[9f60c06] | 1 | .. _magnetism: |
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| 2 | |
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| 3 | Polarisation/Magnetic Scattering |
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[990d8df] | 4 | ================================ |
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[9f60c06] | 5 | |
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[6e0c1df] | 6 | Spherical Polarimetry in SANS: This description allows for freedom in |
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| 7 | field/polarisation axis away from the detector plane |
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| 8 | i.e. allows in-beam direction or AC or rotational fields. |
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[9f60c06] | 9 | |
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[6b86bee] | 10 | For magnetic systems, the scattering length density (SLD = $\beta$) is a combination |
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| 11 | of the nuclear and magnetic SLD. For polarised neutrons, the resulting effective SLD |
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| 12 | depends on the spin state of the neutron before and after being scattered in the sample. |
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| 13 | |
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| 14 | Models in Sasview, which define a SLD parameter, can be evaluated also as magnetic models introducing |
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| 15 | the magnetisation (vector) $\mathbf{M}=M (\sin\theta_M \cos \phi_M, \sin \theta_M \sin \phi_M,\cos\theta_M )$ and the associated magnetic SLD given by |
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| 16 | the simple relation $\beta_M= b_H M$, where $b_H=\dfrac{\gamma r_0}{2\mu_B}=2.7$ fm |
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| 17 | denotes the magnetic scattering length and $M=\lvert \mathbf{M} \rvert$ the magnetisation |
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| 18 | magnitude, where $\gamma = -1.913$ is the gyromagnetic ratio, $\mu_B$ is the |
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| 19 | Bohr magneton, $r_0$ is the classical radius of electron. |
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| 20 | |
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| 21 | It is assumed that the magnetic SLD in each region of the model is uniformly for |
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| 22 | nuclear scattering and has one effective magnetisation orientation |
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| 23 | |
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[6e0c1df] | 24 | The external field $\mathbf{H}=H \mathbf{P}$ coincides with the polarisation axis |
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[6b86bee] | 25 | $\mathbf{P}=(\sin\theta_P \cos \phi_P, \sin \theta_P \sin \phi_P,\cos\theta_P )$ for the neutrons, which is the quantisation axis |
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| 26 | for the Pauli spin operator. |
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| 27 | |
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| 28 | .. figure:: |
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| 29 | mag_img/M_angles_pic.png |
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| 30 | |
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| 31 | .. note:: |
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| 32 | The polarisation axis at the sample position is the determining factor and determines |
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| 33 | the scattering geometry. Before and after the field at the sample position, |
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| 34 | the polarisation turns adiabatically to the guide field of the instrument. |
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| 35 | This operation does not change the observed spin-resolved scattering at the detector. |
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| 36 | Anyway the magnetic field is the vector defining a symmetry axis of the |
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| 37 | system and the magnetisation vector will orient with respect to the field. |
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| 38 | |
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| 39 | |
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| 40 | .. note:: |
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| 41 | For AC oscillating/rotation field varying in space with time, you can coupling the magnetisation |
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| 42 | with the field axis via a constrained fit. This will allow to easily parametrise |
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| 43 | a phase shift of the magnetisation lagging behind a magnetic field varying from time frame to time frame. |
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| 44 | Anyway the magnetic field is the vector defining a symmetry axis of the |
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| 45 | system and the magnetisation vector will most often orient symmetrically with respect to the field. |
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| 46 | |
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| 47 | |
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| 48 | The neutrons are polarised parallel (+) or antiparallel (-) to $\mathbf{P}$. One can |
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| 49 | distinguish 4 spin-resolved cross sections: |
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| 50 | |
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| 51 | * Non-spin-flip (NSF) $(+ +)$ and $(- -)$ |
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| 52 | |
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| 53 | * Spin-flip (SF) $(+ -)$ and $(- +)$ |
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| 54 | |
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| 55 | The spin-dependent magnetic scattering length densities are defined as (see Moon, Riste, Koehler) |
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| 56 | |
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| 57 | .. math:: |
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| 58 | \beta_{M, s_{in} s_{out}} = b_H\sigma \cdot \mathbf{M_\perp} |
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| 59 | |
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| 60 | where $\sigma$ is the Pauli spin, and $s_{in/out}$ describes the spin state of the neutron before and |
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| 61 | after the sample. |
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| 62 | |
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[6e0c1df] | 63 | For magnetic neutron scattering, only the magnetisation component or Halpern-Johnson vector |
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[6b86bee] | 64 | $\mathbf{M_\perp}$ perpendicular to the scattering vector |
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| 65 | $\mathbf{Q}=q \mathbf{n}=q (\cos\theta, \sin \theta,0)$ contributes to the magnetic scattering: |
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| 66 | |
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| 67 | .. math:: |
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| 68 | \mathbf{M_\perp} = \mathbf{n} [\mathbf{n} \cdot \mathbf{M}] -\mathbf{M} |
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| 69 | |
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| 70 | with $\mathbf{n}$ the unit scattering vector and $\theta$ denotes the angle |
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| 71 | between $\mathbf{Q}$ and the x-axis. |
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[9f60c06] | 72 | |
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| 73 | .. figure:: |
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[0cd9158] | 74 | mag_img/mag_vector.png |
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[9f60c06] | 75 | |
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[6b86bee] | 76 | The two NSF cross sections are given by |
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[9f60c06] | 77 | |
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| 78 | .. math:: |
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[6e0c1df] | 79 | I^{\pm\pm} = N^2 \mp \mathbf{P}\cdot(N^{\ast}\mathbf{M_\perp} +N\mathbf{M_\perp}^{\ast}) |
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[6b86bee] | 80 | + (\mathbf{P}\cdot \mathbf{M_\perp})^2 |
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| 81 | |
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| 82 | and the two SF channels: |
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[9f60c06] | 83 | |
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[6b86bee] | 84 | .. math:: |
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| 85 | I^{\pm\mp} = \mathbf{M_\perp}\cdot \mathbf{M_\perp} - (\mathbf{P}\cdot \mathbf{M_\perp})^2 |
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| 86 | \mp i \mathbf{P}\cdot \mathbf{M_\perp} \times \mathbf{M_\perp}^{\ast} |
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[9f60c06] | 87 | |
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[6b86bee] | 88 | with $i=\sqrt{-1}$, and $^{\ast}$ denoting the complex conjugate quantity, and |
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| 89 | $\times$ and $\cdot$ the vector and scalar product, respectively. |
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[4f5afc9] | 90 | |
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[6b86bee] | 91 | The polarisation axis at the sample position is the determining factor and determines |
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| 92 | the scattering geometry. For the NSF scattering the component of the Halpern-Johnson |
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| 93 | vector parallel to $P$ contributes |
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[4f5afc9] | 94 | |
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[6b86bee] | 95 | .. math:: |
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[6e0c1df] | 96 | \mathbf{M}_{\perp,\parallel P } = ( \mathbf{P}\cdot \mathbf{M}_{\perp }) \mathbf{P} |
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| 97 | |
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[9f60c06] | 98 | |
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[6b86bee] | 99 | The component perpendicular to the polarisation gives rise to SF scattering. The perpendicular |
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| 100 | plane is constructed with the two vectors |
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[4f5afc9] | 101 | |
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| 102 | .. math:: |
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[6b86bee] | 103 | \mathbf{M}_{\perp,\perp P } = \mathbf{M}_{\perp } - (\mathbf{P}\cdot \mathbf{M}_{\perp }) \mathbf{P} |
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[6e0c1df] | 104 | |
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[9f60c06] | 105 | |
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[6b86bee] | 106 | and a third vector perpendicular to both $\mathbf{P}$ and $\mathbf{M}_{\perp,\perp P } $ : |
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| 107 | |
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| 108 | .. math:: |
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[6e0c1df] | 109 | \mathbf{O} = \mathbf{M}_{\perp} \times \mathbf{P} - \mathbf{M}_{\perp,\perp P } |
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| 110 | = [\mathbf{n}\cdot(\mathbf{M}\times\mathbf{P})(\mathbf{n}-\mathbf{P}\times\mathbf{n})] |
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[6b86bee] | 111 | |
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[6e0c1df] | 112 | |
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| 113 | For symmetric, collinear spin structures ($\mathbf{M}_{\perp}^{\ast}=\mathbf{M}_{\perp}^{\ast}$), $\mathbf{O}\cdot \mathbf{O}^{\ast}=0$ |
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[6b86bee] | 114 | since $\mathbf{M}_{\perp} \times \mathbf{P} \cdot \mathbf{M}_{\perp} \times \mathbf{P} = \mathbf{M}_{\perp,\perp P }$. |
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[9f60c06] | 115 | |
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| 116 | |
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[6b86bee] | 117 | Depending on the spin state of the |
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[9f60c06] | 118 | neutrons, the scattering length densities, including the nuclear scattering |
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[6e0c1df] | 119 | length density $(\beta{_N})$ are for the non-spin-flip states |
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[9f60c06] | 120 | |
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| 121 | .. math:: |
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[6e0c1df] | 122 | \beta_{\pm\pm} = \beta_N \mp b_H \mathbf{P}\cdot \mathbf{M}_{\perp } |
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| 123 | |
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[9f60c06] | 124 | |
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[6e0c1df] | 125 | and for spin-flip states |
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[9f60c06] | 126 | |
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| 127 | .. math:: |
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[6e0c1df] | 128 | \beta_{\pm\mp} = -b_H (\lvert\mathbf{M}_{\perp,\perp P }\rvert \pm i \mathbf{n}\cdot (\mathbf{M}\times \mathbf{P} (1-\mathbf{P}\cdot\mathbf{n})) |
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[9f60c06] | 129 | |
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| 130 | |
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[6e0c1df] | 131 | |
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| 132 | with |
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| 133 | |
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| 134 | .. math:: |
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| 135 | \lvert\mathbf{M}_{\perp,\perp P }\rvert= (\mathbf{M}_{\perp,\perp P } \cdot \mathbf{M}_{\perp,\perp P })^{1/2} |
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| 136 | =(M_{\perp,x}^2+M_{\perp,y}^2+M_{\perp,z}^2-(M_{\perp,x} P_x+ M_{\perp,y} P_y + M_{\perp,z} P_z )^2 )^{1/2}. |
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[6b86bee] | 137 | |
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| 138 | |
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| 139 | |
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| 140 | |
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| 141 | Every magnetic scattering cross section can be constructed from an incoherent mixture |
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| 142 | of the 4 spin-resolved spin states depending on the efficiency parameters before |
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| 143 | ($u_i$) and after ($u_f$) the sample. For a half-polarised experiment(SANSPOL with $u_f=0.5$) or |
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| 144 | full (longitudinal) polarisation analysis, the accessible spin states are measured |
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| 145 | independently and a simultaneous analysis of the measured states is performed, |
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| 146 | tying all the model parameters together except $u_i$ and $u_f$, which are set based |
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| 147 | on the (known) polarisation efficiencies of the instrument. |
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| 148 | |
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| 149 | .. note:: |
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| 150 | The values of the 'up_frac_i' ($u_i$) and 'up_frac_f' ($u_f$) must be in the range 0 to 1. |
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| 151 | The parameters 'up_frac_i' and 'up_frac_f' can be easily associated to |
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| 152 | polarisation efficiencies 'e_in/out' (of the instrument). Efficiency values range from 0.5 |
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| 153 | (unpolarised beam) to 1 (perfect optics). For 'up_frac_i/f' <0.5 |
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| 154 | a cross section is constructed with the spin reversed/flipped with respect |
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| 155 | to the initial supermirror polariser. The actual polarisation efficiency |
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| 156 | in this case is however 'e_in/out' = 1-'up_frac_i/f'. |
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[9f60c06] | 157 | |
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| 158 | |
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| 159 | |
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| 160 | |
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| 161 | The user input parameters are: |
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| 162 | |
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| 163 | =========== ================================================================ |
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[6b86bee] | 164 | sld_M0 $b_H M_0$ |
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[df87acf] | 165 | sld_mtheta $\theta_M$ |
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| 166 | sld_mphi $\phi_M$ |
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[6b86bee] | 167 | up_frac_i $u_i$ polarisation efficiency *before* the sample |
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| 168 | up_frac_f $u_f$ = polarisation efficiency *after* the sample |
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| 169 | p_theta $\theta_P$ |
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| 170 | p_phi $\phi_P$ |
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[9f60c06] | 171 | =========== ================================================================ |
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| 172 | |
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[6b86bee] | 173 | |
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[9f60c06] | 174 | .. note:: |
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[6b86bee] | 175 | P.S. of Dirk: |
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| 176 | This is the most general description of magnetic SANS ever written and will supersede prior art! |
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| 177 | Works for fully magnetically saturated systems. If you figure out how to |
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| 178 | implement an isotropic ensemble of particle magnetisation ( similar for orientations). |
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| 179 | This is needed to generate two populations with spin pointing in opposite directions in order to describe |
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| 180 | field-dependence correctly, i.e. the different variation of mean magnetisation vs |
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| 181 | square mean quantities. |
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| 182 | With proper generalised orientation distribution, you cover all "normal" use cases |
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| 183 | (except of the fancy stuff one have to simulate). |
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| 184 | |
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| 185 | |
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| 186 | References |
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| 187 | ---------- |
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| 188 | |
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| 189 | .. [#] R. M. Moon and T. Riste and W. C. Koehler, *Phys. Rev.*, 181 (1969) 920. |
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[9f60c06] | 190 | |
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[59485a4] | 191 | *Document History* |
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[990d8df] | 192 | |
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[59485a4] | 193 | | 2015-05-02 Steve King |
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[befe905] | 194 | | 2017-11-15 Paul Kienzle |
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| 195 | | 2018-06-02 Adam Washington |
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[6b86bee] | 196 | | 2019-03-29 Dirk Honecker |
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