Modelling The Asymmetry Spectrum of the Vortex State

In terms of $\lambda$, the experimental raw asymmetry for TF-$\mu ^{+}$SR analysis of the vortex state in an isotropic type-II superconductor can be modelled assuming that the contribution to A(t) from a particular point in the flux lattice is

$$A^{th}_{raw}(t) = \frac{A^{\circ}}{N} \sum_{i=1}^{N} \cos \le . . . . . . \, \frac{K^{2} \lambda^{2}}{(1-b)}} \right] + \theta \right\}$$ (35)

where I have combined Eqs. (3.9), (3.27) and (3.31). Eq. (3.35) applies to an ideal flux lattice. To account for smearing of the field distribution due to flux-lattice disorder and nuclear dipolar fields, Eq. (3.35) must be convoluted with a gaussian distribution in the form of Eq. (3.34):

$$A^{th}_{raw}(t) = \frac{A^{\circ}}{N} e^{- \sigma_{eff}^{2} . . . . . . \, \frac{K^{2} \lambda^{2}}{(1-b)}} \right] + \theta \right\}$$ (36)

Assuming that the flux-line lattice is composed of equilateral triangles as depicted in Fig. 3.1(a), an arbitrary reciprocal-lattice vector $\vec{K} = n \vec{k}_{1} + m \vec{k}_{2}$ can be written in terms of the real lattice vectors $\vec{r}_{1} = L \widehat{x}$and $\vec{r}_{2} = L/2 ( \widehat{x} + \sqrt{3} \widehat{y} )$ so that

$$\vec{K} = \frac{2 \pi}{L} \left[ n \widehat{x} + \frac{1}{\sqrt{3}} (2m-n) \widehat{y} \right]$$ (37)

where

$$L = \sqrt{\frac{\, 2 \, \Phi_{\circ}}{\sqrt{3} \, \overline{B}}}$$ (38)

is the distance between adjacent vortices. Combining Eqs. (3.36) and (3.37) gives

 

$$A^{th}_{raw}(t) = \frac{A^{\circ}}{N_{1} N_{2}} e^{\frac{- \sigma_{eff}^{2} t^{2}}{2}} \sum_{i=0}^{N_{1}} \sum_{j=0}^{N_{2}}$$

$$\times \cos \left\{ \left[ \gamma_{\mu} \overline{B} t \sum_{n,m} . . . . . . L} \right)^2 (n^{2} + m^{2} -mn) \lambda^{2}}{(1-b)}} \right] + \theta \right\}$$ (39)

where $\vec{r}_{ij} = a_{i} \widehat{x} + a_{j} \widehat{y}$ is a vector in real space and the sum over a_i and a_j extends over a single unit cell. Eq. (3.39) can be substituted into Eq. (3.30) and the resulting theoretical corrected asymmetry used to fit the measured asymmetry.