Talk:Unit cell - Revision history

KevinYager: /* Vectors */

2022-11-14T14:26:54Z

‎Vectors

KevinYager: /* Vectors */

2022-11-14T14:14:26Z

‎Vectors

KevinYager: /* Vectors (Wrong?) */

2022-11-14T13:58:21Z

‎Vectors (Wrong?)

KevinYager at 13:30, 14 November 2022

2022-11-14T13:30:16Z

KevinYager at 13:29, 14 November 2022

2022-11-14T13:29:42Z

KevinYager at 16:18, 13 April 2020

2020-04-13T16:18:47Z

KevinYager at 15:24, 18 June 2014

2014-06-18T15:24:49Z

68.194.136.6: Created page with "====TBD: Reciprocal vector components==== : $\begin{alignat}{2} \mathbf{u} & = ... \\ \mathbf{v} & = ... \\ \mathbf{w} & = \frac{\mathbf{a}\times\mathbf{b}}{\mathbf{a}\..."$

2014-06-04T00:21:52Z

Created page with "====TBD: Reciprocal vector components==== :<math> \begin{alignat}{2} \mathbf{u} & = ... \\ \mathbf{v} & = ... \\ \mathbf{w} & = \frac{\mathbf{a}\times\mathbf{b}}{\mathbf{a}\..."

New page

====TBD: Reciprocal vector components====
:<math>
\begin{alignat}{2}
\mathbf{u} & = ... \\
\mathbf{v} & = ... \\
\mathbf{w}
& = \frac{\mathbf{a}\times\mathbf{b}}{\mathbf{a}\cdot (\mathbf{b}\times\mathbf{c}) } \\
& =\frac{1}{V} \mathbf{a}\times\mathbf{b} \\
& =\frac{1}{V} \begin{bmatrix} 0 \\ 0 \\ a b \sin{\gamma} \end{bmatrix} \\
\end{alignat}
</math>

@@ Line 103: / Line 103: @@
      &= \frac{c}{\sin\gamma} \sqrt{1 + 2 \cos\alpha\cos\beta\cos\gamma - \cos^2 \alpha - \cos^2\beta -\cos^2\gamma}\\
      &= c \sqrt{\frac{1 + 2 \cos\alpha\cos\beta\cos\gamma - \cos^2 \alpha - \cos^2\beta -\cos^2\gamma}{\sin^2\gamma} }\\
      &= c \sqrt{\frac{1}{\sin^2\gamma} + \frac{2 \cos\alpha\cos\beta\cos\gamma}{\sin^2\gamma} - \frac{\cos^2 \alpha}{\sin^2\gamma} - \frac{\cos^2\beta}{\sin^2\gamma} - \frac{\cos^2\gamma}{\sin^2\gamma} }\\
 \end{alignat}
 </math>

@@ Line 74: / Line 74: @@
 According to:
 * K. N. Trueblood, H.-B. Bürgi, H. Burzlaff, J. D. Dunitz, C. M. Gramaccioli, H. H. Schulz, U. Shmueli and S. C. Abrahams [https://scripts.iucr.org/cgi-bin/paper?S0108767396005697 Atomic Dispacement Parameter Nomenclature. Report of a Subcommittee on Atomic Displacement Parameter Nomenclature] ''Acta Cryst'' '''1996''', A52, 770-781. [https://doi.org/10.1107/S0108767396005697 doi: 10.1107/S0108767396005697]
 :<math>\begin{array}{l}
 \mathbf{a} = \begin{bmatrix}
@@ Line 92: / Line 93: @@
 = \begin{bmatrix}
 c \cos{\beta} \\
-\frac{c}{\sin \gamma} \left( \cos \alpha - \cos \beta \cos \gamma \right) \\
+c \frac{\cos\alpha -\cos\beta \cos\gamma  }{\sin\gamma} \\
-\frac{c}{\sin \gamma} \sqrt{ 1 - \cos^2 \alpha - \cos^2 \beta - \cos^2 \gamma + 2 \cos \alpha \cos \beta \cos \gamma } \\
+\frac{c}{\sin\gamma} \sqrt{1 + 2 \cos\alpha\cos\beta\cos\gamma - \cos^2 \alpha - \cos^2\beta -\cos^2\gamma}\\
 \end{bmatrix}
 \end{array}
 </math>

@@ Line 20: / Line 20: @@
 </math>
-====Vectors (Wrong?)====
+====Vectors====
 :<math>\begin{array}{l}
 \mathbf{a} = \begin{bmatrix}
@@ Line 45: / Line 46: @@
 </math>
 ====TBD: Reciprocal vector components====

@@ Line 20: / Line 20: @@
 </math>
-===Vectors (Wrong?)===
+====Vectors (Wrong?)====
 :<math>\begin{array}{l}
 \mathbf{a} = \begin{bmatrix}

← Older revision		Revision as of 16:18, 13 April 2020
Line 31:		Line 31:
	\end{alignat}		\end{alignat}
	</math>		</math>
		+
		+	==Calculate q_hkl generally==
		+	<source lang="python">
		+	def q_hkl(self, h, k, l):
		+	"""Determines the position in reciprocal space for the given reflection."""
		+
		+	# The 'unitcell' coordinate system assumes:
		+	# a-axis lies along x-axis
		+	# b-axis is in x-y plane
		+	# c-axis is vertical (or at a tilt, depending on beta)
		+
		+	# Convert from (unitcell) Cartesian to (unitcell) fractional coordinates
		+	reduced_volume = sqrt( 1 - (cos(self.alpha))2 - (cos(self.beta))2 - (cos(self.gamma))*2 + 2cos(self.alpha)cos(self.beta)cos(self.gamma) )
		+	#volume = reduced_volumeself.lattice_spacing_aself.lattice_spacing_b*self.lattice_spacing_c
		+	a = ( self.lattice_spacing_a , \
		+	0.0 , \
		+	0.0 )
		+	b = ( self.lattice_spacing_b*cos(self.gamma) , \
		+	self.lattice_spacing_b*sin(self.gamma) , \
		+	0.0 )
		+	c = ( self.lattice_spacing_c*cos(self.beta) , \
		+	self.lattice_spacing_c( cos(self.alpha) - cos(self.beta)cos(self.gamma) )/( sin(self.gamma) ) , \
		+	self.lattice_spacing_c*reduced_volume/( sin(self.gamma) ) )
		+
		+	# Compute (unitcell) reciprocal-space lattice vectors
		+	volume = np.dot( a, np.cross(b,c) )
		+	u = np.cross( b, c ) / volume # Along qx
		+	v = np.cross( c, a ) / volume # Along qy
		+	w = np.cross( a, b ) / volume # Along qz
		+
		+	qhkl_vector = 2pi( hu + kv + l*w )
		+	qhkl = sqrt( qhkl_vector[0]2 + qhkl_vector[1]2 + qhkl_vector[2]**2 )
		+
		+	return (qhkl, qhkl_vector)
		+
		+
		+
		+	def q_hkl_length(self, h, k, l):
		+
		+	qhkl, qhkl_vector = self.q_hkl(h,k,l)
		+	#qhkl = sqrt( qhkl_vector[0]2 + qhkl_vector[1]2 + qhkl_vector[2]**2 )
		+
		+	return qhkl
		+	</source>

← Older revision		Revision as of 15:24, 18 June 2014
Line 1:		Line 1:
		+	====Extra math expressions====
		+	A given real-space cubic lattice will have dimensions:
		+	:<math>\left(a,b,c\right)</math>
		+	Such that the position of any particular cell within the infinite lattice is:
		+	:<math>\mathbf{r}_{hkl} = \left\langle ah,bk,cl\right\rangle </math>
		+	Where ''h'', ''k'', and ''l'' are indices.
		+	The corresponding inverse-space lattice would be:
		+	:<math>
		+	\mathbf{q}_{hkl} = 2\pi \left\langle \frac{h}{a} , \frac{k}{b} , \frac{l}{c} \right\rangle
		+	</math>
		+	:<math>
		+	q_{hkl} = 2\pi \sqrt{ \left( \frac{h}{a} \right)^2 + \left( \frac{k}{b} \right)^2 + \left( \frac{l}{c} \right)^2 }
		+	</math>
		+	In the case where <math>a=b=c</math>:
		+	:<math>
		+	\begin{alignat}{2}
		+	q_{hkl} & = 2\pi \sqrt{ \left( \frac{h}{a} \right)^2 + \left( \frac{k}{a} \right)^2 + \left( \frac{l}{a} \right)^2 } \\
		+	& = \frac{2\pi}{a} \sqrt{ h^2 + k^2 + l^2 }
		+	\end{alignat}
		+	</math>
		+
	====TBD: Reciprocal vector components====		====TBD: Reciprocal vector components====
	:<math>		:<math>

Talk:Unit cell - Revision history

KevinYager: /* Vectors */

KevinYager: /* Vectors */

KevinYager: /* Vectors (Wrong?) */

KevinYager at 13:30, 14 November 2022

KevinYager at 13:29, 14 November 2022

KevinYager at 16:18, 13 April 2020

KevinYager at 15:24, 18 June 2014

68.194.136.6: Created page with "====TBD: Reciprocal vector components==== : \begin{alignat}{2} \mathbf{u} & = ... \\ \mathbf{v} & = ... \\ \mathbf{w} & = \frac{\mathbf{a}\times\mathbf{b}}{\mathbf{a}\..."

68.194.136.6: Created page with "====TBD: Reciprocal vector components==== : $\begin{alignat}{2} \mathbf{u} & = ... \\ \mathbf{v} & = ... \\ \mathbf{w} & = \frac{\mathbf{a}\times\mathbf{b}}{\mathbf{a}\..."$