Notes on "Influence of Atomic Kinetics on Inverse Bremsstrahlung Heating and Nonlocal Thermal Transport" Paper

XFEL&Plasma

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I. Atomic States and Rate Equation

I.1 Atomic states

The atomic state of an atom (or an ion) is determined by the state of its electrons. We start from the simple case of Hydrogen-like atom.

I.1-1 Atomic state of Hydrogen atom

The atomic state of the hydroden atoms is determined by its only electron. From quantum machenics we know that the electron state is determined by 4 quantum numbers: the principle quantum number $\bar n$, the angular $\bar l$, the megnetic $\bar m_l$ and the spin $\bar m_s$. The states are:

$\bar n$, $\bar l$	$\bar l=0$	$\bar l=1$	$\bar l=2$
$\bar n = 1$	$1 {\rm s}$ states
$\bar n = 2$	$2 {\rm s}$ states	$2 {\rm p}$ states
$\bar n = 3$	$3 {\rm s}$ states	$3 {\rm p}$ states	$3 {\rm d}$ states

$\cdots$

We should note that the components in this table contains more than one states. E.g., 3d contains the states of $\bar m_l$ ranging from $-2$ to $+2$, but in our study this is neglected, we treat these states as a same state.

For Hydrogen atoms, the angular $\bar l$, megnetic $\bar m_l$ and spin $\bar m_s$ quantum numbers do not affect the energy levels (neglecting LS coupling and other higher order effects). But strictly speeking, $\bar l$ will slightly affect the energy in the case of multi-electron ions. That is what we called sub-levels. Later we will not consider $\bar m_l$ and $\bar m_s$ any more.

An electron can be either on this state or that (it can also be at a superposotion state, which in our case is neglected). Therefore, each atomic state corresponds to an electron state.

I.1-2 Atomic states of multi-electron atoms

For multi-electron atoms we should determine each of the electrons. It is our assumption that the megnetic $\bar m$ and spin $\bar m_s$ quantum numbers do not affect the energy. So we only consider electron states determined by $\bar n$ and $\bar l$. Let's take two electrons as an example, we consider only $1 \rm s$ and $2\rm s$ states for electrons , there are three possible states for this atom:
$|1\rangle=|1 \rm s\rangle |1 \rm s\rangle$
$|2\rangle=|1 \rm s\rangle |2 \rm s\rangle$
$|3\rangle=|2 \rm s\rangle |2 \rm s\rangle$

This is interpreted as: state 1 is two electrons both at state $1 \rm s$, state 2 is one electron at state $1 \rm s$ and one at $2 \rm s$, state 3 is two electrons at state $2 \rm s$. Note that the electrons are identical.

Sometimes we should consider more electron states, therefore we might get states like $|2 \rm s\rangle |3 \rm d\rangle$, $|4 \rm p\rangle |5 \rm f\rangle$, etc.

if we have more eletrons we might have (e.g., 4) $|2 \rm p\rangle |3 \rm s\rangle^2 |4 \rm d\rangle$

I.1-3 Atomic state of ions

In a plasma some of the electrons are ionized, therefore the number of the electrons may vary. We take Mg as an example. Neutral Mg has 12 electrons, if 3 of them are ionized, then one possible state could be:
$|1s\rangle^2|2s\rangle^2|2p\rangle^4|3s\rangle^1|{\rm free}\rangle^3$

Given that one can count the number of bound electrons, the $|{\rm free}\rangle$ notation can be dropped.

I.2 Rate Equation

To investigate the population of each states, we should slove the rate equation.

I.2 Rate equation set-up

We are still taking an example of a two-electron atom with 2 electron states: $1 \rm s$ and $2\rm s$. The possible states are:
$|1\rangle=|1 \rm s\rangle |1 \rm s\rangle$
$|2\rangle=|1 \rm s\rangle |2 \rm s\rangle$
$|3\rangle=|2 \rm s\rangle |2 \rm s\rangle$

We denote the population of each state as $N_\bar i$. The transition rate from $\bar i$ to $\bar j$ is $K_{(\bar i\rightarrow\bar j)}$

Now we consider the population of state 1: the atoms are going from $| 1\rangle$ to $| 2\rangle$ and $| 3\rangle$ , while some others are going from $| 2\rangle$ and $| 3\rangle$ to $| 1\rangle$. The $K$ rates are defined as the probability of transition per unit of time, therefore the population from $|\bar i\rangle$ to $|\bar j\rangle$ within time ${\rm d}t$ is $N_{\bar i}K_{(\bar i\rightarrow \bar j)}{\rm d}t$. The population difference ${\rm d}N_1$ within this time interval is

$${\rm d}N_1=-[\#\ {\rm of\ atoms\ from |1\rangle\ to |2\rangle\ and\ |3\rangle}]+[\#\ {\rm of\ atoms\ from |2\rangle\ and |3\rangle\ to\ |3\rangle}]\\=-(N_{1}K_{(1\rightarrow 2)}{\rm d}t+N_{1}K_{(1\rightarrow 3)}{\rm d}t)+N_{2}K_{(2\rightarrow 1)}{\rm d}t+N_{3}K_{(3\rightarrow 1)}{\rm d}t$$

$$\frac{{\rm d}N_1}{{\rm d}t}=-N_1(K_{(1\rightarrow 2)}+K_{(1\rightarrow 3)})+N_2K_{(2\rightarrow 1)}+N_{3}K_{(3\rightarrow 1)}$$

Similarly,

$$\frac{{\rm d}N_2}{{\rm d}t}=N_1K_{(1\rightarrow 2)}-N_2(K_{(2\rightarrow 1)}+K_{(2\rightarrow 3)})+N_{3}K_{(3\rightarrow 2)}$$

$$\frac{{\rm d}N_3}{{\rm d}t}=N_1K_{(1\rightarrow 3)}+N_2K_{(2\rightarrow 3)}-N_{3}(K_{(3\rightarrow 1)}+K_{(3\rightarrow 1)})$$

This can be generalized to a rate equation for more than three states. If we consider a atomic model with $N$ electron states, and the total number of states is $N_0$, then,

$$\frac{{\rm d}N_i}{{\rm d}t}=\sum_{j=1}^{N_0} K_{\bar i,\bar j}N_{\bar j}$$

with

$$K_{\bar i,\bar j}=K_{(\bar i\rightarrow \bar j)} ({\forall\ \bar i\not =\bar j})\\K_{\bar i,\bar i}=-\left(\sum_{\bar j\not =\bar i}K_{(\bar i\rightarrow \bar j)}\right)$$

I.2-2 $K$ rates associated with bound electron transitions

The $K$ rates are closely related to the physical processes. Still taking the 2-electron 3-state atom as an example. The transition from $|1\rangle$ to $|2\rangle$ is just an electron going from $|1{\rm s}\rangle$ to $|2{\rm s}\rangle$. This should be proportional to the rate of electron transition from $|1{\rm s}\rangle$ to $|2{\rm s}\rangle$. This is also related to the occupation number of the electron states. We will simply explain this.

First we only consider processes with one bound electron transition. If the only physical process corresponds to atomic transition from $|\bar i\rangle$ to $|\bar i'\rangle$ is electron transition from $|\bar n,\bar l\rangle$ to $|\bar n', \bar l'\rangle$, and if the rate of one electron transition is $K'_{(|\bar n,\bar l\rangle\rightarrow|\bar n',\bar l'\rangle)}$, then the rate of atomic transition should be proportional to $K'_{(|\bar n,\bar l\rangle\rightarrow|\bar n',\bar l'\rangle)}N_{|\bar n,\bar l\rangle}$, given that the occupation number of electron state $|\bar n,\bar l\rangle$ is $N_{|\bar n,\bar l\rangle}$ (beacuse there are $N_{|\bar n,\bar l\rangle}$ electrons on the state $|\bar n,\bar l\rangle$ in the state $|\bar i\rangle$ ).

But we should also note that the the occupation number of electron state $|\bar n', \bar l'\rangle$ in $|\bar i'\rangle$ is also relevent. As a simple model we set this to be proportional to $1-\frac{N_{|\bar n',\bar l'\rangle}}{2(2\bar l'+1)}$ where $2(2\bar l'+1)$ is the maximum occupation number of the state $|\bar n', \bar l'\rangle$ (The magnetic quntum number $\bar m_{l'}$ range from $-\bar l'$ to $\bar l'$, hence we have $2\bar l' +1$ choices. With each choice the spin quantum number $\bar m_s$ can be $\pm \frac{1}{2}$, therefore 2 states for each $\bar m_l$. therefore we have $2(2\bar l'+1)$ electron states that can be occupied). This is because the occupied states can not be occupied again be the transit electron due to the Pauli exclusion principle. Electron can only transit to the Vancency and that ratio is $1-\frac{N_{|\bar n',\bar l'\rangle}}{2(2\bar l'+1)}$.

If more than one electron transition is invovled then the rate should be the product of the electron transitions.

I.2-3 $K$ rates associated with free electrons

Here we only discussed the situiation of bound electrons. To consider the free electrons we should take ionisation and recombination into account. For instence, if we are investigating a 5 electron atom, and the ion states are:

$|1\rangle=|1{\rm s\rangle}^2|2{\rm s}\rangle$
$|2\rangle=|1{\rm s\rangle}^2|2{\rm s}\rangle^2$

clearly, one free electron is captured by the ion. The rate of this process should be:

$$K= \int_\vec v v\varrho[\vec v]S[\vec v]{\rm d}v_x{\rm d}v_y{\rm d}v_z$$

Where $\varrho[\vec v]$ is the electron distribution fuction and $S[\vec v]$ is the cross section of this process. This could be explained by the following model. Since $\varrho[\vec v]$ is the electron distribution fuction, we have:

$$[{\rm \#\ of\ electrons\ having\ velocity}\ \vec v {\rm \ with\ interval\ d}\vec v{\rm \ per\ unit\ of\ volumn}] \\=varrho[\vec v]{\rm d}v_x{\rm d}v_y{\rm d}v_z$$

The cross section physically denoted the area where the electrons with velocity $\vec v$ are captured. If the electrons all have velocity $\vec v$, then within time interval ${\rm d}t$, the electrons in the volumn of a sylinder with hight $v{\rm d}t$ and Area $S[\vec v]$ will all be captured. Therefore

$$[{\rm \#\ of\ electrons\ }{\rm having\ velocity}\ \vec v {\rm \ with\ interval\ d}\vec v{\rm \ being\ captured\ within\ d}t ] \\=\varrho[\vec v]{\rm d}v_x{\rm d}v_y{\rm d}v_z\ vS[\vec v]{\rm d}t$$

We should take all kinds of electrons into account, therefore we should intergrate it over $\vec v$, and that the rate we want.

I.2-4 Bound transition related to free electrons

We've considered the transition of bound electrons, but we didn't actually discuss its physical process. We get back to the example of ion transition:
$|\bar i\rangle\rightarrow|\bar i'\rangle$
which is just the transition of electron:
$|\bar n,\bar l\rangle\rightarrow|\bar n',\bar l'\rangle$

Suppose the energy of later one is higher, therefore the electron should gether energy to transit. The energy can either come from a photon or an free electron. If a free electron collides with the ion on the state $|\bar i\rangle$ and transfer energy to the electron on $|\bar n,\bar l\rangle$ causing a transition to $|\bar n',\bar l'\rangle$ which leads the ion to state $|\bar i'\rangle$ then it is called a collisional excitation. This is how bound transition related to free electrons. This process also follows the rule derived in the last subsection.

$$K'_{(|\bar n,\bar l\rangle\rightarrow|\bar n',\bar l'\rangle,{\rm CE})}= \int_\vec v v\varrho[\vec v]S_{(|\bar n,\bar l\rangle\rightarrow|\bar n',\bar l'\rangle,{\rm CE})}[\vec v]{\rm d}v_x{\rm d}v_y{\rm d}v_z$$

I.3 Coupled rate equation

The rate equation is so called coupled because $K$ rates are related to the distribution function. Additionally, if a electronsis ionised, recombined, or scattered by the ion, it will still affect the distrion function.

I.3-1 Vlasov-Fokker-Planck equation

The evolution of the distribution function is discibed by the Vlasov-Fokker-Planck equation:

$$\frac{\partial \varrho}{\partial t}+ \vec \nabla \varrho \cdot\frac{\vec p}{m}+\vec \nabla_{\vec p}'\varrho\cdot \vec F=\left(\frac{{\rm d} \varrho}{{\rm d}t}\right)_\rm c$$

Where $\left(\frac{{\rm d} \varrho}{{\rm d}t}\right)_{\rm c}$ is the collisional operator, and

$$\left(\frac{{\rm d} \varrho}{{\rm d}t}\right)_{\rm c}=$$