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# Module weights

source code

 Functions

 calc_sphere_ci(data, diff_data) Weight for spherical diffusion. source code

 calc_spheroid_ci(data, diff_data) Weights for spheroidal diffusion. source code

 calc_spheroid_dci(data, diff_data) Weight gradient for spheroidal diffusion. source code

 calc_spheroid_d2ci(data, diff_data) Weight Hessian for spheroidal diffusion. source code

 calc_ellipsoid_ci(data, diff_data) Weight equations for ellipsoidal diffusion. source code

 calc_ellipsoid_dci(data, diff_data) Weight gradient for ellipsoidal diffusion. source code

 calc_ellipsoid_d2ci(data, diff_data) Weight Hessian for ellipsoidal diffusion. source code
 Variables
__package__ = `'lib.diffusion'`

Imports: sqrt, outer

 Function Details

### calc_sphere_ci(data, diff_data)

source code

Weight for spherical diffusion.

The equation is:

```   c0 = 1.
```

### calc_spheroid_ci(data, diff_data)

source code

Weights for spheroidal diffusion.

The equations are:

```   c-1 = 1/4 (3dz**2 - 1)**2,

c0  = 3dz**2 (1 - dz**2),

c1  = 3/4 (dz**2 - 1)**2,
```

where dz is the direction cosine of the unit bond vector along the z-axis of the diffusion tensor which is calculated as the dot product of the unit bond vector with a unit vector along Dpar.

### calc_spheroid_dci(data, diff_data)

source code

The equations are:

```   dc-1                     ddz
---- =  3dz (3dz**2 - 1) --- ,
dOi                      dOi

dc0                      ddz
---  =  6dz (1 - 2dz**2) --- ,
dOi                      dOi

dc1                     ddz
---  =  3dz (dz**2 - 1) --- ,
dOi                     dOi
```

where the orientation parameter set O is {theta, phi}.

### calc_spheroid_d2ci(data, diff_data)

source code

Weight Hessian for spheroidal diffusion.

The equations are:

```    d2c-1        /             ddz   ddz                      d2dz   \
-------  =  3 |(9dz**2 - 1) --- . ---  +  dz (3dz**2 - 1) ------- | ,
dOi.dOj       \             dOi   dOj                     dOi.dOj /

d2c0         /             ddz   ddz                      d2dz   \
-------  =  6 |(1 - 6dz**2) --- . ---  +  dz (1 - 2dz**2) ------- | ,
dOi.dOj       \             dOi   dOj                     dOi.dOj /

d2c1         /             ddz   ddz                     d2dz   \
-------  =  3 |(3dz**2 - 1) --- . ---  +  dz (dz**2 - 1) ------- | ,
dOi.dOj       \             dOi   dOj                    dOi.dOj /
```

where the orientation parameter set O is {theta, phi}.

### calc_ellipsoid_ci(data, diff_data)

source code

Weight equations for ellipsoidal diffusion.

The equations are:

```   c-2 = 1/4 (d - e),

c-1 = 3dy**2.dz**2,

c0  = 3dx**2.dz**2,

c1  = 3dx**2.dy**2,

c2  = 1/4 (d + e),
```

where:

```   d  = 3(dx**4 + dy**4 + dz**4) - 1,

e  =  1/R [(1 + 3Dr)(dx**4 + 2dy**2.dz**2) + (1 - 3Dr)(dy**4 + 2dx**2.dz**2)
- 2(dz**4 + 2dx**2.dy**2)],
```

and where the factor R is defined as:

```        ___________
R = V 1 + 3Dr**2.
```

dx, dy, and dz are the direction cosines of the XH bond vector along the x, y, and z-axes of the diffusion tensor, calculated as the dot product of the unit bond vector and the unit vectors along Dx, Dy, and Dz respectively.

source code

# Oi partial derivatives

The equations are:

```   dc-2       /       ddx           ddy           ddz \     de
----  =  3 | dx**3 ---  +  dy**3 ---  +  dz**3 --- |  -  --- ,
dOi        \       dOi           dOi           dOi /     dOi

dc-1            /    ddz        ddy \
----  =  6dy.dz | dy ---  +  dz --- | ,
dOi             \    dOi        dOi /

dc0            /    ddz        ddx \
---  =  6dx.dz | dx ---  +  dz --- | ,
dOi            \    dOi        dOi /

dc1            /    ddy        ddx \
---  =  6dx.dy | dx ---  +  dy --- | ,
dOi            \    dOi        dOi /

dc2       /       ddx           ddy           ddz \     de
---  =  3 | dx**3 ---  +  dy**3 ---  +  dz**3 --- |  +  --- ,
dOi       \       dOi           dOi           dOi /     dOi
```

where:

```   de      1 /           /      ddx           /    ddz        ddy \ \
---  =  - | (1 + 3Dr) |dx**3 ---  +  dy.dz | dy ---  +  dz --- | |
dOi     R \           \      dOi           \    dOi        dOi / /

/       ddy           /    ddz        ddx \ \
+ (1 - 3Dr) | dy**3 ---  +  dx.dz | dx ---  +  dz --- | |
\       dOi           \    dOi        dOi / /

/       ddz           /    ddy        ddx \ \ \
- 2 | dz**3 ---  +  dx.dy | dx ---  +  dy --- | | | ,
\       dOi           \    dOi        dOi / / /
```

and where the orietation parameter set O is:

```   O = {alpha, beta, gamma}.
```

# tm partial derivatives

The equations are:

```   dc-2
----  =  0,
dtm

dc-1
----  =  0,
dtm

dc0
---   =  0,
dtm

dc1
---   =  0,
dtm

dc2
---   =  0.
dtm
```

# Da partial derivatives

The equations are:

```   dc-2
----  =  0,
dDa

dc-1
----  =  0,
dDa

dc0
---   =  0,
dDa

dc1
---   =  0,
dDa

dc2
---   =  0.
dDa
```

# Dr partial derivatives

The equations are:

```   dc-2       3 de
----  =  - - ---,
dDr        4 dDr

dc-1
----  =  0,
dDr

dc0
---   =  0,
dDr

dc1
---   =  0,
dDr

dc2      3 de
---   =  - ---,
dDr      4 dDr
```

where:

```   de       1   /                                                                                                \
---  =  ---- | (1 - Dr) (dx**4 + 2dy**2.dz**2) - (1 + Dr) (dy**4 + 2dx**2.dz**2) + 2Dr (dz**4 + 2dx**2.dy**2) | .
dDr     R**3 \                                                                                                /
```

### calc_ellipsoid_d2ci(data, diff_data)

source code

Weight Hessian for ellipsoidal diffusion.

# Oi-Oj partial derivatives

The equations are:

```    d2c-2        /       /     d2dx       ddx   ddx \           /     d2dy       ddy   ddy \           /     d2dz       ddz   ddz \ \       d2e
-------  =  3 | dx**2 | dx ------- + 3 --- . --- |  +  dy**2 | dy ------- + 3 --- . --- |  +  dz**2 | dz ------- + 3 --- . --- | |  -  ------- ,
dOi.dOj       \       \    dOi.dOj     dOi   dOj /           \    dOi.dOj     dOi   dOj /           \    dOi.dOj     dOi   dOj / /     dOi.dOj

d2c-1              /     d2dz       ddz   ddz \              / ddy   ddz     ddz   ddy \             /     d2dy       ddy   ddy \
-------  =  6 dy**2 | dz -------  +  --- . --- |  +  12 dy.dz | --- . ---  +  --- . --- |  +  6 dz**2 | dy -------  +  --- . --- | ,
dOi.dOj             \    dOi.dOj     dOi   dOj /              \ dOi   dOj     dOi   dOj /             \    dOi.dOj     dOi   dOj /

d2c0               /     d2dz       ddz   ddz \              / ddx   ddz     ddz   ddx \             /     d2dx       ddx   ddx \
-------  =  6 dx**2 | dz -------  +  --- . --- |  +  12 dx.dz | --- . ---  +  --- . --- |  +  6 dz**2 | dx -------  +  --- . --- | ,
dOi.dOj             \    dOi.dOj     dOi   dOj /              \ dOi   dOj     dOi   dOj /             \    dOi.dOj     dOi   dOj /

d2c1               /     d2dy       ddy   ddy \              / ddx   ddy     ddy   ddx \             /     d2dx       ddx   ddx \
-------  =  6 dx**2 | dy -------  +  --- . --- |  +  12 dx.dy | --- . ---  +  --- . --- |  +  6 dy**2 | dx -------  +  --- . --- | ,
dOi.dOj             \    dOi.dOj     dOi   dOj /              \ dOi   dOj     dOi   dOj /             \    dOi.dOj     dOi   dOj /

d2c2         /       /     d2dx       ddx   ddx \           /     d2dy       ddy   ddy \           /     d2dz       ddz   ddz \ \       d2e
-------  =  3 | dx**2 | dx ------- + 3 --- . --- |  +  dy**2 | dy ------- + 3 --- . --- |  +  dz**2 | dz ------- + 3 --- . --- | |  +  ------- ,
dOi.dOj       \       \    dOi.dOj     dOi   dOj /           \    dOi.dOj     dOi   dOj /           \    dOi.dOj     dOi   dOj / /     dOi.dOj
```

where:

```     d2e       1 /           /       /     d2dx       ddx   ddx \           /     d2dz       ddz   ddz \
-------  =  - | (1 + 3Dr) | dx**2 | dx ------- + 3 --- . --- |  +  dy**2 | dz -------  +  --- . --- |
dOi.dOj     R \           \       \    dOi.dOj     dOi   dOj /           \    dOi.dOj     dOi   dOj /

/     d2dy       ddy   ddy \            / ddy   ddz     ddz   ddy \ \
+ dz**2 | dy -------  +  --- . --- |  +  2dy.dz | --- . ---  +  --- . --- | |
\    dOi.dOj     dOi   dOj /            \ dOi   dOj     dOi   dOj / /

/       /     d2dy       ddy   ddy \           /     d2dz       ddz   ddz \
+ (1 - 3Dr) | dy**2 | dy ------- + 3 --- . --- |  +  dx**2 | dz -------  +  --- . --- |
\       \    dOi.dOj     dOi   dOj /           \    dOi.dOj     dOi   dOj /

/     d2dx       ddx   ddx \            / ddx   ddz     ddz   ddx \ \
+ dz**2 | dx -------  +  --- . --- |  +  2dx.dz | --- . ---  +  --- . --- | |
\    dOi.dOj     dOi   dOj /            \ dOi   dOj     dOi   dOj / /

/       /     d2dz       ddz   ddz \           /     d2dy       ddy   ddy \
- 2 | dz**2 | dz ------- + 3 --- . --- |  +  dx**2 | dy -------  +  --- . --- |
\       \    dOi.dOj     dOi   dOj /           \    dOi.dOj     dOi   dOj /

/     d2dx       ddx   ddx \            / ddx   ddy     ddy   ddx \ \ \
+ dy**2 | dx -------  +  --- . --- |  +  2dx.dy | --- . ---  +  --- . --- | | |
\    dOi.dOj     dOi   dOj /            \ dOi   dOj     dOi   dOj / / /
```

# Oi-tm partial derivatives

The equations are:

```    d2c-2
-------  =  0,
dOi.dtm

d2c-1
-------  =  0,
dOi.dtm

d2c0
-------  =  0,
dOi.dtm

d2c1
-------  =  0,
dOi.dtm

d2c2
-------  =  0.
dOi.dtm
```

# Oi-Da partial derivatives

The equations are:

```    d2c-2
-------  =  0,
dOi.dDa

d2c-1
-------  =  0,
dOi.dDa

d2c0
-------  =  0,
dOi.dDa

d2c1
-------  =  0,
dOi.dDa

d2c2
-------  =  0.
dOi.dDa
```

# Oi-Dr partial derivatives

The equations are:

```    d2c-2            d2e
-------  =  - 3 -------,
dOi.dDr         dOi.dDr

d2c-1
-------  =  0,
dOi.dDr

d2c0
-------  =  0,
dOi.dDr

d2c1
-------  =  0,
dOi.dDr

d2c2           d2e
-------  =  3 -------,
dOi.dDr       dOi.dDr
```

where:

```    d2e         1   /          /       ddx           /    ddz        ddy \ \
-------  =  ---- | (1 - Dr) | dx**3 ---  +  dy.dz | dy ---  +  dz --- | |
dOi.dDr     R**3 \          \       dOi           \    dOi        dOi / /

/       ddy           /    ddz        ddx \ \
- (1 + Dr) | dy**3 ---  +  dx.dz | dx ---  +  dz --- | |
\       dOi           \    dOi        dOi / /

/       ddz           /    ddy        ddx \ \ \
+ 2Dr | dz**3 ---  +  dx.dy | dx ---  +  dy --- | | |
\       dOi           \    dOi        dOi / / /
```

# tm-tm partial derivatives

The equations are:

```   d2c-2
-----  =  0,
dtm2

d2c-1
-----  =  0,
dtm2

d2c0
----   =  0,
dtm2

d2c1
----   =  0,
dtm2

d2c2
----   =  0.
dtm2
```

# tm-Da partial derivatives

The equations are:

```    d2c-2
-------  =  0,
dtm.dDa

d2c-1
-------  =  0,
dtm.dDa

d2c0
-------  =  0,
dtm.dDa

d2c1
-------  =  0,
dtm.dDa

d2c2
-------  =  0.
dtm.dDa
```

# tm-Dr partial derivatives

The equations are:

```    d2c-2
-------  =  0,
dtm.dDr

d2c-1
-------  =  0,
dtm.dDr

d2c0
-------  =  0,
dtm.dDr

d2c1
-------  =  0,
dtm.dDr

d2c2
-------  =  0.
dtm.dDr
```

# Da-Da partial derivatives

The equations are:

```   d2c-2
------  =  0,
dDa**2

d2c-1
------  =  0,
dDa**2

d2c0
------  =  0,
dDa**2

d2c1
------  =  0,
dDa**2

d2c2
------  =  0.
dDa**2
```

# Da-Dr partial derivatives

The equations are:

```    d2c-2
-------  =  0,
dDa.dDr

d2c-1
-------  =  0,
dDa.dDr

d2c0
-------  =  0,
dDa.dDr

d2c1
-------  =  0,
dDa.dDr

d2c2
-------  =  0.
dDa.dDr
```

# Dr-Dr partial derivatives

The equations are:

```   d2c-2        3  d2e
------  =  - - ------,
dDr**2       4 dDr**2

d2c-1
------  =  0,
dDr**2

d2c0
------  =  0,
dDr**2

d2c1
------  =  0,
dDr**2

d2c2      3  d2e
------  =  - ------,
dDr**2     4 dDr**2
```

where:

```    d2e        1   /                                                                                                                           \
------  =  ---- | (6Dr**2 - 9Dr - 1)(dx**4 + 2dy**2.dz**2) + (6Dr**2 + 9Dr - 1)(dy**4 + 2dx**2.dz**2) - 2(6Dr**2 - 1)(ddz*4 + 2dx**2.dy**2) |
dDr**2     R**5 \                                                                                                                           /
```

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