1. Structure in the mixed phase
Gautam I. Menon
IMSc, Chennai, India

2. Information: Flux-line coordinates as functions of time
The Problem
Describe structure in a compact manner
Correlation functions
Distinguish ordered and disordered states. Also unusual orderings: hexatic
Information: Flux-line coordinates as functions of time

3. Vortex Structures Lines/tilted lines
Pancake vortices in layered systems in fields applied normal to the layers
Josephson vortices in layered systems for fields applied parallel to the planes
Vortex chains and crossing lattices for layered systems in general tilted fields

4. Address via correlation functions
Probability of finding
a “pancake” vortex
a specified distance
away from another
one

5. Correlation Functions
Defines average density at r:
Sum over all particles
A correlation function
Related to the probability of finding a particle at r1, given a distinct particle at r2
The two-point correlation function in a fluid depends only on the relative distance between two points, by rotational and translational invariance.

6. Correlation Functions II
Brackets denote a thermodynamic average
Defines a structure factor
From the previous definition of (r)
In terms of Fourier components of the density

7. Correlation Functions in a Solid
This sum is over lattice sites. It is non-zero only if q=G (a reciprocal lattice vector), in which case it has value N, i.e. (q) = Nq,G
Implies

8. Correlation Functions III
Inserting the definition
In terms of n2

9. Just removes an uninteresting q=0 delta-function
Correlations IV
Defines g(r)
From g(r), S(q)
Just removes an uninteresting q=0 delta-function

10. Why are correlation functions interesting?
Experiments measure them!
Theorists like them ……

11. The generic scattering experiment measures precisely a correlation function
and from there g(r)

12. Physical Picture of g(r)
Area under first peak measures number of neighbours in first coordination shell

13. Scattering

14. Intensities as functions of q

15. Melting from Neutron Scattering
Bragg spots go to rings:
Evidence for a melting
transition
Ling and collaborators

16. The Disordered Superconductor
Larkin/Imry/Ma: No translational long-range order in a crystal with a quenched disordered background.
Natterman/Giamarchi/Le Doussal: This doesn’t preclude a more exotic order, power-law translational correlations
The Bragg Glass

17. Different types of Ordering
What does long range order mean?
What does quasi-long range order mean?
What does short-range order mean?

18. The Bragg Glass proposal
Precise consequence for small angle neutron scattering experiments: S(q) decay about (quasi-) Bragg spots

19. More exotic forms of ordering

20. Hexatics
In 2-d systems, thermal fluctuations destroy crystalline LRO except at T=0. Positional order decays as a power law at low T
But, orientational long-range order can exist at finite but low temperatures

21. Hexatics
In the liquid, short range order in positional and orientational correlations
How do power-law translational order and the orientational long-range order go away as T is increased?
Must be a transition – one or more?

22. Hexatics: Nelson/Halperin
Two transitions out of the low T phase
Intermediate hexatic phase, power-law decay of orientational correlations, short-ranged translational order.
Topological defects: transitions driven by dislocation and disclination unbinding

23. Orientational Correlations
Hexatic

24. Hexatic vs Fluid Structure

28. Muon-Spin Rotation

29. The -SR Method I Positively charged muons from an accelerator
Muons polarized transverse to applied magnetic field. Implanted within the sample

30. What the muons see

31. Muon Spin Rotation II
Muons precess in magnetic field due to vortex lines
Muons are unstable particles. Decay into positrons, anti-neutrinos and gamma rays

32. Muon Spin Rotation III
Muon lifetime » 10-6 s. Muon decay ! positron emitted preferentially with respect to muon polarization. Emitted positron polarization recorded

33. Muon Spin Rotation IV
The Principle: Can reconstruct the local magnetic
field from knowledge of the polarization state
of the muon when it decays
Need to average over a large number of
muons for good statistics
Muons are local probes

34. Muon Spin Rotation V The magnetic field distribution function
Moments of the field distribution function
Moments contain important information, obtain l

35. Muon-Spin Rotation Field at point r Density of vortex lines
In Fourier space. A is the area of the system

36. Muon Spin Rotation II
Flux quantum

37. Muon Spin Rotation VI Assuming a perfect lattice
The sum is over reciprocal lattice vectors of a
triangular lattice

38. Muon-Spin Rotation Spectra
Sonier, Brewer and Kiefl, Rev. Mod. Phys. 72, 769 (2000).
<ΔB>1 λ2 _

39. MgCNi3 The rate of muon depolarisation in zero-field µSR (ZF-µSR)
is a sensitive probe for spontaneous internal magnetic fields.
0.1G
0.05G
MgCNi3
This experiment:
•no spontaneous fields present greater
than ~0.03G above 2.5K

40. Important information about the superconducting gap
MgCNi3
 ns/m*-2
Results:
•Tc=7K
• Functional form implies s-wave gap
Important information about the superconducting gap

41. Results from m-Spin Rotation
Underdoped LSCO, Divakar et al.

42. Muon Spin Rotation LSCO
Why do line-widths increase with field?
Strong disorder in-plane, almost rigid rods
The “true” vortex glass
U.K. Divakar et al. PRL (2004)

43. Phase Behavior from mSR
Probing the
glassy state
and its local
correlations

44. Lee and collaborators

45. Lee and collaborators

46. Lee and collaborators

47. Menon, Drew, Lee, Forgan, Mesot, Dewhurst ++…..

48. Three body correlations in the flux-line glass phase

49. Nontrivial Information about the Nature of superconductivity: Uemura Plot

50. NMR and the Mixed Phase

51. NMR as a Mixed State Probe
Information obtained is virtually identical to that
obtained in Muon-Spin Rotation
But the probe is different

52. NMR as a Vortex Probe I
Interaction of nuclear magnetic moment with local magnetic field splits nuclear energy levels
Nuclear magnetic dipole transitions excited among these levels by applying a RF field of an appropriate frequency.
When the frequency of the RF field is such that the energy is equal to the energy separation between the quantum states of the nuclear spin, energy absorbed. The resulting resonance can be detected.

53. NMR as a Vortex Probe
Since the distances between similar nuclei in a superconductor are small relative to vortex separation, sample n(B) by measuring fields at the sites of nuclei.
Nuclei uniformly distributed, so sampling is volume-weighted.

54. NMR as a Vortex probe III:Method
In “pulsed NMR” observe time-dependent transverse nuclear polarization or ``free induction decay'' of nuclear polarization.
Here an RF pulse is applied to rotate nuclear spins from the direction of the local magnetic field . When the RF field is switched off, nuclear spins perform a free precession around the local field and relax back to their initial direction
The frequency of the nuclear spin precession is a measure of the local field
In this technique, different precession frequencies are observed simultaneously.

55. NMR as a Vortex Probe IV: Limitations
Several limitations and added difficulties associated with the NMR technique which are overcome in a SR experiment.
Because the skin depth of the RF field probe is small, NMR only probes the sample surface. Often the surface has many imperfections, so strong vortex-line pinning and a disordered vortex lattice
The penetration depth of the RF field also limits the range over which the vortex lattice can be sampled. Plus additional sources of broadening.

56. Magnetic Decoration

57. Decoration Experiments
Evaporate magnetic material (fine ferromagnetic grains) onto the surface of the sample
Image
Essmann and Trauble (1968)

60. Decoration Data
MgB2
YBCO

61. Magnetic Decoration
Several issues: Nature of ordering, how good are the lattice which are formed
Hexatic phases
Correlation between top and bottom of the sample – how do vortex lines thread the sample?
Glassy phases, short-range order
Melting? Flux-line movement across short times

62. Delaunay Triangulation
Fasano et al, PRB’02

63. Domain States?

64. Problems? Confined really to low fields
Not bulk, only surface information
Useless for dynamics – only static pictures
Yet .. some indicator of lattice quality
Orientational order at surfaces .. maybe the best way of looking at it

65. Finally ..
The structural probes I talked about all complement each other
Each provides valuable information, yet misses many other important things
Probing at this “mesoscopic” scale is surprisingly difficult, considering that we can image the structure of complex protein molecules to a precision of a few Angstrom ……………… food for thought.