arXiv:0705.0371v1 [cond-mat.stat-mech] 2 May 2007
Magneto-elastic waves in crystals of magnetic molecules
C. Calero, E. M. Chudnovsky, and D. A. Garanin
Department of Physics and Astronomy,
Lehman College, City University of New York,
250 Bedford Park Boulevard West, Bronx, New York 10468-1589, U.S.A.
February 1, 2008
We study magneto-elastic effects in crystals of magnetic molecules. Coupled equations of motion
for spins and sound are derived and the possibility of strong resonant magneto-acoustic coupling
is demonstrated. Dispersion laws for interacting linear sound and spin excitations are obtained
for bulk and surface acoustic waves. We show that ultrasound can generate inverse population of
spin levels. Alternatively, the decay of the inverse population of spin levels can generate ultrasound.
Possibility of solitary waves of the magnetization accompanied by the elastic twists is demonstrated.
PACS numbers: 75.50.Xx, 73.50.Rb, 75.45.+j
Crystals of molecular magnets are paramagnets1,2that
have the ability to maintain macroscopic magnetization
for a long time in the absence of the external magnetic
field. This is a consequence of the magnetic bi-stability
of individual molecules3that in many respects behave as
superparamagnetic particles. The latter is due to a large
spin (e.g., S = 10 for Mn-12 and Fe-8) and high magnetic
anisotropy of the molecules. Together with quantization
of spin energy levels this leads to a distinctive feature
of molecular magnets: A staircase hysteresis curve in a
macroscopic measurement of the magnetization.4
Hybridization of electron paramagnetic resonance
(EPR) with longitudinal ultrasonic waves has been stud-
ied by Jacobsen and Stevens5within a phenomenologi-
cal model of magneto-elastic interaction proportional to
the magnetic field.General theory of magneto-elastic
effects on the phonon dispersion and the sound veloc-
ity in conventional paramagnets has been developed by
Dohm and Fulde.6The advantage of molecular magnets
is that they, unlike conventional paramagnets, can be
prepared in a variety of magnetic states even in the ab-
sence of the magnetic field. Spontaneous transitions be-
tween spin levels in molecular magnets are normally due
to the emission and absorption of phonons. Interactions
of molecular spins with phonons have been studied in the
context of magnetic relaxation,7,8,9,10conservation of an-
gular momentum,11,12,13phonon Raman processes,14and
phonon superradiance.15Parametric excitation of acous-
tic modes in molecular magnets has been studied.16,17
It has been suggested that surface acoustic waves can
produce Rabi oscillations of magnetization in crystals of
molecular magnets.18In this paper we study coupled dy-
namics of paramagnetic spins and elastic deformations at
a macroscopic level.
When considering magneto-elastic waves in paramag-
nets the natural question is why the adjacent spins should
rotate in unison rather than behave independently. In
ferromagnets the local alignment of spins is due to the
strong exchange interaction. Due to this interaction the
length of the local magnetization is a constant through-
out the ferromagnet. We shall argue now that a some-
what similar quantum effect exists in a system of weakly
interacting two-level entities described by a fictitious spin
1/2. Indeed, since any product of Pauli matrices reduces
to a single Pauli matrix σα, interaction of N independent
two-state systems with an arbitrary field A(r) should be
linear on σα,
αAβ(rn) , (1)
where σ(n)describes a two-state system located at a
point r = rn. If A was independent of coordinates, then
the Hamiltonian (1) would reduce to
H = gαβΣαAβ, (2)
is the total fictitious spin of the system. In this case
the interaction Hamiltonian would commute with Σ2,
thus preserving the length of the total fictitious “mag-
netization”. This observation is crucial for understand-
ing Dicke superradiance:19A system of independent two-
state entities behaves collectively in a field whose wave-
length significantly exceeds the size of the system. When
the wavelength of the field is small compared to the size
of the system but large compared to the distance be-
tween the two-state entities, the same argument can be
made about the rigidity of Σ =?σ(n)summed up over
Consequently, the system that has been initially prepared
in a state with all spins up, and then is allowed to evolve
through interaction with a long-wave Bose field, should
conserve the length of the local “magnetization” in the
same way as ferromagnets do.
The relevance of the above argument to the dynam-
ics of magnetic molecules interacting with elastic defor-
mations becomes obvious when only two spin levels are
the distances that are small compared to the wavelength.
important. This is the case when the low-energy dynam-
ics of the molecular magnet is dominated by, e.g., tunnel
split spin-levels or when the magneto-acoustic wave is
generated by a pulse of sound of resonant frequency. Re-
cently, experiments with surface acoustic waves in the
GHz range have been performed in crystals of molecu-
lar magnets.20The existing techniques, in principle, al-
low generation of acoustic frequencies up to 100 GHz.21
This opens the possibility of resonant interaction of gen-
erated ultrasound with spin excitations. In this paper we
study coupled magneto-elastic waves in the ground state
of a crystal of molecular magnets. We derive equations
describing macroscopic dynamics of sound and magne-
tization and show that high-frequency ultrasound inter-
acts strongly with molecular spins when the frequency of
the sound equals the distance between spin levels. We
obtain the dispersion relation for magneto-elastic waves
and show that non-linear equations of motion also pos-
sess solutions describing solitary waves of magnetization
coupled to the elastic twists.
The paper is organized as follows. The model of spin-
phonon coupling is discussed in Section II where coupled
magneto-elastic equation are derived. Linear magneto-
elastic waves are studied in Section III where we obtain
dispersion laws for bulk and surface acoustic waves. Non-
linear solitary waves are studied in Section IV. Sugges-
tions for experiments are made in Section V.
II.MODEL OF MAGNETO-ELASTIC
We consider a molecular magnet interacting with
a local crystal field described by a phenomenological
anisotropy Hamiltonianˆ HA. The spin cluster is assumed
to be more rigid than its elastic environment, so that the
long-wave crystal deformations can only rotate it as a
whole but cannot change its inner structure responsible
for the parameters of the Hamiltonianˆ HA. This approx-
imation should apply to many molecular magnets as they
typically have a compact magnetic core inside a large unit
cell of the crystal. In the presence of deformations of the
crystal lattice, given by the displacement field u(r), local
anisotropy axes defined by the crystal field are rotated
by the angle
2∇ × u(r,t).
As a consequence of the full rotational invariance of the
system (spins + crystal lattice), the rotation of the lattice
is equivalent to the rotation of the operatorˆS in the
opposite direction, which can be performed by the (2S +
1) × (2S + 1) matrix in the spin space,13
Therefore, the total Hamiltonian of a molecular magnet
in the magnetic field B must be written as
ˆR = eiˆS·δφ.(5)
ˆ H = e−iˆS·δφ ˆ HAeiˆS·δφ+ˆ HZ+ˆ Hph,(6)
whereˆ HAis the anisotropy Hamiltonian in the absence
of phonons,ˆ HZ= −gµBB·ˆS is the Zeeman Hamiltonian
andˆ Hphis the Hamiltonian of harmonic phonons. The
angle of rotation produced by the deformation of the lat-
tice is small, so one can expand Hamiltonian (6) to first
order in the angle δφ and obtain
ˆ H ≃ˆ H0+ˆ Hs−ph,(7)
whereˆ H0is the Hamiltonian of non-interacting spins and
ˆ H0=ˆ HS+ˆ Hph,
andˆ Hs−phis the spin-phonon interaction term, given by
ˆ HS=ˆ HA+ˆ HZ, (8)
ˆ Hs−ph= i
A.Coupling of spins to the elastic twists
For certainty, we consider a crystal of molecular mag-
nets with the anisotropy Hamiltonian
ˆ HA= −DˆS2
whereˆV is a small term that does not commute with
theˆSzoperator. This term is responsible for the tunnel
splitting, ∆, of the levels on resonance.
kBT,gµBB ? ∆, when the frequency of the displace-
ment field u(r) satisfies ω ≪ 2DS/?, only the two lowest
states ofˆ HA are involved in the evolution of the sys-
tem. Thus, one can reduce the spin-Hamiltonian of the
molecular magnet to an effective two-state Hamiltonian
in terms of pseudospin-1/2 operators ˆ σi,
2(Wez+ ∆ex) · ˆ σ ,(11)
where ˆ σi are the Pauli matrices in the basis of theˆSz-
states close to the resonance between |S? and |−S?, and
W = ES− E−Sis the energy difference for the resonant
states at ∆ = 0. The non-degenerate eigenfunctions of
√2(C±|S? ∓ C∓| − S?)
In terms of |ψ∓? the Hamiltonian (11) can be written as
W2+ ∆2 ˆ˜ σz, (14)
whereˆ˜ σi are now the Pauli matrices in the new basis
|ψ±?, i.e.,ˆ˜ σz = |ψ+??ψ+| − |ψ−??ψ−|. The projection
of the spin-phonon interaction Hamiltonian (9) onto this
new two-state basis results in
?ψi|ˆ Hs−ph|ψj?|ψi??ψj| = δφzS∆ˆ˜ σy,
withˆ˜ σy= −i|ψ+??ψ−| + i|ψ−??ψ+|. The total Hamilto-
nian (6) of a single molecular magnet becomes
ˆ H(eff)= −1
2b(eff)·ˆ˜ σ +ˆ Hph,
W2+ ∆2ez− 2δφzS∆ey.(16)
Here we have assumed that the perturbation introduced
by the spin-phonon interaction is much smaller than the
perturbationˆV producing the splitting ∆, which will usu-
ally be the case. Note also that ∆ and W can in gen-
eral be made r-dependent to account for possible inho-
mogeneities of the crystal.
When considering magneto-elastic excitations we will
need to know whether they are accompanied by a non-
zero local magnetization of the crystal. For that rea-
son it is important to have the magnetic moment of the
(with g being the gyromagnetic ratio and µB being the
Bohr magneton), in terms of its wave function
|Ψ? = K+|ψ+? + K−|ψ−?,(18)
where K± are arbitrary complex numbers satisfying
|K−|2+|K+|2= 1. With the help of Eq. (12) one obtains
∆?ˆ˜ σx? − W?ˆ˜ σz?
We want to describe our system of N spins in terms of
the spin field
ˆ n(r) =
ˆ˜ σiδ(r − ri),(20)
satisfying commutation relations
[ˆ nα(r), ˆ nβ(r′)] = 2iǫαβγˆ nγ(r)δ(r − r′).
In terms of this field the total Hamiltonian becomes
ˆ H = −1
d3r ˆ n(r) · b(eff)(r) +ˆ Hph.(22)
The classical pseudo-spin field n(r,t) can be defined as
n(r,t) = ?ˆ n(r)?,(23)
where ?...? contains the average over quantum spin states
and the statistical average over spins inside a small vol-
ume around the point r. If the size of that volume is small
compared to the wavelength of the phonon displacement
field, then, as has been discussed in the Introduction,
n2(r) should be approximately constant in time. Accord-
ing to equations (17), (19) and (20), the magnetization
is given by
Mz(r) = gµBS∆nx(r) − W nz(r)
The dynamical equation for the classical pseudo-spin
field n(r,t) is
[ˆ H, ˆ n]
which, with the help of Eq.(21), can be written as
= n(r,t) × b(eff)(r,t).(26)
In this treatment we are making a common assumption
that averaging over spin and phonon states can be done
independently.This approximation is expected to be
good in the long-wave limit.
The dynamical equation for the displacement field is
where σαβ = ∂h/∂eαβ is the stress tensor, eαβ =
∂uα/∂xβis the strain tensor, h is the Hamiltonian den-
sity of the system inˆ H =?d3rh(r), and ρ is the mass
ric part originating from the magneto-elastic interaction
in the Hamiltonian,
density. Note that the stress tensor has an antisymmet-
σαβ = σ(s)
This implies that at each point r there is a torque per
τα(r) = −δαzS∆ny(r),(29)
created by the interaction with the magnetic system.
This effect can be viewed as the local Einstein – de Haas
effect: Spin rotation produces a torque in the crystal lat-
tice due to the necessity to conserve angular momentum.
With the help of equations (16), (22), and (27), using
standard results of the theory of elasticity, one obtains
t)∇α(∇ · u) =S∆
where cland ctare velocities of longitudinal and trans-
verse sound. The source of deformation in the right hand
side of this equation is due to the above-mentioned torque
generated by the spin rotation.
Equations (26) and (30) describe coupled motion of
the pseudospin field n(r,t) and the displacement field
u(r,t). It is easy to see from these equations that in ac-
cordance with the argument presented in the Introduc-
zis independent of time. It may, nev-
ertheless, depend on coordinates, reflecting the structure
of the initial state. In this paper we study cases in which
the crystal of molecular magnets was initially prepared
in the ground state n = n0ezwith n0being the concen-
tration of magnetic molecules. In this case the dynamics
of n(r) described by equations (26) and (30) reduces to
its rotation, with the length of n(r) being a constant n0.
Remarkably, this situation is similar to a ferromagnet,
despite the absence of the exchange interaction.
III.LINEAR MAGNETO-ELASTIC WAVES
For magnetic molecules whose magnetic cores are more
rigid than their environments, only the transverse part
of the displacement field (with ∇ · u(r) = 0) interacts
with the magnetic degrees of freedom. This is a conse-
quence of the fact that the elastic deformation produced
by the rotation of local magnetization is a local twist of
the crystal lattice, required by the conservation of angu-
lar momentum. Let us consider then a transverse plane
wave propagating along the X-axis. From Eqs.(26) and
(30) one obtains
W2+ ∆2− nzS∆∂uy
We shall study linear waves around the ground state
|ψ+? corresponding to nz = n0,nx,y = 0,uy = 0. The
perturbation around this state results in nonzero nx,y
and uy. Linearized equations of motion are
W2+ ∆2− S∆n0∂uy
W2+ ∆2. (32)
For uy,nx,y ∝ exp(iqx − iωt), the above equations be-
FIG. 1: Interacting sound and spin modes. Notice the gap
below spin resonance ω0.
The spectrum of coupled excitations is given by
In the vicinity of the resonance,
one can write
ω = ω0(1 + δ)(36)
with δ to be determined by the dispersion relation. Sub-
stituting equations (35) and (36) into Eq. (34), one ob-
δ = ±
that describes the splitting of two coupled modes at the
resonance. The repulsion of elastic and spin modes is
illustrated in Fig. 1. The relative splitting of the modes
reaches maximum at W = 0 (?ω0= ∆):
where M = ρ/n0 is the mass of the volume containing
one molecule of spin S. Notice also another consequence
of Eq. (34): The presence of the energy gap below ω0=
√W2+ ∆2/? (see Fig. 1). The value of the gap follows
from Eq. (34) at large q. It equals 2δ2ω0. This effect
is qualitatively similar to the one obtained in Ref.
from an ad hoc model of spin-phonon interaction.
contrast with that model our results for the splitting of
the modes and for the gap do not contain any unknown
interaction constants as they are uniquely determined by
the conservation of the total angular momentum (spin +
According to equations (33) and (34) the Fourier trans-
forms of nyand uyare related through
Due to the condition of the elastic theory quy≪ 1, the
absolute value of the ratio ny/n0is generally small, un-
less ω is close to ω0. This means that away from the
resonance the sound cannot significantly change the pop-
ulation of excited spin states. At the magneto-elastic
resonance, substituting equations (36) and (37) into the
above equation, one obtains:
Although this relation is valid only at |ny| ≪ n0, it allows
one to estimate the amplitude of ultrasound that will
significantly affect populations of spin states. We shall
postpone the discussion of this effect until Section V.
Meantime let us compute the magnetization generated
by the linear elastic wave, uy = u0cos[q0(x − ctt)]), in
resonance with our two-state spin system. The last of
Eqs. (32) yields nx= i(ω/ω0)ny. Then, with the help of
Eq. (24) and Eq. (40) one obtains
q0u0cos[q0(x − ctt)]. (41)
So far we have investigated coupled magneto-elastic
waves in the vicinity of the ground state, nz= n0. Eqs.
(31) also allow one to obtain the increment, Γ, of the
decay of the unstable macroscopic state of the crystal,
nz= −n0, in which all molecules are initially in the ex-
cited state |ψ−?. In fact, the result can be immediately
obtained from equations (32) – (34) by replacing n0with
−n0. It is then easy to see from Eq. (34) that in the vicin-
ity of the resonance the frequency acquires an imaginary
part that attains maximum at the resonance where
ω = ω0(1 ± i|δ|). (42)
The mode growing at the rate Γ = ω0|δ| represents the
decay of |ψ−? spin states into |ψ+? spin states, separated
by energy ?ω0. This decay is accompanied by the ex-
ponential growth of the amplitude of ultrasound of fre-
B. Surface waves
Magneto-elastic coupling in crystals of molecular mag-
nets can be studied with the help of surface acoustic
FIG. 2: Geometry of the problem with surface acoustic waves.
waves (see Discussion). To describe the surface waves
we chose a geometry in which the surface of interest
is the XZ-plane and the solid extends to y > 0 with
waves running along the direction that makes an angle
θ with the X-axis, see Fig.2.
that the displacement field u(r,t) and the components
nx(r,t),ny(r,t) have the form
As usual22we assume
A = A0e−αyei(qxx+qzz)e−iωt.(43)
It is convenient to express the components of the dis-
placement field in the coordinate system defined by
(el,et,ep), see Fig.2,
ux = ulcosθ − utsinθ
uy = up
uz = ulsinθ + utcosθ.(44)
Equations of motion for ul, ut, and up follow from Eq.
t(α2− q2)?ut +
2ρ∆q cosθny= 0.
lq2?ul − iαq(c2
tq2?up − iαq(c2
It is easy to see that for θ ?= kπ , k = 0,1,2... and ny?= 0,
the transverse component ut cannot be zero, contrary
to the case of Rayleigh waves. This is the signature of
As in the analysis of bulk waves, we shall study the
linear waves around the ground state corresponding to
the pseudospin field polarized in the Z-direction, nz =
n0,nx,y = 0. The excitations above this state are de-
scribed by Eqs. (26), which become
− i?ωnx = S∆?−α(ulcosθ − utsinθ) − iq?cosθup
−i?ωny = −
Substitution of these two equations into Eqs.(45) leads
to a homogeneous system of algebraic equations for ul,
ut, and up, that have a non-zero solution only if its deter-
minant equals zero. From this condition we obtain three
values of the coefficient α that describe the decay of the
wave away from the surface:
tq2− ω2+ ηq2cos2θ
η + c2
2M[?2ω2− (W2+ ∆2)]. (48)
Note that if there are no spins (S = 0), then α3 = α2
and one obtains decay coefficients for ordinary Rayleigh
The general plane wave solution for the components of
the displacement field can be written as
and i = l,t,p. For each k, the amplitudes u(k)
are related through Eqs.(45) (there are two independent
equations, so we can express, e.g., u(k)
l0).Therefore, there still are three unknowns, say
l0. The boundary conditions for the stress
tensor at the surface, σiy|y=0 = 0, provide a system of
homogeneous equations for u(1)
terminant must be zero to allow for non-trivial solution.
From this last condition we obtain the dispersion relation
for surface magneto-elastic waves:
is the amplitude corresponding to each αk
p0in terms of
l0, whose de-
q2S2∆2ω0cos2θ + 2Mc2
= 0. (50)
This equation should be solved numerically to obtain the
dispersion law for magneto-elastic modes. Qualitatively,
the repulsion of the modes is similar to the one shown in
An interesting feature of Eqs. (31) is the existence of
transverse non-linear plane wave solutions of the form
ui= ui(x − vt),ni= ni(x − vt). For such a choice, Eq.
where ¯ x ≡ x−vt and the constant of integration was put
zero assuming that there is no duy/d¯ x independent from
ny. Substituting this into the equations of motion for n,
Eqs.(31), one obtains
= ny− γnynz
ξ ≡¯ x√W2+ ∆2
t− v2)√W2+ ∆2
The system of Eqs.(52) can be reduced to
nz = C −1
1 − γC +1
where C is a constant of integration. The first integral
of the last differential equation is
2(1 − γC)n2
y+ A ≥ 0, (56)
where A is another integration constant.
We are interested in real bounded solutions of Eq.(55)
with nyvanishing at x−vt → ±∞, so that the integration
constant A must be zero. In this case, for the right hand
side of Eq.(56) to be positive we must have 1 − γC < 0.
Then, the solution of Eq.(55) is
√γC − 1e±√γC−1(ξ−ξ0)
γ + γe±2√γC−1(ξ−ξ0)
From the equations
,nz= C −1
one determines with the help of the condition n2
0that C = ±n0 .
|γ| > 1/n0for the equation (55) to have a solution satisfy-
ing the conditions specified above. Setting the reference
point ξ0= 0 one obtains
Therefore, γ must satisfy
ny(ξ) = ±2
|γ|n0− 1 sech
|γ|n0− 1 ξ
FIG. 3: Magnetization inside the soliton as a function of ξ for
W = 0.
nz(ξ) = ±1 ∓ 2|γ|n0− 1
|γ|n0− 1 ξ
In these formulas, the upper sign corresponds to γ > 0
and the lower sign to γ < 0.
Eq. (60) describes a solitary wave of a characteristic
travelling at a speed v. The parameter γ given by Eq.
(53) is determined by v, which is the only free parameter
of the soliton. The magnetization inside the soliton is
given by Eq. (24) with nx and nz defined by equations
(58) – (60). At, e.g., W = 0
Mz= ∓gµB2S(|γ|n0− 1)
|1 − v2/c2
√W2+ ∆2> 1(63)
requires v to be very close to the speed of sound ct. This
is a consequence of ∆ being very small compared to Mc2
Note that the maximal value of the magnetization inside
|Mz| = gµBS
is, in general, of the order of saturation magnetization
M0= gµBSn0. We should also note that although the
above non-linear solution of the equations of motion for-
mally allows v to be both slightly lower or slightly higher
than ct, the supersonic soliton should be unstable with
respect to Cherenkov radiation of sound waves.
Eq. (38) provides the splitting of the bulk sound fre-
quency in a magnetized crystal of magnetic molecules in
the vicinity of the resonance between sound and spin lev-
els. At a zero field bias (W = 0) the resonant condition,
∆ = ?ctq, should be easily accessible at low ∆. However,
the splitting given by Eq. (38) will be very small unless ∆
is in the GHz range or higher. Such a large ∆ will be also
beneficial for decreasing inhomogeneous broadening of ∆
and for insuring low decoherence of quantum spin states.
Surface acoustic waves can, in principle, be generated up
to 100GHz21. They may also be easier to use for the
observation of the discussed splitting. By order of mag-
nitude it will still be given by Eq. (38). Substituting into
this equation S = 10, ∆ ∼ 0.1K (frequency f in the GHz
be observable if the quality factor of ultrasound in the
GHz range exceeds 100. The magneto-elastic nature of
the splitting can be confirmed through its dependence on
the angle between the wave vector and the easy magne-
tization axis of the crystal, see Sec. III-B. Observation
of the gap, 2δ2ω0, in the excitation spectrum (see Fig.
1) will be more challenging. For practical values of δ the
gap is likely to be small compared to the width of the
spin resonance and the width of the ultrasonic mode in
the GHz range.
Eq. (40) shows that at M ∼ 10−21g and ω0∼ 1010s−1
ultrasound of amplitude u0 ∼ 0.1nm will significantly
affect population of spin levels. Moreover, it will result
in the oscillating magnetization of large amplitude, Eq.
(41). We have also demonstrated that one can prepare
the crystal in the excited spin state and generate ultra-
sound due to the decay of the population of that state.
This result is another confirmation of the phonon laser
effect suggested in Ref. 15. Equations (42) and (38) show
that at ω0 ∼ 1010s−1the amplitude of the sound wave
may grow at the rate as high as Γ ∼ 108s−1. Magneto-
elastic effects studied in this paper should be sensitive
to the decoherence of spin states. However, when the
oscillation of spin population is driven by the external
acoustic wave, the latter should force the phase coherence
upon the spin system. To provide the resonance condi-
tion, the broadening of the level splitting due to disorder
and dipolar fields should be small compared to ∆. If it
is not, the tunnel splitting, ∆, should be increased by
applying a sufficiently large transverse magnetic field.
One fascinating prediction of our theory is the exis-
tence in molecular magnets of solitary waves of the mag-
netization reversal coupled to elastic twists. Such waves
have quantum origin as they are related to the quantum
splitting of spin-up and spin-down states. They can be
ignited in experiment that starts with all molecules in the
ground state. Such a state of the crystal has zero magne-
tization as the molecules are in a superposition of spin-
up and spin-down states. The soliton discussed above
is characterized by a narrow region of a large non-zero
magnetization that propagates through the solid with the
t∼ 105K, one obtains δmax∼ 10−2. This will
8 Download full-text
velocity close to the speed of transverse sound. It can be
generated by, e.g., a localized pulse of the magnetic field
or by a localized mechanical twist, and detected through
local measurements of the magnetization. In general the
width of the soliton, given by Eq. (61), is of order of the
wavelength of sound of frequency
wider solitons are allowed if |γ|n0 → 1. In experiment
this width should depend on the width of the field pulse
or the size of the twisted region that generates the soli-
√W2+ ∆2/?, though
This work has been supported by the NSF Grant No.
1E. M. Chudnovsky and J. Tejada, Lectures on Magnetism
(Rinton Press, 2006).
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and L. J. de Jongh, Phys. Rev. Lett. 93, 117202 (2004).
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