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Excitations of the transversely polarized spin density waves in chromium
ELSEVIER
Physica B 241 243 (1998) 622 624
Excitations of the transversely polarized spin density waves in chromium W.-T. Lee ~'*, S.A. Werner ~, J.A. Fernandez-Baca b, R.S.
Fishman b
"Department q/Physics, Unirersitv o/Missouri, ('olumhia, MO 6521 I, USA hSolid State Dit'ision, Oak Rid£,e Natiomd LahoratopT. Octk Ridge, TN 37,~3 I, U%4
Abstract
We report here the results of an inelastic neutron scattering experiment of the magnetic excitations of the transversely polarized spin density wave (TSDW) in chromium in zero applied field and a 5 T field. Our results show that the mode with spin fluctuations perpendicular to both the ordered moment m and the TSDW ordering vector Q, and the mode with spin fluctuations parallel to m are equally intense. However, the mode with spin fluctuation perpendicular to m. but parallel to Q is the most intense excitation, c 1998 Published by Elsevier Science B.V. All rights reserved. Kevwords." Inelastic neutron scattering; Magnetic excitations; Spin density wave: Chromium; Itinerant magnetism
The magnetism of BCC chromium metal is characterized by its spin density wave (SDW) [1] which can be thought of as an incommensurate sinusoidal modulation of an antiferromagnetic spin arrangement. In a stress-free, single-magnetic-domain, single crystal of chromium, the SDW appears as satellites with ordering vector Q = + {1 -~5, 0, 0) from the nuclear sites in the reciprocal space (Fig. lal. At temperatures between 311 and 122 K, the SDW is polarized transversely to Q (TSDW). This paper reports results of constant energy, inelastic neutron scattering scans at T = 160 K, for AE = 5 and 20 meV, in zero applied field and with a 5 T magnetic field. These experiments are motivated by the recent theoretical work of Fishman *Corresponding aulhor. Fax: hal(a nopt.physics.missouri.edu.
~ I 573 882 636(I: e-mail:
and Liu [2,3], and focus on the relative intensities of the modes associated with the three directions of spin fluctuation, extending the detailed measurements of Lorenzo et al. [4] in the low-temperature LSDW phase and other experiments [5 7]. The measurements were carried out at the High Flux Isotope Reactor at the Oak Ridge National Laboratory. Inelastic neutron scattering probes the target's magnetic fluctuations Am that are perpendicular to the neutron m o m e n t u m transfer K. Three Goldstone magnetic excitation modes are associated with each static polarization m of the T S D W (Fig. lb) [2,3]: two spin wave modes with Am transverse to m in directions perpendicular to Q (T1) and parallel to Q (We), and the phason mode (L) for which Am is parallel to m. We take Q to be along [1 0 0]. Consider the {& 1, 0) satellite with the
0921-4526/98/$19.00 { 1998 Published by Elsevier Science B.V. All rights reserved PII S 0 9 2 1 - 4 5 2 6 1 9 7 ) 0 0 6 6 2 - 5
W.-E Lee et al.
[0 k O]
TSDW
L
(1,1,0)
(_+~,1,0)
@
Q
T=./,L. T1 /
nuclear
ml nuclear
(o,o,o1 Q
TSDW ~. ~
,/Physica
[h 0
I[o o 1] /
0]
0-+&0,0) (a)
(b)
B 241 243 (1998) 622 624
623
10000 Ha) E l a s t i c (i+6 0 0) H=0
I~ H=5T~ 8000 F ~
--''
~,b)~t,i~ ~%,-~;i
mon=2 =6sec
~ ;,ooo
H=5T
4000
3000.~
~ ~ooo ~
g
4000
2000 o
oooo
L , i I tj ,~,!'c'SmeV (1+8,0,0) ' I (d) i 5meV (+&l,O) , • H=0 • H=0 120 F o H 5T mon=300 - - o H=5T i = =9min [
1000 -
0
i 4120
o~ 8
Vig. 1. (a) T S D W satellites in reciprocal space. (b) The polarizations Am of the modes of magnetic excitation (T~, T2, L) associated with the T S D W polarizations m~ and m2.
[ i I i
1
I
r --~--- i (e) 20meV (l_+&O,O) 120
polarization m2 parallel to [0 1 0]; the three modes are: T~, Am along [0 0 1], T2, Am along [1 0 0], and L, Am along [0 1 0]. For the L mode, Am is nearly parallel to the momentum transfer K = (6, 1,0), only the T~ and T 2 modes contribute to the inelastic scattering. Taking into account the modes associated with the polarization m~ parallel to [0 0 1], the inelastic scattering intensity at (6, 1, 0) equals T1 + 2T2 + L. When a 5 T magnetic field is applied along [0 0 1], the TSDW static polarization is confined to be along [0 1 0] [7]. The inelastic scattering intensity at (b, 1, 0) then becomes 2T~ + 2T2. Listed below is a summary of the preceding analysis: Satellite (1-1-3,0,0) (4`6, 1,0)
H = 0 2Tj + 2 L T1 + 2 T z + L
H = 5T 2T~ + 2 L 2T1 + 2 T 2
Using the same polarization selection rules, the elastic scattering at (1 + 6, 0, 0) is expected to be approximately twice that at ( 4- 6, 1, 0) when H = 0. For H = 5 T, the (1 _+ 5, 0, 0) elastic intensities will remain unchanged while (4`6, 1, 0) diminish. The elastic scans shown in Fig. 2a and Fig. 2b are in good agreement with these predictions. The reduction in (_+ 6, 1, 0) under H = 5 T indicates 96% of the polarization is confined to the [0 1 0] direction. The inelastic scans across the satellites (1 4` b, O, O) and (_+ b, l, O) at A E = 5 and 20meV under' zero field and H = 5T are shown in Fig. 2c-Fig. 21".The data for ( 4- b, 1, 0) with H = 0 and H = 5 T are very similar, suggesting that, with-
• <',
H=0 H=5T
mon=300
,o,
0
?
Io
(f) 20meV (_+&l,O) I • -- o
H=0 H=5T
~20
Tj*'%.o 0,9
10 1.1 (h 0 0) (r.Lu.)
-0.1
00 0.0 (h 1 0) (r.l.u.)
Fig. 2. Constant energy scans across the T S D W satellites. The tilled circles are H = 0 data. The opened circles are H = 5 T data. The solid lines are curve fits with Gaussians.
in experimental error, the L and T1 modes are equally intense. The difference between the (1 4` 6, 0, 0) and (_+ 6, 1, 0) shows that the T 2 mode is more intense than both the T1 and L modes. Curve-fitting with Gaussians gives T1 : T2 : L 2 : 3 : 2 at 5meV and T I : T z : L ~ I : 3 : I at 20 meV. Using the )./2 phonon scattering at ( 1 0 0) and (0 1 0), we determined the scattering volume near these two locations differ by less than 10%. These results are in contrast to other recent experiments [8,9] and the theoretical predictions [2,3]. More analysis and measurements may be necessary to resolve these discrepancies. W.-T. Lee and S.A. Werner would like to acknowledge the support of the NSF through Grant No. NSF-PHY 9024608. O R N L is managed by Lockheed Martin Energy Research Corp. for the US Department of Energy under contract number DE-AC05-96OR22464.
624
W.-T Lee et al. / Physica B 241-243 (1998) 622 624
References [1] For a comprehensive review see: E. Fawcett, Rev. Mod. Phys. 60 (1988) 209. [2] R.S. Fishman, S.H. Liu, Phys. Rev. Lett. 76 (1996) 2398. [3] R.S. Fishman, S.H. Liu, Phys. Rev. B 54 (1996) 7233,7252. [4] J.E. Lorenzo, B.J. Sternlieb, G. Shirane, S,A. Werner, Phys. Rev. Lett. 72 (1994) 1762. [5] S.K. Burke, W.G. Stirling, K.R.A. Ziebeck, J.G. Booth, Phys. Rev. Lett. 51 (1983) 494.
[6] T. Fukuda, Y. Endoh, K. Yamada, M. Takeda, S. ltoh, M. Arai, T. Otomo, J. Phys. Soc. Japan 65 (1996) 1418. [7] Y. Tsunoda, R.M. Nicklow, J. Phys.: Condens. Matter 8 (1996) 2655. [8] P. B6ni, B.J. Sternlieb, B. Roessli, J.E. Lorenzo, G. Shirane, S.A. Werner, these proceedings. [9] Y. Endoh, T. Fukuda, K. Nakajima, K. Kakurai, J. Phys. Soc. Japan 66 (1997) 1615.
Physica B 241 243 (1998) 622 624
Excitations of the transversely polarized spin density waves in chromium W.-T. Lee ~'*, S.A. Werner ~, J.A. Fernandez-Baca b, R.S.
Fishman b
"Department q/Physics, Unirersitv o/Missouri, ('olumhia, MO 6521 I, USA hSolid State Dit'ision, Oak Rid£,e Natiomd LahoratopT. Octk Ridge, TN 37,~3 I, U%4
Abstract
We report here the results of an inelastic neutron scattering experiment of the magnetic excitations of the transversely polarized spin density wave (TSDW) in chromium in zero applied field and a 5 T field. Our results show that the mode with spin fluctuations perpendicular to both the ordered moment m and the TSDW ordering vector Q, and the mode with spin fluctuations parallel to m are equally intense. However, the mode with spin fluctuation perpendicular to m. but parallel to Q is the most intense excitation, c 1998 Published by Elsevier Science B.V. All rights reserved. Kevwords." Inelastic neutron scattering; Magnetic excitations; Spin density wave: Chromium; Itinerant magnetism
The magnetism of BCC chromium metal is characterized by its spin density wave (SDW) [1] which can be thought of as an incommensurate sinusoidal modulation of an antiferromagnetic spin arrangement. In a stress-free, single-magnetic-domain, single crystal of chromium, the SDW appears as satellites with ordering vector Q = + {1 -~5, 0, 0) from the nuclear sites in the reciprocal space (Fig. lal. At temperatures between 311 and 122 K, the SDW is polarized transversely to Q (TSDW). This paper reports results of constant energy, inelastic neutron scattering scans at T = 160 K, for AE = 5 and 20 meV, in zero applied field and with a 5 T magnetic field. These experiments are motivated by the recent theoretical work of Fishman *Corresponding aulhor. Fax: hal(a nopt.physics.missouri.edu.
~ I 573 882 636(I: e-mail:
and Liu [2,3], and focus on the relative intensities of the modes associated with the three directions of spin fluctuation, extending the detailed measurements of Lorenzo et al. [4] in the low-temperature LSDW phase and other experiments [5 7]. The measurements were carried out at the High Flux Isotope Reactor at the Oak Ridge National Laboratory. Inelastic neutron scattering probes the target's magnetic fluctuations Am that are perpendicular to the neutron m o m e n t u m transfer K. Three Goldstone magnetic excitation modes are associated with each static polarization m of the T S D W (Fig. lb) [2,3]: two spin wave modes with Am transverse to m in directions perpendicular to Q (T1) and parallel to Q (We), and the phason mode (L) for which Am is parallel to m. We take Q to be along [1 0 0]. Consider the {& 1, 0) satellite with the
0921-4526/98/$19.00 { 1998 Published by Elsevier Science B.V. All rights reserved PII S 0 9 2 1 - 4 5 2 6 1 9 7 ) 0 0 6 6 2 - 5
W.-E Lee et al.
[0 k O]
TSDW
L
(1,1,0)
(_+~,1,0)
@
Q
T=./,L. T1 /
nuclear
ml nuclear
(o,o,o1 Q
TSDW ~. ~
,/Physica
[h 0
I[o o 1] /
0]
0-+&0,0) (a)
(b)
B 241 243 (1998) 622 624
623
10000 Ha) E l a s t i c (i+6 0 0) H=0
I~ H=5T~ 8000 F ~
--''
~,b)~t,i~ ~%,-~;i
mon=2 =6sec
~ ;,ooo
H=5T
4000
3000.~
~ ~ooo ~
g
4000
2000 o
oooo
L , i I tj ,~,!'c'SmeV (1+8,0,0) ' I (d) i 5meV (+&l,O) , • H=0 • H=0 120 F o H 5T mon=300 - - o H=5T i = =9min [
1000 -
0
i 4120
o~ 8
Vig. 1. (a) T S D W satellites in reciprocal space. (b) The polarizations Am of the modes of magnetic excitation (T~, T2, L) associated with the T S D W polarizations m~ and m2.
[ i I i
1
I
r --~--- i (e) 20meV (l_+&O,O) 120
polarization m2 parallel to [0 1 0]; the three modes are: T~, Am along [0 0 1], T2, Am along [1 0 0], and L, Am along [0 1 0]. For the L mode, Am is nearly parallel to the momentum transfer K = (6, 1,0), only the T~ and T 2 modes contribute to the inelastic scattering. Taking into account the modes associated with the polarization m~ parallel to [0 0 1], the inelastic scattering intensity at (6, 1, 0) equals T1 + 2T2 + L. When a 5 T magnetic field is applied along [0 0 1], the TSDW static polarization is confined to be along [0 1 0] [7]. The inelastic scattering intensity at (b, 1, 0) then becomes 2T~ + 2T2. Listed below is a summary of the preceding analysis: Satellite (1-1-3,0,0) (4`6, 1,0)
H = 0 2Tj + 2 L T1 + 2 T z + L
H = 5T 2T~ + 2 L 2T1 + 2 T 2
Using the same polarization selection rules, the elastic scattering at (1 + 6, 0, 0) is expected to be approximately twice that at ( 4- 6, 1, 0) when H = 0. For H = 5 T, the (1 _+ 5, 0, 0) elastic intensities will remain unchanged while (4`6, 1, 0) diminish. The elastic scans shown in Fig. 2a and Fig. 2b are in good agreement with these predictions. The reduction in (_+ 6, 1, 0) under H = 5 T indicates 96% of the polarization is confined to the [0 1 0] direction. The inelastic scans across the satellites (1 4` b, O, O) and (_+ b, l, O) at A E = 5 and 20meV under' zero field and H = 5T are shown in Fig. 2c-Fig. 21".The data for ( 4- b, 1, 0) with H = 0 and H = 5 T are very similar, suggesting that, with-
• <',
H=0 H=5T
mon=300
,o,
0
?
Io
(f) 20meV (_+&l,O) I • -- o
H=0 H=5T
~20
Tj*'%.o 0,9
10 1.1 (h 0 0) (r.Lu.)
-0.1
00 0.0 (h 1 0) (r.l.u.)
Fig. 2. Constant energy scans across the T S D W satellites. The tilled circles are H = 0 data. The opened circles are H = 5 T data. The solid lines are curve fits with Gaussians.
in experimental error, the L and T1 modes are equally intense. The difference between the (1 4` 6, 0, 0) and (_+ 6, 1, 0) shows that the T 2 mode is more intense than both the T1 and L modes. Curve-fitting with Gaussians gives T1 : T2 : L 2 : 3 : 2 at 5meV and T I : T z : L ~ I : 3 : I at 20 meV. Using the )./2 phonon scattering at ( 1 0 0) and (0 1 0), we determined the scattering volume near these two locations differ by less than 10%. These results are in contrast to other recent experiments [8,9] and the theoretical predictions [2,3]. More analysis and measurements may be necessary to resolve these discrepancies. W.-T. Lee and S.A. Werner would like to acknowledge the support of the NSF through Grant No. NSF-PHY 9024608. O R N L is managed by Lockheed Martin Energy Research Corp. for the US Department of Energy under contract number DE-AC05-96OR22464.
624
W.-T Lee et al. / Physica B 241-243 (1998) 622 624
References [1] For a comprehensive review see: E. Fawcett, Rev. Mod. Phys. 60 (1988) 209. [2] R.S. Fishman, S.H. Liu, Phys. Rev. Lett. 76 (1996) 2398. [3] R.S. Fishman, S.H. Liu, Phys. Rev. B 54 (1996) 7233,7252. [4] J.E. Lorenzo, B.J. Sternlieb, G. Shirane, S,A. Werner, Phys. Rev. Lett. 72 (1994) 1762. [5] S.K. Burke, W.G. Stirling, K.R.A. Ziebeck, J.G. Booth, Phys. Rev. Lett. 51 (1983) 494.
[6] T. Fukuda, Y. Endoh, K. Yamada, M. Takeda, S. ltoh, M. Arai, T. Otomo, J. Phys. Soc. Japan 65 (1996) 1418. [7] Y. Tsunoda, R.M. Nicklow, J. Phys.: Condens. Matter 8 (1996) 2655. [8] P. B6ni, B.J. Sternlieb, B. Roessli, J.E. Lorenzo, G. Shirane, S.A. Werner, these proceedings. [9] Y. Endoh, T. Fukuda, K. Nakajima, K. Kakurai, J. Phys. Soc. Japan 66 (1997) 1615.