Category:Forces - Revision history

Liebetreu: Reverted edit by Liebetreu (talk) to last revision by Schlipf

2026-04-15T12:24:13Z

Reverted edit by Liebetreu (talk) to last revision by Schlipf

← Older revision		Revision as of 12:24, 15 April 2026
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	* thermodynamic processes ([[MD\|molecular dynamics]])		* thermodynamic processes ([[MD\|molecular dynamics]])

	For a classical particle, Newton's second law of motion states that the change of motion of an object is proportional to the force acting on the object and oriented in the same direction as the force vector ~~<math>~~\mathbf{F}(t)~~</math>~~. Therefore, the force is defined as the change of particle momentum with time		For a classical particle, Newton's second law of motion states that the change of motion of an object is proportional to the force acting on the object and oriented in the same direction as the force vector $\mathbf{F}(t)$. Therefore, the force is defined as the change of particle momentum with time

	::<math>		::<math>
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	</math>		</math>

	where <math> \mathbf{a}(t) </math> is the acceleration of the particle. Here, the velocity is defined as the change of position with time ~~<math>~~\mathbf{v}(t) = \frac{d\mathbf{r}(t)}{dt}~~</math>~~, <math>\mathbf{r}(t)</math> is the position of the particle, and the momentum <math>\mathbf{p}(t)</math> of the particle is the velocity times the particle mass <math>m</math>: <math>\mathbf{p}(t) = m\mathbf{v}(t).</math>		where <math> \mathbf{a}(t) </math> is the acceleration of the particle. Here, the velocity is defined as the change of position with time $\mathbf{v}(t) = \frac{d\mathbf{r}(t)}{dt}$, <math>\mathbf{r}(t)</math> is the position of the particle, and the momentum <math>\mathbf{p}(t)</math> of the particle is the velocity times the particle mass <math>m</math>: <math>\mathbf{p}(t) = m\mathbf{v}(t).</math>
	{{NB\|mind\|With this equation of motion, the knowledge of some starting conditions <math>\mathbf{r}(0)</math> and <math>\mathbf{v}(0)</math> and an algorithm to compute the forces <math> \mathbf{F} </math> the trajectory <math> \mathbf{r}(t)</math> of a particle can be predicted for all times.}}		{{NB\|mind\|With this equation of motion, the knowledge of some starting conditions <math>\mathbf{r}(0)</math> and <math>\mathbf{v}(0)</math> and an algorithm to compute the forces <math> \mathbf{F} </math> the trajectory <math> \mathbf{r}(t)</math> of a particle can be predicted for all times.}}
	Moreover, one can directly relate the force and the negative gradient of the potential energy. The gradient of the potential energy can be computed from the Lagrangian of the particle system of interest. The Lagrangian for an N particle system is		Moreover, one can directly relate the force and the negative gradient of the potential energy. The gradient of the potential energy can be computed from the Lagrangian of the particle system of interest. The Lagrangian for an N particle system is
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	</math>		</math>

	and is, therefore, the gradient of the force. The force-constant matrix is a fundamental concept in solid-state physics and materials science, especially in the context of understanding the vibrational properties of crystals, i.e., [[phonons]]. It is a generalization of the spring constant ~~<math>~~k~~</math>~~ inside (~~<math>~~\mathbf{F}=k\mathbf{x}~~</math>~~) for the case of 3D crystals. This matrix is used to describe the relationships between atomic displacements and the resulting forces that occur in a crystal. By Fourier transforming the force-constant matrix, the [[Phonons: Theory\|dynamical matrix]] is obtained.		and is, therefore, the gradient of the force. The force-constant matrix is a fundamental concept in solid-state physics and materials science, especially in the context of understanding the vibrational properties of crystals, i.e., [[phonons]]. It is a generalization of the spring constant $k$ inside ($\mathbf{F}=k\mathbf{x}$) for the case of 3D crystals. This matrix is used to describe the relationships between atomic displacements and the resulting forces that occur in a crystal. By Fourier transforming the force-constant matrix, the [[Phonons: Theory\|dynamical matrix]] is obtained.

	By computing the eigenvalues of the dynamical matrix on various reciprocal lattice points, the [[Computing the phonon dispersion and DOS\|phonon-dispersion relation]] can be obtained.		By computing the eigenvalues of the dynamical matrix on various reciprocal lattice points, the [[Computing the phonon dispersion and DOS\|phonon-dispersion relation]] can be obtained.

Liebetreu at 07:32, 15 April 2026

2026-04-15T07:32:08Z

← Older revision		Revision as of 07:32, 15 April 2026
Line 4:		Line 4:
	* thermodynamic processes ([[MD\|molecular dynamics]])		* thermodynamic processes ([[MD\|molecular dynamics]])

	For a classical particle, Newton's second law of motion states that the change of motion of an object is proportional to the force acting on the object and oriented in the same direction as the force vector $\mathbf{F}(t)$. Therefore, the force is defined as the change of particle momentum with time		For a classical particle, Newton's second law of motion states that the change of motion of an object is proportional to the force acting on the object and oriented in the same direction as the force vector <math>\mathbf{F}(t)</math>. Therefore, the force is defined as the change of particle momentum with time

	::<math>		::<math>
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	</math>		</math>

	where <math> \mathbf{a}(t) </math> is the acceleration of the particle. Here, the velocity is defined as the change of position with time $\mathbf{v}(t) = \frac{d\mathbf{r}(t)}{dt}$, <math>\mathbf{r}(t)</math> is the position of the particle, and the momentum <math>\mathbf{p}(t)</math> of the particle is the velocity times the particle mass <math>m</math>: <math>\mathbf{p}(t) = m\mathbf{v}(t).</math>		where <math> \mathbf{a}(t) </math> is the acceleration of the particle. Here, the velocity is defined as the change of position with time <math>\mathbf{v}(t) = \frac{d\mathbf{r}(t)}{dt}</math>, <math>\mathbf{r}(t)</math> is the position of the particle, and the momentum <math>\mathbf{p}(t)</math> of the particle is the velocity times the particle mass <math>m</math>: <math>\mathbf{p}(t) = m\mathbf{v}(t).</math>
	{{NB\|mind\|With this equation of motion, the knowledge of some starting conditions <math>\mathbf{r}(0)</math> and <math>\mathbf{v}(0)</math> and an algorithm to compute the forces <math> \mathbf{F} </math> the trajectory <math> \mathbf{r}(t)</math> of a particle can be predicted for all times.}}		{{NB\|mind\|With this equation of motion, the knowledge of some starting conditions <math>\mathbf{r}(0)</math> and <math>\mathbf{v}(0)</math> and an algorithm to compute the forces <math> \mathbf{F} </math> the trajectory <math> \mathbf{r}(t)</math> of a particle can be predicted for all times.}}
	Moreover, one can directly relate the force and the negative gradient of the potential energy. The gradient of the potential energy can be computed from the Lagrangian of the particle system of interest. The Lagrangian for an N particle system is		Moreover, one can directly relate the force and the negative gradient of the potential energy. The gradient of the potential energy can be computed from the Lagrangian of the particle system of interest. The Lagrangian for an N particle system is
Line 122:		Line 122:
	</math>		</math>

	and is, therefore, the gradient of the force. The force-constant matrix is a fundamental concept in solid-state physics and materials science, especially in the context of understanding the vibrational properties of crystals, i.e., [[phonons]]. It is a generalization of the spring constant $k$ inside ($\mathbf{F}=k\mathbf{x}$) for the case of 3D crystals. This matrix is used to describe the relationships between atomic displacements and the resulting forces that occur in a crystal. By Fourier transforming the force-constant matrix, the [[Phonons: Theory\|dynamical matrix]] is obtained.		and is, therefore, the gradient of the force. The force-constant matrix is a fundamental concept in solid-state physics and materials science, especially in the context of understanding the vibrational properties of crystals, i.e., [[phonons]]. It is a generalization of the spring constant <math>k</math> inside (<math>\mathbf{F}=k\mathbf{x}</math>) for the case of 3D crystals. This matrix is used to describe the relationships between atomic displacements and the resulting forces that occur in a crystal. By Fourier transforming the force-constant matrix, the [[Phonons: Theory\|dynamical matrix]] is obtained.

	By computing the eigenvalues of the dynamical matrix on various reciprocal lattice points, the [[Computing the phonon dispersion and DOS\|phonon-dispersion relation]] can be obtained.		By computing the eigenvalues of the dynamical matrix on various reciprocal lattice points, the [[Computing the phonon dispersion and DOS\|phonon-dispersion relation]] can be obtained.

Schlipf: /* Machine-learned forces */

2025-10-24T11:30:48Z

Machine-learned forces

← Older revision		Revision as of 11:30, 24 October 2025
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	*[[Best practices for machine-learned force fields\|Machine learning best practice]]		*[[Best practices for machine-learned force fields\|Machine learning best practice]]

	It is possible to '''apply external machine-learned models''' during a {{VASP}} run using the Python-[[plugins]] feature. The {{TAG\|PLUGINS/MACHINE_LEARNING}} tag in combination with a ~~Phython~~ script offers a hook to directly supply a machine-learned potential or any other model expressed in terms of an [https://wiki.fysik.dtu.dk/ase/development/calculators.html ASE Calculator].		It is possible to '''apply external machine-learned models''' during a {{VASP}} run using the Python-[[plugins]] feature. The {{TAG\|PLUGINS/MACHINE_LEARNING}} tag in combination with a Python script offers a hook to directly supply a machine-learned potential or any other model expressed in terms of an [https://wiki.fysik.dtu.dk/ase/development/calculators.html ASE Calculator].

	== Applying external forces ==		== Applying external forces ==

Schlipf: /* Applying external forces */

2025-10-24T11:30:00Z

Applying external forces

← Older revision		Revision as of 11:30, 24 October 2025
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	== Applying external forces ==		== Applying external forces ==

	There are multiple ways to run simulations with effective forces acting on the ions. This includes [[selective dynamics]] defined in the {{FILE\|POSCAR}} file, the {{TAG\|LATTICE_CONSTRAINTS}} tag and the {{FILE\|ICONST}} file. To apply static driving forces in a closer sense, the {{TAG\|EFOR}} tag offers direct control. Additionally it is possible to apply dynamically changing forces and stress using the Python-[[plugins]] feature. In particular, the {{TAG\|PLUGINS/FORCE_AND_STRESS}} tag in combination with an ~~appropsiate~~ Python ~~scipt~~ yields direct control at each ionic step.		There are multiple ways to run simulations with effective forces acting on the ions. This includes [[selective dynamics]] defined in the {{FILE\|POSCAR}} file, the {{TAG\|LATTICE_CONSTRAINTS}} tag and the {{FILE\|ICONST}} file. To apply static driving forces in a closer sense, the {{TAG\|EFOR}} tag offers direct control. Additionally it is possible to apply dynamically changing forces and stress using the Python-[[plugins]] feature. In particular, the {{TAG\|PLUGINS/FORCE_AND_STRESS}} tag in combination with an appropriate Python script yields direct control at each ionic step.
	{{NB\|tip\| Depending on the method, an overall '''drift''' may be removed from the forces. See [[#DFT forces]] For instance, for [[structure optimization]] and for the [[Nosé-Hoover thermostat]] drifts are removed, but not for the stochastic [[Langevin thermostat]].}}		{{NB\|tip\| Depending on the method, an overall '''drift''' may be removed from the forces. See [[#DFT forces]] For instance, for [[structure optimization]] and for the [[Nosé-Hoover thermostat]] drifts are removed, but not for the stochastic [[Langevin thermostat]].}}

Huebsch at 09:48, 24 October 2025

2025-10-24T09:48:45Z

← Older revision		Revision as of 09:48, 24 October 2025
Line 85:		Line 85:
	*[[Machine learning force field calculations: Basics\|Machine learning Basics]]		*[[Machine learning force field calculations: Basics\|Machine learning Basics]]
	*[[Best practices for machine-learned force fields\|Machine learning best practice]]		*[[Best practices for machine-learned force fields\|Machine learning best practice]]

			It is possible to '''apply external machine-learned models''' during a {{VASP}} run using the Python-[[plugins]] feature. The {{TAG\|PLUGINS/MACHINE_LEARNING}} tag in combination with a Phython script offers a hook to directly supply a machine-learned potential or any other model expressed in terms of an [https://wiki.fysik.dtu.dk/ase/development/calculators.html ASE Calculator].

	== Applying external forces ==		== Applying external forces ==

	There are multiple ways to run simulations with effective forces acting on the ions. This includes [[selective dynamics]] defined in the {{FILE\|POSCAR}} file, the {{TAG\|LATTICE_CONSTRAINTS}} tag and the {{FILE\|ICONST}} file. To apply driving forces in a closer sense, the {{TAG\|EFOR}} tag offers direct control.		There are multiple ways to run simulations with effective forces acting on the ions. This includes [[selective dynamics]] defined in the {{FILE\|POSCAR}} file, the {{TAG\|LATTICE_CONSTRAINTS}} tag and the {{FILE\|ICONST}} file. To apply static driving forces in a closer sense, the {{TAG\|EFOR}} tag offers direct control. Additionally it is possible to apply dynamically changing forces and stress using the Python-[[plugins]] feature. In particular, the {{TAG\|PLUGINS/FORCE_AND_STRESS}} tag in combination with an appropsiate Python scipt yields direct control at each ionic step.
	{{NB\|tip\| Depending on the method, an overall '''drift''' may be removed from the forces. See [[#DFT forces]] For instance, for [[structure optimization]] and for the [[Nosé-Hoover thermostat]] drifts are removed, but not for the stochastic [[Langevin thermostat]].}}		{{NB\|tip\| Depending on the method, an overall '''drift''' may be removed from the forces. See [[#DFT forces]] For instance, for [[structure optimization]] and for the [[Nosé-Hoover thermostat]] drifts are removed, but not for the stochastic [[Langevin thermostat]].}}

	~~== Controlling forces ==~~

	It is possible to directly manipulate the computed forces during a {{VASP}} run using the Python [[plugins]]. In particular, the {{TAG\|PLUGINS/FORCE_AND_STRESS}} tag and the {{TAG\|PLUGINS/MACHINE_LEARNING}} tag offer a hook to control forces.

	== Related concepts ==		== Related concepts ==
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	</math>		</math>

	and is, therefore, the gradient of the force. The force-constant matrix is a fundamental concept in solid-state physics and materials science, especially in the context of understanding the vibrational properties of crystals, i.e., [[phonons]]. It is a ~~mathematical representation~~ of the ~~interatomic forces and their interactions within a crystal lattice~~. This matrix is used to describe the relationships between atomic displacements and the resulting forces that occur in a crystal. By Fourier transforming the force-constant matrix, the [[Phonons: Theory\|dynamical matrix]] is obtained.		and is, therefore, the gradient of the force. The force-constant matrix is a fundamental concept in solid-state physics and materials science, especially in the context of understanding the vibrational properties of crystals, i.e., [[phonons]]. It is a generalization of the spring constant $k$ inside ($\mathbf{F}=k\mathbf{x}$) for the case of 3D crystals. This matrix is used to describe the relationships between atomic displacements and the resulting forces that occur in a crystal. By Fourier transforming the force-constant matrix, the [[Phonons: Theory\|dynamical matrix]] is obtained.

	By computing the eigenvalues of the dynamical matrix on various reciprocal lattice points, the [[Computing the phonon dispersion and DOS\|phonon-dispersion relation]] can be obtained.		By computing the eigenvalues of the dynamical matrix on various reciprocal lattice points, the [[Computing the phonon dispersion and DOS\|phonon-dispersion relation]] can be obtained.

Schlipf: /* Controling forces */

2025-10-24T09:19:46Z

Controling forces

← Older revision		Revision as of 09:19, 24 October 2025
Line 91:		Line 91:
	{{NB\|tip\| Depending on the method, an overall '''drift''' may be removed from the forces. See [[#DFT forces]] For instance, for [[structure optimization]] and for the [[Nosé-Hoover thermostat]] drifts are removed, but not for the stochastic [[Langevin thermostat]].}}		{{NB\|tip\| Depending on the method, an overall '''drift''' may be removed from the forces. See [[#DFT forces]] For instance, for [[structure optimization]] and for the [[Nosé-Hoover thermostat]] drifts are removed, but not for the stochastic [[Langevin thermostat]].}}

	== ~~Controling~~ forces ==		== Controlling forces ==

	It is possible to directly manipulate the computed forces during a {{VASP}} run using the Python [[plugins]]. In particular, the {{TAG\|PLUGINS/FORCE_AND_STRESS}} tag and the {{TAG\|PLUGINS/MACHINE_LEARNING}} tag offer a hook to control forces.		It is possible to directly manipulate the computed forces during a {{VASP}} run using the Python [[plugins]]. In particular, the {{TAG\|PLUGINS/FORCE_AND_STRESS}} tag and the {{TAG\|PLUGINS/MACHINE_LEARNING}} tag offer a hook to control forces.

Csheldon: /* DFT forces */

2025-10-22T10:15:27Z

DFT forces

← Older revision		Revision as of 10:15, 22 October 2025
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	where <math> \nabla_{A} </math> denotes the gradient with respect to ionic position <math>\mathbf{R}_{A}</math>. The DFT forces will depend on the chosen [[:Category:Exchange-correlation functionals\|exchange-correlation functional]] via the electronic ground-state density <math> n({\bf r}) </math>. Therefore, the choice of the proper exchange-correlation functional for the system of interest is crucial for obtaining proper forces and, hence, the correct material properties.		where <math> \nabla_{A} </math> denotes the gradient with respect to ionic position <math>\mathbf{R}_{A}</math>. The DFT forces will depend on the chosen [[:Category:Exchange-correlation functionals\|exchange-correlation functional]] via the electronic ground-state density <math> n({\bf r}) </math>. Therefore, the choice of the proper exchange-correlation functional for the system of interest is crucial for obtaining proper forces and, hence, the correct material properties.
	{{NB\|important\|The force '''drift''' is printed to the {{FILE\|OUTCAR}} after <code>total drift:</code>. The forces are printed with the force drift removed for static calculations and for the Nose-Hoover thermostat (MD), while for other MD thermostats it is not removed.}}		<!--{{NB\|important\|The force '''drift''' is printed to the {{FILE\|OUTCAR}} after <code>total drift:</code>. The forces are printed with the force drift removed for static calculations and for the Nose-Hoover thermostat (MD), while for other MD thermostats it is not removed.}}-->

	== RPA forces ==		== RPA forces ==

Csheldon: /* DFT forces */

2025-10-22T09:51:53Z

DFT forces

← Older revision		Revision as of 09:51, 22 October 2025
Line 39:		Line 39:

	where <math> \nabla_{A} </math> denotes the gradient with respect to ionic position <math>\mathbf{R}_{A}</math>. The DFT forces will depend on the chosen [[:Category:Exchange-correlation functionals\|exchange-correlation functional]] via the electronic ground-state density <math> n({\bf r}) </math>. Therefore, the choice of the proper exchange-correlation functional for the system of interest is crucial for obtaining proper forces and, hence, the correct material properties.		where <math> \nabla_{A} </math> denotes the gradient with respect to ionic position <math>\mathbf{R}_{A}</math>. The DFT forces will depend on the chosen [[:Category:Exchange-correlation functionals\|exchange-correlation functional]] via the electronic ground-state density <math> n({\bf r}) </math>. Therefore, the choice of the proper exchange-correlation functional for the system of interest is crucial for obtaining proper forces and, hence, the correct material properties.
	{{NB\|important\|The force '''drift''' is printed to the {{FILE\|OUTCAR}} after <code>total drift:</code>. The forces are printed with the force drift removed for static calculations, while, for MD, it is not removed.}}		{{NB\|important\|The force '''drift''' is printed to the {{FILE\|OUTCAR}} after <code>total drift:</code>. The forces are printed with the force drift removed for static calculations and for the Nose-Hoover thermostat (MD), while for other MD thermostats it is not removed.}}

	== RPA forces ==		== RPA forces ==

Huebsch at 08:17, 21 October 2025

2025-10-21T08:17:36Z

Show changes

Csheldon: /* Force constants and phonons */

2025-09-22T12:01:30Z

Force constants and phonons

← Older revision		Revision as of 12:01, 22 September 2025
Line 127:		Line 127:
	</math>		</math>

	and is, therefore, the gradient of the force. The force-constant matrix is a fundamental concept in solid-state physics and materials science, especially in the context of understanding the vibrational properties of crystals. It is a mathematical representation of the interatomic forces and their interactions within a crystal lattice. This matrix is used to describe the relationships between atomic displacements and the resulting forces that occur in a crystal. By Fourier transforming the force-constant matrix, the [[Phonons: Theory\|dynamical matrix]] is obtained. By computing the eigenvalues of the dynamic matrix on various reciprocal lattice points, the phonon dispersion relation can be obtained.		and is, therefore, the gradient of the force. The force-constant matrix is a fundamental concept in solid-state physics and materials science, especially in the context of understanding the vibrational properties of crystals, i.e., [[:Category:Phonons \| phonons]]. It is a mathematical representation of the interatomic forces and their interactions within a crystal lattice. This matrix is used to describe the relationships between atomic displacements and the resulting forces that occur in a crystal. By Fourier transforming the force-constant matrix, the [[Phonons: Theory\|dynamical matrix]] is obtained. By computing the eigenvalues of the dynamic matrix on various reciprocal lattice points, the phonon dispersion relation can be obtained.
	Understanding phonons is essential as they influence materials properties such as the electrical conductivity, thermal conductivity, and mechanical properties of materials.		Understanding phonons is essential as they influence materials properties such as the electrical conductivity, thermal conductivity, and mechanical properties of materials.

← Older revision		Revision as of 09:48, 24 October 2025
Line 85:		Line 85:
	*[[Machine learning force field calculations: Basics\|Machine learning Basics]]		*[[Machine learning force field calculations: Basics\|Machine learning Basics]]
	*[[Best practices for machine-learned force fields\|Machine learning best practice]]		*[[Best practices for machine-learned force fields\|Machine learning best practice]]

			It is possible to '''apply external machine-learned models''' during a {{VASP}} run using the Python-[[plugins]] feature. The {{TAG\|PLUGINS/MACHINE_LEARNING}} tag in combination with a Phython script offers a hook to directly supply a machine-learned potential or any other model expressed in terms of an [https://wiki.fysik.dtu.dk/ase/development/calculators.html ASE Calculator].

	== Applying external forces ==		== Applying external forces ==

	There are multiple ways to run simulations with effective forces acting on the ions. This includes [[selective dynamics]] defined in the {{FILE\|POSCAR}} file, the {{TAG\|LATTICE_CONSTRAINTS}} tag and the {{FILE\|ICONST}} file. To apply driving forces in a closer sense, the {{TAG\|EFOR}} tag offers direct control.		There are multiple ways to run simulations with effective forces acting on the ions. This includes [[selective dynamics]] defined in the {{FILE\|POSCAR}} file, the {{TAG\|LATTICE_CONSTRAINTS}} tag and the {{FILE\|ICONST}} file. To apply static driving forces in a closer sense, the {{TAG\|EFOR}} tag offers direct control. Additionally it is possible to apply dynamically changing forces and stress using the Python-[[plugins]] feature. In particular, the {{TAG\|PLUGINS/FORCE_AND_STRESS}} tag in combination with an appropsiate Python scipt yields direct control at each ionic step.
	{{NB\|tip\| Depending on the method, an overall '''drift''' may be removed from the forces. See [[#DFT forces]] For instance, for [[structure optimization]] and for the [[Nosé-Hoover thermostat]] drifts are removed, but not for the stochastic [[Langevin thermostat]].}}		{{NB\|tip\| Depending on the method, an overall '''drift''' may be removed from the forces. See [[#DFT forces]] For instance, for [[structure optimization]] and for the [[Nosé-Hoover thermostat]] drifts are removed, but not for the stochastic [[Langevin thermostat]].}}

	~~== Controlling forces ==~~

	It is possible to directly manipulate the computed forces during a {{VASP}} run using the Python [[plugins]]. In particular, the {{TAG\|PLUGINS/FORCE_AND_STRESS}} tag and the {{TAG\|PLUGINS/MACHINE_LEARNING}} tag offer a hook to control forces.

	== Related concepts ==		== Related concepts ==
Line 124:		Line 122:
	</math>		</math>

	and is, therefore, the gradient of the force. The force-constant matrix is a fundamental concept in solid-state physics and materials science, especially in the context of understanding the vibrational properties of crystals, i.e., [[phonons]]. It is a ~~mathematical representation~~ of the ~~interatomic forces and their interactions within a crystal lattice~~. This matrix is used to describe the relationships between atomic displacements and the resulting forces that occur in a crystal. By Fourier transforming the force-constant matrix, the [[Phonons: Theory\|dynamical matrix]] is obtained.		and is, therefore, the gradient of the force. The force-constant matrix is a fundamental concept in solid-state physics and materials science, especially in the context of understanding the vibrational properties of crystals, i.e., [[phonons]]. It is a generalization of the spring constant $k$ inside ($\mathbf{F}=k\mathbf{x}$) for the case of 3D crystals. This matrix is used to describe the relationships between atomic displacements and the resulting forces that occur in a crystal. By Fourier transforming the force-constant matrix, the [[Phonons: Theory\|dynamical matrix]] is obtained.

	By computing the eigenvalues of the dynamical matrix on various reciprocal lattice points, the [[Computing the phonon dispersion and DOS\|phonon-dispersion relation]] can be obtained.		By computing the eigenvalues of the dynamical matrix on various reciprocal lattice points, the [[Computing the phonon dispersion and DOS\|phonon-dispersion relation]] can be obtained.