Blaubeuren 2006 Blaubeuren 2006 laxation mechanisms in exchange couple spin systems – I Line broadening and the Kubo-Tomito approach Joachim Deisenhofer Université de Genève Blaubeuren 6th September 2006
Dec 26, 2015
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Relaxation mechanisms in exchange coupled spin systems – I
Line broadening and the Kubo-Tomito approach
Joachim Deisenhofer
Université de Genève
Blaubeuren6th September 2006
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Outline
ESR in exchange-coupled system: definition of the problem
linear response and ESR:
the Kubo-Tomita approach
the effective spin Hamiltonian and the broadening mechanisms
summary
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Motivation - ESR in transition-metal compounds
ESR locally probes the spin of interest
low-dimensional spin systems–S = 1/2 chains: LiCuVO4, CuGeO3, TiOCl - and ladders: NaV2O5 –chains with larger spin PbNi2V2O8 (S = 1), (NH4)2MnF5 (S =2)–2D honeycomb-lattice BaNi2V2O8
metal-insulator-transitions–heavy-fermion-properties in Gd1-xSrxTiO3 –spin-state transitions in GdBaCo2O5+
materials with colossal magnetoresistance effect–magnetic structure in thiospinels FeCr2S4, MnCr2S4
–orbital ordering in La1-xSrxMnO3
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parameters: intensity:local spin susceptibility
resonance field:g = g - 2.0023local symmetry
linewidth H:spin-relaxation,anisotropic interactions
The absorption signal
resBHg
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Why one Lorentzian only? “Exchange narrowing”
isotropic exchange interaction between neighbouring spins causes local field fluctuations
originally Gaussian lineshapes are therefore “narrowed“ in the middle and extendend in the wings
phenomenon similar to “motional narrowing“ in liquids as seen by NMR
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~
What are we measuring? - Definition of the problem
usual case: Faraday configuration;microwave field perpendicular to static magnetic field couples to the spin
Microwave absorption in the cavity is given by
Problem: What is the (linear) response function of the spin system to the microwave perturbation?
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ESR in Linear Response – ingredients
observed quantity:
time-dependent perturbation:
general spin susceptibility:
Heisenberg equation of motion:
Hamiltonian of the spin system:
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The effective spin Hamiltonian
Hamiltonian for strongly correlated spin systems:
H'H i
iii
i Jg 1B SSSH
Zeeman energy
isotropicexchange
additionalinteractions
strong isotropic coupling averages local fields similar to
fast movements of the spins “exchange narrowing“ of the ESR signal
local fluctuating fields local, static resonance shift inhomogenous broadening of
the ESR signal
e.g. crystal fieldanisotropic exchangedipole-dipole interactionhyperfine interaction
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final general expression:
with
lineshape is completely determined by
: resonance line infinitely sharp at ω=ω0 (δ –peak)
assuming a Lorentzian lineshape and
gives linewidth:
and resonance shift:
Where does linear response theory take us?
Oshikawa and Affleck, PRB 65, 134410 (2002)
Nagata and Tatsuke, J. Phys. Soc. Jpn. 32, 337 (1972)
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The Kubo-Tomita formula for the linewidth
T ∞: high-temperature approximation
correlation function decays exponentially:
characteristic time is governed by isotropic exchange J:
equivalent expression:
R. Kubo and K. Tomita, J. Phys. Soc. Jpn. 9, 888 (1954)
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Characteristics of the Kubo-Tomita approach - I
temperature dependence is governed by the static susceptibility
describes many 3D magnetic systems, e.g. manganites
Causa et al., PRB 58, 3233 (1998)
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Characteristics of the Kubo-Tomita approach - II
anisotropies completely contained in the second Moment M2:
remaining tasks:
calculate the second moment for the different contributions to the spin Hamiltonian, find the dominating line-broadening mechanism (and check for the anisotropy)
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The effective spin Hamiltonian
Hamiltonian for strongly correlated spin systems:
H'H i
iii
i Jg 1B SSSH
Zeeman energy
isotropicexchange
additionalinteractions
strong isotropic coupling averages local fields similar to
fast movements of the spins “exchange narrowing“ of the ESR signal
local fluctuating fields local, static resonance shift inhomogenous broadening of
the ESR signal
e.g. crystal fieldanisotropic exchangedipole-dipole interactionhyperfine interaction
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Line broadening mechanisms:
Zero-Field Splitting (crystal field)
Hyperfine interaction
Dipole-dipole interaction
Anisotropic Zeeman interaction
Anisotropic exchange interactions:
Symmetric anisotropic exchange
Antisymmetric anisotropic exchange(Dzyaloshinsky-Moriya interaction)
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Crystal field – zero-field splitting
two independent elements for orthorhombic symmetry:
perturbation theory (spin-orbit coupling perturbs the crystal-field levels):
important source of line broadening and anisotropy (e.g. in manganites)
i
iyixiz SSEDS )( 222ZFSH
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Anisotropic Zeeman interaction
Different Cu sites give rise to superposition of resonance lines, e.g. in CuGeO3 (1D Heisenberg spin-chain)
exchange coupling along the b-directionyields then only one broadened resonance line
Second moment depends on the external field! good candidate for field-dependent linewidth
B. Pilawa, J. Phys.: Cond. Mat. 9, 3779 (1997)
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Line broadening mechanisms:
Zero-Field Splitting (crystal field)
Hyperfine interaction
Dipole-dipole interaction
Anisotropic Zeeman interaction
Anisotropic exchange interactions:
Symmetric anisotropic exchange
Antisymmetric anisotropic exchange(Dzyaloshinsky-Moriya interaction)
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magnetic ions interact indirectly via an intermediate diamagnetic ion (O2-, F1-,..)
potential exchange: describes the self-energy of the charge distribution → ferromagnetic
kinetic exchange:electrons can hop,stabilization of the singletover the triplet state :→ antiferromagnetic
perturbationtreatment:
Reminder: Isotropic superexchange
2
ˆ ˆˆ, with h.c.
1 2 2 , 2 .ab a b ab
V VV t a b
tJ S S J
H H
H H
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Mechanism of anisotropic exchange interaction
This effect adds to the isotropic exchange interaction an anisotropic part (dominant source of anisotropy for S=½ systems!)
free spin couples to the lattice via the spin-orbit interactionHLS=(l·s)
excited orbital states are involved in the exchange process
described as virtual hoppings of electrons via the excited orbital states (additional perturbation term – (LS)-coupling – acts on one site between the orbital levels)
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Theoretical treatment - perturbation theory
4th order: describes 4 virtual electrons hoppings Isotropic superexchange
5th order: 4 hoppings + on-site (LS)-coupling Antisymmetric part of anisotropic exchange = Dzyaloshinsky-Moriya interaction
6th order: 4 hoppings + 2 times on-site (LS)-coupling Symmetric part of anisotropic exchange
= Pseudo-dipol interaction
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Effective spin Hamiltonian of the antisymmetric exchange in form of a cross-product:
direction of D (Dzyaloshinsky-Moriya vector)
Perturbative result:
Antisymmetric part of anisotropic exchange
2j ja b
iD l J
S S
sa sb
ra rb
dVVVVVVVVVVVVVVab a bD r r
j = {x, y, z}, – orbital levels, – energy splitting,lj – operator of the LS-coupling,J – exchange integral.
jiij SSD
sssl baaa
DM
IsoSELSDM
H
HHH
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Symmetric part of anisotropic exchange
Is
2
oSE
( )Г
8
AE LS LS a a a b a a
AE a ab
a b a b
aa
a
b
a b
l s s s l s
S S
lJ S S S S
S S
l
H H H H
H I >
I> I>
1 3 2
I I>
ba
, = {x, y, z};’ – orbital
levels.
Effective exchange constant of the pseudo-dipol interaction is a tensor of second rank and does not allow a simple graphical presentation:
Nonzero elements of can be determined by the product of the matrix elements of the (LS)-coupling and the hopping integrals.
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Uffhh – How to sort out things?
Things are not as bad as they might look like:
Second moments of the broadening mechanisms are known
Geometry and order of magnitude estimations often allow to single out the dominant interactions
Within the KT-approach the temperature dependence is the same for all spin-spin interactions anisotropies are “very helpful”
But… keep in mind:
lots of questions for one single broad absorption line
justification of the KT- approach (dimension)?
possibility of spin-lattice (phonon) relaxation
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summary and outlook
High-temperature KT-approach describes well 3D magnets
Lineshape completely determined by perturbation term
Dominating relaxation mechanism has to explain order of magnitude and anisotropy
Tomorrow:
Success of the KT-approach: the case of manganites
Limits of the KT approach - the case of linear spin chains:e.g. NaV2O5, LiCuVO4, CuGeO3, TiOCl
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Relaxation mechanisms in exchange coupled spin systems – II
Two case files: the manganite system La1-xSrxMnO3 and ESR in spin chain compounds
Joachim Deisenhofer
Université de Genève
Blaubeuren7th September 2006
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outline
Part 1: the manganite system La1-xSrxMnO3
Phase diagram of La1-xSrxMnO3
ESR results
spin relaxation in La0.95Sr0.05MnO3: orbital ordering
Part 2: ESR in 1D Heisenberg spin chains
Magnetic properties of spin S = 1/2 chains
temperature dependence of the ESR linewidth in the model systems NaV2O5, LiCuVO4, CuGeO3
the case of TiOCl
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phase diagram of La1-xSrxMnO3
Hemberger et al., PRB 66, 094410 (2002).
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Jahn-Teller distortion in LaMnO3
cooperative JT-distortions of MnO6 octahedra in the orthorhombic O'-phase up to TJT
750 K
A-type AFM (TN = 140 K):
FM coupling in ac-planes AFM coupling between ac-planes
Huang et al., PRB 55, 14987 (1997)
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estimate for JT? (~0.1eV instead of ~1eV)
orbital excitations (orbitons) in LaMnO3 or multiphonon processes?
OO due to electron-phonon and/or electron-electron interaction?
What is the value of ? Suggestions: 90°, 102°, 106°, 120°
orbital physics in LaMnO3
OO
?
2222
2sin3
2cos yxrzg
xy
z
xy
z
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parameters:intensity:local spin susceptibility
resonance field:g = g - 2.0023local symmetry
linewidth H:spin-relaxation,anisotropic interactions
the ESR Signal
resBHg
3.3 3.4 3.5 3.6
intensity
Hres
linewidth2 H
Abso
rptio
n P
H (kOe)
9.4 GHz LorentzianES
R s
igna
l dP/
dH
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0.00 0.05 0.10 0.15 0.200
200
400
600
800
1000
O"/I / FM O / M
FM
R / MFM
R / MPM
O / I PM
O'/IPM
T (
K)
Sr concentration x
O'/I/CA/A-type
ESR in La1-xSrxMnO3 (0 x 0.2)
Ivanshin et al., PRB 61, 6213 (2000)
phase diagram
Paraskevopoulos et al., J. Phys.:Cond.Mat. 12, 3993 (2000).
linewidth
La0.95Sr0.05MnO3:
still an AFM insulator (TN = 140 K)reduced JT transition temperature TJT = 600 K effective treatment like LaMnO3
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T-dependence of the linewidth
0 200 400 6001.0
1.5
2.0
2.5
ac-plane: FM coupling
b-axis : AFM coupling
O' O
H || a H || c H || b
La0.95
Sr0.05
MnO3
H
(kO
e)
T (K)
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Line broadening mechanisms:
Observed linewidth: ~ 1-2 kOe
• Zero-Field Splitting (crystal field): ~1 kOe
• Anisotropic exchange interactions:
Symmetric anisotropic exchange ~1 Oe
Antisymmetric anisotropic exchange ~1 kOe(Dzyaloshinsky-Moriya interaction)
• Hyperfine interaction: ~10 Oe
• Dipole-dipole interaction ~1 Oe
• Anisotropic Zeeman interaction ~1 Oe
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Crystal field – zero-field splitting
two independent elements for orthorhombic symmetry:
perturbation theory (spin-orbit coupling perturbs the crystal-field levels):
i
iyixiz SSEDS )( 222ZFSH
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Effective spin Hamiltonian of the antisymmetric exchange in form of a cross-product:
direction of D (Dzyaloshinsky-Moriya vector)
Dzyaloshinsky-Moriya interaction
jiij SSD
DMH
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200 400 6000.5
1.0
1.5
2.0
2.5
TJT
= 605 K
T(K)
9.35 GHz
O O'
H || a H || b H || c
La0.95
Sr0.05
MnO3
H (
kOe)
fitting the T-dependence of the linewidth
(E/D)
6(E/D)1)(
),(),(
2
),(
div
N
NCFDregCF
JTDM
CW fTT
Tf
T
T
T
TTTH
3(+1) fit parameters:ZFS = 0.57 kOe
DM = 1.0 kOe
= 0.16 (critical JT-exponent)
+
ZFS-ratio E/D=0.37
Kochelaev et al., Mod. Phys. Lett. B 17, 459 (2003).
JD et al., PRB 68, 214427 (2003).
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consistent ESR fitparameters
ZFS ratio E/D=0.37was obtained from fitting
theg-factor anisotropy
Deisenhofer et al., PRB 68, 214427 (2003). .
100 200 300
1.9
2.0
2.11
2
3
La0.95
Sr0.05
MnO3
gef
f
34 GHz
T (K)
H || a H || b H || c
H
(kO
e)
200 400 6000.5
1.0
1.5
2.0
2.5
TJT
= 605 K
T(K)
9.35 GHz
O O'
H || a H || b H || c
La0.95
Sr0.05
MnO3
H (
kOe)
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zero-field splitting and orbital order
/3tan
sin)/(3-
cos)/-3(2
2
DE
E
D
perturbation theory:
ESR=106
°
ESR
ND=106°
21
221
222
23JT
/2/tan
3
cc
yxcrzc
TQTQg
g
iixiizi
H
neutron diffraction
[Rodriguez-Carvajal et al., PRB 57, R3189 (1998)]
excellentagreement!
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summary - part I
KT-approach allows to describe the temperature dependence and the anisotropy of the linewidth
Dominant contribution are the ZFS and the DM interaction in agreement with neutron diffraction data
type of orbital ordering in La0.95Sr0.05MnO3 has been derived from the analysis of the ESR g-factor and linewidth (ZFS)
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outline
Part 1: the manganite system La1-xSrxMnO3
Phase diagram of La1-xSrxMnO3
ESR results
spin relaxation in La0.95Sr0.05MnO3: orbital ordering
Part 2: ESR in 1D Heisenberg spin chains
Magnetic properties of spin S = 1/2 chains
temperature dependence of the ESR linewidth in the model systems NaV2O5, LiCuVO4, CuGeO3
the case of TiOCl
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Susceptibility of free spins
0 100 200 3000
20
40
60
80
Curie law = M/H ~ 1/T
M = gBS
i
susceptibility
spin Si
magnetic field H
magnetization
(
10-4 e
mu/
mol
)
temperature T(K)
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Interacting spins (3D)
0 100 200 3000
20
40
60
80
-J SiS
j
TN
~1/T
Heisenberg exchange
~ 1/(T - )
Curie-Weiss ~ J
antiferromagnet < 0
order T < TN ~ J
(
10-4 e
mu/
mol
)
temperature T(K)
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Interacting spins (1D)
0 100 200 3000
20
40
susceptibility (S=1/2)Bonner-Fisher(numerical)
Curie-Weiss for T>Jmaximum at T
MAX~J
no magnetic order !
Si
J
TMAX
~(T-)-1
antiferromagnetic chain
(10-4
em
u/m
ol)
temperature T(K)
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Real spin chains
• Only in ideal 1D antiferromagnets no phase transition
• In real systems:
– weak inter-chain coupling not negligible 3D antiferromagnetic order at T << J
– electron-phonon interaction Spin-Peierls transition into dimerized ground state
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Spin-Peierls transition
0 1000
20
= (T )
Si+S
i+1 = 0
T < TSP
dimerization
TSP
(T<TSP
) ~ exp(-/T )
Si
J
TMAX
~J
antiferromagnetic chain
(10-4
em
u/m
ol)
temperature T(K)
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Model systems
LiCuVO4
Cu2+
S = 1/2 chain
J = 40 K
TN = 2.1 K
antiferromagnetic
order
NaV2O5
S = 1/2 per 2 V4.5+
¼-filled ladder
J = 570 K
TCO = 34 K
dimerization
via
charge order
CuGeO3
Cu2+
S = 1/2 chain
J = 120 K
TSP = 14 K
dimerized,
spin-Peierls S = 0
ground state
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Temperature dependence of the ESR linewidth
LiCuVO4 CuGeO3 NaV2O5
0 100 200 3000.0
0.5
1.0
1.5
2.0
TN
H || a H || b H || c
LiCuVO4
H (
kOe
)
T (K)0 100 200 300
0.0
0.5
1.0
1.5
TSP
CuGeO3
H || a H || b H || c
T(K)0 200 400 600
0.0
0.2
0.4
TCO
NaV2O
5
H || a H || b H || c
T(K)
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Universal temperature law
0 100 200 3000.0
0.5
1.0
1.5
2.0
C1 = 60 (5) K
C2 = 15 (5) K
TN
H || a H || b H || c
LiCuVO4
H (
kO
e)
T (K)0 100 200 300
0.0
0.5
1.0
1.5
C1 = 235 (5) K
C2 = 40 (2) K
TSP
CuGeO3
H || a H || b H || c
T(K)0 200 400 600
0.0
0.2
0.4
C1 = 420 (20) K
C2 = 80 (10) K
TCO
NaV2O
5
H || a H || b H || c
T(K)
1
2
( ) ( ) expC
H T HT C
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Limits of the KT-approach
High-temperature approximation fails for T < J (!)
Field theoretical approach (M. Oshikawa and I. Affleck, Phys. Rev. B 65, 134410, 2002)
For temperatures T << J :
H (T ) ~ T for symmetric anisotropic exchange
H (T ) ~ 1/T 2 for antisymmetric DM interaction
in LiCuVO4, CuGeO3 and NaV2O5 symmetric anisotropic
exchange is the dominant relaxation process
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Universal behavior of the linewidth
low temperatures T << J : H (T) ~ T for symmetric anisotropic exchange H (T) ~ 1/T 2 for antisymmetric DM interaction
0 100 200 3000.0
0.5
1.0
1.5
2.0
TN
H || a H || b H || c
LiCuVO4
H (
kO
e)
T (K)0 100 200 300
0.0
0.5
1.0
1.5
TSP
CuGeO3
H || a H || b H || c
T(K)0 200 400 600
0.0
0.2
0.4
TCO
NaV2O
5
H || a H || b H || c
T(K)
What about the antisymmetric interaction? Observation of a low-temperature 1/T2 divergence due to this interaction?
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The system TiOCl
There is no center of inversion between the ions in the Ti-O-layers
antisymmetric anisotropic exchange
[A. Seidel et al., Phys. Rev. B 67, 020405(R) (2003)]
Isotropic exchange constant J = 660 K
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Analysis of the anisotropic exchange mechanisms Dzyaloshinsky-Moriya interaction Pseudo-dipol interaction
D is almost parallel to the b-direction
Dominant component of the tensor of the pseudo-dipol interaction is (aa)
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Temperature dependence of H
60 90 120 1500
100
200
300
400
500TiOCl
a-axis b-axis c-axis
H (
Oe)
T (K)
Tc1
Tc2
2
1
2
( ) ( ) ( ) expDM AE
CJH T K K
T T C
[Oe] KAE (∞) KDM (∞)
H || a 1429 1.397H || b 765 2.319H || c 930 1.344
The temperature and angular dependence of H can be described as a competition of the symmetric and the antisymmetric exchange interactions!
[Zakharov et al., PRB 73, 094452 (2006)]
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Summary
Anisotropic exchange dominates the ESR line broadening in low dimensional S=1/2 transition-metal oxides in contrast to estimations based on the KT-approach
Universal temperature dependence of the ESR linewidth in spin chains with dominant symmetric anisotropic exchange
Interplay of antisymmetric Dzyaloshinsky-Moriya and symmetric anisotropic exchange in TiOCl