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manual_part_2 [2009/08/26 14:57]
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manual_part_2 [2009/08/27 09:54] (current)
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 The weight fraction w<​sub>​p</​sub>​ for phase "​p"​ is calculated as follows: The weight fraction w<​sub>​p</​sub>​ for phase "​p"​ is calculated as follows:
  
-| <​sub>​{{Technical_Reference%20V4-1_files:​image156.gif?​76x70}}</​sub>​   | where   N<​sub>​p</​sub>​ = Number of phases Q<​sub>​p</​sub>​ = S<​sub>​p</​sub>​M<​sub>​p</​sub>​V<​sub>​p</​sub>/​B<​sub>​p</​sub>​ S<​sub>​p</​sub>​ = Rietveld scale factor for phase p. M<​sub>​p</​sub>​ = Unit cell mass for phase p. V<​sub>​p</​sub>​ = Unit cell volume for phase p. B<​sub>​p</​sub>​ = Brindley correction for phase p. |+| <​sub>​{{techref_files:​image156.gif?​76x70}}</​sub>​   | where   N<​sub>​p</​sub>​ = Number of phases Q<​sub>​p</​sub>​ = S<​sub>​p</​sub>​M<​sub>​p</​sub>​V<​sub>​p</​sub>/​B<​sub>​p</​sub>​ S<​sub>​p</​sub>​ = Rietveld scale factor for phase p. M<​sub>​p</​sub>​ = Unit cell mass for phase p. V<​sub>​p</​sub>​ = Unit cell volume for phase p. B<​sub>​p</​sub>​ = Brindley correction for phase p. |
  
 The Brindley correction is a function of //​brindley_spherical_r_cm//​ and the phase and mixture linear absorption coefficients;​ the latter two are in turn functions of //​phase_MAC//​ and //​mixture_MAC//​ respectively,​ or, The Brindley correction is a function of //​brindley_spherical_r_cm//​ and the phase and mixture linear absorption coefficients;​ the latter two are in turn functions of //​phase_MAC//​ and //​mixture_MAC//​ respectively,​ or,
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 **[//​chi2_convergence_criteria//​ !E]** **[//​chi2_convergence_criteria//​ !E]**
  
-Convergence is determined when the change in <​sub>​{{Technical_Reference%20V4-1_files:​image002.gif?​20x23}}</​sub>​ is less than //​chi2_convergence_criteria//​ for three consecutive cycles and when all defined //​stop_when//​ parameter attributes evaluate to true. Example:+Convergence is determined when the change in <​sub>​{{techref_files:​image002.gif?​20x23}}</​sub>​ is less than //​chi2_convergence_criteria//​ for three consecutive cycles and when all defined //​stop_when//​ parameter attributes evaluate to true. Example:
  
 chi2_convergence_criteria =  If(Cycle_Iter < 10, .001, .01); chi2_convergence_criteria =  If(Cycle_Iter < 10, .001, .01);
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 //​lp_search//​ uses a new indexing algorithm that is independent of d-spacing extraction; lp-search-pbso4.inp demonstrates its use. lp_search minimizes on a new figure of merit function that gives a measure of correctness for a particular set of lattice parameters. More specifically the figure of merit function assigns parts of the diffraction pattern to peaks and then sums the absolute values of the products of the diffraction intensities multiplied by the distance to the peaks: //​lp_search//​ uses a new indexing algorithm that is independent of d-spacing extraction; lp-search-pbso4.inp demonstrates its use. lp_search minimizes on a new figure of merit function that gives a measure of correctness for a particular set of lattice parameters. More specifically the figure of merit function assigns parts of the diffraction pattern to peaks and then sums the absolute values of the products of the diffraction intensities multiplied by the distance to the peaks:
  
-<​sub>​{{Technical_Reference%20V4-1_files:​image158.gif?​213x39}}</​sub>​where the summation in ‘j’ is over the calculated Bragg positions 2//​q<​sub>​0</​sub>//​ and the summation in ‘i’ is  over part of the diffraction pattern such that (2//​q//<​sub>​</​sub><​sub>​0,​ j-1</​sub>​ + 2//​q//<​sub>​</​sub>​ <​sub>​0,​ j</​sub>​.)/​2 < 2//​q//<​sub>​i</​sub>​  <  (2//​q//<​sub>​</​sub>​ <​sub>​0,​ j+1</​sub>​ + 2//​q//<​sub>​</​sub><​sub>​0,​ j</​sub>​.)/​2. The method avoids difficulties associated with extracting d-spacings from complex patterns comprising heavily overlapped lines; the primary difficulty being that of ascertaining the number of lines present.+<​sub>​{{techref_files:​image158.gif?​213x39}}</​sub>​where the summation in ‘j’ is over the calculated Bragg positions 2//​q<​sub>​0</​sub>//​ and the summation in ‘i’ is  over part of the diffraction pattern such that (2//​q//<​sub>​</​sub><​sub>​0,​ j-1</​sub>​ + 2//​q//<​sub>​</​sub>​ <​sub>​0,​ j</​sub>​.)/​2 < 2//​q//<​sub>​i</​sub>​  <  (2//​q//<​sub>​</​sub>​ <​sub>​0,​ j+1</​sub>​ + 2//​q//<​sub>​</​sub><​sub>​0,​ j</​sub>​.)/​2. The method avoids difficulties associated with extracting d-spacings from complex patterns comprising heavily overlapped lines; the primary difficulty being that of ascertaining the number of lines present.
  
 //​I_parameter_names_have_hkl//​ gives generated Intensity parameters a name starting with $start_of_parameter_name and ending with the corresponding hkl. //​I_parameter_names_have_hkl//​ gives generated Intensity parameters a name starting with $start_of_parameter_name and ending with the corresponding hkl.
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 Calculates the mass absorption coefficient in cm<​sup>​2</​sup>/​g for a mixture as follows: Calculates the mass absorption coefficient in cm<​sup>​2</​sup>/​g for a mixture as follows:
  
-<​sub>​{{Technical_Reference%20V4-1_files:​image160.gif?​160x45}}</​sub>​where //​w//<​sub>​i</​sub>​ and (//​m/////​r//​)<​sub>​i</​sub>​ is the weight percent and //​phase_MAC//​ of phase i respectively. Errors are reported for //​phase_MAC//​ and //​mixture_MAC//​.+<​sub>​{{techref_files:​image160.gif?​160x45}}</​sub>​where //​w//<​sub>​i</​sub>​ and (//​m/////​r//​)<​sub>​i</​sub>​ is the weight percent and //​phase_MAC//​ of phase i respectively. Errors are reported for //​phase_MAC//​ and //​mixture_MAC//​.
  
 The following example provides phase and mixture mass absorption coefficients. The following example provides phase and mixture mass absorption coefficients.
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 //​phase_penalties//​ for a single hkl is defined as follows: //​phase_penalties//​ for a single hkl is defined as follows:
  
-<​sub>​{{Technical_Reference%20V4-1_files:​image162.gif?​444x50}}</​sub>​where <​sub>​{{Technical_Reference%20V4-1_files:​image164.gif?​21x29}}</​sub>​= assigned phase, <​sub>​{{Technical_Reference%20V4-1_files:​image166.gif?​20x25}}</​sub>​= calculated phase, I<​sub>​c</​sub>​ = calculated intensity and //d// is the reflection d-spacing.  The name N returns the sum of the //​phase_penalties//​ and it can be used in equations and in particular //penalty// equations. <​sub>​{{Technical_Reference%20V4-1_files:​image166.gif?​20x25}}</​sub>​ is calculated from sites identified in [[#​k142|$sites]].+<​sub>​{{techref_files:​image162.gif?​444x50}}</​sub>​where <​sub>​{{techref_files:​image164.gif?​21x29}}</​sub>​= assigned phase, <​sub>​{{techref_files:​image166.gif?​20x25}}</​sub>​= calculated phase, I<​sub>​c</​sub>​ = calculated intensity and //d// is the reflection d-spacing.  The name N returns the sum of the //​phase_penalties//​ and it can be used in equations and in particular //penalty// equations. <​sub>​{{techref_files:​image166.gif?​20x25}}</​sub>​ is calculated from sites identified in [[#​k142|$sites]].
  
-#h, #k, #l are user defined hkls; they are used for formulating the phase penalties. #Re and #Im are the real and imaginary parts of <​sub>​{{Technical_Reference%20V4-1_files:​image164.gif?​21x29}}</​sub>​. An example use of phase penalties (see examples AE14-12.INP and AE5-AUTO.inp) is as follows:+#h, #k, #l are user defined hkls; they are used for formulating the phase penalties. #Re and #Im are the real and imaginary parts of <​sub>​{{techref_files:​image164.gif?​21x29}}</​sub>​. An example use of phase penalties (see examples AE14-12.INP and AE5-AUTO.inp) is as follows:
  
 penalty = pp1; penalty = pp1;
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 **[//​quick_refine//​ !E [//​quick_refine_remove//​ !E]]** **[//​quick_refine//​ !E [//​quick_refine_remove//​ !E]]**
  
-//​quick_refine//​ removes parameters that influence <​sub>​{{Technical_Reference%20V4-1_files:​image002.gif?​20x23}}</​sub>​ in a small manner during a [[#​k144|refinement cycle]]. Use of //​quick_refine//​ speeds up simulated annealing, see for example the macro Auto_T which is used in example AE1-AUTO.INP. All refined parameters are reinstated for refinement at the start of subsequent cycles. Large //​quick_refine//​ values aggressively removes parameters and convergence to low <​sub>​{{Technical_Reference%20V4-1_files:​image002.gif?​20x23}}</​sub>​ maybe hindered. A value of 0.1 works well.+//​quick_refine//​ removes parameters that influence <​sub>​{{techref_files:​image002.gif?​20x23}}</​sub>​ in a small manner during a [[#​k144|refinement cycle]]. Use of //​quick_refine//​ speeds up simulated annealing, see for example the macro Auto_T which is used in example AE1-AUTO.INP. All refined parameters are reinstated for refinement at the start of subsequent cycles. Large //​quick_refine//​ values aggressively removes parameters and convergence to low <​sub>​{{techref_files:​image002.gif?​20x23}}</​sub>​ maybe hindered. A value of 0.1 works well.
  
 //​quick_refine//​ has the following consequences:​ //​quick_refine//​ has the following consequences:​
  
-  * If parameters are not reinstated using //​quick_refine_remove//​ then<​sub>​{{Technical_Reference%20V4-1_files:​image002.gif?​20x23}}</​sub>​ does not get to its lowest possible value for a particular refinement cycle. +  * If parameters are not reinstated using //​quick_refine_remove//​ then<​sub>​{{techref_files:​image002.gif?​20x23}}</​sub>​ does not get to its lowest possible value for a particular refinement cycle. 
   * The degree of parameter randomization increases with increasing values of //​quick_refine//​. Thus randomization should be reduced as //​quick_refine//​ increases. Alternatively //​randomize_on_errros//​ can be used which automatically determines the amount a parameter is randomized.   * The degree of parameter randomization increases with increasing values of //​quick_refine//​. Thus randomization should be reduced as //​quick_refine//​ increases. Alternatively //​randomize_on_errros//​ can be used which automatically determines the amount a parameter is randomized.
  
 //​quick_refine_remove//​removes a parameter from refinement if it evaluates to non-zero or reinstates a parameter if it evaluates to zero. It can be a function of the reserved parameters QR_Removed or QR_Num_Times_Consecutively_Small and additionally global reserved parameter names such as Cycle_Iter, Cycle and T. If //​quick_refine_remove//​ is not defined then the removal scheme of section 5.6 is used and parameters are not reinstated until the next refinement cycle. //​quick_refine_remove//​removes a parameter from refinement if it evaluates to non-zero or reinstates a parameter if it evaluates to zero. It can be a function of the reserved parameters QR_Removed or QR_Num_Times_Consecutively_Small and additionally global reserved parameter names such as Cycle_Iter, Cycle and T. If //​quick_refine_remove//​ is not defined then the removal scheme of section 5.6 is used and parameters are not reinstated until the next refinement cycle.
  
-In most refinements the following will produce close to the lowest <​sub>​{{Technical_Reference%20V4-1_files:​image002.gif?​20x23}}</​sub>​ and in a shorter time period (see for example PAWLEY1.INP). ​+In most refinements the following will produce close to the lowest <​sub>​{{techref_files:​image002.gif?​20x23}}</​sub>​ and in a shorter time period (see for example PAWLEY1.INP). ​
  
 quick_refine .1 quick_refine .1
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 N of //​sites_flatten//​ returns a restraint term that decreases as the sites become coplanar; it is defined as follows: N of //​sites_flatten//​ returns a restraint term that decreases as the sites become coplanar; it is defined as follows:
  
-<​sub>​{{Technical_Reference%20V4-1_files:​image169.gif?​529x47}}</​sub>​ +<​sub>​{{techref_files:​image169.gif?​529x47}}</​sub>​ 
  
 where tol corresponds to //​sites_flatten_tol//,​ n corresponds to the number of sites defined with the //​site_to_restrain//​ keyword, b are Cartesian unit length vectors between the sites and the geometric centre of the sites. where tol corresponds to //​sites_flatten_tol//,​ n corresponds to the number of sites defined with the //​site_to_restrain//​ keyword, b are Cartesian unit length vectors between the sites and the geometric centre of the sites.
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 **[****//​swap_sites//​** **$sites_1 $sites_2]…** **[****//​swap_sites//​** **$sites_1 $sites_2]…**
  
-//​swap_sites//​ attempts to find a lower <​sub>​{{Technical_Reference%20V4-1_files:​image002.gif?​20x23}}</​sub>​ by swapping [[#​k142|sites defined]] in $sites_1 with those defined in $sites_2. The swapping continues until all swaps are performed. Outline of the algorithm:+//​swap_sites//​ attempts to find a lower <​sub>​{{techref_files:​image002.gif?​20x23}}</​sub>​ by swapping [[#​k142|sites defined]] in $sites_1 with those defined in $sites_2. The swapping continues until all swaps are performed. Outline of the algorithm:
  
 While //​swap_site//​ processes still to be processed { While //​swap_site//​ processes still to be processed {
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 Randomize the cation configuration according to //​swap_sites//​. Randomize the cation configuration according to //​swap_sites//​.
  
-Find the local minimum of <​sub>​{{Technical_Reference%20V4-1_files:​image002.gif?​20x23}}</​sub>​+Find the local minimum of <​sub>​{{techref_files:​image002.gif?​20x23}}</​sub>​
  
 k = k + 1 k = k + 1
  
-If <​sub>​{{Technical_Reference%20V4-1_files:​image016.gif?​20x23}}</​sub>​(k+1) > <​sub>​{{Technical_Reference%20V4-1_files:​image016.gif?​20x23}}</​sub>​(k) then reset site positions+If <​sub>​{{techref_files:​image016.gif?​20x23}}</​sub>​(k+1) > <​sub>​{{techref_files:​image016.gif?​20x23}}</​sub>​(k) then reset site positions
  
-While <​sub>​{{Technical_Reference%20V4-1_files:​image016.gif?​20x23}}</​sub>​(k+1) > <​sub>​{{Technical_Reference%20V4-1_files:​image016.gif?​20x23}}</​sub>​(k) or all swap possibility exhausted+While <​sub>​{{techref_files:​image016.gif?​20x23}}</​sub>​(k+1) > <​sub>​{{techref_files:​image016.gif?​20x23}}</​sub>​(k) or all swap possibility exhausted
  
 } }
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 The reserved parameter T returns the current temperature and it can be used in equations and in particular the //​[[#​x000|val_on_continue]]//​ attribute. The first //​temperature//​ defined becomes the starting temperature;​ subsequent //​temperature//​(s) become the current temperature. The reserved parameter T returns the current temperature and it can be used in equations and in particular the //​[[#​x000|val_on_continue]]//​ attribute. The first //​temperature//​ defined becomes the starting temperature;​ subsequent //​temperature//​(s) become the current temperature.
  
-If <​sub>​{{Technical_Reference%20V4-1_files:​image016.gif?​20x23}}</​sub>​ increases relative to a previous cycle then the temperature is advanced to the next //​temperature//​. If <​sub>​{{Technical_Reference%20V4-1_files:​image016.gif?​20x23}}</​sub>​ decreases relative to previous temperatures of lesser values then the current temperature is rewound to a previous temperature such that its previous is of a greater value.+If <​sub>​{{techref_files:​image016.gif?​20x23}}</​sub>​ increases relative to a previous cycle then the temperature is advanced to the next //​temperature//​. If <​sub>​{{techref_files:​image016.gif?​20x23}}</​sub>​ decreases relative to previous temperatures of lesser values then the current temperature is rewound to a previous temperature such that its previous is of a greater value.
  
 //​move_to_the_next_temperature_regardless_of_the_change_in_rwp//​ forces the refinement to move to the next temperature regardless of the change in Rwp from the previous temperature. //​move_to_the_next_temperature_regardless_of_the_change_in_rwp//​ forces the refinement to move to the next temperature regardless of the change in Rwp from the previous temperature.
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 temperature 1 temperature 1
  
-If the current temperature is the last one defined (the fourth one), and <​sub>​{{Technical_Reference%20V4-1_files:​image016.gif?​20x23}}</​sub>​ decreased relative to the second and third //​temperature//​s,​ then the current temperature is set to the second temperature.+If the current temperature is the last one defined (the fourth one), and <​sub>​{{techref_files:​image016.gif?​20x23}}</​sub>​ decreased relative to the second and third //​temperature//​s,​ then the current temperature is set to the second temperature.
  
 The current temperature can be used in all equations using the reserved parameter T, for example: The current temperature can be used in all equations using the reserved parameter T, for example:
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 Change the cation configuration according to //​try_site_patterns//​ Change the cation configuration according to //​try_site_patterns//​
  
-Find the local minimum of <​sub>​{{Technical_Reference%20V4-1_files:​image002.gif?​20x23}}</​sub>​+Find the local minimum of <​sub>​{{techref_files:​image002.gif?​20x23}}</​sub>​
  
-If <​sub>​{{Technical_Reference%20V4-1_files:​image016.gif?​20x23}}</​sub>​(k+1) > <​sub>​{{Technical_Reference%20V4-1_files:​image016.gif?​20x23}}</​sub>​(k) then reset site positions+If <​sub>​{{techref_files:​image016.gif?​20x23}}</​sub>​(k+1) > <​sub>​{{techref_files:​image016.gif?​20x23}}</​sub>​(k) then reset site positions
  
 k = k + 1 k = k + 1
  
-While <​sub>​{{Technical_Reference%20V4-1_files:​image016.gif?​20x23}}</​sub>​(k+1) > <​sub>​{{Technical_Reference%20V4-1_files:​image016.gif?​20x23}}</​sub>​(k) or //k// < //​num_patterns_at_a_time//​+While <​sub>​{{techref_files:​image016.gif?​20x23}}</​sub>​(k+1) > <​sub>​{{techref_files:​image016.gif?​20x23}}</​sub>​(k) or //k// < //​num_patterns_at_a_time//​
  
 } }
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 //​try_site_patterns//​ "Ca Zr" //​num_patterns_at_a_time//​ 3 //​try_site_patterns//​ "Ca Zr" //​num_patterns_at_a_time//​ 3
  
-N<​sub>​CaZr</​sub>​=10 and thus //​try_site_patterns//​ will cycle through the 10 patterns searching for a reduction in <​sub>​{{Technical_Reference%20V4-1_files:​image002.gif?​20x23}}</​sub>​.+N<​sub>​CaZr</​sub>​=10 and thus //​try_site_patterns//​ will cycle through the 10 patterns searching for a reduction in <​sub>​{{techref_files:​image002.gif?​20x23}}</​sub>​.
  
 **[//​user_defined_convolution//​ E //min// E //max// E]...** **[//​user_defined_convolution//​ E //min// E //max// E]...**
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 **[//​weighting//​ !E  [//​recal_weighting_on_iter//​]]** **[//​weighting//​ !E  [//​recal_weighting_on_iter//​]]**
  
-Used for calculating the //xdd// dependent weighting function  in<​sub>​{{Technical_Reference%20V4-1_files:​image016.gif?​20x23}}</​sub>,​ see equation Eq. (5‑2). Can be a function of the reserved parameter names X, Yobs, Ycalc and  SigmaYobs. The default is as follows:+Used for calculating the //xdd// dependent weighting function  in<​sub>​{{techref_files:​image016.gif?​20x23}}</​sub>,​ see equation Eq. (5‑2). Can be a function of the reserved parameter names X, Yobs, Ycalc and  SigmaYobs. The default is as follows:
  
 weighting = 1 / Max(Yobs, 1); weighting = 1 / Max(Yobs, 1);
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 | ***.XY** |   | 2q and intensity data values | | ***.XY** |   | 2q and intensity data values |
 | ***.XYE** |   | 2q, intensity and intensity error values. | | ***.XYE** |   | 2q, intensity and intensity error values. |
- 
-====== 11       Charge-flipping ====== 
- 
-The charge-flipping method of [[http://​scripts.iucr.org/​cgi-bin/​citedin?​search_on=name&​amp;​author_name=Oszl%26aacute;​nyi,​%20G.|Oszlányi]] & [[http://​scripts.iucr.org/​cgi-bin/​citedin?​search_on=name&​amp;​author_name=S%26uuml;​to,​%20A.|Süto]]  (2004) has been implemented using the keywords shown in Table 11‑2.  Also included is the use of the tangent formula (Hauptman & Karle, 1956) within the iterative charge-flipping process. Equations appearing in charge-flipping keywords can be functions of the items shown in Table 11‑1. At the end of a charge flipping process a file with the same name as that given by //​cf_hkl_file//​ is created but with a *.FC extension. Almost all of the charge-flipping keywords can be equations allowing for great flexibility in regards to changing resolution etc… on the fly. Table 11‑3 lists charge-flipping examples found in the CF directory. 
- 
-  
- 
-| **Table** **11****‑1****.** Items that can be used in charge-flipping equations || 
-| Get(Aij) Get(alpha_sum) Get(density) Get(cycles_since_last_best) Get(d_squared_inverse) Get(initial_phase) Get(iters_since_last_best) Get(F000) Get(max_density) Get(max_density_at_cycle_iter_0) Get(num_reflections_above_d_min) Get(phase_difference) Get(r_factor_1),​ Get(r_factor_2) Get(threshold) |   These are updated internally each charge-flipping iteration or cycle or when needed. | 
-| **Reserved parameter names** Cycle_Iter, Cycle, Iter, D_spacing |   | 
-| **Macros** |   | 
-| Ramp, Ramp_Clamp, Cycle_Ramp,​Tangent,​ Restart_CF, Pick, Pick_Best | See TOPAS.INC for details. | 
-| Out_for_cf(file) : Outputs the A matrix from a Pawley refinement for use in charge flipping; uses //​[[#​cf_e26|cf_in_A_matrix]]//​. See example CF-CIME-PAWLEY.INP. |   | 
- 
-  
- 
-\\ |\\ **Table 11‑2****.** Keywords that can be used in charge-flipping. || 
-| //​charge_flipping//​ | **Default** | 
-|       //​[[#​k000|a]]//​[[#​k000| ]][[#​k000|!E ]]//​[[#​k000|b]]//​[[#​k000| !E ]]//​[[#​k000|c]]//​[[#​k000| !E []]//​[[#​k000|al]]//​[[#​k000| !E] []]//​[[#​k000|be]]//​[[#​k000| !E] []]//​[[#​k000|ga]]//​[[#​k000| !E]]] | //al// = //be// = //ga// = 90 | 
-|       [[#​cf_e1|[]]//​[[#​cf_e1|cf_hkl_file]]//​[[#​cf_e1| $file]]] |   | 
-|       [[#​cf_e26|[]]//​[[#​cf_e26|cf_in_A_matrix]]//​[[#​cf_e26| $file]][[#​cf_e26|]]] |   | 
-|             [//​scale_Aij//​ !E] | Get(Aij)%%^%%2 | 
-|       [[#​cf_e3|[]]//​[[#​cf_e3|break_cycle_if_true]]//​[[#​cf_e3| !E]]] |   | 
-|       [[#​cf_e38|[]]//​[[#​cf_e38|delete_observed_reflections]]//​[[#​cf_e38| !E]]] |   | 
-|       [[#​cf_e36|[]]//​[[#​cf_e36|extend_calculated_sphere_to]]//​[[#​cf_e36| !E]]] |   | 
-|       [[#​cf_e4|[]]//​[[#​cf_e4|f_atom_type]]//​[[#​cf_e4| $type ]]//​[[#​cf_e4|f_atom_quantity]]//​[[#​cf_e4| !E]…]] |   | 
-|       [[#​cf_e34|[]]//​[[#​cf_e34|find_origin]]//​[[#​cf_e34| !E]]] | 1 | 
-|       [[#​cf_e5|[]]//​[[#​cf_e5|fraction_density_to_flip]]//​[[#​cf_e5| !E]]] | 0.75 | 
-|       [[#​cf_e6|[]]//​[[#​cf_e6|fraction_reflections_weak]]//​[[#​cf_e6| !E]]] | 0 | 
-|       [[#​cf_e7|[]]//​[[#​cf_e7|min_d]]//​[[#​cf_e7| !E]]] | 0 | 
-|       [[#​cf_e8|[]]//​[[#​cf_e8|min_grid_spacing]]//​[[#​cf_e8| !E]]] |   | 
-|       [[#​cf_e30|[]]//​[[#​cf_e30|neutron_data]]//​[[#​cf_e30|]]] |   | 
-|       [[#​cf_e16|[]]//​[[#​cf_e16|space_group]]//​[[#​cf_e16| $]]] | //P1// | 
-|       [[#​cf_e27|[]]//​[[#​cf_e27|use_Fc]]//​[[#​cf_e27|]]] |   | 
-| **__Electron density perturbations__** |   | 
-| [[#​cf_e24|[]]//​[[#​cf_e24|flip_equation]]//​[[#​cf_e24| !E]]] |   | 
-| [[#​cf_e33|[]]//​[[#​cf_e33|flip_regime_2]]//​[[#​cf_e33| !E]]] |   | 
-| [[#​cf_e40|[]]//​[[#​cf_e40|flip_regime_3]]//​[[#​cf_e40| !E]]] |   | 
-| [[#​cf_e44|[]]//​[[#​cf_e44|histogram_match_scale_fwhm]]//​[[#​cf_e44| !E]]] |   | 
-| [//​hm_size_limit_in_fwhm//​ !E] | 1 | 
-| [//​hm_covalent_fwhm//​ !E] | 1 | 
-| [[#​cf_e43|[]]//​[[#​cf_e43|pick_atoms]]//​[[#​cf_e43| $atoms]]]... |   | 
-| [//​activate//​ !E] | 1 | 
-| [//​choose_from//​ !E] |   | 
-| [//​choose_to//​ !E] |   | 
-| [//​choose_randomly//​ !E] |   | 
-| [//omit// !E] |   | 
-| [//​displace//​ !E] |   | 
-| [//insert// !E] |   | 
-| [[#​cf_e42|[]]//​[[#​cf_e42|scale_density_below_threshold]]//​[[#​cf_e42| !E]]] |   | 
-| [[#​cf_e15|[]]//​[[#​cf_e15|symmetry_obey_0_to_1]]//​[[#​cf_e15| !E]]] |   | 
-| **__Phase perturbations__** |   | 
-|       [[#​cf_e23|[]]//​[[#​cf_e23|add_to_phases_of_weak_reflections]]//​[[#​cf_e23| !E]]] |   | 
-|       [[#​cf_e11|[]]//​[[#​cf_e11|randomize_phases_on_new_cycle_by]]//​[[#​cf_e11| !E]]] | 0 | 
-|       [[#​cf_e13|[]]//​[[#​cf_e13|set_initial_phases_to]]//​[[#​cf_e13| $file]]] |   | 
-|             [//​modify_initial_phases//​ !E] |   | 
-| [[#​cf_e14|[]]//​[[#​cf_e14|tangent_num_h_read]]//​[[#​cf_e14| !E]]] |   | 
-|       [//​tangent_num_k_read//​ !E] |   | 
-|       [//​tangent_num_h_keep//​ !E] |   | 
-|       [//​tangent_max_triplets_per_h//​ !E] | 30 | 
-|       [//​tangent_min_triplets_per_h//​ !E] | 1 | 
-|       [//​tangent_scale_difference_by//​ !E] | 1 | 
-| **__Miscellaneous__** |   | 
-|       [[#​cf_e28|[]]//​[[#​cf_e28|apply_exp_scale]]//​[[#​cf_e28| !E]]] | 1 | 
-|       [[#​cf_e45|[]]//​[[#​cf_e45|correct_for_atomic_scattering_factors]]//​[[#​cf_e45| !E]]] | 1 | 
-|       [[#​cf_e32|[]]//​[[#​cf_e32|correct_for_temperature_effects]]//​[[#​cf_e32| !E]]] | 1 | 
-|       [[#​k162|[]]//​[[#​k162|hkl_plane]]//​[[#​k162| $hkl]…]] |   | 
-|       [[#​cf_e9|[]]//​[[#​cf_e9|randomize_initial_phases_by]]//​[[#​cf_e9| !E]]] | Rand(-180,​180) | 
-|       [[#​cf_e10|[]]//​[[#​cf_e10|scale_E]]//​[[#​cf_e10| !E]]] | 1 | 
-|       [[#​cf_e12|[]]//​[[#​cf_e12|scale_F]]//​[[#​cf_e12| !E]]] | 1 | 
-|       [[#​cf_e39|[]]//​[[#​cf_e39|scale_F000]]//​[[#​cf_e39| !E]]] | 0 | 
-|       [[#​cf_e37|[]]//​[[#​cf_e37|scale_weak_reflections]]//​[[#​cf_e37| !E]]] |   | 
-|       [[#​cf_e35|[]]//​[[#​cf_e35|user_threshold]]//​[[#​cf_e35| !E]]] |   | 
-|       [[#​cf_21|[]]//​[[#​cf_21|verbose]]//​[[#​cf_21| #]]] | 1 | 
-| **__GUI Related__** | __ __ | 
-|       [[#​cf_e22|[]]//​[[#​cf_e22|add_to_cloud_N]]//​[[#​cf_e22| !E []]//​[[#​cf_e22|add_to_cloud_when]]//​[[#​cf_e22| !E]]]] | __ __ | 
-|       [[#​cf_e31|[]]//​[[#​cf_e31|pick_atoms_when]]//​[[#​cf_e31| !E]]] | __ __ | 
-|       [[#​cf_e19|[]]//​[[#​cf_e19|view_cloud]]//​[[#​cf_e19| !E]]] | 1 | 
- 
-  
- 
-===== 11.1  Charge-flipping usage ===== 
- 
-CF works particularlly well on data at good resolution (<1Å resolution). For data at poor resolution or for difficult structures then inclusion of the tangent formula can facilitate solution and sharpen electron densities, see example CF-1A7Y.INP. Powder diffraction data usually fall under the poor resolution/​data quality category and as such additional symmetry restraints using //​symmetry_obey_0_to_1//​ can sharpen electron densities. Example CF-ALVO4.INP demonstrates the use of the tangent formula on powder data. 
- 
-The choice and amount of perturbation necessary for finding a solution are important considerations. Not enough perturbation leads to the system being trapped within a local parameter space; too much perturbation may lead to a solution not being found and in addition contrast in //​R//​-factors prior to and at convergence are diminished leading to difficult to identify solutions. Many of the examples in the CF directory uses the Ramp macro to gradually vary control parameters, here are some examples: 
- 
-fraction_density_to_flip = Ramp(0.85, 0.8, 100); 
- 
-fraction_reflections_weak = Ramp(0.5, 0, 100); 
- 
-flip_regime_2 = Ramp(1, 0, 200); 
- 
-flip_regime_3 = Ramp(0.25, 0.5, 200);   
- 
-symmetry_obey_0_to_1 = Ramp(0.5, 1, 100); 
- 
-tangent_scale_difference_by = Ramp(0, 1, 100); 
- 
-Choosing control parameters in this manner gradually decreases perturbation allowing for solutions to be found and identified. This is similar to a simulated annealing process where temperatures start at high values and then progressively lowered. 
- 
-==== 11.1.1          Perturbations ==== 
- 
-Perturbations can be categorized as being of either phase, structure factor intensity or electron density perturbations as shown in Table 11‑2. There are two built in flipping regimes, //​flip_regime_2//​ and //​flip_regime_3//,​ and one user defined regime //​flip_equation//​. Only one can be used and they all modify the electron density. In the absence of a flipping regime the following is used: 
- 
-| <​sub>​{{Technical_Reference%20V4-1_files:​image171.gif?​175x48}}</​sub>​ | (11‑1) | 
- 
-where //d// corresponds to the electron density threshold. 
- 
-Using the tangent formula on either difficult structures or on data at poor resolution often leads to uranium atom solutions. Uranium atom solutions can be avoided by modifying the electron density using a flipping regime that dampens high electron densities or by using //​pick_atoms//​. 
- 
-Using a large number of triplets per //​E<​sub>​h</​sub>//​ value (a value for //​tangent_max_triplets_per_h//​ greater than 100) reduces perturbation,​ increases occurrences of uranium atom solutions and increases the chances of finding a solution after an initial phase randomization. A large number of triplets would typically be used for poor resolution data; correspondingly a flipping regime that avoids uranium atom solutions should be chosen. 
- 
-Perturbations mostly increase randomness in the system with the exceptions of the tangent formula, //​scale_density_below_threshold//​ and //​histogram_match_scale_fwhm//​. 
- 
-==== 11.1.2          The Ewald sphere, weak reflections and CF termination ==== 
- 
-By default CF uses the minimum observed //d// spacing to define the Ewald sphere; alternatively //min_d// can be used. The Ewald sphere can be increased using //​[[#​cf_e36|extend_calculated_sphere_to]]//;​ this inserts missing reflections and gives them the status of “weak” reflections. Weak reflections are also inserted for missing reflections within the Ewald sphere. Weak reflection phases and structure factors can be modified using //​scale_weak_reflections//​ and //​add_to_phases_of_weak_reflections//​. 
- 
-Reflections that have zero intensities according to the space group are not included in CF; correspondingly the number of observed reflections removed are reported. Structure factor intensities  within a family of reflections are determined by averaging the observed structure factors intensities. This averaging is also performed on calculated intensities each CF iteration for weak reflections. 
- 
-Changing the space group is possible; changing the space group to a higher symmetry from that as implied in the input hkl file often makes sense. Changing the space group to a lower symmetry implies less symmetry and is useful for checking whether a significantly better //​R//​-factor is realized. 
- 
-Typically a fraction of observed reflections are given the status of “weak” using //​fraction_reflections_weak//​. When a solution is found and CF terminates a *.FC file is saved; this file comprises structures factors that produced the best //​R//​-factor. A new CF process can be initiated with phase information saved in the *.FC file using the Restart_CF macro. To further complete the structure the new CF process may for example reduce perturbations in order to sharpen the electron density. 
- 
-==== 11.1.3          Powder data considerations ==== 
- 
-For powder data it is usually best to maximize the number of constraints due to poor data quality; it is also best to use *.A files as generated by a Pawley refinement and to then use //​cf_in_A_matrix//​. No weak observed reflections within the observed Ewald sphere should be assigned by setting //​fraction_reflections_weak//​ to zero. Instead weak reflections can be included by extending the Ewald sphere with something like: 
- 
-extend_calculated_sphere_to 1 
- 
-add_to_phases_of_weak_reflections = 90 Ramp(1, 0, 100); 
- 
-If the Ewald sphere is extended such that the weak reflections are many then some of these weak reflections could well be of high intensity. Subsequently offsetting high intensity weak reflections by a constant could lead to too much perturbation and thus the following may be preferential:​ 
- 
-extend_calculated_sphere_to 1 
- 
-add_to_phases_of_weak_reflections = Rand(-180,​180) Ramp(1, 0, 100); 
- 
-In a Pawley refinement the calculated intensities at the low //​d//​-spcaing edge of are often in error to a large extent; it is therefore best to remove these reflections using //​delete_observed_reflections//,​ for example example: 
- 
-delete_observed_reflections = D_spacing < 1.134; 
- 
-A typical first try INP file template for powders is as follows: 
- 
-macro Nr { 100 } 
- 
-charge_flipping 
- 
-cf_in_A_matrix PAWLEY_FILE.A 
- 
-space_group $ 
- 
-a # b # c al # be # ga # 
- 
-delete_observed_reflections = D_spacing < #; 
- 
-extend_calculated_sphere_to # 
- 
-add_to_phases_of_weak_reflections = 90 Ramp(1, 0, Nr); 
- 
-flip_regime_2 = Ramp(1, 0, Nr); 
- 
-symmetry_obey_0_to_1 = Ramp(0.5, 1, Nr); 
- 
-Tangent(.3, 30) 
- 
-min_grid_spacing .3 
- 
-load f_atom_type f_atom_quantity { … } 
- 
-==== 11.1.4          The algorithm of Oszlányi & Süto (2005) and F000 ==== 
- 
-Normalized structure factors enhances the chances of finding a solution ([[http://​scripts.iucr.org/​cgi-bin/​citedin?​search_on=name&​amp;​author_name=Oszl%26aacute;​nyi,​%20G.|Oszlányi]] & [[http://​scripts.iucr.org/​cgi-bin/​citedin?​search_on=name&​amp;​author_name=S%26uuml;​to,​%20A.|Süto]],​ 2006) and are realized by inclusion of //​f_atom_type//​’s and when //​correct_for_temperature_effects//​ is non-zero. Example CF-1A7Y-GABOR.INP implements the algorithm of [[http://​scripts.iucr.org/​cgi-bin/​citedin?​search_on=name&​amp;​author_name=Oszl%26aacute;​nyi,​%20G.|Oszlányi]] & [[http://​scripts.iucr.org/​cgi-bin/​citedin?​search_on=name&​amp;​author_name=S%26uuml;​to,​%20A.|Süto]] (2005) with normalized structure factors. In the original algorithm the amount of charge flipped is a function of the maximum electron density; this can be realized using: 
- 
-user_threshold = 0.2 Get(max_density_at_cycle_iter_0);​ 
- 
-Get(max_density_at_cycle_iter_0) is a different value at the start of each CF process as phases are chosen randomly. An alternative means of defining the threshold is: 
- 
-fraction_density_to_flip 0.83 
- 
-The CF process is sensitive to the threshold value. A value of 0.83 for //​fraction_density_to_flip//​ is optimum for //1a7y// and produces a solution in ~1000 iterations.  A solution is not found however at 0.75 or 0.85. To overcome this sensitivity the //​fraction_density_to_flip//​ parameter could be ramped as a function of iteration from a high value to a low value, or, 
- 
-fraction_density_to_flip = Ramp(0.85, 0.8, 100); 
- 
-Implementation of such a ramp solves 1a7y in ~2000 iterations. 
- 
-F000 is allowed to float when //​scale_F000//​ is set to 1. In the [[http://​scripts.iucr.org/​cgi-bin/​citedin?​search_on=name&​amp;​author_name=Oszl%26aacute;​nyi,​%20G.|Oszlányi]] & [[http://​scripts.iucr.org/​cgi-bin/​citedin?​search_on=name&​amp;​author_name=S%26uuml;​to,​%20A.|Süto]] (2005) algorithm a floating F000 produces the best results for some structures but not for others (see section 11.2.3). 
- 
-When the electron density is perturbed then a floating F000 often produces unfavourable oscillations in //​R//​-factors. In general the electron density is best left unperturbed when //​scale_F000//​ is non-zero. 
- 
-Example CF-1A7Y-GABOR.INP does not seem to solve at a lower resolution, try for example: 
- 
-delete_observed_reflections = D_spacing < 1.1; 
- 
-On the other hand when //​scale_F000//​ is zero then electron density perturbations are possible; CF_1A7Y.INP solves //1a7y// at 1.1 Angstrom (ie. include “//​delete_observed_reflections//​ = D_spacing < 1.1”);  CF_1A7Y.INP uses //​flip_regime_2//​ and the tangent formula. 
- 
-===== 11.2  Charge-flipping Investigations / Tutorials ===== 
- 
-The effects of CF keywords can be investigated by inclusion/​exclusion of keywords or by changing equations. This section lists some investigative examples and highlights the use of keywords necessary to solve particular examples found in the CF directory. 
- 
- 
-==== 11.2.1            Preventing uranium atom solutions using pick_atoms ==== 
- 
-Example CF-1A7Y-OMIT.INP uses //​pick_atoms//​ to modify the peaks of the highest intensity atoms as follows: 
- 
-  
- 
-pick_atoms * 
- 
-choose_to 5 
- 
-omit = Rand(1, 1.1); 
- 
-This example additionally uses the tangent formula and //1a7y// solves in ~100 iterations and with a large contrast in //​R//​-factors before and at converegnce. Another means to modify the peaks are: 
- 
-pick_atoms * 
- 
-choose_to 5 
- 
-insert = Rand(-.1, 1); 
- 
-The //insert// case is slightly slower than the //omit// case as the 5 atoms are first omitted before insertion. Each case however solves //1a7y// in a similar number of iterations. 
- 
-Example CF-1A7Y-NO-TANGENT.INP is similar but without the tangent formula, //1a7y// in this case solves in ~1000 iterations. 
- 
-==== 11.2.2          The tangent formula on powder data ==== 
- 
-In CF-ALVO4.INP comment out the Tangent line as follows: 
- 
-‘ Tangent(.5, 50) 
- 
-Run CF-ALVO4.INP and turn on Octahedra viewing in the OpenGL window. Visual inspection of picked atoms should show electron densities that are not recognizable as correct solutions. 
- 
-Include the Tangent line and rerun; after a minute or two and at the bottom of the Ramps visual inspection of picked atoms should show a well defined solution. 
- 
-Thus use of the tangent formula assists in solving CF-ALVO4.INP. 
- 
-==== 11.2.3          Pseudo symmetry – 441 atom oxide ==== 
- 
-CF works particularly well on pseudo symmetric structures ([[http://​scripts.iucr.org/​cgi-bin/​citedin?​search_on=name&​amp;​author_name=Oszl%26aacute;​nyi,​%20G.|Oszlányi]] //et el.,// 2006). Example CF-PN-02.INP is an oxide structure that contains 441 atom in the asymmetric unit (Lister //et el.,// 2004); run CF to convergence. Pick atoms and turn on Octahedra viewing; all polyhedra should be well formed. Thus CF works extremely fast and trivializes the solving of such structures. The contents of the INP file is as follows: 
- 
-charge_flipping 
- 
-cf_hkl_file 020pn.hkl 
- 
-space_group Pn 
- 
-a 24.1332 
- 
-b 19.5793 
- 
-c 25.1091 
- 
-be 99.962 
- 
-fraction_reflections_weak 0.4 
- 
-symmetry_obey_0_to_1 .3 
- 
-Tangent(.25,​ 30) 
- 
-load f_atom_type f_atom_quantity 
- 
-{ 
- 
-MO = 42 2; 
- 
-P  = (126 - 42) 2; 
- 
-O  = (441 - 126) 2; 
- 
-} 
- 
-The tangent formula is used to assist //​symmetry_obey_0_to_1//​ and to assist in finding the solution faster; it is not necessary for this example however. The [[http://​scripts.iucr.org/​cgi-bin/​citedin?​search_on=name&​amp;​author_name=Oszl%26aacute;​nyi,​%20G.|Oszlányi]] & [[http://​scripts.iucr.org/​cgi-bin/​citedin?​search_on=name&​amp;​author_name=S%26uuml;​to,​%20A.|Süto]] (2005) algorithm can be used by replaceing //​symmetry_obey_0_to_1//​ and the Tangent line with the following: 
- 
-scale_F000 1 
- 
-fraction_reflections_weak .4 
- 
-add_to_phases_of_weak_reflections 90 
- 
-user_threshold = 0.15 Get(max_density_at_cycle_iter_0);​ 
- 
-Slow convergence is then observed and the reason is the use of F000. This is opposite to the case of //1a7y// in CF-1A7Y-GABOR.INP where F000 is necessary. Setting //​scale_F000//​ to zero greatly increases the rate of convergence. 
- 
-==== 11.2.4          Origin finding and symmetry_obey_0_to_1 ==== 
- 
-When //​symmetry_obey_0_to_1 is// defined origin finding is performed each iteration of charge flipping. Symmetry elements of the space group are used in finding an origin. On finding an origin the electron density is shifted to a position that best matches the symmetry of the space group. Additionally a restraint is placed on the electron density pixels forcing symmetry to be obeyed. 
- 
-Run CF-AE14.INP to convergence;​ notice the //P-1// symmetry. Remove //​symmetry_obey_0_to_1//​ and run to convergence;​ the origin should now be arbitrary. 
- 
-==== 11.2.5          symmetry_obey_0_to_1 on poor resolution data ==== 
- 
-Run CF-AE5.INP until a solution is found; terminate CF, this saves the phase information to the file AE5.FC. 
- 
-Copy AE5.FC to AE5-SAVE.FC. 
- 
-Place the following lines into the file CF-AE5-POOR.INP 
- 
-set_initial_phases_to ae5-save.fc 
- 
-randomize_initial_phases_by 0 
- 
-This simply starts CF with optimum phase values. Also include the following line: 
- 
-symmetry_obey_0_to_1 .75 
- 
-Run CF-AE5-POOR.INP;​ the atom positions after picking should visually produce the correct result. Comment out //​symmetry_obey_0_to_1//​ and rerun CF-AE5-POOR.INP. //​R//​-factors should diverge and picked atoms should show a non-solution. Thus //​symmetry_obey_0_to_1//​ assists in solving CF-AE5-POOR.INP. 
- 
-Include //​symmetry_obey_0_to_1//​ and remove //​set_initial_phases_to//​ and //​randomize_initial_phases_by//​ and then rerun CF-AE5-POOR.INP. A solution should be obtained in a few minutes. Note that in this example the default flipping regime leads to regular occurrences of uranium atom solutions; this can be trivially ascertained by viewing the electron density. To reduce the occurrences of uranium atom solutions the following flipping regime is used: 
- 
-flip_regime_3 0.5 
- 
-==== 11.2.6          Sharpening clouds - extend_calculated_sphere_to ==== 
- 
-Example CF-AE9-POOR.INP demonstrates the limit to which the present CF implementation can operate. Single crystal data is purposely chosen to isolate resolution effects and not intensity errors. The tangent formula is critical where without it the CF process is extremely perturbed and unstable. “flip_regime_3 .5” is used due to occurrences of uranium atom solutions. 
- 
-There are no ramps, instead the CF process is restarted when the //​R//​-factor fails to decrease for 100 consecutive iterations, or, 
- 
-break_cycle_if_true = Get(iters_since_last_best) > 100; 
- 
-randomize_phases_on_new_cycle_by = Rand(-180, 180); 
- 
-Half of the observed reflections are considered weak and additionally missing reflections up to 1 Angstrom are included and considered weak using: 
- 
-fraction_reflections_weak 0.5 
- 
-extend_calculated_sphere_to 1 
- 
-The intensities of weak reflections are left untouched and instead a Pi/2 phase shift is randomly applied to ~30% of weak reflections as follows: 
- 
-add_to_phases_of_weak_reflections = If(Rand(0, 1) < .3, 90, 0); 
- 
-A //​symmetry_obey_0_to_1//​ of 0.7 is used not merely to find an origin but rather to prevent the electron density from straying. 
- 
-Run CF-AE9-POOR.INP and a solution should be clearly recognizable after a few minutes. Change/​remove keywords and rerun to view effects. Examples CF-CIME-POOR.INP and CF-AE5-POOR.INP are similar. 
- 
-==== 11.2.7          A difficult powder, CF-SUCROSE.INP ==== 
- 
-CF-SUCROSE.INP without //​scale_density_below_threshold=0//​ exhibits large oscillations in //​R//​-factors resulting in difficult to identify solutions; this can be prevented by increasing the amount of charge flipped and including //​scale_density_below_threshold=0//,​  for example 
- 
-fraction_density_to_flip 0.83 
- 
-scale_density_below_threshold 0 
- 
-When //​scale_density_below_threshold=0//​ is used the percentage of charge that is less than the threshold before the application of //​scale_density_below_threshold//​ is reported; the difference between this reported value and (1-//​fraction_density_to_flip//​) gives the amount of flipped pixels that survived //​scale////​_density_below_threshold//​. At //​fraction_density_to_flip//​ of 0.83 approximately 23% of pixels survives //​scale_density_below_threshold=0//​ which in effect means that only 23% of pixels are actually flipped out of the original 83%. 
- 
-//​picked_atoms//​ is used as a perturbation where 30% of atoms are omitted using: 
- 
-pick_atoms * 
- 
-activate = Cycle_Iter == 0; 
- 
-insert = If(Rand(0, 1) > 0.3, 10, 0); 
- 
-Note that atoms are inserted at an intensity that is 10 times the average intensity. This increases the weight of inserted atoms relative to electron density noise. It also initially gives more weight to weak reflections. 
- 
-Use of //​scale_density_below_threshold//​ often results in CF requiring more interations to solution; a solution however is preferable to no solution. 
- 
-==== 11.2.8          Increasing contrast in R-factors ==== 
- 
-The act of flipping introduces an appreciable amount of unwanted high frequencies in the structure factors. This effect can be reduced by dampening high frequencies using //​apply_exp_scale//​ which is ON by default. //​apply_exp_scale changes R//-factors and not phases, directions taken by CF are unchanged. 
- 
-Run CF-1A7Y.INP until convergence. The difference in //​R//​-factors before and at convergence should be ~0.39 (ie. 0.81 and 0.42). Turn OFF //​apply_exp_scale//​ by including the following line: 
- 
-apply_exp_scale 0 
- 
-Rerun CF-1A7Y.INP until convergence. The difference in //​R//​-factors before and at convergence should now be ~0.29 (ie. 0.81 and 0.52). Thus //​apply_exp_scale//​ increases contrast in //​R//​-factors. Note that most of the increase seems to be realized from d-spacings less than 1 Angstrom. 
- 
-===== 11.3  Charge-flipping Examples ===== 
- 
-| **Table ****11****‑3****.** Examples found in the CF directory, Number of atoms corresponds to the number of non-hydrogen atoms within the asymmetric unit. ||| 
-| **Single crystal data** | **Number of atoms in asymmetric unit** | **Space group** | 
-| cf-1a7y.inp cf-1a7y-gabor.inp cf-1a7y-omit.inp cf-1a7y-no-tangent.inp | 314 | //P1// | 
-| cf-ae14.inp | 43 | //P-1// | 
-| cf-ae5.inp cf-ae5-poor.inp | 23 | //C2/c// | 
-| cf-ae9.inp cf-ae9-poor.inp | 53 | //P-1// | 
-| cf-gebaa.inp | 17 | //​P4<​sub>​1</​sub>​2<​sub>​1</​sub>​2//​ | 
-| cf-pn-02.inp | 441 | //Pn// | 
-| cf-ylidm.inp | 17 | //​P2<​sub>​1</​sub>​2<​sub>​1</​sub>​2<​sub>​1</​sub>//​ | 
-| **Powder data** |   | // // | 
-| cf-alvo4.inp,​ cf-alvo4-pawley.inp | 18 | //P-1// | 
-| cf-cime-pawley.inp cf-cime.inp,​ cf-cime-histo.inp cf-cime-poor.inp,​ cf-cime-poor-histo.inp | 17 | //​P2<​sub>​1</​sub>/​a//​ | 
-| cf-sucrose.inp,​ cf-sucrose-pawley.inp | 23 | //​P2<​sub>​1</​sub>//​ | 
- 
-  
- 
-===== 11.4  Keywords in detail ===== 
- 
-**[//​add_to_cloud_N//​ !E]** 
- 
-**[//​add_to_cloud_when//​ !E]** 
- 
-The current cloud is added to the GUI cloud creating a running average cloud for display purposes. //​add_to_cloud_N//​ corresponds to the number of most recent clouds to include in the running average. //​add_to_cloud_when//​ determines when the current cloud is to be included in the running average; here’s an example: 
- 
-add_to_cloud_N 10 
- 
-add_to_cloud_when = Mod(Cycle_Iter,​ 2); 
- 
-Averaged clouds eliminate noise and is effective if the cloud remains stationery which is generally the case. Note that //​add_to_phases_of_weak_reflections//​ can produce translations of the cloud and should not be included when averaging clouds. 
- 
-**[//​add_to_phases_of_weak_reflections//​ !E]** 
- 
-Allows for modification to phases of weak reflections. For example, to add p/2 to the phases of weak reflections then the following could be used: 
- 
-add_to_phases_of_weak_reflections 90 
- 
-When //​add_to_phases_of_weak_reflections//​ is defined then the intensities of weak reflections are not set to zero; instead they are left untouched meaning that their intensities are set to the values as determined by the inverse Fourier transform. See also //​[[#​cf_e37|scale_weak_reflections]]//​. 
- 
-**[//​apply_exp_scale//​ !E]** 
- 
-Determines //a// and //b// each CF iteration such that the following is a minimum: 
- 
-//​R//​-factor = ∑| //a// Exp(//b// / D_spacing^2) //Fc// -- //Fo// | 
- 
-where //Fc// and //Fo// are the calculated and observed moduli respectively. Use of //​apply_exp_scale//​ corrects //​R//​-factors in case of an incorrect temperature factor correction as applied when normalizing structure factors. Use of //​apply_exp_scale//​ typically increases the difference between //​R//​-factors prior to and at convergence. //​apply_exp_scale//​ is used by default, setting it to zero prevents its use. 
- 
-** [//​cf_hkl_file//​ $file]** 
- 
-Defines the input hkl file. 
- 
-**[//​cf_in_A_matrix//​ $file [//​scale_Aij//​ !E] ]** 
- 
-Data input is from a file created using //​out_A_matrix//​ from a previous Pawley refinement. The correlations in $file are used to partition intensities during each iteration of charge-flipping. This partitioning is applied to structure factors as used by CF and as used by the tangent formula. 
- 
-//​scale_Aij//​ can be used to modify the A matrix off-diagonal coefficients,​ here are some examples: 
- 
-scale_Aij = Get(Aij); 
- 
-scale_Aij = Get(Aij)^2; ‘ the default 
- 
-scale_Aij = 0; ‘ Equivalent to using a Pawley generated hkl file 
- 
-CF on powder data can also be initiated using standard hkl files. 
- 
-**[//​break_cycle_if_true//​ !E]** 
- 
-Interrupts charge flipping to execute //​randomize_phases_on_new_cycle_by. Cycle_Iter//​ is set to zero and //Cycle// is incremented//​.//​ 
- 
-**[//​correct_for_atomic_scattering_factors//​ !E]** 
- 
-Structure factors are normalized when non-zero and when //​f_atom_type//​’s are defined. By default structure factors are normalized. 
- 
-**[//​correct_for_temperature_effects//​ !E]** 
- 
-Attempts to remove isotropic temperature effects from the structure factors. //​correct_for_temperature_effects//​ is ON by default, setting it to zero will prevent the correction. Normalized structure factors are realized when //​correct_for_temperature_effects//​ is //ON// and the unit cell contents is defined using //​f_atom_type//​ and //​f_atom_quantity//​. 
- 
-**[//​delete_observed_reflections//​ !E]** 
- 
-Reflections are deleted before entering CF according to //​delete_observed_reflections//;​ it can be a function of D_spacing, for example: 
- 
-delete_observed_reflections = D_spacing < 1.1; 
- 
-Once deleted, observed reflections cannot be reinstated by changing //min_d//. 
- 
-**[//​extend_calculated_sphere_to//​ !E]** 
- 
-Used to sharpen electron density clouds by filling in missing reflections;​ added reflections are given the status of “weak”. //​extend_calculated_sphere_to//​ can be used in conjunction with //​scale_weak_reflections//​ and //​add_to_phases_of_weak_reflections//​ to modify “weak” reflection magnitudes and phases respectively (see section 11.2.6); here’s an example: 
- 
-extend_calculated_sphere_to 1 
- 
-add_to_phases_of_weak_reflections = If(Rand(0, 1) < .3, 90, 0); 
- 
-**[//​f_atom_type//​ $type //​f_atom_quantity//​ !E]…** 
- 
-Defines atom types and number of atoms within the unit cell; used by the tangent formula in determining //​E<​sub>​h</​sub>//​ values and by the OpenGL display for picking atoms. For the tangent formula then realtive quantities are important. 
- 
-**[//​find_origin//​ !E]** 
- 
-If defined and non-zero then origin finding is turned ON. //​symmetry_obey_0_to_1//​ defines //​find_origin//​ by default. //​symmetry_obey_0_to_1//​ can be used without //​find_origin//​ by defining and setting //​find_origin//​ to zero. 
- 
-**[//​flip_equation//​ !E]** 
- 
-Allwows for a user defined flip; here’s an example: 
- 
-flip_equation = 
- 
-If(Get(density) < Get(threshold),​ -Get(density),​ Get(density));​ 
- 
-**[//​flip_regime_2//​ !E]** 
- 
-The elctron density is modified according to Eq. (11‑1) and then further modified using: 
- 
-<​sub>​{{Technical_Reference%20V4-1_files:​image173.gif?​264x28}}</​sub>//​flip_regime_2//​ is typically ramped from 1 to 0. 
- 
-**[//​flip_regime_3//​ !E]** 
- 
-The elctron density is modified according to Eq. (11‑1) and then further modified using: 
- 
-<​sub>​{{Technical_Reference%20V4-1_files:​image175.gif?​393x51}}</​sub>​A value of 0.5 for //​flip_regime_3//​ introduces little perturbation whilst reducing the occurance of uranium atom solutions. It is recommended that //​flip_regime_3//​ be used in cases where //​flip_regime_2//​ produces uranium atom solutions. An additional perturbation,​ such as “//​add_to_phases_of_weak_reflections//​=90;​” may be necessary. 
- 
-**[//​fraction_density_to_flip//​ !E]** 
- 
-The amount of charge flipped is fractionally based. A value of 0.6, for example, sets the threshold //d// such that the sign of the lowest 60% of charge is changed. Get(//​threshold//​) can be used to retrieve //d//. 
- 
-**[//​fraction_reflections_weak//​ !E]** 
- 
-Defines the fraction of observed reflections to flag as “weak”. When //​scale_weak_reflections//,​ //​add_to_phases_of_weak_reflections//​ and //​extend_calculated_sphere_to//​ are all not defined then intensities of weak reflections are set to zero effectively removing them from the charge flipping process. Otherwise intensities of weak reflections are not set to zero; instead they are left untouched prior to //​scale_weak_reflections//​ and //​add_to_phases_of_weak_reflections//​ and space group family averaging. 
- 
-**[//​histogram_match_scale_fwhm//​ !E]** 
- 
-**[//​hm_size_limit_in_fwhm//​ !E]** 
- 
-**[//​hm_covalent_fwhm//​ !E]** 
- 
-An implementation of Histogram Matching (Baerlocher //et al.,// 2007) where the distribution of pixels within the unit cell is restrained to one that mathes Gaussian atoms with intensities corresponding to the atoms defined by //​f_atom_type//​‘s. The Histogram matching operation is performed when //​ ////​histogram_match_scale_fwhm//​ evaluates to a non-zero value. Subsequently the full width at half maximum (FWHM) of the Gaussians (obtained from the file ATOM_RADIUS.DEF) is scaled by //​histogram_match_scale_fwhm. hm_size_limit_in_fwhm//​ corresponds to the extent to which the Gaussians are calculated in units of FWHM. Covalent radii is used if //​hm_covalent_fwhm evaluates// to a non-zero value otherwise ionic radii is used. An example use is as follows: 
- 
-histogram_match_scale_fwhm = If(Mod(Cycle_Iter,​ 3), 0, 1); 
- 
-hm_size_limit_in_fwhm 1 
- 
-hm_covalent_fwhm 1 
- 
-Reported on is the fraction of pixels modified; values of 1 for both //​histogram_match_scale_fwhm//​ and //​hm_size_limit_in_fwhm//​ seem optimal where typically ~15 to 20% of pixels are modified. Use of histogram matching should produce //​R//​-factors at convergence that are equal to or than less //​R//​-factors produced when not using histogram matching. Histogram matching sharpens the electron density cloud for data  at poor resolution (see examples CF-CIME-HISTO.INP and CF-CIME-POOR-HISTO.INP). 
- 
-** [//​min_d//​ !E]** 
- 
-Determines in Angstroms the resolution of observed reflections to work with; only observed reflections with a d-spacing above //min_d// are considered. //min_d// is evaluated each CF iteration. Get(//​num_observed_reflections_above_d_min//​) is updated when a change in //min_d// is detected. See also //​[[#​cf_e36|extend_calculated_sphere_to]]//​ and //​[[#​cf_e38|delete_observed_reflections]]//​. 
- 
-**[//​min_grid_spacing//​ !E]** 
- 
-If defined then the grid spacing used is set to the smaller of //min_d///2 and //​min_grid_spacing//;​ useful for obtaining many grid points for graphical purposes. 
- 
-**[//​neutron_data//​]** 
- 
-Signals that the input data is of neutron type. Used in the picking of atoms and additionally //​E<​sub>​h</​sub>//​ values are not corrected from any defined //​f_atom_type//​ and //​f_atom_quantity//​ keywords. 
- 
-**[//​pick_atoms//​ $atoms]****...** 
- 
-**[//​activate//​ !E]** 
- 
-**[//​choose_from//​ !E]** 
- 
-**[//​choose_to//​ !E]** 
- 
-**[//​choose_randomly//​ !E]** 
- 
-**[//omit// !E]** 
- 
-**[//​displace//​ !E]** 
- 
-**[//​insert//​ !E]** 
- 
-//​pick_atoms//​ modifies the electron density based on picked atoms. $atom corresponds to the atom types to be operated on; it can contain the wild card character ‘*’ and the negation character ‘!’, see section 7.6 for details. The operations of //​pick_atoms//​ are invoked when //​activate//​ evaluates to a non-zero value; for example, 
- 
-pick_atoms “O C” 
- 
-activate = Mod(Cycle_Iter,​ 20) == 0; 
- 
-The picking routine will attempts to locate the atom types found in $atom based on the intensities of picked atoms and the scattering power of the atoms defined in //​f_atom_type//​. For example, 
- 
-load f_atom_type f_atom_quantity { Ca 2 O 10 C 12 } 
- 
-pick_atoms “O C” 
- 
-Here 2 Ca atoms are first picked and then 10 O atoms and then 12 C atoms. The picked atoms operated on will be the O and C atoms with the Ca atoms ignored. 
- 
-//​choose_from//​ and //​choose_to//​ can be used to limit the number of atoms operated on. Note, that picked atoms within a particular //​pick_atoms//​ are sorted in decreasing intensity order. For example, to not operate on the first thee O atoms and the last 2 C atoms then the following could be used: 
- 
-choose_from 4 
- 
-choose_to 20 
- 
-//​choose_randomly//​ further reduces the atoms operated on and is exected after //​choose_from//​ and //​choose_to//​. 
- 
-//omit// removes operated on atoms from the electron density. Atoms can be partially removed by setting //omit// to values less than 1. Values greater than 1 can also be used, the effect is to change the sign of the electron density. //omit// operating on a few of the highest intensity atoms is an extremely effective means of preventing the occurance of uranium atom solutions, see CF-1A7Y-OMIT.INP;​ for example: 
- 
-pick_atoms * 
- 
-choose_to 5 
- 
-omit = Rand(1, 1.1); 
- 
-Omitting atoms randomly is a technique referred to as “random omit maps“ in ShelXD, [[http://​scripts.iucr.org/​cgi-bin/​citedin?​search_on=name&​amp;​author_name=Schneider,​%20T.R.|(Schneider]] and [[http://​scripts.iucr.org/​cgi-bin/​citedin?​search_on=name&​amp;​author_name=Sheldrick,​%20G.M.|Sheldrick]],​ 2002). 
- 
-//insert// inserts operated on atoms; a value of 1 inserts the atoms with an intensity that is equal to the average of the picked atoms. Values of less than 1 descreases the intensity of the inserted atoms. When //insert// is defined then //omit// is internally defined if it does not already exist. Thus, atoms are removed before insertion by default. 
- 
-//​displace//​ (in Angstroms) displaces atom positions from their picked positions; it is evaluated before //insert//. For example, to randomly displace atoms by 0.3 Angstroms then the following could be used: 
- 
-displace = Rand(0.4, 0.6); 
- 
-insert 1 
- 
-There can be more than one occurance of //​pick_atoms//,​ for example to limit uranium atom solutions then the follow can be used: 
- 
-pick_atoms * 
- 
-choose_to 5 
- 
-insert = Rand(-.1, 1); 
- 
-To further randomly remove ~33% of atoms then the following could additionally be used: 
- 
-break_cycle_if_true = Get(iters_since_last_best) > 10; 
- 
-pick_atoms * 
- 
-activate = Cycle_Iter == 0; 
- 
-insert = If(Rand(0, 1) > 0.33, 10, 0); 
- 
-Note that in this example atoms are inserted at ten times the average picked intensity; this simply gives more weight to picked atoms relative to electron density noise. Additionaly weak reflections are also given more weighting. 
- 
-**[//​pick_atoms_when//​ !E]** 
- 
-Atoms are picked in the OpenGL display when //​pick_atoms_when//​ evaluates to a non-zero value; here’s an example: 
- 
-pick_atoms_when = Mod(Cycle_Iter + 1, 10) == 0; 
- 
-Note that picking can be manually initiated from the Cloud dialog of the OpenGL display. A text description of picked atoms can be obtained by opening the “Temporary output” text window of the OpenGL window. 
- 
-**[//​randomize_initial_phases_by//​ !E]** 
- 
-Initializes  phases. To start a process with already saved phase information then the following could be used: 
- 
-set_initial_phases_to aleady_saved.fc 
- 
-randomize_initial_phases_by 0 ' this has a default of Rand(-180,​180) 
- 
-**[//​randomize_phases_on_new_cycle_by//​ !E]** 
- 
-randomize_phases_on_new_cycle_by = Rand(-180, 180); ‘ an example 
- 
-**[//​scale_density_below_threshold//​ !E]** 
- 
-Electron density pixels that are less than the threshold value are scaled by //​scale////​_density_below_threshold//​. Values for //​scale////​_density_below_threshold//​ thata re less than 1 tends to sharpen the electron density and to reduce large oscillations in //​R//​-factors;​ the latter occurs for bad data, see example CF-SUCROSE.INP. A value of zero for //​scale////​_density_below_threshold//​ results in “low density elimination“ simlar to that of Shiono & Woolfson (1992). 
- 
-**[//​scale_E//​ !E]** 
- 
-Normalized structure factors (//​E<​sub>​h</​sub>//​ values) are a function of //​correct_for_temperature_effects//​ and unit cell contents. //scale_E// allows for an additional scaling of //​E<​sub>​h</​sub>//​ values. 
- 
-**[//​scale_F//​ !E]** 
- 
-CF works with normalized structure factors by default. //scale_F// is an additional scaling of structure factors. The defualt //scale_F// broadens electron density peaks to avoid pixilation effects and is given by: 
- 
-scale_F = Exp(-0.2 Get(d_squared_inverse));​ 
- 
-**[//​scale_F000//​ !E]** 
- 
-Scale should be set to 1 for compliance with the algorithm of [[http://​scripts.iucr.org/​cgi-bin/​citedin?​search_on=name&​amp;​author_name=Oszl%26aacute;​nyi,​%20G.|Oszlányi]] & [[http://​scripts.iucr.org/​cgi-bin/​citedin?​search_on=name&​amp;​author_name=S%26uuml;​to,​%20A.|Süto]] (2004). When //​scale_F000//​ is non_zero then modifcations to the electron density produces unfavourable effects. 
- 
-**[//​scale_weak_reflections//​ !E]** 
- 
-By default weak reflection structure factors are set to zero; however when either //​scale_weak_reflections//​ or //​add_to_phases_of_weak_reflections//​ is defined then weak reflections structure factors are instead modified accordingly,​ for example: 
- 
-//​scale_weak_reflections = Rand(-0.2, 0.4);// 
- 
-//​scale_weak_reflections//​ or //​add_to_phases_of_weak_reflections//​ can be a function of D_spacing. 
- 
-**[//​set_initial_phases_to//​ $file]** 
- 
-**[//​modify_initial_phases//​ !E]** 
- 
-Sets initial phases to those appearing in $file. Typically $file corresponds to a *.FC file saved in a previous charge-flipping process. //​modify_initial_phases//​ is executed each iteration of CF; it can be used to restrain the phases of $file. For example, 
- 
-modify_initial_phases = 
- 
-Get(initial_phase) + Min(Abs(Get(phase_difference)),​ 45); 
- 
-where phase_difference corresponds to the difference between the current phase and the initial phase; it has a value between ±90º. //​modify_initial_phases//​ can be used to constrain phases to those as determined by HRTEM (Baerlocher //et al.//, 2007)//.// 
- 
- 
-**[//​space_group//​ $]** 
- 
-If defined then the //​cf_hkl_file//​ is assumed to comprise merged hkls corresponding to the defined space group; otherwise the //​cf_hkl_file//​ is assumed to be of space group type //P1.// 
- 
-**[//​symmetry_obey_0_to_1//​ !E]** 
- 
-If a space group is defined then symmetry is adhered to according to //​symmetry_obey_0_to_1//​.  //​symmetry_obey_0_to_1//​ can be through of as a real space electron density restraint; its value should range between 0 and 1. If 1 then symmetry is obeyed 100%. //​symmetry_obey_0_to_1//​ is implemented follows: 
- 
-For a particular set of equivalent grid points as determined by the equivalent positions of the space group an average density //​r//<​sub>​avg</​sub>​ is obtained. The electron densities on the grid points are then adjusted as follows: 
- 
-//​r//<​sub>​new</​sub>​ = //r// (1 - symmetry_obey_0_to_1) + symmetry_obey_0_to_1 //​r//<​sub>​avg</​sub>​ 
- 
-The text output '​symmetry error' as displayed when //​symmetry_obey_0_to_1//​ is used is defined as follows: 
- 
-'​symmetry error' = <​sub>​{{Technical_Reference%20V4-1_files:​image177.gif?​127x29}}</​sub>​ 
- 
-where the summation is over all of the electron density grid points. 
- 
-//​symmetry_obey_0_to_1//​ defines //​find_origin//​ by default. //​find_origin//​ is applied before //​symmetry_obey_0_to_1//​. //​find_origin//​ shifts the electron density such that an approximate error in '​symmetry error' is minimized; thus //​find_origin//​ assists in the //​symmetry_obey_0_to_1//​ restraint. 
- 
-**[//​tangent_num_h_read//​ !E]** 
- 
-**[//​tangent_num_k_read//​ !E]** 
- 
-**[//​tangent_num_h_keep//​ !E]** 
- 
-**[//​tangent_max_triplets_per_h//​ !E]** 
- 
-**[//​tangent_min_triplets_per_h//​ !E]** 
- 
-**[//​tangent_scale_difference_by//​ !E]** 
- 
-//​tangent_num_h_read//​ and //​tangent_num_k_read//​ defines the number of highest //h// and highest //k// reflections to read in determining triplets. 
- 
-//​tangent_num_h_keep//​ defines the number of highest //h// reflections to include for tangent formula updating. 
- 
-//​tangent_max_triplets_per_h//​ and //​tangent_min_triplets_per_h//​ defines the maximum and minium number of triplets per reflection //h//. Reflections with less that //​tangent_min_triplets_per_h//​ are not included for tangent formula updating. 
- 
-//​tangent_scale_difference_by//​ corresponds to //S// in the following: 
- 
-| <​sub>​{{Technical_Reference%20V4-1_files:​image179.gif?​210x25}}</​sub>​ <​sub>​{{Technical_Reference%20V4-1_files:​image181.gif?​76x25}}{{Technical_Reference%20V4-1_files:​image183.gif?​48x24}}</​sub>​ <​sub>​{{Technical_Reference%20V4-1_files:​image185.gif?​215x36}}</​sub>​  <​sub>​{{Technical_Reference%20V4-1_files:​image187.gif?​219x36}}</​sub>​ <​sub>​{{Technical_Reference%20V4-1_files:​image189.gif?​113x27}}</​sub>,​ <​sub>​{{Technical_Reference%20V4-1_files:​image191.gif?​100x31}}</​sub>​ | (1) | 
- 
-**[//​user_threshold//​ !E]** 
- 
-By default Get(//​threshold//​) is determined using //​fraction_density_to_flip//​. When defined then //​user_threshold//​ overrides //​fraction_density_to_flip//​. Electron density pixels are normalized to have a maximum value of 1, thus typical values for //​user_threshold//​ range between 0 and 0.1. 
- 
-**[//​use_Fc//​]** 
- 
-Sets initial phases to those saved in a previous *.FC file. The FC file used corresponds to the same name as the data file, defined using //​cf_hkl_file//​ or //​cf_in_A_matrix//,​ but with a FC extension. //use_Fc// is similar to //​set_initial_phases_to//​ except that the file used is implied. 
- 
-**[//​verbose//​ #]** 
- 
-A value of 1 outputs text in a verbose manner. A value of 0 outputs text only when a //​R//​-factor less that a previous value is encountered within a particular Cyle. 
- 
-**[//​view_cloud//​ !E]** 
- 
-Informs a detected GUI to display the electron density. Here are some examples: 
- 
-view_cloud 1 ' Update cloud every charge-flipping iteration 
- 
-view_cloud = Mod(Cycle_Iter,​ 10) == 0; 
- 
-  
- 
-====== 12       Indexing ====== 
- 
-The following algorithm is based on the iterative method of Coelho (2003). Unlike [[#​k029|lp_serach]] it requires the extraction of d-spacings. The INDEXING directory contains example INP files, for example: 
- 
-index_zero_error 
- 
-try_space_groups "2 75" 
- 
-load index_d { 
- 
-   8.912 
- 
-   7.126 
- 
-   4.296 
- 
-   … 
- 
-} 
- 
-Individual space groups can be tried or for simplicity all of the Bravais lattices can be tried by placing them in the INP file using the standard macros as follows: 
- 
-Bravais_Cubic_sgs 
- 
-Bravais_Trigonal_Hexagonal_sgs 
- 
-Bravais_Tetragonal_sgs 
- 
-Bravais_Orthorhombic_sgs 
- 
-Bravais_Monoclinic_sgs 
- 
-Bravais_Triclinic_sgs 
- 
-On termination of Indexing a *.NDX file is created with a name corresponding to the name of the INP file and placed in the same directory as the INP file. The *.NDX file contains solutions found as well as a detailed summary of the best 20 solutions. Here’s an example of an NDX file: 
- 
-' Indexing method - Alan Coelho (2003), J. Appl. Cryst. 36, 86-95 
- 
-' Time: 2.015 seconds 
- 
-  
- 
-     '​Sg     Status UNI      Vol       Gof     Zero      Lps... 
- 
-  
- 
-Indexing_Solutions_With_Zero_Error_2 { 
- 
-  
- 
-   0) P42/​nmc    3   0    1187.321    38.82   0.0000    11.1924  ... 
- 
-   1) P42/​nmc    3   0    1187.057    38.64   0.0000    11.1896  ...   
- 
-   2) P42/​nmc    3   0    1187.458    38.61   0.0000    11.1914  ...   
- 
-... 
- 
-} 
- 
-%%/*%% 
- 
-%%======================================================================%% 
- 
-   0) P-1         0     985.652    30.80   0.0111     7.0877  ...   
- 
-  
- 
-   h   k   l       dc       do    do-dc     2Thc     2Tho   2Tho-2Thc 
- 
-   0   0   1   15.857   15.830   -0.027    5.569    5.578    0.009 
- 
-   0   1   0    8.765    8.750   -0.015   10.084   10.101    0.017 
- 
-   0   0   2    7.928    7.910   -0.018   11.151   11.177    0.026 
- 
-   0   1   1    7.788    7.780   -0.008   11.352   11.364    0.012 
- 
-   0  -1   1    7.559    7.560    0.001   11.698   11.696   -0.002 
- 
-... 
- 
-%%*/%% 
- 
-===== 12.1  Reprocessing solutions - DET files ===== 
- 
-Details of solutions can be obtained at a later stage by including solution lines found in the NDX file into the INP file. For example, supposing details of solutions 50 and 51 were sought then the following (see example INDEXING\EX10.INP) could be used: 
- 
-index_lam 1.540596 
- 
-index_zero_error      
- 
-try_space_groups 2 
- 
-  
- 
-Indexing_Solutions_With_Zero_Error_2 { 
- 
-  50) P-1        1   0    2064.788     9.74   0.0000   ... 
- 
-  51) P-1        3   0    3128.349     9.61   0.0115   ... 
- 
-} 
- 
-load index_d { 
- 
-   15.83 good 
- 
-    8.75 
- 
-    7.91 
- 
-    ... 
- 
-} 
- 
-After running this INP file a *.DET file is created containing details of the supplied solutions. 
- 
-===== 12.2  Keywords and data structures ===== 
- 
-The data structures for indexing are as follows: 
- 
-Tindexing 
- 
-[[#​i1|[]]//​[[#​i1|index_lam]]//​[[#​i1| ]][[#​i1| !E1.540596]]] 
- 
-[[#​i2|[]]//​[[#​i2|index_min_lp]]//​[[#​i2| !E2] ]][[#​i2| ]][[#​i2|[]]//​[[#​i2|index_max_lp]]//​[[#​i2| !E]]] 
- 
-[[#​i3|[]]//​[[#​i3|index_max_Nc_on_No]]//​[[#​i3| !E5]]] 
- 
-[[#​i4|[]]//​[[#​i4|index_max_number_of_solutions]]//​[[#​i4| #3000]]] 
- 
-[[#​i5|[]]//​[[#​i5|index_max_th2_error]]//​[[#​i5| !E0.05]]] 
- 
-[[#​i6|[]]//​[[#​i6|index_max_zero_error]]//​[[#​i6| #0.2]]] 
- 
-[[#​i7|[]]//​[[#​i7|index_th2]]//​[[#​i7| ]][[#​i7| !E | ]]//​[[#​i7|index_d]]//​[[#​i7| !E]…]] 
- 
-[//​index_I//​  E1 [good]] 
- 
-[[#​i8|[]]//​[[#​i8|index_x0]]//​[[#​i8| !E]]] 
- 
-[[#​i9|[]]//​[[#​i9|index_zero_error]]//​[[#​i9|]]] 
- 
-[[#​i11|[]]//​[[#​i11|seed]]//​[[#​i11|]]] 
- 
-[[#​i12|[]]//​[[#​i12|try_space_groups]]//​[[#​i12| $]...]] 
- 
-[//​x_angle_scaler//​ #0.1] 
- 
-[//​x_scaler//​ #] 
- 
-Values for most keywords are automatically determined or have default values (appearing as numbers above) adequate for difficult indexing problems. In the following example from UPPW (service provided by Armel Le Bail to the SDPD mailing list at [[http://​sdpd.univ-lemans.fr/​uppw/​|http://​sdpd.univ-lemans.fr/​uppw/​]]) only a few keywords are necessary. Also note the use of the //dummy// keyword; this allows for the exclusion of 2q and I values without having to edit the columns of data. 
- 
-  
- 
-seed 
- 
-index_lam  0.79776 
- 
-index_zero_error    
- 
-index_max_Nc_on_No 6 
- 
-try_space_groups 3 
- 
-load index_th2 dummy dummy index_I dummy  { 
- 
-    ' d (A)  2Theta     Height      Area     FWHM 
- 
-     1.724  26.50645    2758.3   23303.7    0.0450 
- 
-     2.646  17.27733  150393.8  747063.6    0.0250 
- 
-     3.235  14.13204   98668.8  493153.7    0.0250 
- 
-     3.417  13.37776   11102.6   53185.0    0.0250 
- 
-     5.190   8.80955     782.7    3910.9    0.0250 
- 
-     ... 
- 
-} 
- 
-===== 12.3  Keywords in detail ===== 
- 
-**[//​index_lam//​  !E1.540596]** 
- 
-Defines the wavelength in Å. 
- 
-**[//​index_min_lp//​ !E2.5] [//​index_max_lp//​ !E]** 
- 
-Defines the minimum and maximum allowed lattice parameters. Typically the maximum is determined automatically. 
- 
-**[//​index_max_Nc_on_No//​ !E5]** 
- 
-Determines the maximum ratio of the number of calculated to observed lines. The value of 6 allows for up to 83% of missing lines. 
- 
-**[//​index_max_number_of_solutions//​ #1000]** 
- 
-The number of best solutions to keep. 
- 
-**[//​index_max_th2_error//​ !E0.05]** 
- 
-Used for determining impurity lines (un-indexed lines UNI in *.NDX). Large values, 1 for example, forces the consideration of more observed input lines. For example if it is know that there are none or maybe just one impurity line then a large value for //​index_max_th2_error//​ will speed up the indexing procedure. 
- 
-**[//​index_max_zero_error//​ !E0.2]** 
- 
-Excludes solutions with zero errors greater than //​index_max_zero_error.//​ 
- 
-**[//​index_th2//​  !E | //index_d// !E]…** 
- 
-**[//​index_I//​  E1 [good]]** 
- 
-//​index_th2//​ or //index_d// defines a reflection entry in 2q degrees or d-spacing in Å. 
- 
-//index_I// is typically set to the area under the peak; it is used to weight the reflection. 
- 
-//​g////​ood//​ signals that the corresponding d-spacing is not an impurity line. A single use of //good// on a large d-spacing decreases the number of possible solutions and hence speeds up the indexing process (see examples INDEXING\EX10.INP). 
- 
-**[//​index_x0//​ !E]** 
- 
-Defines //​X<​sub>​hh</​sub>//​ in the reciprocal lattice equation: 
- 
-<​sub>​{{Technical_Reference%20V4-1_files:​image193.gif?​319x50}}</​sub>​In a triclinic lattice the highest d-spacing can probably be indexed as 100 or 200 etc. Thus 
- 
-index_x0 = 1%%/​(%%d<​sub>​max</​sub>​)^2;​ 
- 
-speeds up the indexing process (if, in this case, the first line can be indexed as 100) and additionally the chances of finding the correct solution is enhanced. Example EX13.INP demonstrates this. Note that if the data is in 2Th degrees then the following can be used: 
- 
-index_x0 = (2 Sin(2Th<​sub>​min</​sub>​ Pi/360) / wavelength))^2;​ 
- 
-The two macros Index_x0_from_d and Index_x0_from_th2 simplify the use of //​index_x0//​. 
- 
-**[//​index_zero_error//​]** 
- 
-Includes a zero error. 
- 
-**[//​seed//​]** 
- 
-Seeds the random number generator. 
- 
-**[//​try_space_groups//​ $]...** 
- 
-**[//​x_angle_scaler//​ #0.1]** 
- 
-**[//​x_scaler//​ #]** 
- 
-Defines the space groups to be searched. The macros Bravais_Cubic_sgs etc... (see TOPAS.INC) defines lowest symmetry Bravais space groups. It is almost always sufficient to use only these. Higher symmetry space groups for the Bravais lattices corresponding to the 10 best solutions is subsequently searched. Here are some examples of using //​try_space_groups//​. 
- 
-| **Search** | **Use** | 
-| Primitive monoclinic | try_space_groups 3 | 
-| The two monoclinic Bravais lattices of lowest symmetry. | Bravais_Monoclinic_sgs | 
-| C-centered monoclinic of lowest symmetry. | try_space_groups 5 | 
-| All orthorhombic space groups individually. | Unique_Orthorhombic_sgs | 
- 
-Below is a list showing which space groups have identical hkls in regards to powder data. 
- 
-//x_scaler is a// scaling factor used for determining the number of steps to search in parameter space. //​x_scaler//​ needs to be less than 1. Increasing //​x_scaler//​ searches parameter space in finer detail. Default values are as follows: 
- 
-Cubic                            0.99 
- 
-Hexagonal/​Trigonal         0.95 
- 
-Tetragonal                     0.95 
- 
-Orthorhombic                 0.89 
- 
-Monoclinic                     0.85 
- 
-Triclinic                         0.72 
- 
-//​x_angle_scaler//​ is a scaling factor for determining the number of angular steps for monoclinic and triclinic space groups. Small values, 0.05 for example, increases the number of angular steps. The dult value of 0.1 is usually sufficient. 
- 
-===== 12.4  Identifying dominant zones ===== 
- 
-Here are two example output lines from an NDX file. 
- 
-0) P42/nmc  3   0    1187.124    38.82   0.0000    11.1904    11.1904     9.4799     90.000     90.000     90.000 ' ===  24  19 
- 
-6) P-421c   3   0    1187.124    35.67   0.0000    11.1904    11.1904     9.4799     90.000     90.000     90.000 ' ===  24  19 
- 
-Ø       The 1<​sup>​st</​sup>​ column corresponds to the rank of the solution. 
- 
-Ø       The 2<​sup>​nd</​sup>​ corresponds to the space group. ​ 
- 
-Ø       The 3<​sup>​rd</​sup>​ corresponds to the Status of the solution with meaning of the number as follows: 
- 
-|   | Status 1: | Weighting applied as defined in Coelho (2003) | 
-|   | Status 2: | Zero error attempt applied | 
-|   | Status 3: | Zero error attempt successful and impurity lines removal attempt successful | 
-|   | Status 4: | Impurity line(s) removed | 
- 
-Ø       The 4<​sup>​th</​sup>​  column corresponds to the number of un-indexed lines. 
- 
-Ø       The 5<​sup>​th</​sup>​ column corresponds to the volume of the lattice. 
- 
-Ø       The 6<​sup>​th</​sup>​ corresponds to the goodness of fit value. 
- 
-Ø       The 7<​sup>​th</​sup>​ corresponds to the zero error if //​index_zero_error//​ is included. 
- 
-Ø       Columns 8 to 13 contains the lattice parameters. 
- 
-The last 2 columns contain the number of non-zero h<​sup>​2</​sup>​ + k<​sup>​2</​sup>​ + h k and l<​sup>​2</​sup>​ values used in the indexed lines. These represent the hkl coefficient for X0 and X1 respectively for Trigonal/​Hexagonal systems. When one of these numbers are zero then the corresponding lattice parameters is not represented and the number is therefore displayed as the negative number of --999. This facility is particularly useful for identifying dominant zones. For example, if the smallest lattice parameter is 3Å and the smallest d-spacings is 4Å then it is impossible to determine the small lattice parameter. In these cases values of --999 will be obtained. 
- 
-The following table gives the hkl coefficients corresponding to the X<​sub>​nn</​sub>​ reciprocal lattice parameters for the 7 crystal systems. 
- 
-| ** ** | X0 | X1 | X2 | X3 | X4 | X5 | 
-| Cubic | h<​sup>​2</​sup>​+k<​sup>​2</​sup>​+l<​sup>​2</​sup>​ |   |   |   |   |   | 
-| Hexagonal Trigonal | h<​sup>​2</​sup>​+k<​sup>​2</​sup>​+h k | l<​sup>​2</​sup>​ |   |   |   |   | 
-| Tetragonal | h<​sup>​2</​sup>​+k<​sup>​2</​sup>​ | l<​sup>​2</​sup>​ |   |   |   |   | 
-| Orhtorhombic | h<​sup>​2</​sup>​ | k<​sup>​2</​sup>​ | l<​sup>​2</​sup>​ |   |   |   | 
-| Monoclinic | h<​sup>​2</​sup>​ | k<​sup>​2</​sup>​ | l<​sup>​2</​sup>​ | h l |   |   | 
-| Triclinic | h<​sup>​2</​sup>​ | k<​sup>​2</​sup>​ | l<​sup>​2</​sup>​ | h k | h l | k l | 
- 
-===== 12.5  %%***%% Probable causes of Failure %%***%% ===== 
- 
-The most probable cause of failure is the inclusion of too many d-spacings. If it is assumed that the smallest lattice parameter is greater than 3Å then it is problematic to include d-spacings with values less than about 2.5Å when there are already 30 to 40 reflections with d values greater than 2.5Å. Some of the problems caused by very low d-spacings are: 
- 
-Ø       The number of calculated lines increases dramatically and thus //​index_max_Nc_on_No//​ will need to be increased. 
- 
-Ø       The low d-spacings are probably inaccurate due to peak overlap at the high angles they are observed at. 
- 
-A situation where it is necessary to include low d-spacings is when there are only a few d-spacings available as in higher symmetry lattices. 
- 
-===== 12.6  Unique space group hkls in Powder diffraction ===== 
- 
-| **Space group numbers with** **identical hkls** | **Space group symbols with** **identical hkls** | 
-| **Triclinic** || 
-| 1 2 | P1 P-1 | 
-| **Monoclinic** || 
-| 9 15 | Cc C2/c | 
-| 5 8 12 | C2 Cm C2/m | 
-| 14 | P21/c | 
-| 7 13 | Pc P2/c | 
-| 4 11 | P21 P21/m | 
-| 3 6 10 | P2 Pm P2/m | 
-| **Orthorhombic** || 
-| 70 | Fddd | 
-| 43 | Fdd2 | 
-| 22 42 69 | F222 Fmm2 Fmmm | 
-| 68 | Ccca | 
-| 73 | Ibca | 
-| 37 66 | Ccc2 Cccm | 
-| 45 72 | Iba2 Ibam | 
-| 41 64 | Aba2 Cmca | 
-| 46 74 | Ima2 Imma | 
-| 36 40 63 | Cmc21 Ama2 Cmcm | 
-| 39 67 | Abm2 Cmma | 
-| 20 | C2221 | 
-| 23 24 44 71 | I222 I212121 Imm2 Immm | 
-| 21 35 38 65 | C222 Cmm2 Amm2 Cmmm | 
-| 52 | Pnna | 
-| 56 | Pccn | 
-| 60 | Pbcn | 
-| 61 | Pbca | 
-| 48 | Pnnn | 
-| 54 | Pcca | 
-| 50 | Pban | 
-| 33 62 | Pna21 Pnma | 
-| 34 58 | Pnn2 Pnnm | 
-| 32 55 | Pba2 Pbam | 
-| 30 53 | Pnc2 Pmna | 
-| 29 57 | Pca21 Pbcm | 
-| 27 49 | Pcc2 Pccm | 
-| 31 59 | Pmn21 Pmmn | 
-| 26 28 51 | Pmc21 Pma2 Pmma | 
-| 19 | P212121 | 
-| 18 | P21212 | 
-| 17 | P2221 | 
-| 16 25 47  | P222 Pmm2 Pmmm | 
-| **Tetragonal** || 
-| 142 | I41/acd | 
-| 110 | I41cd | 
-| 141 | I41/amd | 
-| 109 122 | I41md I-42d | 
-| 108 120 140 | I4cm I-4c2 I4/mcm | 
-| 88 | I41/a | 
-| 80 98 | I41 I4122 | 
-| 79 82 87 97 107 119 121 139  | I4 I-4 I4/m I422 I4mm I-4m2 I-42m I4/mmm | 
-| 130 | P4/ncc | 
-| 126 | P4/nnc | 
-| 133 | P42/nbc | 
-| 103 124 | P4cc P 4/mcc | 
-| 104 128 | P4nc P 4/mnc | 
-| 106 135 | P42bc P 42/mbc | 
-| 137 | P42/nmc | 
-| 138 | P42/ncm | 
-| 134 | P42/nnm | 
-| 125 | P4/nbm | 
-| 114 | P-421c | 
-| 105 112 131 | P42mc P-42c P42/mmc | 
-| 102 118 136 | P42nm P-4n2 P42/mnm | 
-| 101 116 132 | P42cm P-4c2 P42/mcm | 
-| 100 117 127 | P4bm P-4b2 P4/mbm | 
-| 86 | P42/n | 
-| 85 129 | P4/n P4/nmm | 
-| 92 96 | P41212 P43212 | 
-| 94 | P42212 | 
-| 76 78 91 95 | P41 P43 P4122 P4322 | 
-| 77 84 93 | P42 P 42/m P4222 | 
-| 90 113 | P4212 P-421m | 
-| 75 81 83 89 99 111 115 123   | P4 P-4 P4/m P422 P4mm P-42m P-4m2 P4/mmm | 
-| **Trigonal & Hexagonal** || 
-| 161 167 | R3c R-3c | 
-| 146 148 155 160 166 | R3 R-3 R32 R3m R-3m | 
-| 184 192 | P6cc P6/mcc | 
-| 159 163 186 190 194 | P31c P-31c P63mc P-62c P63/mmc | 
-| 158 165 185 188 193 | P3c1 P-3c1 P63cm P-6c2 P63/mcm | 
-| 169 170 178 179 | P61 P65 P6122 P6522 | 
-| 144 145 151 152 153 154 171 172 180 181   | P31 P32 P3112 P3121 P3212 P3221 P62 P64 P6222 P6422 | 
-| 173 176 182 | P63 P63/m P6322 | 
-|  143 147 149 150 156 157 162 164 168 174 175 177 183 187 189 191 | P3 P-3 P312 P321 P3m1 P31m P-31m P-3m1 P6 P-6 P6/m P622 P6mm P-6m2 P-62m P6/mmm | 
-| **Cubic** || 
-| 228 | Fd-3c | 
-| 219 226 | F-43c Fm-3c | 
-| 203 227 | Fd-3 Fd-3m | 
-| 210 | F4132 | 
-| 196 202 209 216 225    | F23 Fm-3 F432 F-43m Fm-3m | 
-| 230 | Ia-3d | 
-| 220 | I-43d | 
-| 206 | Ia-3 | 
-| 214 | I4132 | 
-| 197 199 204 211 217 229 | I23 I213 Im-3 I432 I-43m Im-3m | 
-| 222 | Pn-3n | 
-| 218 223 | P-43n Pm-3n | 
-| 201 224 | Pn-3 Pn-3m | 
-| 205 | Pa-3 | 
-| 212 213 | P4332 P4132 | 
-| 198 208 | P213 P4232 | 
-| 195 200 207 215 221    | P23 Pm-3 P432 P-43m Pm-3m | 
- 
-===== 12.7  Equations in Indexing - Background ===== 
- 
-**//a//**, **//b//** and **//c//** lattice vectors can be converted to Cartesian coordinates with **//a//** collinear with the Cartesian //x// axis and **//b//** coplanar with the Cartesian //x//-//y// plane as follows: 
- 
-| **//a//** = //​a//<​sub>​x</​sub>​ i **//b//** = //​b//<​sub>​x</​sub>​ **i** + //​b//<​sub>​y</​sub>​ **j** **//c//** = //​c//<​sub>​x</​sub>​ **i** + c<​sub>​y</​sub>​ **j  +** //​c//<​sub>​z</​sub>​ **k** | (12‑1) | 
- 
-where 
- 
-//​a//<​sub>​x</​sub>​ = //a// 
- 
-//​b//<​sub>​x</​sub>​ = //b// cos(g),   //​b//<​sub>​y</​sub>​ = //b// sin(g) 
- 
-//​c//<​sub>​x</​sub>​ = //c// cos(//​b//​),​   //​c//<​sub>​y</​sub>​ =  //c// (cos(//a//) -- cos(//b//) cos(g)) / sin(g),   //​c//<​sub>​z</​sub><​sup>​2</​sup>​ = //​c//<​sup>​2</​sup>​ - (//​c//<​sub>​x</​sub>​)<​sup>​2</​sup>​-- (//​c//<​sub>​y</​sub>​)<​sup>​2</​sup>​ 
- 
-//a//, //b//, //c// are the lattice parameters and //a, b,// g the lattice angles. The reciprocal lattice vectors **//A//**, **//B//**, and **//C//** calculated from the lattice vectors of Eq. (12‑1) become: 
- 
-**//A//**= //​A//<​sub>​x</​sub>​ **i** + //​A//<​sub>​y</​sub>​ **j  +** //​A//<​sub>​z</​sub>​**k** 
- 
-**//B//** = //​B//<​sub>​y</​sub>​ **j** + //​B//<​sub>​z</​sub>​ **k** 
- 
-**//C//** = //​C//<​sub>​z</​sub>​ 
- 
-The equation relating a particular d-spacing //​d//<​sub>​hkl</​sub>​ to a particular hkl in terms of the reciprocal lattice parameters is: 
- 
-  
- 
-| <​sub>​{{Technical_Reference%20V4-1_files:​image195.gif?​368x25}}</​sub>​ | (12‑2) | 
- 
-where 
- 
-<​sub>​{{Technical_Reference%20V4-1_files:​image197.gif?​119x27}}</​sub><​sub>​{{Technical_Reference%20V4-1_files:​image199.gif?​89x27}}</​sub><​sub>​{{Technical_Reference%20V4-1_files:​image201.gif?​59x25}}</​sub><​sub>​{{Technical_Reference%20V4-1_files:​image203.gif?​160x25}}</​sub><​sub>​{{Technical_Reference%20V4-1_files:​image205.gif?​93x24}}</​sub><​sub>​{{Technical_Reference%20V4-1_files:​image207.gif?​92x24}}</​sub>​====== 13       Batch mode operation – TC.EXE ====== 
- 
-The command line program tc.exe provides for batch mode operation. Running tc.exe without arguments displays help information. Running an INP file is as follows: 
- 
-tc pbso4 
- 
-Macros can be passed to the command line. One use for this is to pass a file name to an INP file as follows: 
- 
-1)   Create a TEMPLATE.INP file with the required refinement details, this should look something like the following: 
- 
-xdd FILE 
- 
-etc... 
- 
-2)   TEMPLATE.INP is fed to tc.exe by command line and the word FILE (within TEMPLATE.INP) is expanded to whatever the macro on the command line is. For example, 
- 
-tc ...\file_directory\TEMPLATE.INP "macro FILE { file.xy }" 
- 
-The macro called FILE is described on the command line within quotation marks. On running tc.exe the word '​FILE'​ occurring in TEMPATE.INP is expanded to '​file.xy'​. Note that more than one macro can be described on the command line. 
- 
-To process a whole directory of data files, say *.XY file for example, then: 
- 
-1)       From the file directory execute the DOS command: 
- 
-dir *.xy > ...\main_ta_directory\XY.BAT 
- 
-      The XY.BAT file will then reside in the main TA directory. 
- 
-2)   Edit ...\main_ta_directory\XY.BAT to look like the following: 
- 
-tc ...\file_directory\template "macro FILE { file1.xy }" 
- 
-copy ...\file_directory\template.out ...\file_directory\file1.out 
- 
-tc ...\file_directory\template.inp "macro FILE { file2.xy }" 
- 
-copy ...\file_directory\template.out ...\file_directory\file2.out 
- 
-etc.... 
- 
-After each run of tc.exe a TEMPLATE.OUT file is created containing refined results. This file is copied to another file "​file1.out",​ "​file2.out"​ etc... in order to save it from being overwritten. 
- 
-After running XY.BAT a number of *.OUT files is created one for each *.XY file. 
- 
-In summary TC.EXE receives TEMPLATE.INP to process. Words occurring in TEMPLATE.INP are expanded depending on the macros described on the command line. 
- 
-  
- 
-====== 14       References ====== 
- 
-**Baerlocher,​ C; Gramm, F.; Massüger, L; McCusker, L; He, Z; Hovmöller, S & Zou, X. (2007).** SCIENCE VOL 315 23 FEBRUARY 2007 
- 
-**Baerlocher,​ C.; McCusker, L.B.; & Palatinus, L. (2007)**.  Z.  Kristallogr. 222, 47-53 
- 
-**Balzar, D. (1999).** Microstructure Analysis from Diffraction,​ edited by R. L. Snyder, H. J. Bunge, and J. Fiala, International Union of Crystallography,​ 1999. “//​Voigt-function model in diffraction line-broadening analysis//​” 
- 
-**Bergmann, J., Kleeberg, R., Haase, A. & Breidenstein,​ B. (2000).** Mat. Sci. Forum, **347-349**,​ 303-308. “//​Advanced Fundamental Parameters Model for Improved Profile Analysis”.//​ 
- 
-**Brindley, G.W. (1945)**//​.//​ Phil. Mag., **36**, 347-369. “//The effect of grain or particle size on X-ray reflections from mixed powders and alloys, considered in relation to the quantitative determination of crystalline substances by X-ray methods//​” 
- 
-**Broyden, C.G**., J. Inst. Maths. Applics., Vol. 6, pp 76-90, 1970. "The Convergence of a Class of Double-rank Minimization Algorithms,"​ 
- 
-**Cagliotti,​ G., Paoletti, A. & Ricci, F.P (1958).** Nucl. Inst., **3**, 223-228. “//Choice of collimators for a crystal spectrometer for neutron diffraction”//​ 
- 
-**Coelho, A. A. (2005)**. //J. Appl. Cryst.// 38, 455-461. "A bound constrained conjugate gradient solution method as applied to crystallographic refinement problems"​ 
- 
-**Coelho, A. A,. (2003)**. //J. Appl. Cryst.//, **36**, 86--95. “Indexing of powder diffraction patterns by iterative use of singular value decomposition”. 
- 
-**Coelho, A. A. & Cheary, R. W. (1997).** Computer Physics Communications,​ **104**, 15-22. “//A fast and simple method for calculating electrostatic potentials”//​ 
- 
-**Cheary, R.W. & Coelho, A.A. (1998a).** J. Appl. Cryst., **31**, 851-861. //“Axial divergence in a conventional X‑ray powder diffractometer I. Theoretical foundations”//​ 
- 
-**Cheary, R.W. & Coelho, A.A. (1998b).** J. Appl. Cryst., **31**, 862-868. “//Axial divergence in a conventional X‑ray powder diffractometer II. Implementation and comparison with experiment//​” 
- 
-**Baerlocher,​ C; Gramm, F.; Massüger, L; McCusker, L; He, Z; Hovmöller, S & Zou, X. (2007)**. SCIENCE VOL 315 23 FEBRUARY 2007. 
- 
-**Chernick, M.R. (1999)**. //Bootstrap Methods, A Practitioner’s Guide//, Wiley, New York.  
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