 104A (1984) 487. [2] J. Callaway and A.K. Chatterjee, J. Phys. F8 (1978) 2569. [3] T. Izuyama, D.J. Kim and R. Kubo, J. Phys. Soc. Japan 18 (1963) 1025. [4] P. R6sibois and C. Piette, Phys. Rev. Lett. 24 (1970) 514.
340
4 November 1985
[5] R.D. Lowde and C.G. Windsor, Adv. Phys. 19 (1970) 813. [6] P. Bbni, G. Shirane, J.P. Wicksted and C. Stassis, Phys. Rev. B31 (1985) 4597. [7] J.P. Wicksted, P. B6ni and G. Shirane, Phys. Rev. B30 (1984) 3655. [8] P. B6ni, B.H. Grief, G. Shirane and Y. Ishikawa, preprint. [9] H.A. Mook and J.W. Lynn, J. Appl. Phys. 57 (1985) 3006.