Merge branch 'main' of https://github.com/PmagPy/Essentials-JupyterBook

Author: Swanson-Hysell

book/chapters/appendix.md

@@ -95,7 +95,7 @@ Paleomagnetism is famous for its use of a large number of incomprehensible acron
 | $\mathbf{H}$ | Magnetic field: [](#sect:H) |
 | $H_{cr}$ | Coercivity of remanence; field required to reduce saturation IRM to zero: [](#sect:flipping) |
 | $H_c$ | Coercivity; the magnetic field required to change the magnetic moment of a particle from one easy axis to another: [](#sect:flipping) |
-| $k$ | Boltzmann's constant (1.381 x 10$^{-23}$ JK$^{-1}$): [](#sect:para) |
+| $k_B$ | Boltzmann's constant (1.381 x 10$^{-23}$ JK$^{-1}$): [](#sect:para) |
 | $K_i$ | AMS measurement: [](#app:K15) |
 | $K_u$ | Constant of uniaxial anisotropy energy: [](#sect:K1) and [](#sect:shape) |
 | $\mathbf{m}$ | Magnetic moment: [](#sect:moment) |
@@ -156,7 +156,7 @@ Paleomagnetism is famous for its use of a large number of incomprehensible acron
 Here we derive the Langevin function for a paramagnetic substance with magnetic moments $m$ in an applied field $H$ at temperature $T$. If we make the assumption that there is no preferred alignment within the substance, we can assume that the number of moments ($n(\alpha)$) between angles $\alpha$ and $\alpha + d\alpha$ with respect to $\mathbf{H}$ is proportional to the solid angle $\sin\alpha d\alpha$ and the probability density function, i.e.
 
 $$
-n(\alpha) d\alpha \propto \exp \bigl({ -E_m\over {kT}} \bigr) \sin \alpha d\alpha,
+n(\alpha) d\alpha \propto \exp \bigl({ -E_m\over {k_BT}} \bigr) \sin \alpha d\alpha,
 $$ (eq:nalpha)
 
 where $E_m$ is the magnetic energy. When we measure the induced magnetization, we really measure only the component of the moment parallel to the applied field, or $n(\alpha) m \cos\alpha$. The net induced magnetization $M_I$ of a population of particles with volume $v$ is therefore:
@@ -178,10 +178,10 @@ $$
 $$
 
 $$
-= { {\int_0^{\pi} e^{(m\mu_o H \cos \alpha )/kT}\cos \alpha \sin \alpha d\alpha}\over { \int_0^{\pi} e^{(m\mu_o H\cos \alpha )/kT}\sin \alpha d\alpha}}.
+= { {\int_0^{\pi} e^{(m\mu_o H \cos \alpha )/k_BT}\cos \alpha \sin \alpha d\alpha}\over { \int_0^{\pi} e^{(m\mu_o H\cos \alpha )/k_BT}\sin \alpha d\alpha}}.
 $$
 
-By substituting $a=m\mu_oH/kT$ and $\cos \alpha =x$, we write
+By substituting $a=m\mu_oH/k_BT$ and $\cos \alpha =x$, we write
 
 $$
 {M\over {M_s}} = N { {\int_{-1}^{1} e^{a x}xdx} \over {\int_{-1}^1 e^{a x}dx} } = \bigl( { {e^{a} + e^{-a}} \over {e^{a} - e^{-a}} } - {1\over{a} } \bigr),
@@ -199,7 +199,7 @@ $$ (eq:Langapp)
 The derivation of superparamagnetism follows closely that of paramagnetism whereby the probability of finding a magnetization vector an angle $\alpha$ away from the direction of the applied field is given by:
 
 $$
-n(\alpha )d\alpha = 2\pi n_o e^{({{M_sBv\cos \alpha}\over {kT}})}\sin \alpha d\alpha.
+n(\alpha )d\alpha = 2\pi n_o e^{({{M_sBv\cos \alpha}\over {k_BT}})}\sin \alpha d\alpha.
 $$ (eq:ccnalpha)
 
 The total magnetization contributed by the $N$ moments is:
@@ -215,10 +215,10 @@ $$
 $$
 
 $$
-= N { {\int_0^{\pi} e^{(M_sBv\cos \alpha )/kT}\cos \alpha \sin \alpha d\alpha}\over { \int_0^{\pi} e^{(M_sBv\cos \alpha )/kT}\sin \alpha d\alpha}}.
+= N { {\int_0^{\pi} e^{(M_sBv\cos \alpha )/k_BT}\cos \alpha \sin \alpha d\alpha}\over { \int_0^{\pi} e^{(M_sBv\cos \alpha )/k_BT}\sin \alpha d\alpha}}.
 $$
 
-By substituting $a= M_sBv/kT$ and $\cos \alpha =x$, and remembering [Equation %s](#eq:Langapp), we can write:
+By substituting $a= M_sBv/k_BT$ and $\cos \alpha =x$, and remembering [Equation %s](#eq:Langapp), we can write:
 
 $$
 {M\over {M_s}} = N { {\int_1^{-1} e^{a x}xdx} \over {\int_1^{-1} e^{a x}dx} } = N\mathcal{L} (a).

book/chapters/chapter3.md

@@ -132,7 +132,7 @@ We learned in [Chapter 1](#chap:physics) that the proportionality between induce
 :alt: Two plots: (a) Langevin function M/Ms rising from zero and saturating near 1 versus a, and (b) M/Ms versus T showing inverse-temperature Curie law decay.
 :width: 100%
 
-a) Paramagnetic magnetization (obtained from the Langevin function $\mathcal{L}(a)$ versus $a= mB/kT$.) b) Paramagnetic magnetization as a function of temperature (Curie Law).
+a) Paramagnetic magnetization (obtained from the Langevin function $\mathcal{L}(a)$ versus $a= mB/k_BT$.) b) Paramagnetic magnetization as a function of temperature (Curie Law).
 :::
 
 (sect:para)=
@@ -154,14 +154,14 @@ $$ (eq:Em)
 
 Magnetic energy is at a minimum when the magnetic moment is lined up with the magnetic field.
 
-4. There is competition between the magnetic energy $E_m$ and the thermal energy $kT$ where $k$ is Boltzmann's constant (1.38 × 10$^{-23}$ m$^2$ kg s$^{-2}$ K$^{-1}$) and $T$ is temperature in kelvin.
+4. There is competition between the magnetic energy $E_m$ and the thermal energy $k_BT$ where $k_B$ is Boltzmann's constant (1.38 × 10$^{-23}$ m$^2$ kg s$^{-2}$ K$^{-1}$) and $T$ is temperature in kelvin.
 
 Consider an atomic magnetic moment, ($m$ = 2$m_b$ = 1.85 × 10$^{-23}$ Am$^2$), in a magnetic field of 10$^{-2}$ T, (for reference, the largest geomagnetic field at the surface is about 65 $\mu$T — see [Chapter 2](#chap:geomag)). The aligning energy is therefore $mB$ = 1.85 × 10$^{-25}$ J. However, thermal energy at 300K (traditionally chosen as a temperature close to room temperature providing easy arithmetic) is Boltzmann's constant times the temperature, or about 4 × 10$^{-21}$ J. So thermal energy is several orders of magnitude larger than the aligning energy and the net magnetization is small even in this rather large (compared to the Earth's field) magnetizing field.
 
 Using the principles of statistical mechanics, we find that the probability density of a particular magnetic moment having a magnetic energy of $E_m$ is given by:
 
 $$
-P(E) \propto \exp (-E_m/kT).
+P(E) \propto \exp (-E_m/k_BT).
 $$ (eq:PE)
 
 From this we see that the degree of alignment depends exponentially on the ratio of magnetic energy to thermal energy. The degree of alignment with the magnetic field controls the net magnetization $M$. When spins are completely aligned, the substance has a *saturation magnetization* $M_s$. The probability density function leads directly to the following relation (derived in [](#app:langevin)):
@@ -170,18 +170,18 @@ $$
 {M\over {M_s}} = \left[\coth a - {1\over{a}}\right]=\mathcal{L}(a).
 $$ (eq:Lang)
 
-where $a=mB/kT$. The function enclosed in square brackets is known as the *Langevin function* ($\mathcal{L}$).
+where $a=mB/k_BT$. The function enclosed in square brackets is known as the *Langevin function* ($\mathcal{L}$).
 
-[Equation %s](#eq:Lang) is plotted in [](#fig:para)a and predicts several intuitive results: 1) $M = 0$ when $B=0$ and 2) $M/M_s = 1$ when the applied magnetic field is infinite. Furthermore, $M$ is some 90% of $M_s$ when $mB$ is some 10-20 times $kT$. When $kT>> mB$, $\mathcal{L}(a)$ is approximately linear with a slope of $\sim 1/3$. At room temperature and fields up to many tesla, $\mathcal{L}(a)$ is approximately $mB/3kT$. If the moments are unpaired spins ($m=m_b$), then the maximum magnetization possible ($M_s$) is given by the number of moments $N$, their magnitude ($m_b$) normalized by the volume of the material $v$ or $M_s=Nm_b/v$, and
+[Equation %s](#eq:Lang) is plotted in [](#fig:para)a and predicts several intuitive results: 1) $M = 0$ when $B=0$ and 2) $M/M_s = 1$ when the applied magnetic field is infinite. Furthermore, $M$ is some 90% of $M_s$ when $mB$ is some 10-20 times $k_BT$. When $k_BT>> mB$, $\mathcal{L}(a)$ is approximately linear with a slope of $\sim 1/3$. At room temperature and fields up to many tesla, $\mathcal{L}(a)$ is approximately $mB/3k_BT$. If the moments are unpaired spins ($m=m_b$), then the maximum magnetization possible ($M_s$) is given by the number of moments $N$, their magnitude ($m_b$) normalized by the volume of the material $v$ or $M_s=Nm_b/v$, and
 
 $$
-{M\over{M_s}} \simeq { { m_b \mu_o }\over {3kT} } H .
+{M\over{M_s}} \simeq { { m_b \mu_o }\over {3k_BT} } H .
 $$
 
 Please note that we have neglected all deviations from isotropy including quantum mechanical effects as well as crystal shape, lattice defects, and state of stress. These complicate things a little, but to first order the treatment followed here provides a good approximation. We can rewrite the above equation as:
 
 $$
-{M\over H} = {{m_b\mu_o}\over {3kT}}\cdot M_s = {{Nm_b^2\mu_o}\over{3kv}}\cdot {1\over T} = \chi_p.
+{M\over H} = {{m_b\mu_o}\over {3k_BT}}\cdot M_s = {{Nm_b^2\mu_o}\over{3k_Bv}}\cdot {1\over T} = \chi_p.
 $$ (eq:chip)
 
 To first order, paramagnetic susceptibility $\chi_p$ is positive, larger than diamagnetism and inversely proportional to temperature. This inverse T dependence (see [](#fig:para)b) is known as Curie's law of paramagnetism. The paramagnetic susceptibility of, for example, biotite is 790 × 10$^{-9}$ m$^3$ kg$^{-1}$, or about three orders of magnitude larger than quartz (and of the opposite sign!).
@@ -227,16 +227,16 @@ $$
 H_{tot} = H + H_w = H + \beta M,
 $$
 
-where $H$ is the external field. By analogy to paramagnetism, we can substitute $a=\mu_om_b(H_{tot})/kT)$ for $H$ in the Langevin function:
+where $H$ is the external field. By analogy to paramagnetism, we can substitute $a=\mu_om_b(H_{tot})/k_BT)$ for $H$ in the Langevin function:
 
 $$
-{M \over {M_s}} {= \mathcal{L} \left({ {\mu_o m_b(H+\beta M)}\over{kT} } \right)}.
+{M \over {M_s}} {= \mathcal{L} \left({ {\mu_o m_b(H+\beta M)}\over{k_BT} } \right)}.
 $$ (eq:Mweiss)
 
 For temperatures above the Curie temperature $T_c$ (i.e. $T-T_c>0$) there is by definition no internal field, hence $\beta M$ is zero. Substituting $N m_b/v$ for $M_s$, and using the low-field approximation for $\mathcal{L} (a)$, [Equation %s](#eq:Mweiss) can be rearranged to get:
 
 $$
-{M\over H} = { {\mu_o N m_b^2}\over {v3k(T-T_c)} } \equiv \chi_f.
+{M\over H} = { {\mu_o N m_b^2}\over {v3k_B(T-T_c)} } \equiv \chi_f.
 $$ (eq:curieweiss)
 
 [Equation %s](#eq:curieweiss) is known as the Curie-Weiss law and governs ferromagnetic susceptibility above the Curie temperature (dashed line in [](#fig:MsT)).
@@ -252,13 +252,13 @@ Various data sets for the behavior of $M_s(T)$ for magnetite.
 Below the Curie temperature $H_w>>H$; we can neglect the external field $H$ and get:
 
 $$
-{M\over {M_s}} = \mathcal{L} \left( {{\mu_o m_b \beta M}\over{kT}}\right).
+{M\over {M_s}} = \mathcal{L} \left( {{\mu_o m_b \beta M}\over{k_BT}}\right).
 $$
 
 Substituting again for $M_s$ and rearranging, we get:
 
 $$
-{M\over M_s} = \mathcal{L} \left( {{Nm_b^2 \beta}\over{vkT}}\cdot {M\over M_s} \right) = \mathcal{L} \left( {T_c \over T} \cdot {M\over M_s} \right),
+{M\over M_s} = \mathcal{L} \left( {{Nm_b^2 \beta}\over{vk_BT}}\cdot {M\over M_s} \right) = \mathcal{L} \left( {T_c \over T} \cdot {M\over M_s} \right),
 $$ (eq:Mferro)
 
 where $T_c$ is the Curie temperature and is given by:

book/chapters/chapter4.md

@@ -680,7 +680,7 @@ Another common method for imaging magnetic domains employs a technique known as
 We have gone some way toward answering the questions posed at the beginning of the chapter. We see now that anisotropy energy, with contributions from crystal structure, shape and stress, inhibits changes in the magnetic direction thereby offering a possible mechanism whereby a given magnetization could be preserved for posterity. We also asked the question of what allows the magnetization to come into equilibrium with the applied magnetic field in the first place; this question requires a little more work to answer. The key to this question is to find some mechanism which allows the moments to "jump over" magnetic anisotropy energy barriers. One such mechanism is thermal energy $E_T$, which was given in [Chapter 3](#chap:inducedremanent) as:
 
 $$
-E_T = kT.
+E_T = k_BT.
 $$
 
 We know from statistical mechanics that the probability $P$ of finding a grain with a given thermal energy sufficient to overcome some anisotropy energy $E_a$ and change from one easy axis to another is $P=\exp (-E_a/E_T )$. Depending on the temperature, such grains may be quite rare, and we may have to wait some time $t$ for a particle to work itself up to jumping over the energy barrier.
@@ -691,15 +691,25 @@ $$
 M(t) = M_o \exp \left(\frac{-t}{\tau}\right),
 $$ (eq:MvT)
 
-where $t$ is time and $\tau$ is an empirical constant called the *relaxation time*. Relaxation time is the time required for the remanence to decay to $1/e$ of $M_o$. This equation is the essence of what is called *Néel theory* (see, e.g., {cite}`neel1955`). The value of $\tau$ depends on the competition between magnetic anisotropy energy and thermal energy. It is a measure of the probability that a grain will have sufficient thermal energy to overcome the anisotropy energy and switch its moment. Therefore in zero external field:
+where $t$ is time and $\tau$ the *relaxation time*. Relaxation time is the time required for the remanence to decay to $1/e$ of $M_o$. This equation is the essence of what is called *Néel theory* (see {cite}`neel1955`). 
+
+:::{figure} ../figures/chapter4/neel_exponential_decay.png
+:name: fig:neel_relaxation
+:alt: Exponential decay curve of normalized magnetization M/Mo versus time normalized by relaxation time. Dashed lines mark the point where magnetization has decayed to Mo/e at time t equal to tau.
+:width: 70%
+
+Magnetic relaxation in an assemblage of single domain ferromagnetic grains. The initial magnetization $M_o$ decays exponentially, falling to $1/e$ (~37%) of its original strength in time $\tau$. For example, for grains with a relaxation time ($\tau$) of 100 s, 63% of the original magnetization is lost within 100 seconds.
+:::
+
+The value of the relaxation time ($\tau$) depends on the competition between magnetic anisotropy energy and thermal energy. It is a measure of the probability that a grain will have sufficient thermal energy to overcome the anisotropy energy and switch its moment. Therefore in zero external field:
 
 $$
-\tau = \frac{1}{C} \exp \frac{Kv}{kT},
+\tau = \frac{1}{C} \exp \frac{Kv}{k_BT},
 $$ (eq:tau)
 
 where $C$ is a frequency factor with a value of something like $10^{10}$ s$^{-1}$. The anisotropy energy is given by the dominant anisotropy parameter $K$ (either $K_u, K_1$, or $\lambda$) times the grain volume $v$.
 
-Thus, the relaxation time is proportional to anisotropy constant and volume, and is inversely related to temperature. Relaxation time $\tau$ varies rapidly with small changes in $v$ and $T$. To see how this works, we can take $K_u$ for slightly elongate cuboids of magnetite (length to width ratio of 1.3 to 1) and evaluate relaxation time as a function of particle width (see [](#fig:tauvd)). There is a sharp transition between grains with virtually no stability ($\tau$ is on the order of seconds) and grains with stabilities of billions of years.
+Thus, the relaxation time is proportional to anisotropy constant and volume, and is inversely related to temperature. In [Chapter 7](#chap:remanence), we will see how these relationships are central to rock's acquiring (and losing) magnetization. Relaxation time $\tau$ varies rapidly with small changes in $v$ and $T$. To see how this works, we can take $K_u$ for slightly elongate cuboids of magnetite (length to width ratio of 1.3 to 1) and evaluate relaxation time as a function of particle width (see [](#fig:tauvd)). There is a sharp transition between grains with virtually no stability ($\tau$ is on the order of seconds) and grains with stabilities of billions of years.
 
 :::{figure} ../figures/chapter4/tauvd.png
 :name: fig:tauvd

book/chapters/chapter5.md

@@ -171,40 +171,40 @@ The most direct way to measure coercivity of remanence is the *backfield experim
 (sect:SP)=
 ### Superparamagnetic particles
 
-In superparamagnetic (SP) particles, the total magnetic energy $E_t=\epsilon_tv$ (where $v$ is volume) is balanced by thermal energy $kT$. This behavior can be modeled using statistical mechanics in a manner similar to that derived for paramagnetic grains in [Chapter 3](#chap:inducedremanent). In fact,
+In superparamagnetic (SP) particles, the total magnetic energy $E_t=\epsilon_tv$ (where $v$ is volume) is balanced by thermal energy $k_BT$. This behavior can be modeled using statistical mechanics in a manner similar to that derived for paramagnetic grains in [Chapter 3](#chap:inducedremanent). In fact,
 
 $$
 \frac{M}{M_s} = N\left(\coth \gamma - \frac{1}{\gamma}\right),
 $$ (eq:Lang1)
 
-where $\gamma = M_sBv/(kT)$ and $N$ is the number of particles of volume $v$, is a reasonable approximation. The end result, [Equation %s](#eq:Lang1), is the familiar Langevin function from our discussion of paramagnetic behavior (see [Chapter 3](#chap:inducedremanent)); hence the term "superparamagnetic" for such particles.
+where $\gamma = M_sBv/(k_BT)$ and $N$ is the number of particles of volume $v$, is a reasonable approximation. The end result, [Equation %s](#eq:Lang1), is the familiar Langevin function from our discussion of paramagnetic behavior (see [Chapter 3](#chap:inducedremanent)); hence the term "superparamagnetic" for such particles.
 
 :::{figure} ../figures/chapter5/loops.png
 :name: fig:loops
 :alt: Two-panel figure: (a) Langevin function curve of M/Ms versus gamma showing S-shaped reversible magnetization with B90 marked; (b) log-scale plot of B90 versus particle size d, showing steep increase below about 10 nm.
 :width: 80%
 
-a) The contribution of SP particles with saturation magnetization $M_s$ and cubic edge length $d$. $\gamma = BM_s d^3/kT$. There is no hysteresis. b) The field at which the magnetization reaches 90% of the maximum $B_{90}$ is when $M_s d^3/kT\simeq 10$. [Figure from {cite}`tauxe1996`.]
+a) The contribution of SP particles with saturation magnetization $M_s$ and cubic edge length $d$. $\gamma = BM_s d^3/k_BT$. There is no hysteresis. b) The field at which the magnetization reaches 90% of the maximum $B_{90}$ is when $M_s d^3/k_BT\simeq 10$. [Figure from {cite}`tauxe1996`.]
 :::
 
 The contribution of SP particles for which the Langevin function is valid for given values of $M_s$ and $d$ is shown in [](#fig:loops)a. The field at which the population reaches 90% saturation $B_{90}$ occurs at $\gamma \sim 10$. Assuming particles of magnetite ($M_s$ = 480 kAm$^{-1}$) and room temperature ($T=300$ K), $B_{90}$ can be evaluated as a function of $d$ (see [](#fig:loops)b). Because of its inverse cubic dependence on $d$, $B_{90}$ rises sharply with decreasing $d$ and is hundreds of tesla for particles a few nanometers in size, approaching paramagnetic values. $B_{90}$ is a quick guide to the SP slope (the SP susceptibility $\chi_{sp}$) contributing to the hysteresis response and was used by {cite}`tauxe1996` as a means of explaining distorted loops sometimes observed for populations of SD/SP mixtures. $B_{90}$ (and $\chi_{sp}$) is very sensitive to particle size with very steep slopes for the particles at the SP/SD threshold. The exact threshold size is still rather controversial, but {cite}`tauxe1996` argue that it is ~20 nm.
 
 For low magnetic fields, the Langevin function can be approximated as $\sim \frac{1}{3} \gamma$. So we have:
 
 $$
-\frac{M}{M_s} = \frac{1}{3} \frac{M_sBv}{kT}.
+\frac{M}{M_s} = \frac{1}{3} \frac{M_sBv}{k_BT}.
 $$
 
 If we substitute $\mu_o H$ for $B$ and rearrange this equation, we can get the superparamagnetic susceptibility $\chi_{sp}$ as:
 
 $$
-\frac{M}{H} = \frac{\mu_o M_s^2v}{3kT}.
+\frac{M}{H} = \frac{\mu_o M_s^2v}{3k_BT}.
 $$ (eq:chiSP)
 
 We can rearrange [Equation %s](#eq:tau) in [Chapter 4](#chap:anisotropy) to solve for the volume at which a uniaxial grain passes through the superparamagnetic threshold:
 
 $$
-v_b = \frac{kT \ln (C\tau)}{K_u}.
+v_b = \frac{k_BT \ln (C\tau)}{K_u}.
 $$
 
 Finally, we can substitute this volume into [Equation %s](#eq:chiSP) as the maximum volume of an SP grain, giving us: