Materials Science and Engineering/Equations/Kinetics

Mathematical Foundation

Time-Dependent Field

 ${\frac {dc}{dt}}=\nabla c\cdot {\overrightarrow {v}}+{\frac {\partial c}{\partial t}}$

${\overrightarrow {v}}({\overrightarrow {r}})$: Velocity

$c({\overrightarrow {r}},t)$: Time-Dependent Field

Accumulation

Rate of accumulation is the negative of the divergence of the flux of the quantity plus the rate of production

 ${\frac {\partial c_{i}}{\partial t}}=-\nabla \cdot {\overrightarrow {J_{i}}}+{\dot {\rho _{i}}}$

${\dot {\rho _{i}}}({\overrightarrow {r}})$: Rate of production of the density of $i$ in $\Delta V$

$\nabla \cdot {\overrightarrow {J_{i}}}$: The divergence of ${\overrightarrow {J_{i}}}$

 ${\dot {M_{i}}}=\int _{\Delta V}-\nabla \cdot {\overrightarrow {J_{i}}}+{\dot {\rho _{i}}}\,dx$

${\dot {M_{i}}}$: Rate at which $i$ flows through area $\Delta {\overrightarrow {A}}$

Divergence Theorem

 $\int _{\mathrm {B} (\Delta V)}{\overrightarrow {J}}\cdot {\dot {n}}\,dA=\int _{\Delta V}\nabla \cdot {\overrightarrow {J}}\,dV$

$\mathrm {B} (\Delta V)$: Oriented surface around a volume

General Set of Linear Equations

 ${\begin{alignedat}{7}M_{11}x_{1}&&\;+\;&&M_{12}x_{2}&&\;+\cdots +\;&&M_{1n}x_{n}&&\;=\;&&&y_{1}\\M_{21}x_{1}&&\;+\;&&M_{22}x_{2}&&\;+\cdots +\;&&M_{2n}x_{n}&&\;=\;&&&y_{2}\\\vdots \;\;\;&&&&\vdots \;\;\;&&&&\vdots \;\;\;&&&&&\;\vdots \\M_{m1}x_{1}&&\;+\;&&M_{m2}x_{2}&&\;+\cdots +\;&&M_{mn}x_{n}&&\;=\;&&&y_{m}\\\end{alignedat}}$

The vector equation is equivalent to a matrix equation of the form

 $M\mathbf {x} =\mathbf {y}$

where M is an m×n matrix, x is a column vector with n entries, and y is a column vector with m entries.

$M={\begin{bmatrix}M_{11}&M_{12}&\cdots &M_{1n}\\M_{21}&M_{22}&\cdots &M_{2n}\\\vdots &\vdots &\ddots &\vdots \\M_{m1}&M_{m2}&\cdots &M_{mn}\end{bmatrix}},\quad \mathbf {x} ={\begin{bmatrix}x_{1}\\x_{2}\\\vdots \\x_{n}\end{bmatrix}},\quad \mathbf {y} ={\begin{bmatrix}y_{1}\\y_{2}\\\vdots \\y_{m}\end{bmatrix}}$

Eigenvalue Equation

 ${\underline {M}}{\overrightarrow {e}}=\lambda {\overrightarrow {e}}$

${\underline {M}}$: $nxn$ square matrix or tensor

${\overrightarrow {e}}$: eigenvector (special vector)

$\lambda$: eigenvalue (special scalar multiplier)

Transformation of Rank-Two Tensor

 $\left[{\mbox{diagonalized matrix}}\right]=\left[{\mbox{eigenvector column matrix}}\right]^{-1}\left[{\mbox{square matrix}}\right]\left[{\mbox{eigenvector column matrix}}\right]$

Irreversible Thermodynamics

Differential Change in Entropy

 $TdS=du-\sum _{j}\Psi _{j}d\zeta _{j}\;$

$\sum _{j}\Psi _{j}d\zeta _{j}=-Pdv+\phi dq+\gamma dA+\mu _{1}dc_{1}+\cdots \;$

Entropy Production

 $T{\dot {\sigma }}=-{\frac {\overrightarrow {J_{Q}}}{T}}\cdot \nabla T-\sum _{j}{\overrightarrow {J_{i}}}\cdot \nabla \Psi _{j}$

${\dot {\sigma }}$: Rate of entropy-density creation

${\overrightarrow {J_{Q}}}$: Flux of heat

${\overrightarrow {J_{i}}}$: Conjugate force

$\nabla \Psi _{j}$: Conjugate flux

Empirical Force-Flux Law

Fourier's

 ${\overrightarrow {J_{Q}}}=-K\nabla T$

Modified Fick's

 ${\overrightarrow {J_{i}}}=-M_{i}c_{i}\nabla \mu _{i}$

Ohm's

 ${\overrightarrow {J_{q}}}=-\rho \nabla \phi$

Basic Postulate of Irreversible Thermodynamics

The local generation of entropy, ${\dot {\sigma }}$ is nonnegative

 ${\dot {\sigma }}={\frac {\partial s}{\partial t}}+\nabla \cdot {\overrightarrow {J_{Q}}}\geq 0$

Coupling Between Forces and Fluxes

 ${\begin{alignedat}{7}{\frac {\partial J_{Q}}{\partial F_{Q}}}F_{Q}&&\;+\;&&{\frac {\partial J_{Q}}{\partial F_{q}}}F_{q}&&\;+\cdots +\;&&{\frac {\partial J_{Q}}{\partial F_{N_{c}}}}F_{N_{c}}&&\;=\;&&&J_{Q}(F_{Q},F_{q},F_{1},F_{2},...,F_{N_{c}})\\{\frac {\partial J_{q}}{\partial F_{Q}}}F_{Q}&&\;+\;&&{\frac {\partial J_{q}}{\partial F_{q}}}F_{q}&&\;+\cdots +\;&&{\frac {\partial J_{q}}{\partial F_{N_{c}}}}F_{N_{c}}&&\;=\;&&&J_{q}(F_{Q},F_{q},F_{1},F_{2},...,F_{N_{c}})\\\vdots \;\;\;&&&&\vdots \;\;\;&&&&\vdots \;\;\;&&&&&\;\vdots \\{\frac {\partial J_{N_{c}}}{\partial F_{Q}}}F_{Q}&&\;+\;&&{\frac {\partial J_{N_{c}}}{\partial F_{q}}}F_{q}&&\;+\cdots +\;&&{\frac {\partial J_{N_{c}}}{\partial F_{N_{c}}}}F_{N_{c}}&&\;=\;&&&J_{N_{c}}(F_{Q},F_{q},F_{1},F_{2},...,F_{N_{c}})\\\end{alignedat}}$

Abbreviated form:

 $J_{\alpha }=\sum _{\beta }L_{\alpha \beta }F_{\beta }\;$

$L_{\alpha \beta }={\frac {\partial J_{\alpha }}{\partial F_{\beta }}}$

Force-Flux Relations with Constrained Extensive Quantities

 $TdS=du+dw-\sum _{i=1}^{N_{c}-1}\left(\mu _{i}-\mu _{N_{c}}\right)dc_{i}$

${\overrightarrow {F_{i}}}=-\nabla \left(\mu _{i}-\mu _{N_{c}}\right)$

Diffusion Potential

 ${\overrightarrow {F_{1}}}=-\nabla \Phi _{1}$

Onsager Symmetry Principle

 $L_{\alpha \beta }=L_{\beta \alpha }\;$

 ${\frac {\partial J_{\alpha }}{\partial F_{\beta }}}={\frac {\partial J_{\beta }}{\partial F_{\alpha }}}$

Driving Forces and Fluxes

Diffusion in Absence of Chemical Effects

• Components diffuse in chemically homogeneous material
• Diffusion measured with radioactive tracer
• Fick's law flux equation derived when self-diffusion occurs by the vacancy-exchange mechanism.
• The crystal is network-constrained
• There are three components:
  • Inert atoms
  • Radioactive atoms
  • Vacancies
• C-frame: single reference frame
• Vacancies assumed to be in equilibrium throughout
• Raoultian behavior

 $J_{^{*}1}^{C}=-kT\left[{\frac {L_{11}}{c_{1}}}-{\frac {L_{1^{*}1}}{c_{^{*}1}}}\right]{\frac {\partial c_{^{*}1}}{\partial x}}$

 $J_{^{*}1}^{C}=-^{*}D{\frac {\partial c_{^{*}1}}{\partial x}}$

Diffusion of i in Chemically Homogeneous Binary Solution

 $J_{^{*}1}^{C}=-kT\left[{\frac {L_{11}}{c_{1}}}-{\frac {L_{1^{*}1}}{c_{^{*}1}}}\right]{\frac {\partial c_{^{*}1}}{\partial x}}$
 $J_{^{*}1}^{C}=-^{*}D_{1}{\frac {\partial c_{^{*}1}}{\partial x}}$

Diffusion of Substitutional Particles in Concentration Gradient

• Constraint associated with vacancy mechanism: ${\overrightarrow {J_{1}^{c}}}+{\overrightarrow {J_{2}^{c}}}+{\overrightarrow {J_{v}^{c}}}=0$
  • Difference in fluxes of the two substitutional species requires net flux of vacancies.
• Gibbs-Duhem relation: $c_{1}{\frac {\partial \mu _{1}}{\partial x}}+c_{2}{\frac {\partial \mu _{2}}{\partial x}}+c_{v}{\frac {\partial \mu _{v}}{\partial x}}$
• Chemical potential gradients related to concentration gradients: $\mu _{i}=\mu _{i}^{\circ }+kT\ln(\gamma _{i}<\Omega >c_{i})$

Flux is proportional to the concentration gradient

 $J_{1}^{c}=-kT\left[{\frac {L_{11}}{c_{1}}}-{\frac {L_{12}}{c_{2}}}\right]\left[1+{\frac {\partial \ln \gamma _{1}}{\partial \ln c_{1}}}+{\frac {\partial \ln <\Omega >}{\partial \ln c_{1}}}\right]{\frac {\partial c_{1}}{\partial x}}$
 $J_{1}^{c}=-D_{1}{\frac {\partial c_{1}}{\partial x}}$

Assumptions that simplify $D_{1}\;$

• Concentration-independent average site volume $<\Omega >\;$
• The coupling (off-diagonal) terms, $L_{12}/c_{2}\;$ and $L_{1^{*}1}/c_{^{*}1}\;$, are small compared with the direct term $L_{11}/c_{2}\;$

 $D_{1}\approx \left[1+{\frac {\partial \ln \gamma _{1}}{\partial \ln c_{1}}}\right]{}^{*}D_{1}$

Diffusion in a Volume-Fixed (V-Frame)

• Velocity of a local C-frame with respect to the V-frame: velocity of any inert marker with respect to the V-frame
• Flux of 1 in the V-frame:

 $J_{1}^{v}=-[c_{1}\Omega _{1}D_{2}+c_{2}+\Omega _{2}D_{1}]{\frac {\partial c_{1}}{\partial x}}$

• The interdiffusivity, $c_{1}\Omega _{1}D_{2}+c_{2}+\Omega _{2}D_{1}\;$, can be simplified through $\Omega _{1}=\Omega _{2}=<\Omega >\;$
• The L-frame and the V-frame are the same

Diffusion of Interstitial Particles in Concentration Gradient

• ${\overrightarrow {J_{1}^{c}}}=L_{11}{\overrightarrow {F_{1}}}$
• ${\overrightarrow {J_{1}^{c}}}=-L_{11}\nabla \Phi _{1}$
• ${\overrightarrow {J_{1}^{c}}}=-L_{11}\nabla \mu _{1}$
• $\mu _{1}=\mu _{1}^{\circ }+kT\ln(K_{1}c_{1})$
• $\nabla \mu _{1}={\frac {kT}{c_{1}}}\nabla c_{1}$

 ${\overrightarrow {J_{1}^{c}}}=-L_{11}{\frac {kT}{c_{1}}}\nabla c_{1}$

• Evaluate $L_{11}$ by substitution of interstitial mobility, $M_{1}$
  • ${\overrightarrow {v_{1}^{c}}}=-M_{1}\nabla \mu _{1}$
  • ${\overrightarrow {v_{1}^{c}}}={\frac {-M_{1}kT}{c_{1}}}\nabla c_{1}$
  • ${\overrightarrow {J_{1}^{c}}}={\overrightarrow {v_{1}^{c}}}c_{1}$

 ${\overrightarrow {J_{1}^{c}}}=-M_{1}kT\nabla c_{1}$

Diffusion of Charged Ions in Ionic Conductors

• ${\overrightarrow {J_{1}}}=L_{11}{\overrightarrow {F_{1}}}$
• ${\overrightarrow {J_{1}}}=-L_{11}\nabla \Phi _{1}$
• ${\overrightarrow {J_{1}}}=-L_{11}\nabla (\mu _{1}+q_{1}\phi )$

 ${\overrightarrow {J_{1}}}=-D_{1}\nabla c_{1}-{\frac {D_{1}c_{1}q_{1}}{kT}}\nabla (\phi )$

• ${\overrightarrow {E}}=-\nabla \phi \;$: Electric field
• Absence of concentration gradient:
  • ${\overrightarrow {J_{q}}}=q_{1}{\overrightarrow {J_{1}}}$
  • ${\overrightarrow {J_{q}}}=-{\frac {D_{1}c_{1}q_{1}^{2}}{kT}}\nabla (\phi )$

• Electrical conductivity:
  • $\rho ={\frac {D_{1}c_{1}q_{1}^{2}}{kT}}$

Electromigration in Metals

• Two fluxes when electric field is applied to a dilute solution of interstitial atoms in metal
  • $J_{q}\;$: Flux of conjuction electrons
  • $J_{1}\;$: Flux of interstitials
• $F_{q}=E\;$
• $F_{q}=-\nabla \phi$

 ${\overrightarrow {J_{1}}}=-L_{11}\nabla \mu _{1}+L_{1q}{\overrightarrow {E}}$
 ${\overrightarrow {J_{1}}}=-D_{1}\left(\nabla c_{1}-{\frac {c_{1}\beta }{kT}}{\overrightarrow {E}}\right)$

Mass Diffusion in Thermal Gradient

• Interstitial flux with thermal gradient where both heat flow and mass diffusion of interstitial component occurs:

 ${\overrightarrow {J_{1}}}=-L_{11}\nabla \mu _{1}-{\frac {L_{1q}}{T}}\nabla T$
 ${\overrightarrow {J_{1}}}=-D_{1}\nabla c_{1}-{\frac {D_{1}c_{1}Q_{1}^{\mbox{trans}}}{kT^{2}}}\nabla T$

Mass Diffusion Driven by Capillarity

• The system consists of two network-constrained components:
  • Host atoms
  • Vacancies
• No mass flow within the crystal (the crystal C-frame is also the V-frame)
• Constant temperature and no electric field
• ${\overrightarrow {J_{A}}}=L_{AA}{\overrightarrow {F_{A}}}$
• ${\overrightarrow {J_{A}}}=-L_{AA}\nabla \Phi _{A}$
• ${\overrightarrow {J_{A}}}=-L_{AA}\nabla (\mu _{A}-\mu _{v})$
• ${\overrightarrow {J_{V}}}=-{\overrightarrow {J_{A}}}$

Fick's Second Law

Diffusion Equation in the General Form

 ${\frac {\partial c}{\partial t}}={\dot {n}}-\nabla \cdot {\overrightarrow {J}}$

${\dot {n}}$: source or sink term

${\overrightarrow {J}}$: any flux in a V-frame

Fick's Second Law

 ${\frac {\partial c}{\partial t}}=\nabla \cdot {\overrightarrow {J}}$
 ${\frac {\partial c}{\partial t}}=\nabla \cdot (D\nabla c)$

Linearization of Diffusion Equation

 ${\frac {\partial c}{\partial t}}=D_{o}\nabla ^{2}c$

Heat Equation

 ${\frac {\partial h}{\partial t}}=-\nabla \cdot {\overrightarrow {J_{Q}}}$
 $c_{P}{\frac {\partial T}{\partial t}}=-\nabla \cdot (-K\nabla T)$
 ${\frac {\partial T}{\partial t}}=\nabla \cdot \left({\frac {K}{c_{P}}}\nabla T\right)$
 $\nabla \cdot (\kappa \nabla T)$

$h\;$: enthalpy density

$c_{P}\;$: heat capacity

$K/c_{p}=\kappa \;$: thermal diffusivity

Constant Diffusivity

${\frac {\partial c}{\partial t}}=D\nabla ^{2}c$

One-Dimensional Diffusion Along x from an Initial Step Function

$c(x,t)={\bar {c}}+{\frac {\Delta c}{2}}{\mbox{erf}}\left({\frac {x}{\sqrt {4Dt}}}\right)$

Localized Source

 $c(x,t)={\frac {c_{o}\Delta x}{\sqrt {4\pi Dt}}}e^{-x^{2}/(4Dt)}$
 $c(x,t)={\frac {n_{d}}{\sqrt {4\pi Dt}}}e^{-x^{2}/(4Dt)}$

• Source strength, $n_{d}=\int _{-\infty }^{\infty }c(x)dx$

Diffusivity as a Function of Concentration

 ${\frac {\partial c}{\partial t}}=\nabla \cdot [D(c)\nabla c]$

• Interdiffusivity: $D(c_{1})=-{\frac {1}{2\tau }}{\frac {dx}{dc_{1}}}\int _{c_{1}^{R}}^{c_{1}}x(c')dc'$

Diffusivity as a Function of Time

 ${\frac {\partial c}{\partial t}}=\nabla \cdot [D(t)\nabla c]$
 ${\frac {\partial c}{\partial t}}=D(t)\nabla ^{2}c]$

• Change of variable: $\tau _{D}=\int _{0}^{t}D(t')dt'$
• Transformed equation: ${\frac {\partial c}{\partial \tau _{D}}}=\nabla ^{2}c$
• Solution:

$c(x,\tau _{D})={\bar {c}}+{\frac {\Delta c}{2}}{\mbox{erf}}\left({\frac {x}{\sqrt {4\tau _{D}}}}\right)$

$c(x,t)={\bar {c}}+{\frac {\Delta c}{2}}{\mbox{erf}}\left({\frac {x}{\sqrt {4\int _{0}^{t}D(t')dt'}}}\right)$

Diffusivity as a Function of Direction

 ${\overrightarrow {J}}=-\mathbf {D} \nabla c$

• The diagonal elements of ${\hat {\mathbf {D} }}$ are the eigenvalues of $\mathbf {D}$, and the coordinate system of ${\hat {\mathbf {D} }}$ defines the principal axes.

• • ${\frac {\partial c}{\partial t}}=-\nabla \cdot {\overrightarrow {J}}$
  • ${\frac {\partial c}{\partial t}}=\nabla \cdot {\hat {\mathbf {D} }}\nabla c$

• Relation of $\mathbf {D}$ and ${\hat {\mathbf {D} }}$:

 ${\hat {\mathbf {D} }}={\underline {R}}\mathbf {D} {\underline {R}}^{-1}$

Steady-State Solutions

Harmonic Functions

 $\nabla ^{2}c=0$

One Dimension

 $J=-D{\frac {dc}{dx}}$
 $J=D{\frac {c^{0}-c^{L}}{L}}$

Cylindrical Shell

• Laplacian Operator: $\nabla ^{2}c={\frac {1}{r}}{\frac {\partial }{\partial r}}\left(r{\frac {\partial c}{\partial r}}\right)+{\frac {1}{r^{2}}}{\frac {\partial ^{2}c}{\partial \theta ^{2}}}+{\frac {\partial ^{2}c}{\partial x^{2}}}$
• Integrate Twice and Apply the Boundary Conditions:

 $c(r)=c^{\mbox{in}}-{\frac {c^{\mbox{in}}-c^{\mbox{out}}}{\ln(r^{\mbox{out}}/r^{\mbox{in}})}}\ln \left({\frac {r}{r^{\mbox{in}}}}\right)$

Spherical Shell

• Laplacian operator in spherical coordinates

$\nabla ^{2}c={\frac {1}{r^{2}}}\left[{\frac {\partial }{\partial r}}\left(r^{2}{\frac {\partial c}{\partial r}}\right)+{\frac {1}{\sin {\theta }}}{\frac {\partial }{\partial \theta }}\left(\sin \theta {\frac {\partial c}{\partial \theta }}\right)+{\frac {1}{\sin ^{2}\theta }}{\frac {\partial ^{2}c}{\partial \phi ^{2}}}\right]$

Variable Diffusivity

• Steady-state conditions
• $D$ varies with position

 $\partial \cdot (D\nabla c)=0$

• Solution is obtained by integration:

 $c(x)=c(x_{1})+a_{1}\int _{x_{1}}^{x}{\frac {d\zeta }{D(\zeta )}}$

Infinite Media with Instantaneous Localized Source

 $c(x,y,z,t)={\frac {n_{d_{x}}}{\sqrt {4\pi Dt}}}e^{-x^{2}/(4Dt)}\times {\frac {n_{d_{y}}}{\sqrt {4\pi Dt}}}e^{-y^{2}/(4Dt)}\times {\frac {n_{d_{z}}}{\sqrt {4\pi Dt}}}e^{-z^{2}/(4Dt)}$

Solutions with the Error Function

• Uniform distribution of point, line, or plana source placed along $x>0$
• Contribution at a general position $x$ from the source:

$c_{\zeta }(x,t)={\frac {c_{o}d\zeta }{\sqrt {4\pi Dt}}}e^{-(\zeta -x)^{2}/(4Dt)}$

• Integral over all sources:

$c(x,t)={\frac {c_{o}}{\sqrt {4\pi Dt}}}\int _{0}^{\infty }e^{-(\zeta -x)^{2}/(4Dt)}d\zeta$

 $c(x,t)={\frac {c_{o}}{2}}+{\frac {c_{o}}{2}}{\mbox{erf}}\left({\frac {x}{2{\sqrt {Dt}}}}\right)$

Method of Separation of Variables

• System : Three Dimensions, $(x,y,z)$
• Equation : ${\frac {dc}{dt}}=D\nabla ^{2}c$
• Solution : $c(r,\theta ,z,t)=R(r)\Theta (\theta )Z(z)T(t)\;$

Method of Laplace Transforms

• Laplace transform of a function $f(x,t)\;$

 $L\{f(x,t)\}={\hat {f}}(x,p)$
 $L\{f(x,t)\}=\int _{0}^{\infty }e^{-pt}f(x,t)dt$

Atomic Models of Diffusion

Model of One-Particle with Step Potential-Energy Wells

 $\mathrm {T} '={\sqrt {\frac {kT}{2\pi m}}}{\frac {1}{L^{\mbox{well}}}}e^{-(E^{A}-E^{\mbox{well}})/(kT)}$
 $\mathrm {T} '={\sqrt {\frac {kT}{2\pi m}}}{\frac {1}{L^{\mbox{well}}}}e^{-(E^{m})/(kT)}$

Model of One-Particle with Step Potential-Energy Wells

 $\mathrm {T} '={\frac {1}{2\pi }}{\sqrt {\frac {\beta }{m}}}e^{-(E^{A}-E^{\mbox{well}})/(kT)}$
 $\mathrm {T} '=\nu e^{-E^{m}/(kT)}$

Many-Body Model

 $\mathrm {T} '=\nu e^{-G^{m}/(kT)}$

Diffusion as Series of Discrete Jumps

 $<R^{2}(N_{\tau })>=N_{\tau }<r^{2}>+2<(\sum _{j=1}^{N_{\tau }-1}\sum _{i=1}^{N_{\tau -j}}|{\overrightarrow {r_{i}}}||{\overrightarrow {r}}_{i+j}|\cos \theta _{i,i+j}>$

Diffusivity and Mean-Square Particle Displacement

 $<R^{2}(\tau )>=6D\tau \;$

Relation of Macroscopic Diffusivity and Microscopic Jump Parameters

 $D={\frac {\mathrm {T} <r^{2}>}{2}}$

Diffusion and Correlated Jumps

• Correlation factor:

 $\mathbf {f} =1+{\frac {2}{N_{\tau }<r^{2}>}}<(\sum _{j=1}^{N_{\tau }-1}\sum _{i=1}^{N_{\tau -j}}|{\overrightarrow {r_{i}}}||{\overrightarrow {r}}_{i+j}|\cos \theta _{i,i+j}>$

• Macroscopic Diffusivity and Microscopic Parameters:

 $D={\frac {<r^{2}>N_{\tau }}{6\tau }}\mathbf {f}$
 $D={\frac {\mathrm {T} \tau <r^{2}>}{6\tau }}\mathbf {f}$
 $D={\frac {\mathrm {T} <r^{2}>}{6}}\mathbf {f}$

Atomic Models of Diffusivity

Metals

Correlation Factor

 $\mathbf {f} ={\frac {1+<\cos \theta >}{1-<\cos \theta >}}$
 $\mathbf {f} \approx {\frac {z-1}{z+1}}$

Isotope Effect

 ${\frac {^{*}D(mass1)}{^{*}D(mass2)}}={\frac {\mathrm {T} _{1}}{\mathrm {T} _{2}}}={\frac {\mathrm {T} '_{1}}{\mathrm {T} '_{2}}}={\frac {\nu _{1}}{\nu _{2}}}={\frac {m_{1}}{m_{2}}}$