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Chapters:

  1. Introduction
  2. The muon
  3. Muon production
  4. Spin polarization
  5. Detect the µ spin
  6. Implantation
  7. Paramagnetic species
  8. A special case: a muon with few nuclei
  9. Magnetic materials
  10. Relaxation functions
  11. Superconductors
  12. Mujpy
  13. Mulab
  14. Musite?
  15. More details

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DiffusionOnCubic

< Use of Trim.SP and SRIM | Index | Simulations of diffusion on a cubic lattice >

Local index:

Imagine a muon diffusing among sites experiencing a purely isotropic hyperfine coupling, within a simple cubic lattice of muon sites, with lattice parameter a.

The diffusion is with unitary jumps {1,0,0}, with 6 possibilities: {$(\pm1,0,0),(0,\pm1,0)$} or {$(0,0,\pm1)$}. This generates path hierarchies labelled by the number of jumps, n. Each path corresponds to a final position {j',k',l'}, at a distance

{$$d_{j^\prime k^\prime l^\prime} = a\sqrt{j^{\prime2}+k^{\prime2}+l^{\prime2}} $$}

with a recursively computable multiplicity

{$$m(n;j^\prime,k^\prime,l^\prime)=m_{jkl}\cdot m(n-1;j,k,l). $$}

The multiplicity factor {$m_{jkl}$} are generated by a simple finite set of rules (see below), the total multiplicity {$m(n;j,k,l)$} can be stored in a sparse matrix, the final position {j,k,l} can be stored in a list of lists (each for each n value) of lists (of coordinate triplets) the first time it is encountered.

The recursive procedure can generate the new position form each starting position at the previous step, by adding the unitary jumps: {j',k',l'}={j,k,l}+{1,0,0}. It can then store the results at iteration n as a list of final positions, {j',k',l'}, the multiplicity m(n-1;j,k,l) of their starting positions {j,k,l} in a new list or directory of the hierarchy of steps, and check that the total number of generated final positions with their multiplicity adds up to M(n)=6n .

The rules to generate the new positions {j',k',l'}={j,k,l}+{1,0,0} and the corresponding factor {$m_{j^\prime k^\prime l^\prime}$} are independent of n. They are written out below for j, but must be applied to k and l as well.

  • if {$j= 0$} then {j',k',l'} = {1,k,l} with multiplicity {$m_{1 k l} =2$}
  • if {$j\ne 0$} then {j',k',l'} = {j+1,k,l} with multiplicity {$m_{j+1 k l}=1$}, plus {j',k',l'} = {j-1,k,l} with multiplicity {$m_{j-1 k l}=1$}

This can produce more versions of a given final position, and their factors must be added. Equivalent final sites must be recognized to generate the final list, with unique final positions and their factor, properly added.

The first few steps yield

n

final position

distance

1st seen

factor

Total

0:

{0,0,0}

{$d_{000} = 0$}

here

{$m_{000}=1$}

{$M(0)=1$}

1:

{0,0,0} + {1,0,0}

{1,0,0}

{$d_{100} = a$}

here

{$m_{100}=6,~m(1;1,0,0)=6$}

{$M(1)=6$}

(here {$m$} is the sum of six factors 1 generated by the rules)

2:

{1,0,0} + {1,0,0}

0:

{$m(2;0,0,0)=m(1;1,0,0)\cdot1=6$}

{2,0,0}

{$d_{200} = 2a$}

here

{$m_{200}=m(2;2,0,0)=m(1;1,0,0)\cdot1=6$}

{1,1,0}

{$d_{110} = a\sqrt2$}

here

{$m_{110}=m(2;1,1,0)=m(1;1,0,0)\cdot4=24$}

{$M(2)=36$}

3:

{0,0,0}+{1,0,0}

1:

{$d_{100}=a$}

{$m(3;1,0,0)=m(2;0,0,0)\cdot6=36$}

{2,0,0}+{1,0,0}

1:

{$d_{100}=a$}

{$m(3;1,0,0)=m(3;1,0,0)+m(2;2,0,0)\cdot1=42$}

{3,0,0}

{$d_{300}=3a$}

here

{$m_{300}=m(3;3,0,0)=m(2;2,0,0)\cdot1=6$}

{2,1,0}

{$d_{210}=a\sqrt 5$}

here

{$m(3;2,1,0)=m(2;2,0,0)\cdot4=24$}

{1,1,0}+{1,0,0}

1:

{$d_{100}=a$}

{$m(3;1,0,0)=m(3;1,0,0)+m(2;1,1,0)\cdot2=90$}

{$d_{210}=a\sqrt 5$}

{$m(3;2,1,0)=m(3;2,1,0)+m(2;1,1,0)\cdot2=72$}

{$m_{210}=m(3;2,1,0)=72$}

{1,1,1}

{$d_{111}=a\sqrt 3$}

here

{$m_{111}=m(3;1,1,1)=m(2;1,1,0)\cdot2=48$}

{$M(3)=216$}

The mean travelled distance after n jumps is

{$$\langle r(n)\rangle = 0,\frac 6 6 1 ,\frac {12+24\sqrt 2}{36}, \frac{6(15+8\sqrt 3 +12\sqrt5+1\cdot3)}{6\cdot 36}= 0, 1, 1.276, 1.630$$}

less then {$\sqrt n = 0, 1, 1.414, 1.732$}

Local field average

The simulation in the figure assumes a random walk of the type described above in a Néel ground state. The dynamics is that of a jump motion: the residence time is equal to the inverse of the jump frequency, wuch longer than almost instantaneous jump time. Each site is surrounded by opposite field sites, hence the local field experienced during residence times is equal in modulus {$B_0$} and alternating in sign. Dynamically it produces an exact replica of the BPP model. Hence the {$T_1^{-1}$} rate in zero external field is approximated by a Lorentzian

{$$\frac 1 {T_1} = \frac {\gamma^2 B_0^2\tau}{1+{(\gamma B_0\tau)^2}}$$}

The average field vs. number of jumps, n, is an oscillating function

{$$ \langle B(n)\rangle = 1, 0, \frac 1 3, 0 \frac 1 5, \cdots, 0,\frac 1 {n+1}\quad (n=0,1,2,\cdots)$$}

and, averaging over 2 jumps, one gets {$\langle B_2(2n)\rangle = 1/(4n+2),~(n =0,1,2,\cdots)$} (solid green line).

Random walk in Néel ground state, with {$B_0=10$} for display convenience.Montecarlo is the symbols, green for field, blue for mean distance. The red csolid line is the first few graph calculations reported above.

Similar simulations can be produced for an A-type AF structure with alternating field sign along, say, the z axis. In this case the field sign must be computed as +1 at even z coordinates and -1 at odd z coordinates.

The result is plotted on the left with an apparent asymptote (dashed red line) at 1/3 the initial field value.

Random walk in type-A antiferromagnet. {$B_0=10$}

Relaxations: self correlation

The self correlation of the field in the Néel state is immediate. As we said the field reverses sign at each jump, with a sequence {$B_0(1,-1,1,-1,1,-1,\cdots)$}, realizing the prototype of the Bloembergen Purcell Pound model. Assuming a jump probablity per unit time {$p=\tau^{-1}$} the self correlation of the local field at the (diffusing) muon site is exponential,

{$$\langle B_0 B(t) \rangle = B_0^2 e^{-t/\tau}$$}

and the relaxation is proportional to its Fourier transform component at the local field frequency {$\omega=\gamma\langle B\rangle$}, an expression similar to the BPP one

{$$\frac 1 {T_1}=\frac {\gamma^2 B_0^2\tau}{1+\omega^2\tau^2}$$}

If the magnetic structure is different from that of a Néel state we can generate a map, calculate the average fields at each site with the graph expansion used in the local field section and calculate correlations form there.

The calculation is simple if the field map is collinear (same direction at all sites), like the in the Néel case, where the quantization axis is global and so is the distinction between secular and non secular local interaction. In non collinear case, after each jump the secular and non secular parts change. Hence also the definition of average, should change (this discussion must anticipate also the mean field).

Think of a long wavelength incommensurate helical state of wavevector {$\mathbf q$} in the simple cubic case. The field {$\mathbf B(n)=B\hat b$} rotates (only for steps along {$\mathbf q$}) of a fixed angle. The classical spin dynamics is a precession of {$\mathbf S_\perp =\mathbf S - \mathbf S \cdot \hat b\,\hat b$}. A simulation starts from the average spin after n-1 jumps, reads the map of positions and fields at these positions after the n-th jump, and calculates the spin precession for time {$t$}, taking the average with an exponential probability {$\exp(-t/\tau)$} that the jump happens at time {$t$}.

The spin at t=0, along {$\hat z$}, selects secular directions {$\hat x,\hat y$}. The average spin before jump 1 is calculated by rotation matrices, see next page. The process can be iterated.

< Use of Trim.SP and SRIM | Index | Simulations of diffusion on a cubic lattice >

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Page last modified on December 08, 2016, at 05:52 PM