Clarification of theta/phi definitions for CMC anisotropy
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Tong Zhao
Aug 22, 2025, 3:14:10 AMAug 22
to Vampire Users
Hi,
I’m using the constrained Monte Carlo (cmc-anisotropy) program to evaluate temperature‑dependent anisotropy in 2D materials. I’m unsure about the exact definitions of the constraint angles:
Is theta (θ) the azimuthal angle in the xy plane from +x, and phi (φ) the polar angle from +z?
My current input (from the workshop example) is:
#------------------------------------------
# Simulation attributes:
#------------------------------------------
######################################
sim:minimum-temperature = 0
sim:maximum-temperature = 2000
sim:temperature-increment = 100
######################################
sim:equilibration-time-steps = 5000
sim:loop-time-steps = 5000
sim:constraint-angle-theta-minimum=0.0
sim:constraint-angle-theta-maximum=0.0
sim:constraint-angle-theta-increment=10.0
sim:constraint-angle-phi-minimum=0.0
sim:constraint-angle-phi-maximum=90.0
sim:constraint-angle-phi-increment=15.0
sim:save-checkpoint=continuous
sim:save-checkpoint-rate=1
#sim:load-checkpoint=continue
#------------------------------------------
# Program and integrator details
#------------------------------------------
sim:integrator = constrained-monte-carlo
sim:program = cmc-anisotropy
#------------------------------------------
# data output
#------------------------------------------
output:temperature
output:constraint-theta
output:constraint-phi
output:mean-magnetisation-length
output:mean-total-torque
output:anisotropy-energy
output:mean-anisotropy-energy
# Simulation attributes:
#------------------------------------------
######################################
sim:minimum-temperature = 0
sim:maximum-temperature = 2000
sim:temperature-increment = 100
######################################
sim:equilibration-time-steps = 5000
sim:loop-time-steps = 5000
sim:constraint-angle-theta-minimum=0.0
sim:constraint-angle-theta-maximum=0.0
sim:constraint-angle-theta-increment=10.0
sim:constraint-angle-phi-minimum=0.0
sim:constraint-angle-phi-maximum=90.0
sim:constraint-angle-phi-increment=15.0
sim:save-checkpoint=continuous
sim:save-checkpoint-rate=1
#sim:load-checkpoint=continue
#------------------------------------------
# Program and integrator details
#------------------------------------------
sim:integrator = constrained-monte-carlo
sim:program = cmc-anisotropy
#------------------------------------------
# data output
#------------------------------------------
output:temperature
output:constraint-theta
output:constraint-phi
output:mean-magnetisation-length
output:mean-total-torque
output:anisotropy-energy
output:mean-anisotropy-energy
My understanding from the manual is that constraint-phi is measured from the z‑axis and constraint-theta from the x‑axis (i.e., φ polar, θ azimuth). Thus, fixing θ=0° and sweeping φ=0→90° would tilt M from +z to +x. Could someone please confirm that this is the correct interpretation for cmc-anisotropy?
Any pointers would be greatly appreciated. Thanks in advance!
gabo...@gmail.com
Aug 22, 2025, 10:11:49 AMAug 22
to Vampire Users
If it helps, do you have "Initialising spins to new constraint direction (phi, theta)" in your log file?
If so, when I ran the input and uni.mat from [1], I noticed it outputted that in the log file as shown below:
username@computername:~$ mkdir cmc-anisotropy
username@computername:~$ cd cmc-anisotropy/
username@computername:~/cmc-anisotropy$ wget https://github.com/richard-evans/vampire-workshop/raw/refs/heads/master/input-files/2_constrained_montecarlo_and_dipole_fields/2a_uniaxial/input
...
username@computername:~/cmc-anisotropy$ wget https://github.com/richard-evans/vampire-workshop/raw/refs/heads/master/input-files/2_constrained_montecarlo_and_dipole_fields/2a_uniaxial/uni.mat
...
username@computername:~/cmc-anisotropy$ vampire-serial
_
(_)
__ ____ _ _ __ ___ _ __ _ _ __ ___
\ \ / / _` | '_ ` _ \| '_ \| | '__/ _ \
\ V / (_| | | | | | | |_) | | | | __/
\_/ \__,_|_| |_| |_| .__/|_|_| \___|
| |
|_|
Version 7.0.0 Aug 4 2025 22:39:21
Git commit: 525bc27ee44c525aee229570f30f3d4c61d54f66
Licensed under the GNU Public License(v2). See licence file for details.
Developers: Richard F L Evans, Sarah Jenkins, Andrea Meo,
Daniel Meilak, Andrew Naden, Matthew Ellis,
Oscar Arbelaez, Sam Morris, Rory Pond, Weijia Fan,
Phanwadee Chureemart, Pawel Sobieszczyk, Joe Barker,
Thomas Ostler, Andreas Biternas, Roy W Chantrell,
Wu Hong-Ye, Razvan Ababei, Sam Westmoreland,
Milton Persson
Compiler Flags:
Vampire includes a copy of the qhull library from C.B. Barber and The
Geometry Center and may be obtained via http from www.qhull.org.
================================================================================
Fri Aug 22 05:56:27 2025
================================================================================
Initialising system variables
Creating system
Generating neighbour list..........done!
Copying system data to optimised data structures.
Using generic/normalised form of exchange interaction with 16 total interactions.
Number of atoms generated: 6750
Starting Simulation with Program CMC-Anisotropy...
Simulation run time [s]: 1897.09
Constrained Monte Carlo statistics:
Total moves: 9.9225e+09
47.6124% Accepted
37.1643% Rejected (Energy)
15.2233% Rejected (Sphere)
Simulation ended gracefully.
username@computername:~/cmc-anisotropy$ grep constraint log
22-08-2025 [05:56:27] Initialising spins to new constraint direction (phi, theta) 0 , 0
22-08-2025 [06:00:58] Initialising spins to new constraint direction (phi, theta) 15 , 0
22-08-2025 [06:05:29] Initialising spins to new constraint direction (phi, theta) 30 , 0
22-08-2025 [06:10:01] Initialising spins to new constraint direction (phi, theta) 45 , 0
22-08-2025 [06:14:32] Initialising spins to new constraint direction (phi, theta) 60 , 0
22-08-2025 [06:19:02] Initialising spins to new constraint direction (phi, theta) 75 , 0
22-08-2025 [06:23:33] Initialising spins to new constraint direction (phi, theta) 90 , 0
username@computername:~$ cd cmc-anisotropy/
username@computername:~/cmc-anisotropy$ wget https://github.com/richard-evans/vampire-workshop/raw/refs/heads/master/input-files/2_constrained_montecarlo_and_dipole_fields/2a_uniaxial/input
...
username@computername:~/cmc-anisotropy$ wget https://github.com/richard-evans/vampire-workshop/raw/refs/heads/master/input-files/2_constrained_montecarlo_and_dipole_fields/2a_uniaxial/uni.mat
...
username@computername:~/cmc-anisotropy$ vampire-serial
_
(_)
__ ____ _ _ __ ___ _ __ _ _ __ ___
\ \ / / _` | '_ ` _ \| '_ \| | '__/ _ \
\ V / (_| | | | | | | |_) | | | | __/
\_/ \__,_|_| |_| |_| .__/|_|_| \___|
| |
|_|
Version 7.0.0 Aug 4 2025 22:39:21
Git commit: 525bc27ee44c525aee229570f30f3d4c61d54f66
Licensed under the GNU Public License(v2). See licence file for details.
Developers: Richard F L Evans, Sarah Jenkins, Andrea Meo,
Daniel Meilak, Andrew Naden, Matthew Ellis,
Oscar Arbelaez, Sam Morris, Rory Pond, Weijia Fan,
Phanwadee Chureemart, Pawel Sobieszczyk, Joe Barker,
Thomas Ostler, Andreas Biternas, Roy W Chantrell,
Wu Hong-Ye, Razvan Ababei, Sam Westmoreland,
Milton Persson
Compiler Flags:
Vampire includes a copy of the qhull library from C.B. Barber and The
Geometry Center and may be obtained via http from www.qhull.org.
================================================================================
Fri Aug 22 05:56:27 2025
================================================================================
Initialising system variables
Creating system
Generating neighbour list..........done!
Copying system data to optimised data structures.
Using generic/normalised form of exchange interaction with 16 total interactions.
Number of atoms generated: 6750
Starting Simulation with Program CMC-Anisotropy...
Simulation run time [s]: 1897.09
Constrained Monte Carlo statistics:
Total moves: 9.9225e+09
47.6124% Accepted
37.1643% Rejected (Energy)
15.2233% Rejected (Sphere)
Simulation ended gracefully.
username@computername:~/cmc-anisotropy$ grep constraint log
22-08-2025 [05:56:27] Initialising spins to new constraint direction (phi, theta) 0 , 0
22-08-2025 [06:00:58] Initialising spins to new constraint direction (phi, theta) 15 , 0
22-08-2025 [06:05:29] Initialising spins to new constraint direction (phi, theta) 30 , 0
22-08-2025 [06:10:01] Initialising spins to new constraint direction (phi, theta) 45 , 0
22-08-2025 [06:14:32] Initialising spins to new constraint direction (phi, theta) 60 , 0
22-08-2025 [06:19:02] Initialising spins to new constraint direction (phi, theta) 75 , 0
22-08-2025 [06:23:33] Initialising spins to new constraint direction (phi, theta) 90 , 0
I found in the source code (cmc.cpp [2]), the lines:
// Output message showing constraint direction re-initialisation
zlog << zTs() << "Initialising spins to new constraint direction (phi, theta) " << sim::constraint_phi << " , " << sim::constraint_theta << std::endl;
// Initialise all spins along the constraint direction.
double sx=sin(sim::constraint_phi*M_PI/180.0)*cos(sim::constraint_theta*M_PI/180.0);
double sy=sin(sim::constraint_phi*M_PI/180.0)*sin(sim::constraint_theta*M_PI/180.0);
double sz=cos(sim::constraint_phi*M_PI/180.0);
zlog << zTs() << "Initialising spins to new constraint direction (phi, theta) " << sim::constraint_phi << " , " << sim::constraint_theta << std::endl;
// Initialise all spins along the constraint direction.
double sx=sin(sim::constraint_phi*M_PI/180.0)*cos(sim::constraint_theta*M_PI/180.0);
double sy=sin(sim::constraint_phi*M_PI/180.0)*sin(sim::constraint_theta*M_PI/180.0);
double sz=cos(sim::constraint_phi*M_PI/180.0);
The above source code lines look like the spherical coordinate equations at [3] but with ρ = 1.
Using a few different values for the equations:
For φ = 0 degrees and θ = 0 degrees,
Sx = sin(φ*π/180)*cos(θ*π/180) = sin(0*π/180)*cos(0*π/180) = 0
Sy = sin(φ*π/180)*sin(θ*π/180) = sin(0*π/180)*sin(0*π/180) = 0
Sz = cos(φ*π/180) = cos(0*π/180) = 1
Sx = sin(φ*π/180)*cos(θ*π/180) = sin(0*π/180)*cos(0*π/180) = 0
Sy = sin(φ*π/180)*sin(θ*π/180) = sin(0*π/180)*sin(0*π/180) = 0
Sz = cos(φ*π/180) = cos(0*π/180) = 1
For the equations, I found on a webpage at [4] that ρ = 1 (or r = 1) can be selected. However, since it hard to see with ρ = 1, ρ = 4 was selected:
For φ = 45 degrees and θ = 0 degrees,
Sx = sin(φ*π/180)*cos(θ*π/180) = sin(45*π/180)*cos(0*π/180) = 0.707
Sy = sin(φ*π/180)*sin(θ*π/180) = sin(45*π/180)*sin(0*π/180) = 0
Sz = cos(φ*π/180) = cos(45*π/180) = 0.707
For φ = 90 degrees and θ = 0 degrees,
Sx = sin(φ*π/180)*cos(θ*π/180) = sin(90*π/180)*cos(0*π/180) = 1
Sy = sin(φ*π/180)*sin(θ*π/180) = sin(90*π/180)*sin(0*π/180) = 0
Sz = cos(φ*π/180) = cos(90*π/180) = 0
Sx = sin(φ*π/180)*cos(θ*π/180) = sin(90*π/180)*cos(0*π/180) = 1
Sy = sin(φ*π/180)*sin(θ*π/180) = sin(90*π/180)*sin(0*π/180) = 0
Sz = cos(φ*π/180) = cos(90*π/180) = 0
[3] https://sites.millersville.edu/bikenaga/calculus3/spherical-coordinates/spherical-coordinates.html
Gavin
VAMPIRE user
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