Changeset ac60a39 in sasmodels for doc/guide/magnetism/magnetism.rst

Timestamp:

Nov 20, 2017 9:33:17 AM (6 years ago)

Author:

Paul Kienzle <pkienzle@…>

Branches:

master, core_shell_microgels, magnetic_model, ticket-1257-vesicle-product, ticket_1156, ticket_1265_superball, ticket_822_more_unit_tests

Children:

1f159bd

Parents:

4f5afc9 (diff), 146793b (diff)
Note: this is a merge changeset, the changes displayed below correspond to the merge itself.
Use the (diff) links above to see all the changes relative to each parent.

Message:

Merge branch 'master' into ticket-776-orientation

File:

• doc/guide/magnetism/magnetism.rst (modified) (6 diffs)

doc/guide/magnetism/magnetism.rst

@@ -5 +5 @@
 
 Models which define a scattering length density parameter can be evaluated
- as magnetic models. In general, the scattering length density (SLD =
- $\beta$) in each region where the SLD is uniform, is a combination of the
- nuclear and magnetic SLDs and, for polarised neutrons, also depends on the
- spin states of the neutrons.
+as magnetic models. In general, the scattering length density (SLD =
+$\beta$) in each region where the SLD is uniform, is a combination of the
+nuclear and magnetic SLDs and, for polarised neutrons, also depends on the
+spin states of the neutrons.
 
 For magnetic scattering, only the magnetization component $\mathbf{M_\perp}$
 perpendicular to the scattering vector $\mathbf{Q}$ contributes to the magnetic
 scattering length.
-
 
 .. figure::
@@ -28 +27 @@
 is the Pauli spin.
 
-Assuming that incident neutrons are polarized parallel (+) and anti-parallel (-)
-to the $x'$ axis, the possible spin states after the sample are then
+Assuming that incident neutrons are polarized parallel $(+)$ and anti-parallel
+$(-)$ to the $x'$ axis, the possible spin states after the sample are then:
 
-No spin-flips (+ +) and (- -)
+* Non spin-flip $(+ +)$ and $(- -)$
 
-Spin-flips    (+ -) and (- +)
+* Spin-flip $(+ -)$ and $(- +)$
+
+Each measurement is an incoherent mixture of these spin states based on the
+fraction of $+$ neutrons before ($u_i$) and after ($u_f$) the sample,
+with weighting:
+
+.. math::
+    -- &= ((1-u_i)(1-u_f))^{1/4} \\
+    -+ &= ((1-u_i)(u_f))^{1/4} \\
+    +- &= ((u_i)(1-u_f))^{1/4} \\
+    ++ &= ((u_i)(u_f))^{1/4}
+
+Ideally the experiment would measure the pure spin states independently and
+perform a simultaneous analysis of the four states, tying all the model
+parameters together except $u_i$ and $u_f$.
 
 .. figure::
@@ -41 +54 @@
 $\phi$ and $\theta_{up}$, respectively, then, depending on the spin state of the
 neutrons, the scattering length densities, including the nuclear scattering
-length density ($\beta{_N}$) are
+length density $(\beta{_N})$ are
 
 .. math::
     \beta_{\pm\pm} =  \beta_N \mp D_M M_{\perp x'}
-    \text{ when there are no spin-flips}
+    \text{ for non spin-flip states}
 
 and
@@ -51 +64 @@
 .. math::
     \beta_{\pm\mp} =  -D_M (M_{\perp y'} \pm iM_{\perp z'})
-    \text{ when there are}
+    \text{ for spin-flip states}
 
 where
 
 .. math::
-    M_{\perp x'} = M_{0q_x}\cos(\theta_{up})+M_{0q_y}\sin(\theta_{up}) \\
-    M_{\perp y'} = M_{0q_y}\cos(\theta_{up})-M_{0q_x}\sin(\theta_{up}) \\
-    M_{\perp z'} = M_{0z} \\
-    M_{0q_x} = (M_{0x}\cos\phi - M_{0y}\sin\phi)\cos\phi \\
-    M_{0q_y} = (M_{0y}\sin\phi - M_{0x}\cos\phi)\sin\phi
+    M_{\perp x'} &= M_{0q_x}\cos(\theta_{up})+M_{0q_y}\sin(\theta_{up}) \\
+    M_{\perp y'} &= M_{0q_y}\cos(\theta_{up})-M_{0q_x}\sin(\theta_{up}) \\
+    M_{\perp z'} &= M_{0z} \\
+    M_{0q_x} &= (M_{0x}\cos\phi - M_{0y}\sin\phi)\cos\phi \\
+    M_{0q_y} &= (M_{0y}\sin\phi - M_{0x}\cos\phi)\sin\phi
 
 Here, $M_{0x}$, $M_{0x}$, $M_{0z}$ are the x, y and z components
@@ -66 +79 @@
 
 .. math::
-    M_{0x} = M_0\cos\theta_M\cos\phi_M \\
-    M_{0y} = M_0\sin\theta_M \\
-    M_{0z} = -M_0\cos\theta_M\sin\phi_M
+    M_{0x} &= M_0\cos\theta_M\cos\phi_M \\
+    M_{0y} &= M_0\sin\theta_M \\
+    M_{0z} &= -M_0\cos\theta_M\sin\phi_M
 
 and the magnetization angles $\theta_M$ and $\phi_M$ are defined in
@@ -76 +89 @@
 
 ===========   ================================================================
- M0_sld        = $D_M M_0$
- Up_theta      = $\theta_\mathrm{up}$
- M_theta       = $\theta_M$
- M_phi         = $\phi_M$
- Up_frac_i     = (spin up)/(spin up + spin down) neutrons *before* the sample
- Up_frac_f     = (spin up)/(spin up + spin down) neutrons *after* the sample
+ M0:sld       $D_M M_0$
+ mtheta:sld   $\theta_M$
+ mphi:sld     $\phi_M$
+ up:angle     $\theta_\mathrm{up}$
+ up:frac_i    $u_i$ = (spin up)/(spin up + spin down) *before* the sample
+ up:frac_f    $u_f$ = (spin up)/(spin up + spin down) *after* the sample
 ===========   ================================================================
 
 .. note::
-    The values of the 'Up_frac_i' and 'Up_frac_f' must be in the range 0 to 1.
+    The values of the 'up:frac_i' and 'up:frac_f' must be in the range 0 to 1.
 
 *Document History*
 
 | 2015-05-02 Steve King
-| 2017-05-08 Paul Kienzle
+| 2017-11-15 Paul Kienzle