Particle in a spherically symmetric potential

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Particle_in_a_spherically_symmetric_potential 

Hydrogen atomic orbitals of different energy levels. The more opaque areas are where one is most likely to find an electron at any given time.

In quantum mechanics, a particle in a spherically symmetric potential is a system with a potential that depends only on the distance between the particle and a center. This label is how isolated atoms are described, as well as playing a central role as a first approximation to the formation of chemical bonds.

In the general case, the dynamics of a particle in a spherically symmetric potential are governed by a Hamiltonian of the following form:

$${\displaystyle \hat{H}=\frac{{\hat{p}}^2}{2m_0}+V(r)}$$

where ${\displaystyle m_0}$ is the mass of the particle, ${\displaystyle \hat{p}}$ is the momentum operator, and the potential ${\displaystyle V(r)}$ depends only on ${\displaystyle r}$, the modulus of the radius vector. The possible quantum states of the particle can be found by using the above Hamiltonian to solve the Schrödinger equation for its eigenvalues and their corresponding eigenstates, which are wave functions.

To describe these systems, it is convenient to use spherical coordinates, ${\displaystyle r}$, ${\displaystyle \theta }$ and ${\displaystyle \varphi }$ - the time-independent Schrödinger equation for the system is then separable. This means solutions to the angular dimensions of the equation can be found independently of the radial dimension. This leaves an ordinary differential equation in terms only of the radius, ${\displaystyle r}$, which determines the eigenstates for the particular potential, ${\displaystyle V(r)}$.

Structure of the eigenfunctions

The eigenstates of the system have the form:

$${\displaystyle \psi (r,\theta ,\varphi )=R(r)\mathrm{\Theta }(\theta )\mathrm{\Phi }(\varphi )}$$

in which the spherical angles ${\displaystyle \theta }$ and ${\displaystyle \varphi }$ represent the polar and azimuthal angle, respectively. The last two factors of ${\displaystyle \psi }$ are often grouped together as spherical harmonics, so that the eigenfunctions take the form:

$${\displaystyle \psi (r,\theta ,\varphi )=R(r)Y_{\ell m}(\theta ,\varphi ).}$$

The differential equation which characterizes the function ${\displaystyle R(r)}$ is called the radial equation.

Derivation of the radial equation

The kinetic energy operator in spherical polar coordinates is:

$${\displaystyle \frac{{\hat{p}}^2}{2m_0}=-\frac{{\hslash }^2}{2m_0}{\mathrm{\nabla }}^2=-\frac{{\hslash }^2}{2m_0{r}^2}\left[\frac{\mathrm{\partial }}{\mathrm{\partial }r}\left(r^2\frac{\mathrm{\partial }}{\mathrm{\partial }r}\right)-{\hat{L}}^2\right].}$$

The spherical harmonics satisfy

$${\displaystyle {\hat{L}}^2{Y}_{\ell m}(\theta ,\varphi )\equiv \left\{-\frac{1}{{\sin }^2\theta }\left[\sin \theta \frac{\mathrm{\partial }}{\mathrm{\partial }\theta }\left(\sin \theta \frac{\mathrm{\partial }}{\mathrm{\partial }\theta }\right)+\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{\varphi }^2}\right]\right\}{Y}_{\ell m}(\theta ,\varphi )=\ell (\ell +1)Y_{\ell m}(\theta ,\varphi ).}$$

Substituting this into the Schrödinger equation we get a one-dimensional eigenvalue equation,

$${\displaystyle \frac{1}{r^2}\frac{d}{dr}\left(r^2\frac{dR}{dr}\right)-\frac{\ell (\ell +1)}{r^2}R+\frac{2m_0}{{\hslash }^2}\left[E-V(r)\right]R=0.}$$

This equation can be reduced to an equivalent 1-D Schrödinger equation by substituting ${\displaystyle R(r)=u(r)/r}$, where ${\displaystyle u(r)}$ satisfies

$${\displaystyle \frac{d^{2}u}{d{r}^2}+\frac{2m_0}{{\hslash }^2}\left[E-V_{\mathrm{e}\mathrm{f}\mathrm{f}}(r)\right]u=0}$$

which is precisely the one-dimensional Schrödinger equation with an effective potential given by

$${\displaystyle V_{\mathrm{e}\mathrm{f}\mathrm{f}}(r)=V(r)+\frac{{\hslash }^2\ell (\ell +1)}{2m_0{r}^2},}$$

where ${\displaystyle r\in [0,\mathrm{\infty })}$. The correction to the potential V(r) is called the centrifugal barrier term.

If $\underset{r\to 0}{lim}{r}^{2}V(r)=0$, then near the origin, ${\displaystyle R\sim r^{\ell }}$.

Solutions for potentials of interest

There are five cases of special importance:

1. ${\displaystyle V(r)=0}$, or solving the vacuum in the basis of spherical harmonics, which serves as the basis for other cases.
2. ${\displaystyle V(r)=V_0}$ (finite) for ${\displaystyle r<r_0}$ and zero elsewhere.
3. ${\displaystyle V(r)=0}$ for ${\displaystyle r<r_0}$ and infinite elsewhere, the spherical equivalent of the square well, useful to describe bound states in a nucleus or quantum dot.
4. ${\displaystyle V(r)\propto r^2}$for the three-dimensional isotropic harmonic oscillator.
5. ${\displaystyle V(r)\propto \frac{1}{r}}$ to describe bound states of hydrogen-like atoms.

The solutions are outlined in these cases, which should be compared to their counterparts in cartesian coordinates, cf. particle in a box.

Vacuum case states

Let us now consider ${\displaystyle V(r)=0}$. Introducing the dimensionless variables

$${\displaystyle \rho \stackrel{\mathrm{d}\mathrm{e}\mathrm{f}}{=}kr,k\stackrel{\mathrm{d}\mathrm{e}\mathrm{f}}{=}\sqrt{\frac{2m_{0}E}{{\hslash }^2}},}$$

the equation becomes a Bessel equation for ${\displaystyle J(\rho )\stackrel{\mathrm{d}\mathrm{e}\mathrm{f}}{=}\sqrt{\rho }R(r)}$:

$${\displaystyle {\rho }^2\frac{d^{2}J}{d{\rho }^2}+\rho \frac{dJ}{d\rho }+\left[{\rho }^2-{\left(\ell +\frac{1}{2}\right)}^2\right]J=0}$$

where regular solutions for positive energies are given by so-called Bessel functions of the first kind ${\displaystyle J_{\ell +1/2}(\rho )}$ so that the solutions written for ${\displaystyle R(r)}$ are the so-called spherical Bessel function

$R(r)=j_l(kr)\stackrel{\mathrm{d}\mathrm{e}\mathrm{f}}{=}\sqrt{\pi /(2kr)}{J}_{\ell +1/2}(kr)$.

The solutions of the Schrödinger equation in polar coordinates in vacuum are thus labelled by three quantum numbers: discrete indices ℓ and m, and k varying continuously in ${\displaystyle [0,\mathrm{\infty })}$:

$${\displaystyle \psi (\mathbf{r})=j_{\ell }(kr)Y_{\ell m}(\theta ,\varphi )}$$

These solutions represent states of definite angular momentum, rather than of definite (linear) momentum, which are provided by plane waves ${\displaystyle \exp (i\mathbf{k}\cdot \mathbf{r})}$.

Sphere with finite "square" potential

Consider the potential ${\displaystyle V(r)=V_0}$ for ${\displaystyle r<r_0}$ and ${\displaystyle V(r)=0}$ elsewhere - that is, inside a sphere of radius ${\displaystyle r_0}$ the potential is equal to ${\displaystyle V_0}$ and it is zero outside the sphere. A potential with such a finite discontinuity is called a square potential.

We first consider bound states, i.e., states which display the particle mostly inside the box (confined states). Those have an energy ${\displaystyle E}$ less than the potential outside the sphere, i.e., they have negative energy. Also worth noticing is that unlike Coulomb potential, featuring an infinite number of discrete bound states, the spherical square well has only a finite (if any) number because of its finite range.

The resolution essentially follows that of the vacuum case above with normalization of the total wavefunction added, solving two Schrödinger equations — inside and outside the sphere — of the previous kind, i.e., with constant potential. The following constraints must hold for a normalizable, physical wavefunction:

1. The wavefunction must be regular at the origin.
2. The wavefunction and its derivative must be continuous at the potential discontinuity.
3. The wavefunction must converge at infinity.

The first constraint comes from the fact that Neumann and Hankel functions are singular at the origin. The physical requirement that ${\displaystyle \psi }$ must be defined everywhere selected Bessel function of the first kind over the other possibilities in the vacuum case. For the same reason, the solution will be of this kind inside the sphere:

$${\displaystyle R(r)=A{j}_{\ell }\left(\sqrt{\frac{2m_0(E-V_0)}{{\hslash }^2}}r\right),r<r_0.}$$

Note that for bound states, ${\displaystyle V_0<E<0}$. Bound states bring the novelty as compared to the vacuum case now that ${\displaystyle E<0}$. This, along with the third constraint, selects the Hankel function of the first kind as the only converging solution at infinity (the singularity at the origin of these functions does not matter since we are now outside the sphere):

$${\displaystyle R(r)=B{h}_{\ell }^{(1)}\left(i\sqrt{\frac{-2m_{0}E}{{\hslash }^2}}r\right),r>r_0}$$

The second constraint on continuity of ${\displaystyle \psi }$ at ${\displaystyle r=r_0}$ along with normalization allows the determination of constants ${\displaystyle A}$ and ${\displaystyle B}$. Continuity of the derivative (or logarithmic derivative for convenience) requires quantization of energy.

Sphere with infinite "square" potential

In case where the potential well is infinitely deep, so that we can take ${\displaystyle V_0=0}$ inside the sphere and ${\displaystyle \mathrm{\infty }}$ outside, the problem becomes that of matching the wavefunction inside the sphere (the spherical Bessel functions) with identically zero wavefunction outside the sphere. Allowed energies are those for which the radial wavefunction vanishes at the boundary. Thus, we use the zeros of the spherical Bessel functions to find the energy spectrum and wavefunctions. Calling ${\displaystyle u_{\ell ,k}}$ the k^th zero of ${\displaystyle j_{\ell }}$, we have:

$${\displaystyle E_{kl}=\frac{u_{\ell ,k}^2{\hslash }^2}{2m_0{r}_0^2}}$$

so that the problem is reduced to the computations of these zeros ${\displaystyle u_{\ell ,k}}$, typically by using a table or calculator, as these zeros are not solvable for the general case.

In the special case ${\displaystyle \ell =0}$ (spherical symmetric orbitals), the spherical Bessel function is $j_0(x)=\frac{\sin x}{x}$, which zeros can be easily given as ${\displaystyle u_{0,k}=k\pi }$. Their energy eigenvalues are thus:

$${\displaystyle E_{k0}=\frac{(k\pi {)}^2{\hslash }^2}{2m_0{r}_0^2}=\frac{k^2{h}^2}{8m_0{r}_0^2}}$$

3D isotropic harmonic oscillator

The potential of a 3D isotropic harmonic oscillator is

$${\displaystyle V(r)=\frac{1}{2}{m}_0{\omega }^2{r}^2.}$$

An N-dimensional isotropic harmonic oscillator has the energies

$${\displaystyle E_n=\hslash \omega \left(n+\frac{N}{2}\right)\text{with}n=0,1,\dots ,\mathrm{\infty },}$$

i.e., ${\displaystyle n}$ is a non-negative integral number; ${\displaystyle \omega }$ is the (same) fundamental frequency of the ${\displaystyle N}$ modes of the oscillator. In this case ${\displaystyle N=3}$, so that the radial Schrödinger equation becomes,

$${\displaystyle \left[-\frac{{\hslash }^2}{2m_0}\frac{d^2}{d{r}^2}+\frac{{\hslash }^2\ell (\ell +1)}{2m_0{r}^2}+\frac{1}{2}{m}_0{\omega }^2{r}^2-\hslash \omega \left(n+\frac{3}{2}\right)\right]u(r)=0.}$$

Introducing

$${\displaystyle \gamma \equiv \frac{m_0\omega }{\hslash }}$$

and recalling that ${\displaystyle u(r)=rR(r)}$, we will show that the radial Schrödinger equation has the normalized solution,

$${\displaystyle R_{n\ell }(r)=N_{n\ell }{r}^{\ell }{e}^{-\frac{1}{2}\gamma r^2}{L}_{\frac{1}{2}(n-\ell )}^{\left(\ell +\frac{1}{2}\right)}(\gamma r^2),}$$

where the function ${\displaystyle L_k^{(\alpha )}(\gamma r^2)}$ is a generalized Laguerre polynomial in ${\displaystyle y{r}^2}$ of order ${\displaystyle k}$.

The normalization constant ${\displaystyle N_{nl}}$ is,

$${\displaystyle N_{n\ell }={\left[\frac{2^{n+\ell +2}{\gamma }^{\ell +\frac{3}{2}}}{{\pi }^{\frac{1}{2}}}\right]}^{\frac{1}{2}}{\left[\frac{\left[\frac{1}{2}(n-\ell )\right]!\left[\frac{1}{2}(n+\ell )\right]!}{(n+\ell +1)!}\right]}^{\frac{1}{2}}.}$$

The eigenfunction ${\displaystyle R_{nl}(r)}$ is associated with energy ${\displaystyle E_n}$, where

$${\displaystyle \ell =n,n-2,\dots ,{\ell }_{min}\text{with}{\ell }_{min}=\{\begin{array}{ll}1& \text{if}n\text{odd}\\ 0& \text{if}n\text{even}\end{array}}$$

This is the same result as the quantum harmonic oscillator, with ${\displaystyle \gamma =2\nu }$.

Derivation

First we transform the radial equation by a few successive substitutions to the generalized Laguerre differential equation, which has known solutions: the generalized Laguerre functions. Then we normalize the generalized Laguerre functions to unity. This normalization is with the usual volume element r² dr.

First we scale the radial coordinate

$${\displaystyle y=\sqrt{\gamma }r\text{with}\gamma \equiv \frac{m_0\omega }{\hslash },}$$

and then the equation becomes

$${\displaystyle \left[\frac{d^2}{d{y}^2}-\frac{\ell (\ell +1)}{y^2}-y^2+2n+3\right]v(y)=0}$$

with $v(y)=u\left(y/\sqrt{\gamma }\right)$.

Consideration of the limiting behavior of v(y) at the origin and at infinity suggests the following substitution for v(y),

$${\displaystyle v(y)=y^{\ell +1}{e}^{-y^2/2}f(y).}$$

This substitution transforms the differential equation to

$${\displaystyle \left[\frac{d^2}{d{y}^2}+2\left(\frac{\ell +1}{y}-y\right)\frac{d}{dy}+2n-2\ell \right]f(y)=0,}$$

where we divided through with ${\displaystyle y^{\ell +1}{e}^{-y^2/2}}$, which can be done so long as y is not zero.

Transformation to Laguerre polynomials

If the substitution ${\displaystyle x=y^2}$ is used, ${\displaystyle y=\sqrt{x}}$, and the differential operators become

$${\displaystyle \frac{d}{dy}=\frac{dx}{dy}\frac{d}{dx}=2y\frac{d}{dx}=2\sqrt{x}\frac{d}{dx},}$$

and

$${\displaystyle \frac{d^2}{d{y}^2}=\frac{d}{dy}\left(2y\frac{d}{dx}\right)=4x\frac{d^2}{d{x}^2}+2\frac{d}{dx}.}$$

The expression between the square brackets multiplying ${\displaystyle f(y)}$ becomes the differential equation characterizing the generalized Laguerre equation (see also Kummer's equation):

$${\displaystyle x\frac{d^{2}g}{d{x}^2}+\left(\left(\ell +\frac{1}{2}\right)+1-x\right)\frac{dg}{dx}+\frac{1}{2}(n-\ell )g(x)=0}$$

with ${\displaystyle g(x)\equiv f(\sqrt{x})}$.

Provided ${\displaystyle k\equiv (n-\ell )/2}$ is a non-negative integral number, the solutions of this equations are generalized (associated) Laguerre polynomials

$${\displaystyle g(x)=L_k^{\left(\ell +\frac{1}{2}\right)}(x).}$$

From the conditions on ${\displaystyle k}$ follows: (i) ${\displaystyle n\ge \ell }$ and (ii) ${\displaystyle n}$ and ${\displaystyle l}$ are either both odd or both even. This leads to the condition on ${\displaystyle l}$ given above.

Recovery of the normalized radial wavefunction

Remembering that ${\displaystyle u(r)=rR(r)}$, we get the normalized radial solution

$${\displaystyle R_{n\ell }(r)=N_{n\ell }{r}^{\ell }{e}^{-\frac{1}{2}\gamma r^2}{L}_{\frac{1}{2}(n-\ell )}^{\left(\ell +\frac{1}{2}\right)}(\gamma r^2).}$$

The normalization condition for the radial wavefunction is

$${\displaystyle {\int }_0^{\mathrm{\infty }}{r}^2|R(r){|}^{2}dr=1.}$$

Substituting ${\displaystyle q=\gamma r^2}$, gives ${\displaystyle dq=2\gamma rdr}$ and the equation becomes

$${\displaystyle \frac{N_{n\ell }^2}{2{\gamma }^{\ell +\frac{3}{2}}}{\int }_0^{\mathrm{\infty }}{q}^{\ell +\frac{1}{2}}{e}^{-q}{\left[L_{\frac{1}{2}(n-\ell )}^{\left(\ell +\frac{1}{2}\right)}(q)\right]}^{2}dq=1.}$$

By making use of the orthogonality properties of the generalized Laguerre polynomials, this equation simplifies to

$${\displaystyle \frac{N_{n\ell }^2}{2{\gamma }^{\ell +\frac{3}{2}}}\cdot \frac{\mathrm{\Gamma }\left[\frac{1}{2}(n+\ell +1)+1\right]}{\left[\frac{1}{2}(n-\ell )\right]!}=1.}$$

Hence, the normalization constant can be expressed as

$${\displaystyle N_{n\ell }=\sqrt{\frac{2{\gamma }^{\ell +\frac{3}{2}}\left(\frac{n-\ell }{2}\right)!}{\mathrm{\Gamma }\left(\frac{n+\ell }{2}+\frac{3}{2}\right)}}}$$

Other forms of the normalization constant can be derived by using properties of the gamma function, while noting that ${\displaystyle n}$ and ${\displaystyle l}$ are both of the same parity. This means that ${\displaystyle n+l}$ is always even, so that the gamma function becomes

$${\displaystyle \mathrm{\Gamma }\left[\frac{1}{2}+\left(\frac{n+\ell }{2}+1\right)\right]=\frac{\sqrt{\pi }(n+\ell +1)!!}{2^{\frac{n+\ell }{2}+1}}=\frac{\sqrt{\pi }(n+\ell +1)!}{2^{n+\ell +1}\left[\frac{1}{2}(n+\ell )\right]!},}$$

where we used the definition of the double factorial. Hence, the normalization constant is also given by

$${\displaystyle N_{n\ell }={\left[\frac{2^{n+\ell +2}{\gamma }^{\ell +\frac{3}{2}}\left[\frac{1}{2}(n-\ell )\right]!\left[\frac{1}{2}(n+\ell )\right]!}{{\pi }^{\frac{1}{2}}(n+\ell +1)!}\right]}^{\frac{1}{2}}=\sqrt{2}{\left(\frac{\gamma }{\pi }\right)}^{\frac{1}{4}}(2\gamma {)}^{\frac{\ell }{2}}\sqrt{\frac{2\gamma (n-\ell )!!}{(n+\ell +1)!!}}.}$$

Hydrogen-like atoms

Main article: hydrogen-like atom

A hydrogenic (hydrogen-like) atom is a two-particle system consisting of a nucleus and an electron. The two particles interact through the potential given by Coulomb's law:

$${\displaystyle V(r)=-\frac{1}{4\pi {\epsilon }_0}\frac{Z{e}^2}{r}}$$

where

• ε₀ is the permittivity of the vacuum,
• Z is the atomic number (eZ is the charge of the nucleus),
• e is the elementary charge (charge of the electron),
• r is the distance between the electron and the nucleus.

In order to simplify the Schrödinger equation, we introduce the following constants that define the atomic unit of energy and length:

$${\displaystyle E_{\text{h}}=\mu {\left(\frac{e^2}{4\pi {\epsilon }_0\hslash }\right)}^2\text{and}{a}_0=\frac{4\pi {\epsilon }_0{\hslash }^2}{\mu e^2}.}$$

where ${\displaystyle \mu \approx m_e}$ is the reduced mass in the ${\displaystyle m_e\ll m_{nucleus}}$ limit. Substitute ${\displaystyle y=Zr/a_0}$ and ${\displaystyle W=E/(Z^2{E}_{\text{h}})}$ into the radial Schrödinger equation given above. This gives an equation in which all natural constants are hidden,

$${\displaystyle \left[-\frac{1}{2}\frac{d^2}{d{y}^2}+\frac{1}{2}\frac{\ell (\ell +1)}{y^2}-\frac{1}{y}\right]u_{\ell }=W{u}_{\ell }.}$$

Two classes of solutions of this equation exist: (i) ${\displaystyle W}$ is negative, the corresponding eigenfunctions are square-integrable and the values of ${\displaystyle W}$ are quantized (discrete spectrum). (ii) ${\displaystyle W}$ is non-negative. Every real non-negative value of ${\displaystyle W}$ is physically allowed (continuous spectrum), the corresponding eigenfunctions are non-square integrable. Considering only class (i) solutions restricts the solutions to wavefunctions which are bound states, in contrast to the class (ii) solutions that are known as scattering states.

For class (i) solutions with negative W the quantity ${\displaystyle \alpha \equiv 2\sqrt{-2W}}$ is real and positive. The scaling of ${\displaystyle y}$, i.e., substitution of ${\displaystyle x\equiv \alpha y}$ gives the Schrödinger equation:

$${\displaystyle \left[\frac{d^2}{d{x}^2}-\frac{\ell (\ell +1)}{x^2}+\frac{2}{\alpha x}-\frac{1}{4}\right]u_{\ell }=0,\text{with }x\ge 0.}$$

For ${\displaystyle x\to \mathrm{\infty }}$ the inverse powers of x are negligible and the normalizable (and therefore, physical) solution for large ${\displaystyle x}$ is ${\displaystyle \exp [-x/2]}$. Similarly, for ${\displaystyle x\to 0}$ the inverse square power dominates and the physical solution for small ${\displaystyle x}$ is x^ℓ+1. Hence, to obtain a full range solution we substitute

$${\displaystyle u_l(x)=x^{\ell +1}{e}^{-x/2}{f}_{\ell }(x).}$$

The equation for ${\displaystyle f_l(x)}$ becomes,

$${\displaystyle \left[x\frac{d^2}{d{x}^2}+(2\ell +2-x)\frac{d}{dx}+(n-\ell -1)\right]f_{\ell }(x)=0\text{with}n=(-2W{)}^{-1/2}=\frac{2}{\alpha }.}$$

Provided ${\displaystyle n-\ell -1}$ is a non-negative integer, this equation has polynomial solutions written as

$${\displaystyle L_k^{(2\ell +1)}(x),k=0,1,\dots ,}$$

which are generalized Laguerre polynomials of order ${\displaystyle k}$. The energy becomes

$${\displaystyle W=-\frac{1}{2n^2}\text{with}n\equiv k+\ell +1.}$$

The principal quantum number ${\displaystyle n}$ satisfies ${\displaystyle n\ge \ell +1}$. Since ${\displaystyle \alpha =2/n}$, the total radial wavefunction is

$${\displaystyle R_{n\ell }(r)=\frac{u(r)}{r}=N_{n\ell }{\left(\frac{2Zr}{n{a}_0}\right)}^{\ell }{e}^{-\frac{Zr}{n{a}_0}}{L}_{n-\ell -1}^{(2\ell +1)}\left(\frac{2Zr}{n{a}_0}\right),}$$

with normalization which absorbs extra terms from $\frac{1}{r}$

$${\displaystyle N_{n\ell }={\left[{\left(\frac{2Z}{n{a}_0}\right)}^3\cdot \frac{(n-\ell -1)!}{2n[(n+\ell )!{]}^3}\right]}^{\frac{1}{2}},}$$

via

$${\displaystyle {\int }_0^{\mathrm{\infty }}{x}^{2\ell +2}{e}^{-x}{\left[L_{n-\ell -1}^{(2\ell +1)}(x)\right]}^{2}dx=\frac{2n(n+\ell )!}{(n-\ell -1)!}.}$$

The corresponding energy is

$${\displaystyle E=-\frac{Z^2}{2n^2}{E}_{\text{h}},n=1,2,\dots .}$$

Covert racism

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Covert_racism 

Covert racism is a form of racial discrimination that is disguised and subtle, rather than public or obvious. Concealed in the fabric of society, covert racism discriminates against individuals through often evasive or seemingly passive methods. Covert, racially biased decisions are often hidden or rationalized with an explanation that society is more willing to accept. These racial biases cause a variety of problems that work to empower the suppressors while diminishing the rights and powers of the oppressed. Covert racism often works subliminally, and much of the discrimination is done subconsciously.

History in the U.S.

With the history of racial inequality in the United States, racism has long been an issue. The enslavement of millions of blacks along with the huge influx of immigrants throughout its history resulted in great diversity but also racial segregation. With the abolition of slavery, different forms of segregation were implemented, including Jim Crow laws and later American social and political structures, which led to segregation within cities and the suburbanization of the working and middle classes. As overt racial discrimination became illegal and less apparent, the idea of the nation homogenizing became popular. As the U.S. accepted more immigrants from different cultures, various "melting pots" of unity were seen to arise. Along with this, ideologies posited that every group of immigrants goes through the same discrimination. Many groups eventually assimilate, but racism still exists, often in different forms and to different degrees.

When black G.I.s returned home from the Vietnam War, they were denied the money promised to them to support their education and help them buy homes. While only 9.5% of soldiers serving in Vietnam were black, they comprised nearly 20% of front-line troops and 25% or more of airborne divisions. Black servicemen were twice as likely to re-enlist in the Navy, Marine Corps, and Air Force and three times as likely to re-enlist in the Army as their white counterparts, not for any sense of adventure but because they found the monetary rewards to be promising and they were treated more equally.

De cardio racism

De cardio racism is a term coined by John L. Jackson Jr. Jackson states that the rise of de cardio racism can be traced to the aftermath of the legislative and political success of the Civil Rights Movement. Those obvious victories, he suggests, did not lead the US into a racial color-blind society but rather shifted racist beliefs to the "hearts" of Americans. To Jackson, this reimagines the manifestation of racism—away from blatant Jim Crow-era discrimination and towards covert, deniable subtleties. Jackson argues, "De cardio racism is about what the law can't touch, what won't be easily proved or disproved, what can't be simply criminalized and deemed unconstitutional. It is racism that is most terrifying because it is hidden, secret, papered over with public niceties and politically correct jargon." In this new era, Jackson states that we must develop new tools to detect racial dog whistles and to penetrate the "hearts" of those who harbor racist beliefs without becoming distracted by empty rhetoric. As compared to the past, where activists confronted a clear and tangible discriminatory foe, Jackson states that modern activists must contend with an opponent parroting the rhetoric of racial equality but not backing it up with any action. The racism is hidden, covert, and tucked away in a part of Americans' hearts that is hard to reach. He argues that they must navigate the head-scratching silence of white Americans who engaged in violent racist attacks just a generation ago. In Jackson's words, "De Cardio Racism asks: where did all of yesterday's racial wolves go, and why do all these sheep seem to be standing around licking their chops?"

De facto racism

De facto racism refers to racism "in fact", or as a matter of circumstance and personal choice. In the post-Civil Rights Era, governments and institutions could no longer overtly discriminate based on race in law, regulation, or policy. Modern de facto racism refers to forms of discrimination that may not be overt but still produce disparate outcomes. Pertinent examples include continued school segregation. Although de jure racist practices of housing segregation were outlawed by the passage of the 1968 Fair Housing Act, de facto racist practices such as "redlining" and "white flight" largely trapped black households in the inner cities and saw white families move to the suburbs. Other de facto racist policies, like basing school funding off property taxes, have led to disparate investment in public schools.

De facto racism can also exist as the lingering remnants of centuries of de jure racism. Past generations of black families' inability to become homeowners or attend well-funded schools has hindered long-term social mobility. As a result, black families are not able to cultivate generational wealth or pass on social or cultural capital to their children. This has led to a persistent wealth gap between black and white families. In 2022, white households held on average $252,000 more than the average black family.

Covert racism and colonialism

After the departure of Western imperial powers from their former colonies, many nations in the Global South were left with odious debt, crumbling institutions, economic decline, and a social morass. No longer in control of their former colonies, some people in the West looked for ways to explain the lasting impact of their colonial rule. Covert concepts that blamed the culture of former colonies replaced clear, moral, and scientific boundaries between the colonizers and colonies. Notable examples include "cultural deficits" and "deficient human capital". To explain this rhetoric, some scholars argue that the moral separation did not go away but changed forms as fatalistic concepts like a "culture of poverty" turned attention away from structural inequities left from colonial rule and put the onus on former colonies.

The white savior complex

Enlightenment colonialism refers to a set of cultural norms that served as a justificatory premise for colonial rule. Enlightenment colonialism comes from the philosophy of the White Man's Burden and the suppositious moral responsibility of white people to "save" the "uncivilized" people in the Global South. The fundamental premise rests on the cultural inferiority of former colonies compared to the "civilized" culture of Western powers. Economic exploitation, slavery, and oppression were justified by the promise of introducing the tenets of a "civilized society" as defined by the fundamental elements of Western culture.

Emerging from decades of Enlightenment colonialism came the White Savior Complex. Although "white saviors" may not overtly impose Western cultural norms, what is similar about both philosophies is the premise of cultural inferiority and the erasure of the agency of former colonies. White Saviorism assumes that the people impacted by the problem, usually in developing nations, are incapable of creating their own solutions and that white people must intervene to save them. Critics of white saviorism argue that it perpetuates a subtle dependency from the survivor to the savior. Developing countries become reliant on charity from Western charities, which puts the recipient in a subservient position to the giver. White saviors have also been criticized for prioritizing their values over the experience of the local community. However, many argue that Western charity is a net good and that it is the responsibility of wealthy nations in the west to support low-income nations in the global south. Nonetheless, many developmental economists have pointed out the tangible importance of including the community in humanitarian solutions.

Covert racism and language

Covert racism in language, or coded racism, is the deployment of common stereotypes or tropes to elucidate a racially charged idea. Rather than expressly perpetuating racist tropes, covert linguistic racism is seen as rational or "common sense", and many aren't aware of its impact. The term microaggressions has been used to describe the deployment of covert racism in language that usually represents an unconscious bias. The term was invented by Chester M. Pierce in 1970 to describe insults he witnessed non-black Americans inflicting on black students. Although the speaker does not state an expressed racial opinion, they often reinforce stereotypes that make the recipient feel outside the norm; a recipient of a microaggression feels as though their individuality becomes lost to an underlying stereotype. Notable examples include asking a person of color where they learned to speak English or presuming that a black person is dangerous. In all cases, the speaker generalizes or idealizes a quality that simultaneously degrades a minority group. However, some scholars have criticized the term for its lack of scientific evidence, reliance on subjective evidence, and elevation of victimhood by focusing on a term connoting harm.

Racial stereotypes

Racial or cultural stereotyping refers to generalizing a group based on a simplified set of norms, behaviors, or characteristics. Although stereotypes can take on positive or negative connotations, they can be intrinsically harmful. The three main arguments are that they reduce an individual to a set of oversimplified characteristics, leave no room for change, and insist the stereotypes are natural and/or biological. Some scholars suggest that stereotyping is a method to reinforce entrenched power structures by making arbitrary social divisions seem "natural" and inevitable. Stereotypes also erase one's individuality and impose rigid norms, leading to frequent tension between one's identity and their perceived social role.

Racial stereotypes have been a powerful tool to lock people of color out of valuable socioeconomic resources, oftentimes inflicting self-destructive behavior. A 2010 study found that victims of stereotyping were more likely to be aggressive, lack self-control, and overindulge in carbohydrate-heavy foods. Furthermore, there is also research showing that individuals who are stereotyped are more likely to behave according to the stereotype, further reinforcing its validity and increasing its pull on targeted groups.

The data around racial stereotypes paints a clear picture of their pervasiveness and utility among white people against people of color. A study by Laura Green of Virginia Commonwealth University showed that 58.9 percent of black and white subjects endorsed at least one stereotypical difference in inborn ability. Additionally, whites are 10 times more likely to be seen as superior in artistic ability and abstract thinking ability, and African-Americans are 10 times more likely to be seen as superior in athletic ability and rhythmic ability. Further, 49 percent of subjects endorsed stereotypical differences in physical characteristics, such as blacks experience less physical pain than whites and have thicker skulls and skin.

Covert racism within politics

Covert racism within politics can be observed in various ways. The process begins with the demonization of certain ethnic groups by associating them with a negative aspect of society, which can be called racialization. Politicians can then perpetuate these negative associations to promote fear of a race and, in turn, offer protection through policy. This strategy is not new to the 20th century and can be observed globally. The way in which politicians throughout history have used covert racism tactically to promote their own agenda is through media outlets. Covert racism in politics has also been deployed to define ethnic groups as monoliths. Political parties define broad-based and universal narratives to appeal to a specific racial voting block. Critics argue that this approach prioritizes stereotypes about the community rather than appealing to their diverse ideological backgrounds. Some individuals have reported feeling taken for granted by politicians because they are assumed to vote a particular way based on their race.

Dog whistling

Dog whistling refers to racially coded political messaging that disparages or warns about specific social groups. The term derives from a physical "dog whistle"—an instrument that produces a sound undetectable to humans but bothersome to dogs. Likewise, what makes a dog whistle covert is that it does not expressly state a racist idea but a coded racial message that maintains a sense of plausible deniability. Compared to other forms of coded racism in language, a dog whistle usually has an expressed political goal. Most often, it is used as a method of fearmongering to drive voter turnout or advocate for specific policies.

"Thugs"

One way in which politicians use covert racism is with keywords like "thugs". Globally, the term thugs has become synonymous with young black people, with its roots coming from British rule and roots in the Hindi language, where the word was used in association with "scoundrel" or "a deceiver". The term is often used to describe a thief or deviant. Thugs are usually closely associated with violence, gangs, and crime. The word gained popularity in the black community and mainstream culture during the rise of rap in the 80s and 90s and the War on Crime. In the era of political dog whistles, many world leaders used the term to disparage criminals and castigate the black community. Some recent examples are the usage of the word "thugs" when describing Afro-Trinidadian impoverished males in the ports of Spain, President Donald Trump describing protestors as "thugs" following the murder of George Floyd, and the British Prime Minister responding to additional protests following George Floyd by saying "racist thuggery", will be answered by “the force of law”. Other words like ghetto, hood, and sketchy are used by politicians in a similar way to represent blackness without explicitly stating it.

Covert racism and the police

Covert racism in policing refers to the subtle and often unconscious biases that police officers may hold towards people of certain racial or ethnic groups. Many scholars argue that covert racism in policing persists as modern police forces move away from openly racist practices. Covert racial disparities in policing are seen in various ways, including by surveilling a perimeter, searching individuals, and conducting traffic stops across the United States. For example, a research team at Stanford University compiled and analyzed a dataset of nearly 100 million traffic stops. They found that after sunset, the "veil of darkness" resulted in black drivers being stopped fewer times than before sunset. Their findings indicated that police stops and search decisions were affected by racial biases. Stop and frisk has also been defined as a form of covert racism in policing. Although many police departments use stop and frisk, the NYPD came under intense scrutiny for its tactics in the mid-2000s. Critics claim that stop and frisk led to the explicit targeting of young men of color by the police. At its peak in 2011, the NYPD reported making nearly 700,000 stops. A disproportionate number of the stops were targeted towards young men of color—over 90%, with over 70% later being found innocent. The use of stop and frisk was deemed unconstitutional in 2013 by a New York court. However, many police officers argue that there have been substantive steps to root out racist practices. Chairs of the UK Black Police Associations stated that "open expressions of racism" have largely disappeared. The New York City Police Department and other departments have also shifted to community-based policing models, hiring more officers of color and imposing implicit-bias training to root out racial tendencies.

Some argue that covert racism manifests in how police officers are treated within the department. There is evidence that many minority officers are eliminated during initial training, given assignments that do not advance their careers, and even given unjust evaluations and discipline.

Covert racism and the media

Political advertising

Covert racism in political advertising can take the form of racial priming or racial cues, which is when political leaders speak about certain topics and subtly link them to racial groups without explicitly referencing race. These cues bypass any conscious thought of race or racism and therefore do not explicitly violate the 'norm of equality'. Some examples include Ronald Reagan nurturing implicit ties between subjects such as 'big government' with the demands of minorities for equality to imply that they were asking for special treatment, or George Bush's "Revolving Door" advertisement that included a photo of Willie Norton, said to strengthen subtle ties between black people and increasing crime rates. These strategies can have the potential to simplify political decision-making by activating racial thinking as a vital factor.

Both anger and fear have been proposed as the emotional responses that prompt racial thinking in the viewers of these kinds of political advertisements, with fear stemming from white people having to share resources they see as scarce with minorities or anger that manifests as blaming minorities for social issues.

Even in more recent US elections, racial cues can be seen in political advertising. Some examples include Mitt Romney's 'Obama isn't Working' slogan in the 2012 elections, emphasizing the implicit use of the stereotype of black people being lazy to undermine Barack Obama's campaign and black people in general and imply that they were gaining advantages at the expense of white people. However, in research surrounding race and the 2016 elections, some findings observe that more explicit racism, as opposed to implicit racial cues, has become more effective in driving white voters' choices.

News

One way covert racism in the news has been said to manifest is as "white normative objectivity". White normative objectivity is described as centering the white lens when reporting about marginalized communities, reinforcing stereotypes, and downplaying the effects of structural racism. Scholars have observed that it can emphasize the otherness of communities of color through stories chosen for shock value or sensation while ignoring the humanity of these communities. White normative objectivity has also been criticized for embracing a form of neutrality, or "both-sides-ism", that obscures harm from dominant society by individualizing institutional inequalities and lacking any critique of power dynamics that historically favor white people. The Missouri School of Journalism outlined several criticisms of news coverage of minorities, saying that most news channels do not show people of color within the context of their communities, portray them as either enablers, criminals, or victims, and do not show how the entire community is impacted by the crime. Furthermore, there is an overemphasis on atypical behavior that portrays communities of color in a constant state of crisis and not enough coverage of the average day-to-day lifestyles in their communities.

Covert racism has also been said to manifest in the lack of diversity within the newsroom, with a 2022 survey saying that 52% of US journalists think their newsroom lacks racial diversity. Though minorities are not legally barred from the newsroom, many diversity surveys of newsrooms show that white men are still disproportionately represented both as journalists and as higher management, seeing as in 2017, it was found that minorities only made up about 16.55% of the workforce in news media organizations, and 85.1% of TV news directors were white. Retention of minority reporters is also still an issue. Many reporters of color cite the lack of diversity as negatively affecting their time in the newsroom, with many speaking about their experience with microaggressions, a lack of promotions, and pay gaps.

The manner in which racism is discussed in news coverage can also contribute to the "debatability" or "plausible deniability" of racism. Currently, overt forms of racism are looked down upon in greater society, so criticizing minority groups usually happens within coded language and actions that can be denied to be truly racist. In covering stories about race, some methods by which the debatability of racism is perpetuated include avoidance of any pointed language about race or racism, denying or debating what actions can be counted as racism, and confining 'true' racism to a historical time period.

Other forms of racism that are deployed in the news are symbolic racism and nationalist language, where minorities are criticized due to their actions as opposed to their identity and are indirectly referenced as parties that cannot uphold the values of greater society and must be defended against.

Covert racism and education

Covert racism in the education system is shown both academically and through disciplinary measures. Discipline policies, including mandatory, zero-tolerance suspensions or expulsions for minor offenses, fuel the school-to-prison pipeline. According to the National School Boards Association (NBSA), white students are statistically more likely to be placed in an academic track. Of the black students who take the SAT, less than 60% are enrolled in an academic track, whereas about 90% of white students taking the SAT are on the academic track.

Such concealed acts of racism impact education by lowering expectations, lacking rigor in curricula, lacking experienced teachers, making fallacious assumptions about students’ intelligence, and raising issues in public policy. Despite the lack of explicit racism, covert racism continues to thwart the possibility of a successful future for black students. As stated by NSBA President Charlie Wilson, these practices and policies are critical factors that lead to gaps in the quality of education between black and white students.

Studies have also shown stark resource disparities in predominantly black or Latino schools and white suburban schools. It has been shown that schools with predominantly black students have scarce textbooks, and students are not allowed to take them home. In some states, like Mississippi, the textbooks and resources are so outdated that many of the new events in history have not yet been added to them. Bathroom access has also been cited as an issue in predominantly black and Latino schools. At some schools, there was a lack of bathrooms, resulting in infestations with vermin.

Many black activists have cited the lack of a well-resourced education as a stain on America's meritocracy. Without equality of condition or opportunity, some argue that black students lack the resources necessary to accrue financial and cultural capital.

Covert racism following major events

Major traumatic events involving different ethnic groups result in an increase in both overt and covert racism. Following traumatic events, there is a compulsion for individuals to continue to associate the past with the present. In the example of 9/11, the United States experienced an increase in exceptionalism as well as the creation of the idea of Arab people as terrorists and menacing people. To this day, the association of Arab people with this event has resulted in de cardio racism, as observed through the media and polls showing that half of polled young Arab Americans had experienced discrimination (whether that be de jure or de cardio) after 9/11. Major events also include events that span over longer periods; as an example, the prohibition of interracial marriage in the United States, which after legal changes still caused interracial couples to face hostility through both covert and overt racism, as seen with "preserve the race" language. Another example is slavery in the Caribbean and how the slot that slaves socially took up in the context of white Europeans still permeates the modern day, creating covert racism and resulting in differing work opportunities.

Plate reconstruction

From Wikipedia, the free encyclopedia
https://en.wikipedia.org/wiki/Plate_reconstruction

Plate reconstruction is the process of reconstructing the positions of tectonic plates relative to each other (relative motion) or to other reference frames, such as the Earth's magnetic field or groups of hotspots, in the geological past. This helps determine the shape and make-up of ancient supercontinents and provides a basis for paleogeographic reconstructions.

Defining plate boundaries

Earthquake epicenters 1963–98

An important part of reconstructing past plate configurations is to define the edges of areas of the lithosphere that have acted independently at some time in the past.

Present plate boundaries

Most present plate boundaries are easily identifiable from the pattern of recent seismicity. This is now backed up by the use of geodetic data, such as GPS/GNSS, to confirm the presence of significant relative movement between plates.

Past plate boundaries

Identifying past (but now inactive) plate boundaries within current plates is generally based on evidence for an ocean that has now closed up. The line where the ocean used to be is normally marked by pieces of the crust from that ocean, included in the collision zone, known as ophiolites. The line across which two plates became joined to form a single larger plate, is known as a suture.

In many orogenic belts, the collision is not just between two plates, but involves the sequential accretion of smaller terranes. Terranes are smaller pieces of continental crust that have been caught up in an orogeny, such as continental fragments or island arcs.

Reference frames

Plate motions, both those observable now and in the past, are referred ideally to a reference frame that allows other plate motions to be calculated. For example, a central plate, such as the African plate, may have the motions of adjacent plates referred to it. By composition of reconstructions, additional plates can be reconstructed to the central plate. In turn, the reference plate may be reconstructed, together with the other plates, to another reference frame, such as the Earth's magnetic field, as determined from paleomagnetic measurements of rocks of known age. A global hotspot reference frame has been postulated (see, e.g., W. Jason Morgan) but there is now evidence that not all hotspots are necessarily fixed in their locations relative to one another or the Earth's spin axis. However, there are groups of such hotspots that appear to be fixed within the constraints of available data, within particular mesoplates.

Euler poles

The movement of a rigid body, such as a plate, on the surface of a sphere can be described as rotation about a fixed axis (relative to the chosen reference frame). This pole of rotation is known as an Euler pole. The movement of a plate is completely specified in terms of its Euler pole and the angular rate of rotation about the pole. Euler poles defined for current plate motions can be used to reconstruct plates in the recent past (few million years). At earlier stages of Earth's history, new Euler poles need to be defined.

Estimating past plate motions

Ages of oceanic lithosphere

In order to move plates backward in time it is necessary to provide information on either relative or absolute positions of the plates being reconstructed such that an Euler pole can be calculated. These are quantitative methods of reconstruction.

Geometric matching of continental borders

Certain fits between continents, particularly that between South America and Africa, were known long before the development of a theory that could adequately explain them. The reconstruction before Atlantic rifting by Bullard based on a least-squares fitting at the 500 fathom contour still provides the best match to paleomagnetic pole data for the two sides from the middle of Paleozoic to Late Triassic.

Plate motion from magnetic stripes

Plate reconstructions in the recent geological past mainly use the pattern of magnetic stripes in oceanic crust to remove the effects of seafloor spreading. The individual stripes are dated from magnetostratigraphy so that their time of formation is known. Each stripe (and its mirror image) represents a plate boundary at a particular time in the past, allowing the two plates to be repositioned relative to one another. The oldest oceanic crust is Jurassic, providing a lower age limit of about 175 Ma for the use of such data. Reconstructions derived in this way are only relative.

Plate reconstructions from paleomagnetism

Paleomagnetic data: Sampling

Paleomagnetic data are obtained by taking oriented samples of rocks and measuring their remanent magnetizations in the laboratory. Good quality data can be recovered from different rock types. In igneous rocks, magnetic minerals crystallize from the melt, and when the rock is cooled below their Curie temperature, it acquires a thermoremanent magnetization (TRM) in the direction of the Earth's magnetic field. In sedimentary rocks, magnetic grains will align their magnetic moments with the direction of the magnetic field during or soon after the deposition, resulting in a detrital or post-detrital remanent magnetization (DRM). A common difficulty with the use of clastic sediments for defining directions of the magnetic field in the past is that the direction of DRM may rotate toward the bedding plane due to the compaction of sediment, resulting in an inclination, which is shallower than the inclination of the field during the deposition. The inclination flattening error can nevertheless be estimated and corrected for through re-deposition experiments, measurements of magnetic anisotropy, and the use of theoretical models for the dispersion of paleomagnetic directions. Metamorphic rocks are not normally used for paleomagnetic measurements due to the complexities related to the acquisition of remanence, uncertainties in magnetization age, and high magnetic anisotropy.

A typical paleomagnetic study would sample a large number of independent rock units of similar age at nearby locations and collect multiple samples from each unit in order to estimate measurement errors and assess how well the obtained paleomagnetic dataset samples geomagnetic secular variation. Progressive demagnetization techniques are used to identify secondary magnetization components (e.g., magnetic overprints that could have been imparted on the rock due to chemical alteration or reheating) and to isolate the primary magnetization, which records the direction of the magnetic field at the time when the rock was formed. Various rock-magnetic and paleomagnetic tests are normally performed to establish the primary nature of the isolated remanent magnetization. The recovered paleomagnetic directions are used to derive paleomagnetic poles, which provide constrains on the latitudinal position of the crustal block from which the rock samples were taken, and its original orientation with respect to the lines of longitude.

Good quality paleomagnetic data are available from the Global Paleomagnetic Database, which is accessible from the World Data Center A in the US at Boulder, Colorado.

Paleomagnetic poles

A paleomagnetic pole is defined by taking the average direction of the primary remanent magnetization for the sampled rocks (expressed as the mean declination and inclination) and calculating the position of a geomagnetic pole for the field of a geocentric magnetic dipole that would produce the observed mean direction at the sampled locality in its present geographic coordinates. An alternative way of defining paleomagnetic poles is to calculate a virtual geomagnetic pole (VGP) for each individual rock unit and then estimate the mean location for all VGPs. Fisher statistics on the sphere is normally used to obtain the mean direction of magnetization, or the mean VGP location, and to estimate their uncertainties. Both approaches are used in paleomagnetic studies, but it has been recognized that averaging directions instead of full remanence vectors can lead to biased estimates of the mean direction of the paleomagnetic field, so that the calculation of paleomagnetic poles by averaging VGPs is currently the preferred technique.

Applications to paleogeographic reconstructions

Paleogeographic reconstruction of the Pangea supercontinent at the Permo-Triassic Boundary (250 Ma). Top panel: Synthetic APWP for Africa (the south paleomagnetic poles are shown with their 95% uncertainty ovals). The red dot highlights the 250 Ma paleomagnetic pole. APWP data are from Torsvik et al. (2012). Middle panel: All continents are assembled in the Pangea configuration at 250 Ma using the estimates of their relative motions, with Africa kept fixed in its present position. The red triangle shows the position of the Euler pole and the red arrow indicates the rotation that would reconstruct the paleomagnetic pole to the south geographic pole. Bottom panel: The Euler rotation has been applied to Pangea, which is now reconstructed paleogeographically. The longitude is arbitrary set to minimize the longitudinal motion of Africa since 250 Ma.

Paleomagnetic studies of geologically recent lavas (Pliocene to Quaternary, 0-5 Ma) indicate that when the geomagnetic field is averaged on time scales of tens of thousands to millions of years – over a time period long enough to fully sample geomagnetic secular variation, the time-averaged field can be accurately approximated by the field of a geocentric axial dipole (GAD) – that is, a magnetic dipole placed in the center of the Earth and aligned with the Earth's rotation axis. Hence, if a paleomagnetic dataset has sampled enough time to average secular variation, the paleomagnetic pole derived from it can be interpreted as an estimate for the location of the geographic pole with respect to the sampling locality fixed in the present geographic position.

The difference between the paleomagnetic pole and the present geographic pole reflects the paleogeographic position of the crustal block containing the sampled area at the time when the studied rocks were formed, including its original latitude (paleolatitude) and orientation. Under the assumption that the mean paleomagnetic direction corresponds to that of the GAD field, the paleolatitude of the sampling location (λ) can be derived from the inclination (I) of the mean direction using a simple equation:

${\displaystyle \tan (\lambda )=\frac{1}{2}\cdot \tan (I)}$

The mean declination (D) gives the sense and amount of rotation about a vertical axis passing through the sampling area, which needs to be applied to restore its original orientation with respect to the lines of longitude. The paleolatitude for any specific location belonging to the same crustal block can be computed as 90° minus the angular distance between this location and the paleomagnetic pole, and the local vertical axis rotation can be estimated by computing the declination expected from the position of the pole. Thus, a paleomagnetic pole defines the paleo-latitudinal position and orientation of the entire tectonic block at a specific time in the past. However, because the GAD field is azimuthally symmetric about the Earth's rotation axis, the pole does not set any constraint on the absolute longitude. From the perspective of paleomagnetic directions, the GAD field has the same values of inclination and declination along a line of constant latitude at all longitudes, so that any conceivable longitude would be an equally viable option for the reconstruction of a tectonic element if its paleogeographic position is constrained by paleomagnetic data alone.

Considering that a paleomagnetic pole approximates the position of the geographic pole with respect to the continent or geologic terrane from which it was determined, the paleolatitude and orientation can be restored by finding a rotation (Euler pole and rotation angle) that reconstructs the paleomagnetic pole to the geographic pole, and applying this rotation to the continent or terrane. By doing so, the crustal block and its paleomagnetic pole are reconstructed using the same Euler rotation, so that they do not move relative to each other, the paleomagnetic pole is placed at the geographic pole, and the crustal block is correctly restored in latitude and orientation (i.e., with respect to the geographic pole). Noting that a further rotation around the geographic pole will only change the longitude of the block, but its latitude and orientation with respect to the lines of longitude will not be affected, the absolute paleolongitude cannot be determined in reconstructions based on paleomagnetism. However, relative longitudes of different crustal blocks can be defined using other types of geological and geophysical data constraining relative motions of tectonic plates, including the histories of seafloor spreading recorded my marine magnetic anomalies, matching of continental borders and geologic terranes, and paleontological data.

Apparent polar wander paths

Main article: Apparent polar wander

Poles from different ages in a single continent, lithospheric plate, or any other tectonic block can be used to construct an apparent polar wander path (APWP). If paths from adjacent crustal fragments are identical, this is taken to indicate that there has been no relative movement between them during the period covered by the path. Divergence of APW paths indicates that the areas in question have acted independently in the past with the point of divergence marking the time at which they became joined. Combined or synthetic APWPs can be constructed by rotating paleomagnetic poles from different plates into the reference frame fixed to a single plate, using estimates of relative plate motions. For the times postdating the assembly of Pangea (320 Ma), synthetic APWPs are often constructed in the reference frame fixed to the African plate because Africa has occupied a central position in the Pangea configuration and has been dominantly surrounded by spreading ridges after the Pangea breakup, which commenced in the early Jurassic (ca. 180 Ma).

Longitude constraints

For a single lithospheric plate, the APWP reflects the motion of the plate with respect to the geographic pole (changes in latitude) and changes of its orientation with respect to paleomeridians. The longitudes of paleogeographic reconstructions based on APWPs are uncertain, but it has been argued that the uncertainty can be minimized by selecting a reference plate that is expected to move the least in longitude from the consideration of the plate tectonics theory and by linking the reconstructions of the remaining plates to this reference plate using the estimates of relative plate motion. For example, and it was shown that assuming no significant longitudinal motion of Africa since the time of the Pangea assembly results in a reasonable plate tectonic scenario, in which no large, coherent east-west motions of the continental lithosphere are observed in paleogeographic reconstructions.

APWPs can be interpreted as records of a combined signal from two sources of plate motion: (1) motion of lithospheric plates with respect to the Earth's mantle and (2) motion of the entire solid Earth (mantle and lithosphere) with respect to the Earth's rotation axis. The second component is commonly referred to as true polar wander (TPW) and on geologic time scales results from gradual redistribution of mass heterogeneities due to convective motions in the Earth's mantle. By comparing plate reconstructions based on paleomagnetism with reconstructions in the mantle reference frame defined by hotspots for the last 120 Ma, the TPW motions can be estimated, which allows tying paleogeographic reconstructions to the mantle and hence constraining them in paleolongitude. For the earlier times in the Mesozoic and Paleozoic, TPW estimates can be obtained through the analysis of coherent rotations of the continental lithosphere, which allows linking the reconstructed paleogeography to the large-scale structures in the lower mantle, commonly referred to as Large Low Shear-wave Velocity Provinces (LLSVPs). It has been argued that the LLSVPs have been stable over at least the past 300 Ma, and possibly longer, and that the LLSVP margins have served as generation zones for the mantle plumes responsible for eruptions of Large Igneous Provinces (LIPs) and kimberlites. Correlating the reconstructed locations of LIPs and kimberlites with the margins of LLSVPs using the estimated TPW rotations makes it possible to develop a self-consistent model for plate motions relative to the mantle, true polar wander, and the corresponding changes of paleogeography constrained in longitude for the entire Phanerozoic, although the origin and long-term stability of LLSVPs are the subject of the ongoing scientific debate.

Apparent polar wander paths geometric parameterizations

Paleomagnetic Euler poles derived by geometrizing apparent polar wander paths (APWPs) potentially allows constraining paleolongitudes from paleomagnetic data. This method could extend absolute plate motion reconstructions deeply into the geologic history as long as there are reliable APWPs.

Hotspot tracks

The Hawaiian-Emperor seamount chain

The presence of chains of volcanic islands and seamounts interpreted to have formed from fixed hotspots allows the plate on which they sit to be progressively restored so that a seamount is moved back over the hotspot at its time of formation. This method can be used back to the Early Cretaceous, the age of the oldest evidence for hotspot activity. This method gives an absolute reconstruction of both latitude and longitude, although before about 90 Ma there is evidence of relative motion between hotspot groups.

Slab constraints

Once oceanic plates subduct in the lower mantle (slabs), they are assumed to sink in a near-vertical manner. With the help of seismic wave tomography, this can be used to constrain plate reconstructions at first order back to the Permian.

Other evidence for past plate configurations

Reconstruction of eastern Gondwana showing position of orogenic belts

Some plate reconstructions are supported by other geological evidence, such as the distribution of sedimentary rock types, the position of orogenic belts and faunal provinces shown by particular fossils. These are semi-quantitative methods of reconstruction.

Sedimentary rock types

Some types of sedimentary rock are restricted to certain latitudinal belts. Glacial deposits for instance are generally confined to high latitudes, whereas evaporites are generally formed in the tropics.

Faunal provinces

Oceans between continents provide barriers to plant and animal migration. Areas that have become separated tend to develop their own fauna and flora. This is particularly the case for plants and land animals but is also true for shallow water marine species, such as trilobites and brachiopods, although their planktonic larvae mean that they were able to migrate over smaller deep water areas. As oceans narrow before a collision occurs, the faunas start to become mixed again, providing supporting evidence for the closure and its timing.

Orogenic belts

When supercontinents break up, older linear geological structures such as orogenic belts may be split between the resulting fragments. When a reconstruction effectively joins up orogenic belts of the same age of formation, this provides further support for the reconstruction's validity.