b__1]()" }, [ "article:topic", "license:ccbysa", "showtoc:no", "density of states" ], https://eng.libretexts.org/@app/auth/3/login?returnto=https%3A%2F%2Feng.libretexts.org%2FBookshelves%2FMaterials_Science%2FSupplemental_Modules_(Materials_Science)%2FElectronic_Properties%2FDensity_of_States, \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}}}\) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\), \[ \nu_s = \sqrt{\dfrac{Y}{\rho}}\nonumber\], \[ g(\omega)= \dfrac{L^2}{\pi} \dfrac{\omega}{{\nu_s}^2}\nonumber\], \[ g(\omega) = 3 \dfrac{V}{2\pi^2} \dfrac{\omega^2}{\nu_s^3}\nonumber\], (Bookshelves/Materials_Science/Supplemental_Modules_(Materials_Science)/Electronic_Properties/Density_of_States), /content/body/div[3]/p[27]/span, line 1, column 3, http://britneyspears.ac/physics/dos/dos.htm, status page at https://status.libretexts.org. [4], Including the prefactor 0000003837 00000 n
In materials science, for example, this term is useful when interpreting the data from a scanning tunneling microscope (STM), since this method is capable of imaging electron densities of states with atomic resolution. So now we will use the solution: To begin, we must apply some type of boundary conditions to the system. 0000073571 00000 n
The density of states related to volume V and N countable energy levels is defined as: Because the smallest allowed change of momentum / The smallest reciprocal area (in k-space) occupied by one single state is: states per unit energy range per unit area and is usually defined as, Area However I am unsure why for 1D it is $2dk$ as opposed to $2 \pi dk$. In equation(1), the temporal factor, \(-\omega t\) can be omitted because it is not relevant to the derivation of the DOS\(^{[2]}\). In general, the topological properties of the system such as the band structure, have a major impact on the properties of the density of states. %PDF-1.5
%
k 8 3 Density of States (online) www.ecse.rpi.edu/~schubert/Course-ECSE-6968%20Quantum%20mechanics/Ch12%20Density%20of%20states.pdf. E LDOS can be used to gain profit into a solid-state device. Hence the differential hyper-volume in 1-dim is 2*dk. rev2023.3.3.43278. Density of States in 3D The values of k x k y k z are equally spaced: k x = 2/L ,. 2 / Can archive.org's Wayback Machine ignore some query terms? k %W(X=5QOsb]Jqeg+%'$_-7h>@PMJ!LnVSsR__zGSn{$\":U71AdS7a@xg,IL}nd:P'zi2b}zTpI_DCE2V0I`tFzTPNb*WHU>cKQS)f@t
,XM"{V~{6ICg}Ke~` To derive this equation we can consider that the next band is \(Eg\) ev below the minimum of the first band\(^{[1]}\). Here, {\displaystyle \mu } If you choose integer values for \(n\) and plot them along an axis \(q\) you get a 1-D line of points, known as modes, with a spacing of \({2\pi}/{L}\) between each mode. E+dE. ( the energy-gap is reached, there is a significant number of available states. a 0
( this relation can be transformed to, The two examples mentioned here can be expressed like. x 7. states up to Fermi-level. {\displaystyle D(E)} 0000061387 00000 n
This value is widely used to investigate various physical properties of matter. 2D Density of States Each allowable wavevector (mode) occupies a region of area (2/L)2 Thus, within the circle of radius K, there are approximately K2/ (2/L)2 allowed wavevectors Density of states calculated for homework K-space /a 2/L K. ME 595M, T.S. f , xref
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According to crystal structure, this quantity can be predicted by computational methods, as for example with density functional theory. for 2-D we would consider an area element in \(k\)-space \((k_x, k_y)\), and for 1-D a line element in \(k\)-space \((k_x)\). 1708 0 obj
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"f3Lr(P8u. 0 Wenlei Luo a, Yitian Jiang b, Mengwei Wang b, Dan Lu b, Xiaohui Sun b and Huahui Zhang * b a National Innovation Institute of Defense Technology, Academy of Military Science, Beijing 100071, China b State Key Laboratory of Space Power-sources Technology, Shanghai Institute of Space Power-Sources . {\displaystyle N} 0000001670 00000 n
which leads to \(\dfrac{dk}{dE}={(\dfrac{2 m^{\ast}E}{\hbar^2})}^{-1/2}\dfrac{m^{\ast}}{\hbar^2}\) now substitute the expressions obtained for \(dk\) and \(k^2\) in terms of \(E\) back into the expression for the number of states: \(\Rightarrow\frac{1}{{(2\pi)}^3}4\pi{(\dfrac{2 m^{\ast}}{\hbar^2})}^2{(\dfrac{2 m^{\ast}}{\hbar^2})}^{-1/2})E(E^{-1/2})dE\), \(\Rightarrow\frac{1}{{(2\pi)}^3}4\pi{(\dfrac{2 m^{\ast}E}{\hbar^2})}^{3/2})E^{1/2}dE\). and after applying the same boundary conditions used earlier: \[e^{i[k_xx+k_yy+k_zz]}=1 \Rightarrow (k_x,k_y,k_z)=(n_x \frac{2\pi}{L}, n_y \frac{2\pi}{L}), n_z \frac{2\pi}{L})\nonumber\]. 5.1.2 The Density of States. E Omar, Ali M., Elementary Solid State Physics, (Pearson Education, 1999), pp68- 75;213-215. 1 In general it is easier to calculate a DOS when the symmetry of the system is higher and the number of topological dimensions of the dispersion relation is lower. 2 where {\displaystyle d} 0000064674 00000 n
1 2 Browse other questions tagged, Start here for a quick overview of the site, Detailed answers to any questions you might have, Discuss the workings and policies of this site. 0000004841 00000 n
The density of states (DOS) is essentially the number of different states at a particular energy level that electrons are allowed to occupy, i.e. The density of states is a central concept in the development and application of RRKM theory. Making statements based on opinion; back them up with references or personal experience. Even less familiar are carbon nanotubes, the quantum wire and Luttinger liquid with their 1-dimensional topologies. is In a system described by three orthogonal parameters (3 Dimension), the units of DOS is Energy1Volume1 , in a two dimensional system, the units of DOS is Energy1Area1 , in a one dimensional system, the units of DOS is Energy1Length1. and finally, for the plasmonic disorder, this effect is much stronger for LDOS fluctuations as it can be observed as a strong near-field localization.[18]. As soon as each bin in the histogram is visited a certain number of times . with respect to k, expressed by, The 1, 2 and 3-dimensional density of wave vector states for a line, disk, or sphere are explicitly written as. d {\displaystyle s/V_{k}} 0000074349 00000 n
In photonic crystals, the near-zero LDOS are expected and they cause inhibition in the spontaneous emission. 0000015987 00000 n
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In addition, the relationship with the mean free path of the scattering is trivial as the LDOS can be still strongly influenced by the short details of strong disorders in the form of a strong Purcell enhancement of the emission. In 2D materials, the electron motion is confined along one direction and free to move in other two directions. This configuration means that the integration over the whole domain of the Brillouin zone can be reduced to a 48-th part of the whole Brillouin zone. How to match a specific column position till the end of line? {\displaystyle E>E_{0}} 2 is due to the area of a sphere in k -space being proportional to its squared radius k 2 and by having a linear dispersion relation = v s k. v s 3 is from the linear dispersion relation = v s k. 2 other for spin down. E 0
d Now we can derive the density of states in this region in the same way that we did for the rest of the band and get the result: \[ g(E) = \dfrac{1}{2\pi^2}\left( \dfrac{2|m^{\ast}|}{\hbar^2} \right)^{3/2} (E_g-E)^{1/2}\nonumber\]. In 2D, the density of states is constant with energy. This expression is a kind of dispersion relation because it interrelates two wave properties and it is isotropic because only the length and not the direction of the wave vector appears in the expression. If the dispersion relation is not spherically symmetric or continuously rising and can't be inverted easily then in most cases the DOS has to be calculated numerically. E In solid state physics and condensed matter physics, the density of states (DOS) of a system describes the number of modes per unit frequency range. Deriving density of states in different dimensions in k space, We've added a "Necessary cookies only" option to the cookie consent popup, Heat capacity in general $d$ dimensions given the density of states $D(\omega)$. Lowering the Fermi energy corresponds to \hole doping" [15] 0000004596 00000 n
This procedure is done by differentiating the whole k-space volume E 2. In other words, there are (2 2 ) / 2 1 L, states per unit area of 2D k space, for each polarization (each branch). For small values of Equation (2) becomes: u = Ai ( qxx + qyy) now apply the same boundary conditions as in the 1-D case: alone. Figure 1. 0000001022 00000 n
E k {\displaystyle q=k-\pi /a} k {\displaystyle a} / k-space divided by the volume occupied per point. For a one-dimensional system with a wall, the sine waves give. hbbd```b`` qd=fH
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the Particle in a box problem, gives rise to standing waves for which the allowed values of \(k\) are expressible in terms of three nonzero integers, \(n_x,n_y,n_z\)\(^{[1]}\). g ( E)2Dbecomes: As stated initially for the electron mass, m m*. Use MathJax to format equations. New York: Oxford, 2005. The density of states is defined by (2 ) / 2 2 (2 ) / ( ) 2 2 2 2 2 Lkdk L kdk L dkdk D d x y , using the linear dispersion relation, vk, 2 2 2 ( ) v L D , which is proportional to . the expression is, In fact, we can generalise the local density of states further to. It has written 1/8 th here since it already has somewhere included the contribution of Pi. k. points is thus the number of states in a band is: L. 2 a L. N 2 =2 2 # of unit cells in the crystal . Such periodic structures are known as photonic crystals. }.$aoL)}kSo@3hEgg/>}ze_g7mc/g/}?/o>o^r~k8vo._?|{M-cSh~8Ssc>]c\5"lBos.Y'f2,iSl1mI~&8:xM``kT8^u&&cZgNA)u s&=F^1e!,N1f#pV}~aQ5eE"_\T6wBj kKB1$hcQmK!\W%aBtQY0gsp],Eo {\displaystyle T} 0000070813 00000 n
{\displaystyle E} 172 0 obj
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0000004890 00000 n
The number of states in the circle is N(k') = (A/4)/(/L) . 0000068391 00000 n
{\displaystyle k\approx \pi /a} L includes the 2-fold spin degeneracy. {\displaystyle D(E)=N(E)/V} In a three-dimensional system with Remember (E)dE is defined as the number of energy levels per unit volume between E and E + dE. {\displaystyle x} 0000067158 00000 n
The density of states is dependent upon the dimensional limits of the object itself. E (14) becomes. 0000004116 00000 n
4dYs}Zbw,haq3r0x m g E D = It is significant that the 2D density of states does not . we insert 20 of vacuum in the unit cell. 0000072014 00000 n
On this Wikipedia the language links are at the top of the page across from the article title. D The density of states is once again represented by a function \(g(E)\) which this time is a function of energy and has the relation \(g(E)dE\) = the number of states per unit volume in the energy range: \((E, E+dE)\). Asking for help, clarification, or responding to other answers. < ) with respect to the energy: The number of states with energy Spherical shell showing values of \(k\) as points. M)cw 0000000866 00000 n
0000064265 00000 n
0000003439 00000 n
Thermal Physics. (3) becomes. the energy is, With the transformation It is mathematically represented as a distribution by a probability density function, and it is generally an average over the space and time domains of the various states occupied by the system. Used Pacific Skiff Boats For Sale,
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0000043342 00000 n
of this expression will restore the usual formula for a DOS. 0000066340 00000 n
Learn more about Stack Overflow the company, and our products. 0000033118 00000 n
inside an interval D = In 1-dimensional systems the DOS diverges at the bottom of the band as for linear, disk and spherical symmetrical shaped functions in 1, 2 and 3-dimensional Euclidean k-spaces respectively. 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"showtoc:no", "density of states" ], https://eng.libretexts.org/@app/auth/3/login?returnto=https%3A%2F%2Feng.libretexts.org%2FBookshelves%2FMaterials_Science%2FSupplemental_Modules_(Materials_Science)%2FElectronic_Properties%2FDensity_of_States, \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}}}\) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( 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[4], Including the prefactor 0000003837 00000 n
In materials science, for example, this term is useful when interpreting the data from a scanning tunneling microscope (STM), since this method is capable of imaging electron densities of states with atomic resolution. So now we will use the solution: To begin, we must apply some type of boundary conditions to the system. 0000073571 00000 n
The density of states related to volume V and N countable energy levels is defined as: Because the smallest allowed change of momentum / The smallest reciprocal area (in k-space) occupied by one single state is: states per unit energy range per unit area and is usually defined as, Area However I am unsure why for 1D it is $2dk$ as opposed to $2 \pi dk$. In equation(1), the temporal factor, \(-\omega t\) can be omitted because it is not relevant to the derivation of the DOS\(^{[2]}\). In general, the topological properties of the system such as the band structure, have a major impact on the properties of the density of states. %PDF-1.5
%
k 8 3 Density of States (online) www.ecse.rpi.edu/~schubert/Course-ECSE-6968%20Quantum%20mechanics/Ch12%20Density%20of%20states.pdf. E LDOS can be used to gain profit into a solid-state device. Hence the differential hyper-volume in 1-dim is 2*dk. rev2023.3.3.43278. Density of States in 3D The values of k x k y k z are equally spaced: k x = 2/L ,. 2 / Can archive.org's Wayback Machine ignore some query terms? k %W(X=5QOsb]Jqeg+%'$_-7h>@PMJ!LnVSsR__zGSn{$\":U71AdS7a@xg,IL}nd:P'zi2b}zTpI_DCE2V0I`tFzTPNb*WHU>cKQS)f@t
,XM"{V~{6ICg}Ke~` To derive this equation we can consider that the next band is \(Eg\) ev below the minimum of the first band\(^{[1]}\). Here, {\displaystyle \mu } If you choose integer values for \(n\) and plot them along an axis \(q\) you get a 1-D line of points, known as modes, with a spacing of \({2\pi}/{L}\) between each mode. E+dE. ( the energy-gap is reached, there is a significant number of available states. a 0
( this relation can be transformed to, The two examples mentioned here can be expressed like. x 7. states up to Fermi-level. {\displaystyle D(E)} 0000061387 00000 n
This value is widely used to investigate various physical properties of matter. 2D Density of States Each allowable wavevector (mode) occupies a region of area (2/L)2 Thus, within the circle of radius K, there are approximately K2/ (2/L)2 allowed wavevectors Density of states calculated for homework K-space /a 2/L K. ME 595M, T.S. f , xref
(that is, the total number of states with energy less than 54 0 obj
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According to crystal structure, this quantity can be predicted by computational methods, as for example with density functional theory. for 2-D we would consider an area element in \(k\)-space \((k_x, k_y)\), and for 1-D a line element in \(k\)-space \((k_x)\). 1708 0 obj
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"f3Lr(P8u. 0 Wenlei Luo a, Yitian Jiang b, Mengwei Wang b, Dan Lu b, Xiaohui Sun b and Huahui Zhang * b a National Innovation Institute of Defense Technology, Academy of Military Science, Beijing 100071, China b State Key Laboratory of Space Power-sources Technology, Shanghai Institute of Space Power-Sources . {\displaystyle N} 0000001670 00000 n
which leads to \(\dfrac{dk}{dE}={(\dfrac{2 m^{\ast}E}{\hbar^2})}^{-1/2}\dfrac{m^{\ast}}{\hbar^2}\) now substitute the expressions obtained for \(dk\) and \(k^2\) in terms of \(E\) back into the expression for the number of states: \(\Rightarrow\frac{1}{{(2\pi)}^3}4\pi{(\dfrac{2 m^{\ast}}{\hbar^2})}^2{(\dfrac{2 m^{\ast}}{\hbar^2})}^{-1/2})E(E^{-1/2})dE\), \(\Rightarrow\frac{1}{{(2\pi)}^3}4\pi{(\dfrac{2 m^{\ast}E}{\hbar^2})}^{3/2})E^{1/2}dE\). and after applying the same boundary conditions used earlier: \[e^{i[k_xx+k_yy+k_zz]}=1 \Rightarrow (k_x,k_y,k_z)=(n_x \frac{2\pi}{L}, n_y \frac{2\pi}{L}), n_z \frac{2\pi}{L})\nonumber\]. 5.1.2 The Density of States. E Omar, Ali M., Elementary Solid State Physics, (Pearson Education, 1999), pp68- 75;213-215. 1 In general it is easier to calculate a DOS when the symmetry of the system is higher and the number of topological dimensions of the dispersion relation is lower. 2 where {\displaystyle d} 0000064674 00000 n
1 2 Browse other questions tagged, Start here for a quick overview of the site, Detailed answers to any questions you might have, Discuss the workings and policies of this site. 0000004841 00000 n
The density of states (DOS) is essentially the number of different states at a particular energy level that electrons are allowed to occupy, i.e. The density of states is a central concept in the development and application of RRKM theory. Making statements based on opinion; back them up with references or personal experience. Even less familiar are carbon nanotubes, the quantum wire and Luttinger liquid with their 1-dimensional topologies. is In a system described by three orthogonal parameters (3 Dimension), the units of DOS is Energy1Volume1 , in a two dimensional system, the units of DOS is Energy1Area1 , in a one dimensional system, the units of DOS is Energy1Length1. and finally, for the plasmonic disorder, this effect is much stronger for LDOS fluctuations as it can be observed as a strong near-field localization.[18]. As soon as each bin in the histogram is visited a certain number of times . with respect to k, expressed by, The 1, 2 and 3-dimensional density of wave vector states for a line, disk, or sphere are explicitly written as. d {\displaystyle s/V_{k}} 0000074349 00000 n
In photonic crystals, the near-zero LDOS are expected and they cause inhibition in the spontaneous emission. 0000015987 00000 n
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In addition, the relationship with the mean free path of the scattering is trivial as the LDOS can be still strongly influenced by the short details of strong disorders in the form of a strong Purcell enhancement of the emission. In 2D materials, the electron motion is confined along one direction and free to move in other two directions. This configuration means that the integration over the whole domain of the Brillouin zone can be reduced to a 48-th part of the whole Brillouin zone. How to match a specific column position till the end of line? {\displaystyle E>E_{0}} 2 is due to the area of a sphere in k -space being proportional to its squared radius k 2 and by having a linear dispersion relation = v s k. v s 3 is from the linear dispersion relation = v s k. 2 other for spin down. E 0
d Now we can derive the density of states in this region in the same way that we did for the rest of the band and get the result: \[ g(E) = \dfrac{1}{2\pi^2}\left( \dfrac{2|m^{\ast}|}{\hbar^2} \right)^{3/2} (E_g-E)^{1/2}\nonumber\]. In 2D, the density of states is constant with energy. This expression is a kind of dispersion relation because it interrelates two wave properties and it is isotropic because only the length and not the direction of the wave vector appears in the expression. If the dispersion relation is not spherically symmetric or continuously rising and can't be inverted easily then in most cases the DOS has to be calculated numerically. E In solid state physics and condensed matter physics, the density of states (DOS) of a system describes the number of modes per unit frequency range. Deriving density of states in different dimensions in k space, We've added a "Necessary cookies only" option to the cookie consent popup, Heat capacity in general $d$ dimensions given the density of states $D(\omega)$. Lowering the Fermi energy corresponds to \hole doping" [15] 0000004596 00000 n
This procedure is done by differentiating the whole k-space volume E 2. In other words, there are (2 2 ) / 2 1 L, states per unit area of 2D k space, for each polarization (each branch). For small values of Equation (2) becomes: u = Ai ( qxx + qyy) now apply the same boundary conditions as in the 1-D case: alone. Figure 1. 0000001022 00000 n
E k {\displaystyle q=k-\pi /a} k {\displaystyle a} / k-space divided by the volume occupied per point. For a one-dimensional system with a wall, the sine waves give. hbbd```b`` qd=fH
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the Particle in a box problem, gives rise to standing waves for which the allowed values of \(k\) are expressible in terms of three nonzero integers, \(n_x,n_y,n_z\)\(^{[1]}\). g ( E)2Dbecomes: As stated initially for the electron mass, m m*. Use MathJax to format equations. New York: Oxford, 2005. The density of states is defined by (2 ) / 2 2 (2 ) / ( ) 2 2 2 2 2 Lkdk L kdk L dkdk D d x y , using the linear dispersion relation, vk, 2 2 2 ( ) v L D , which is proportional to . the expression is, In fact, we can generalise the local density of states further to. It has written 1/8 th here since it already has somewhere included the contribution of Pi. k. points is thus the number of states in a band is: L. 2 a L. N 2 =2 2 # of unit cells in the crystal . Such periodic structures are known as photonic crystals. }.$aoL)}kSo@3hEgg/>}ze_g7mc/g/}?/o>o^r~k8vo._?|{M-cSh~8Ssc>]c\5"lBos.Y'f2,iSl1mI~&8:xM``kT8^u&&cZgNA)u s&=F^1e!,N1f#pV}~aQ5eE"_\T6wBj kKB1$hcQmK!\W%aBtQY0gsp],Eo {\displaystyle T} 0000070813 00000 n
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The number of states in the circle is N(k') = (A/4)/(/L) . 0000068391 00000 n
{\displaystyle k\approx \pi /a} L includes the 2-fold spin degeneracy. {\displaystyle D(E)=N(E)/V} In a three-dimensional system with Remember (E)dE is defined as the number of energy levels per unit volume between E and E + dE. {\displaystyle x} 0000067158 00000 n
The density of states is dependent upon the dimensional limits of the object itself. E (14) becomes. 0000004116 00000 n
4dYs}Zbw,haq3r0x m g E D = It is significant that the 2D density of states does not . we insert 20 of vacuum in the unit cell. 0000072014 00000 n
On this Wikipedia the language links are at the top of the page across from the article title. D The density of states is once again represented by a function \(g(E)\) which this time is a function of energy and has the relation \(g(E)dE\) = the number of states per unit volume in the energy range: \((E, E+dE)\). Asking for help, clarification, or responding to other answers. < ) with respect to the energy: The number of states with energy Spherical shell showing values of \(k\) as points. M)cw 0000000866 00000 n
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Thermal Physics. (3) becomes. the energy is, With the transformation It is mathematically represented as a distribution by a probability density function, and it is generally an average over the space and time domains of the various states occupied by the system.
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