Depinning of Driven Systems: Time-resolved x-ray...
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Depinning of Driven Systems: Time-resolved x-ray scattering measurements of the charge-density waves in NbSe3
Joel Brock, Cornell University (KITP Pattern Formation Conference 8/19/03) 1
KITP August 19, 20031
Brock Research Group
Depinning of Driven Systems: Time-resolved x-ray scattering measurements of the
charge-density waves in NbSe 3
Joel D. Brock
Professor and DirectorSchool of Applied & Engineering Physics
Director
G-line Division, Cornell High Energy Synchrotron Source (CHESS)Cornell University, Ithaca, NY
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Acknowledgements
Kristin Ringland 1, Robin Sampson, Jun-Dar Su, and Yanping Li 2
School of Applied & Engineering PhysicsCornell University, Ithaca, NY 14853
S.G. Lemay 3, and R.E. ThorneDepartment of Physics
Cornell University, Ithaca, NY 14853
Mark SuttonDepartment of Physics
McGill University, Montreal, CANADA
1. McKinsey & Company, Stockholm, Sweden
2. Applied Materials Inc., Santa Clara, CA
3. Delft University of Technology, Delft, The Nether lands
Depinning of Driven Systems: Time-resolved x-ray scattering measurements of the charge-density waves in NbSe3
Joel Brock, Cornell University (KITP Pattern Formation Conference 8/19/03) 2
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Outline
• Intro to CDW Physics/Phenomenology• Intro to Modern X-ray Technology
• Properties of NbSe 3
• Experimental Details – Sample quality• Kinetics of Relaxation from Sliding State• Structure of the Sliding & Zero-field cooled
States• Low Temperature Phase Diagram• Coherent X-ray Measurements: “Speckle”• Summary
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CDW Basics
• At T=0, Electronic energy is lowered by opening an energy gap at the Fermi surface.
• Periodic potential required to do this can be supplied by a longitudinal lattice-density wave (LDW).
• Elastic energy cost for LDW is less than electronic energy gain
In Quasi-One-Dimensional Metals...
( )( )
( ) ( )( )rrbkuru
rrbk
F
F
rrrrr
rr
φ
φρρ
+⋅=
+⋅∆+=
ˆ2sin
ˆ2cos
0
Depinning of Driven Systems: Time-resolved x-ray scattering measurements of the charge-density waves in NbSe3
Joel Brock, Cornell University (KITP Pattern Formation Conference 8/19/03) 3
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Sliding CDWs
Since the wavelength of a CDW is (usually) incommen surate with the periodicity of the lattice, there is no el astic restoring force. In an ideal crystal, the CDW can “slide” rigi dly without friction.
Defects in the crystal lattice provide “friction,” cr eating a threshold field.
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General Predictions
• At zero field, periodicity of lattice →→→→ quasi long-range translational order
• Far below threshold: thermal activation →→→→ creep
• Topological defects →→→→plastic flow
• Near elastic depinning threshold: similar to standard critical phenomena (velocity is order parameter).
• Far above threshold: recovery of perfect lattice
L. Balents and M.P.A. Fisher, PRL, 75, 4270 (1995)
Depinning of Driven Systems: Time-resolved x-ray scattering measurements of the charge-density waves in NbSe3
Joel Brock, Cornell University (KITP Pattern Formation Conference 8/19/03) 4
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X-ray scattering is nearly ideal probe
• non-perturbative• direct interpretation• structural measurement
( ) ( ) ( )[ ] ( ) ( )∫Ω
−⋅−±⋅Ω
= rdeeeuqJtqS titrirqbkGi Frrrr rrrrr
,0,ˆ22
01
1, φφ
( ) ( ) ( ) ( ) ( )[ ]2,0,2
1,
2
1,0, ttxtxgtitxi eeee
φφφφ −−−− ≈=
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Synchrotron Radiation1 / γElectron orbit
Radius, R(2πR ~ 750 m)
Radiation Cone
Dipole radiation pattern of accelerated charges is “folded” forward by special relativity, producing “search light” radiation pattern.
Characteristic opening angle is given by
γ-1 = (E/mc2)-1 ~ 10-4 ~ 0.1 mrad
Depinning of Driven Systems: Time-resolved x-ray scattering measurements of the charge-density waves in NbSe3
Joel Brock, Cornell University (KITP Pattern Formation Conference 8/19/03) 5
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Increase Power with Wiggler Source
Top View
Side View
N NN NS S SS
N S N S N S N S
1/γ
K/γ
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Why bother?
At comparable energy and angular resolution…• Rotating anode generator: ~10 7 x-rays/second• Synchrotron source: 10 12 – 1015 x-rays/second
Not your father’s x-ray source!
Depinning of Driven Systems: Time-resolved x-ray scattering measurements of the charge-density waves in NbSe3
Joel Brock, Cornell University (KITP Pattern Formation Conference 8/19/03) 6
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Brock Research Group
X-ray Synchrotron Sources
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• monoclinic unit cell
• whisker axis || to b, b *
• width || c
• thickness || a*
• TP1 ≈ 145 K, TP2 ≈ 59 K
• Q1 ≈ (0 0.243 0)
• Q2 ≈ (0.5 0.26 0.5)
NbSe3 Properties
Hodeau et al., J. Phys. C 11, 4117 (1978)
a = 10.009 Åb = 3.4805 Å
β = 109.47
c = 15.629 Å
Depinning of Driven Systems: Time-resolved x-ray scattering measurements of the charge-density waves in NbSe3
Joel Brock, Cornell University (KITP Pattern Formation Conference 8/19/03) 7
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NbSe3 Properties
E-T Phase Diagram Reciprocal Space
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Experimental Geometry
• Beryllium Top Hat– x-ray transparent– contains He exchange gas
• Alumina Substrate– supports NbSe 3 whisker– four point connections
• Transmission Scattering Geometry
Depinning of Driven Systems: Time-resolved x-ray scattering measurements of the charge-density waves in NbSe3
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Experimental Geometry
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Sample Characterization
To study ‘intrinsic’ CDW properties, we need to make sure that there are no sample defects that might create extrinsic effects. Therefore, the criteria for choosing samples are:
• Very regular cross sections• Darwin limited (020) Bragg
peaks• Excellent mode-locking
characteristics
Depinning of Driven Systems: Time-resolved x-ray scattering measurements of the charge-density waves in NbSe3
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Observe Predicted Recovery of QLRO?
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Diffraction Topography
Y. Li, et al., PRL 83, 3514 (1999).
Use diffraction topography to image phase shifts across boundaries
Depinning of Driven Systems: Time-resolved x-ray scattering measurements of the charge-density waves in NbSe3
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Due to grain structure of sample!
E>ET
CDW motion:
grainboundary
Crystal width = 30 µm
Y. Li, et al., PRL 83, 3514 (1999)
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Shear in Sliding CDWs
Y. Li, et al., PRL 83, 3514 (1999)
Depinning of Driven Systems: Time-resolved x-ray scattering measurements of the charge-density waves in NbSe3
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Time-resolved Experiments
Weak scattering signal requires signal averaging or time-periodic signal.
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“Typical” Pinning Data
• Current is on for t < 0• Current is off for t > 0• CDW satellite responds by
– sharpening– (possibly) small rotation– integrated intensity remains
constant
K.L. Ringlandet al., PRL 82(9), 1923-26 (1999).
Depinning of Driven Systems: Time-resolved x-ray scattering measurements of the charge-density waves in NbSe3
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Fukuyama-Lee-Rice Theory
Microscopic Model (based on Fröhlich Hamiltonian)
• P.A. Lee, T.M. Rice and P.W. Andersen. “Conductivity from charge or spin density waves.” Solid State Comm., 14, 703 (1974).
Phase only approximation (FLR)
• H. Fukuyama. “Pinning in Peierls-Fröhlich state and conductivity.” J. Phys. Soc. Japan, 41(2), 513 (1976).
• H. Fukuyama and P.A. Lee. “Dynamics of charge density wave. I. Impurity pinning in a single chain.” Phys. Rev. B, 17(2), 535 (1978).
• P.A. Lee and T.M. Rice. “Electric field depinning of charge density waves.” Phys. Rev. B, 19(8), 3970 (1979).
( )[ ] ( ) ( ) ( )∫ ∑
−+⋅−+∇= φφρφφ ErQRrVrdtrH
ii
KFLR CDW
rrrrrrcos, 2
2
( ) ( )ziex φψ=Ψ
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Solution to Equation of Motion
( ) ( )( ) ( )2
42
2
222 22
1
~
0,~
,~ DtqDtq e
qD
qVeqtq −− −+=
r
rr φφ
( ) ( )∫∞
≅r
tqqdtrg/1
23 ,
~,
rrr φ
B. Jancovici, Phys. Rev. Lett., 19, 20 (1967).
( )rVDt
r+∇=∂ φφ 2• Use FLR Hamiltonian• Ignore thermal noise• E=0• Spatial Fourier transform• Integrate EOM• Assume δδδδ-correlated,
quenched random-field• Integrate to obtain g(r,t)
Depinning of Driven Systems: Time-resolved x-ray scattering measurements of the charge-density waves in NbSe3
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Approximate Form for g(x,t)
( ) 2
/1
3 ,),( tkkdtxgx
rr
φ∫∞
≅ ( )
≤≥
≅1//
1//, µ
µ
ξξ
txforx
txforxtxg
f
i
Numerical Integration of FLR Solution Approximation to FLR Solution
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Approximate form for S(q,t)
Piece-wise integration using approximate form for g(x,t)
( ) ( ) ( ) if rrqQGi
t
rrqQGit
eerdeerdtqS ξξ
µ
µ
/3/
0
3,rrrrvrrrrv rrr −⋅−±
∞−⋅−±
∫∫ +≈
( ) ( )( )[ ] ( )
( ) 2/12/1 /22
/22 1
11
, ττ
ξξ
ξξ t
i
it
f
f eGq
eGq
tqS −
⊥⊥
−
⊥⊥ −++−
−+≈r
Depinning of Driven Systems: Time-resolved x-ray scattering measurements of the charge-density waves in NbSe3
Joel Brock, Cornell University (KITP Pattern Formation Conference 8/19/03) 14
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Approximate form for S(q,t)
• µ ≠ ½, but keep α = ½• initial (sliding) state not Lorentzian• integrate over broad directions of
resolution function
( ) ( )( )[ ] ( ) ( ) µµ ττ
ξξ //
220,1
1, tt
f
f etqSeGq
tqS −⊥
−
⊥⊥⊥ =+−
−+≈
Simple “stretched exponential” kinetics
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Results of Analysis
K.L. Ringlandet al., PRL 82(9), 1923-26 (1999).
• Solid lines are best fit results using only data from t > τ
• 2 parameter fit (µ and τ)
• χ2 ≈ 1.5
• ~ 2 × 104 degrees of freedom.• Arrows indicate times for the
“slices” depicted.
Depinning of Driven Systems: Time-resolved x-ray scattering measurements of the charge-density waves in NbSe3
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Results of Analysis
• µ is constant within fitting errors
• µ is not consistent with the value 1/2
• doping with tantalum does not have a statistically significant effect
• τ varies by nearly 6 orders of magnitude.This suggests an activated process with an activation energy barrier roughly equal to the CDW gap
K.L. Ringlandet al., PRL 82(9), 1923-26 (1999).
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Structure of Sliding State
• Expect g(x) ≈ x2α
• Sliding state has αααα = 0.8±±±±0.01• pinned state has αααα = 0.6 ±±±±0.01• ξξξξsliding < ξξξξpinned < ξξξξzero-field
K.L. Ringlandet al., PRB 61, 4405 (2000)
Depinning of Driven Systems: Time-resolved x-ray scattering measurements of the charge-density waves in NbSe3
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Structure of Sliding State (cont.)
αααα and ξξξξ are insensitive to the value of the current, to the temperature or to the position on the sample.
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Structure of Sliding State (cont.)
• Prediction is that system becomes more ordered at higher drives.
• Observation is that system disorders when it depins and does not reorder at accessible drives.
Depinning of Driven Systems: Time-resolved x-ray scattering measurements of the charge-density waves in NbSe3
Joel Brock, Cornell University (KITP Pattern Formation Conference 8/19/03) 17
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Relationship to Zero-Field Cooled State
• Zero-field Cooled• New length scale required
to describe short length scale behavior
• FLR Model is accurate
( )ξ
axaxg
−+≅22
Brock et al., PRL 73, 3588 (1994)
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Low Temperature Behavior:Switching
Depinning of Driven Systems: Time-resolved x-ray scattering measurements of the charge-density waves in NbSe3
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Coherent Flow in Region II!
Lemay, et al. PRL, 83, 2793 (1999)
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Coherent X-rays and Speckle: Equilibrium Dynamics
Depinning of Driven Systems: Time-resolved x-ray scattering measurements of the charge-density waves in NbSe3
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Field Broadening
Q1 CDW, T = 100K
Top: I = 0
Center: I = .8×It
Bottom: I =2 × I t
Averaged over 200 frames and 5 sec each.
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Autocorrelation Function Results
E < ET: StaticE ≥ ET: StaticE ≈ ET
*: DynamicE >> ET
*: “Fast”
Suggests that we’ve got phase diagram “backwards.”
Static (creep) → Adiabatic (elastic) → Plastic
E-T Phase Diagram
Depinning of Driven Systems: Time-resolved x-ray scattering measurements of the charge-density waves in NbSe3
Joel Brock, Cornell University (KITP Pattern Formation Conference 8/19/03) 20
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Summary
Elastic/coherent/adiabatic flow
Plastic flow
Slow fluctuations
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So what should we do?
Need to expand our description of pinning/depinning• New models
– Amplitude fluctuations may be important– Elastic interactions may not be sufficient
• More data– Nail down XPCS data– X-rays measures lattice-distortion wave. Transport measures
conduction electron density wave.
Depinning of Driven Systems: Time-resolved x-ray scattering measurements of the charge-density waves in NbSe3
Joel Brock, Cornell University (KITP Pattern Formation Conference 8/19/03) 21
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Viscoelastic Coupling
M.C. Marchetti, et al., PRL 85, 1104 (2000).
Replace elastic interaction of FLR model by couplings to the local velocity field that are non-local in time.
( )( )
( ) ( )[ ] ( )iiijji
t
jii uFFsusustCdsu ++−−= ∑ ∫ &&&
, 0,0γ
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Modified Swift-Hohenberg Equation
M. Karttunen, et al., PRL 83, 3518 (1999)
Include both amplitude and phase fluctuations.
∑+−
∂∂+
∂∂+−=
∂∂+
∂∂
i
iimpcc V
yq
xq
xE
t3
2
22
2
2
22 4 ψψεψψ
Dislocations proliferate near depinning transition. The dynamical generation of dislocations can make the system more disordered above threshold.
Depinning of Driven Systems: Time-resolved x-ray scattering measurements of the charge-density waves in NbSe3
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Resonant Enhancement
Continuum
K
L
MN
Resonant Scattering
•Use resonant x-ray scattering to sample (diverging) unoccupied density of states of conduction band at 2kF.
•Polarization analysis to suppress scattering from LDW
This September at CHESS
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“The Crew”
Rob ThorneMark SuttonYanping LiJoel B rock
Depinning of Driven Systems: Time-resolved x-ray scattering measurements of the charge-density waves in NbSe3
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Effect of Slit Sizes
Speckle patterns taken on Q1 CDW satellite with vertical slits sizes at45 mm (top), 30 mm (center), 12.5 mm (bottom)
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Finite Size Oscillations Demonstrate Coupling
Y. Li, et al., PRB 63, 041103(R) (2001)
Depinning of Driven Systems: Time-resolved x-ray scattering measurements of the charge-density waves in NbSe3
Joel Brock, Cornell University (KITP Pattern Formation Conference 8/19/03) 24
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Low Temperature - switching - regime
Y. Li, et al., PRB 63, 041103(R) (2001)
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ObservationsNon-Ohmic conductivity
• P. Monceau, N.P. Ong, A.M. Portis, A. Meerschaut, and J. Rouxel. “Electric field breakdown of charge-density-wave-induced anomalies in NbSe3.” Phys. Rev. Lett., 37(10), 602 (1976).
• R.M. Fleming and C.C. Grimes. “Sliding-mode conductivity in NbSe3: Observation of a threshold electric field and conduction noise.” Phys. Rev. Lett., 42, 1423 (1979).
• H. Matsukawa and H. Takayama. “Numerical study of statics and dynamics of charge-density-waves based on Fukuyama-Lee-Rice model.” Physica B, 143, 80 (1986).
Mode-Locking• J.C. Gill. “Transient non-linear phenomena and metastable states of charge-density waves in niobium triselenide.”
Solid State Comm., 39, 1203 (1981).
• M. Sherwin and A. Zettl. “Complete charge-density-wave mode locking and freeze-out of fluctuations in NbSe3..” Phys. Rev. B., 32(8) 5536 (1985).
• S.N. Coppersmith and P.B. Littlewood. “Interference phenomena and mode locking in the model of deformable sliding charge-density waves.” Phys. Rev. Lett., 57(15), 1927 (1986).
Narrow-Band Noise• R.M. Fleming and C.C. Grimes. “Sliding-mode conductivity in NbSe3: Observation of a threshold electric field and
conduction noise.” Phys. Rev. Lett., 42, 1423 (1979).
• S. Bhattacharya, M.J. Higgins and J.P. Stokes. “Harmonic generation and scaling behavior in sliding-charge-density-wave conductors.” Phys. Rev. Lett., 63(14), 1503 (1989).
• P. Sibanl and P.B. Littlewood. “Critical dynamics of a pinned elastic medium in two and three dimensions: A model for charge-density waves.” Phys. Rev. Lett., 64(11), 1305 (1990).
Depinning of Driven Systems: Time-resolved x-ray scattering measurements of the charge-density waves in NbSe3
Joel Brock, Cornell University (KITP Pattern Formation Conference 8/19/03) 25
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Brock Research Group
Rotations of Q
• Frequently, when CDW is sliding, the position of the satellite peak “rotates” away from the symmetric E=0 position
• The 1+Q satellite rotates in the opposite sense as the 1-Q satellite.
• The amount of rotation does not depend on position in the whisker.
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Rotation of Q (cont.)
• The rotation occurs rapidly as the E-field is turned on
• The form of this rotation is similar to that of the phase-slip induced strain and the phase-slip voltage, V PS
DiCarlo et al., PRL 70(6) 845-8 (1994)
Depinning of Driven Systems: Time-resolved x-ray scattering measurements of the charge-density waves in NbSe3
Joel Brock, Cornell University (KITP Pattern Formation Conference 8/19/03) 26
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Brock Research Group
Relationship to Surface Scattering/Growth
CDWs Surface Scattering/Growth
Structure Factor ( ) ( ) ( ) ( )trgrqQGi eerduqJtqS ,32
1,rrrrrrrrr −⋅−±
∫⋅∝ ( )( )
( ) ( )trgqrqGi
nqG
zeerdtqS ,3
ˆ
2
2
1,
rrrr
rr
rr −⋅−
⋅− ∫∝
CorrelationFunction ( ) ( ) ( )[ ] 2
,0,2
1, ttrtrg
rrr φφ −≡ ( ) ( ) ( )[ ] 2,0,
21
, thtrhtrgrrr
−≡
Equation ofMotion
( )trHFLR
t ,rη
δφδφ +Γ−=∂ ( )tr
Ht ,
rηδφδφ +Γ−=∂
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Opening angle of Synchrotron Radiation
On axis
∆t = (c-v)∆t’/c
Off axis
∆t = (c-v cosα)∆t’/c
( )ttt ′∆
+≅′∆
−
−−≅∆
2
22
2 2
1
21
2
111
γαγα
γ
A
B C
c∆t’
v∆t’
α
Depinning of Driven Systems: Time-resolved x-ray scattering measurements of the charge-density waves in NbSe3
Joel Brock, Cornell University (KITP Pattern Formation Conference 8/19/03) 27
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Brock Research Group
Spectrum of Synchrotron Radiation
If T is period of revolution,
∆t =[γ-1/2π] T = 1/(γω0)
Observer “sees” an interval γ2 shorter
∆t = 1/(γ3ω0)
Inverse of pulse width gives frequency spread, so high cut-off frequency is
ωc ≡ (3/2)γ3ω0
Ec [KeV] = 0.665 Ee2[GeV] B[T]
1/γγγγωωωω0
R
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Sub-Threshold Quenches
• Clearly, phase-slip is occurring and FLR model is not able to describe the dynamics of pinning.
• Quenches from I < IT should avoid this difficulty.
• Preliminary data suggest that µ =½ for sub-threshold quenches --consistent with FLR prediction.
• τ appears to be constant -- also consistent with FLR -- but data is not good enough yet.