#### 2.1. Equations of Motion and Pseudopotential

If the total number of ions in the trap is

N, then, in SI units, the equation of motion of ion number

i in an ideal 3DPT in the presence of the other

$N-1$ ions is [

16]

where

$\Gamma $ is the damping constant,

${U}_{0}$ and

${V}_{0}$ are the dc and ac voltages applied to the trap, respectively,

${r}_{0}$ and

${z}_{0}$ are the radius of the ring-electrode and the distance from the trap’s center to the end-cap electrodes, respectively, and

${\u03f5}_{0}$ is the permittivity of the vacuum. Damping does not have any visible effect on the shapes of the loading curves to be discussed in

Section 2.4 nor does it alter the shapes of the ion densities discussed in

Section 2.5 (see also [

13]). Therefore, we do not consider damping and, in what follows, set

$\Gamma =0$. Collisions with background gas are also not included in our simulations since experimentally it is always possible to reduce the rate of these collisions to negligible levels.

Introducing the dimensionless trap control parameters [

16]

the unit of length [

17]

and the unit of time

turns (

1) (with

$\Gamma =0$) into the set of dimensionless equations

which, together with the loading process, form the basis of our molecular-dynamics simulations (see

Section 3). The pseudopotential is obtained by averaging over the fast oscillations of the rf trap field. It is given approximately, but explicitly, by [

16]

where

${\mu}_{x}$ and

${\mu}_{z}$ are the dimensionless pseudo-oscillator frequencies

In SI units, we have [

16]

We are exclusively focusing on the case of a spherically symmetric pseudopotential, obtained for

which results in

We now assume that the trap is filled homogeneously with charge of charge-density

${\rho}^{pp}$. In pseudo-potential approximation we compute

${\rho}^{pp}$ in the following way. Denoting by

${Q}_{\mathrm{encl}}$ the enclosed charge, an ion a distance

r away from the trap’s center experiences a repulsive Coulomb force

which needs to be counterbalanced by the trap force

Thus, from

${F}_{\mathrm{Coul}}={F}_{\mathrm{trap}}$, we obtain, in SI units,

In dimensionless units, we define

${\widehat{\rho}}^{pp}$ as the number of ions per unit volume,

${l}_{0}^{3}$, that is,

Since any physical trap is limited in size by electrodes, other absorbing structures or dynamical processes such as the onset of chaos [

14], or higher-order rf multipoles [

18], we define the effective trap radius

${R}_{\mathrm{cut}}$ in SI units, beyond which ions are considered absorbed, that is, lost from the trap. Defining the dimensionless trap radius,

the expected number of ions in the trap in the saturated regime is

It is convenient to relate the actual number,

${N}_{s}$, of ions in the saturated trap to the respective number of ions in the corresponding saturated pseudopotential. For this purpose we define the ratio

#### 2.2. Trapped Electrons?

The ionization process in the loading region at r∼0 generates electrons at near zero kinetic energy inside of the trap. What is their fate? Are they going to be trapped or expelled from the trap?

An electron generated inside of the trap is under the influence of the trap field and the field produced by the space charge of the trapped ions. Assuming a completely filled psudopotenital (see

Section 2.1), the magnitude of the confining force acting on an electron due to space charge is the same as

${F}_{\mathrm{Coul}}$ in (

13), that is,

For electrons, trap fields that confine ions may be considered as very slowly varying. This is also clear from the fact that the value of the trap control-parameter

q for electrons,

${q}_{e}$, is related to the value of the ionic trap control-parameter,

q, according to

which corresponds to a strongly de-confining trap. Therefore, we may consider the trap fields acting on the electrons as quasi static, and consider only the trap field with maximum amplitude as being relevant for deciding the question of whether electrons are confined or de-confined in the combined fields of trap and ionic space charge. According to (

1) the maximum trap force acting on an electron in

x direction is

The maximal force occurs if

$x=r$. In this case, with

$a={q}^{2}/2$, we have

The ratio of the confining force,

${F}_{\mathrm{conf}}$, and the de-confining force,

${F}_{\mathrm{max}}$, is

since for stable 3DPT operation we need

$q<1$ [

16]. This shows that under the assumption of a filled pseudo-potential, the confining force due to the ionic space charge is always smaller than the maximum of the de-confining force. This implies that as soon as they are created, the electrons are swept out of the trap within at most half of a cycle of the applied rf trap field.

In our derivation of the inequality (

24) we considered only the force in

x direction. However, since the force in

y direction is the same as the force in

x direction, and since, according to (

1), the force in

z direction in a 3DPT is twice as strong as the force in

x direction, this does not change the result that the electrons are lost from the trap essentially as fast as they are created. Thus, in both, actual trap-loading experiments and in our numerical simulations, it is safe to completely neglect the influence of the electrons generated in the photoionization process. However, the inequality (

24) is not very strong in the sense that an ion plasma with a density exceeding the pseudopotential equilibrium density by a factor larger than

$2/\beta =(4/q)+1$, the factor 2 accounting for the twice stronger de-confining force in

z direction, is capable of trapping electrons. As shown in

Section 2.5, such densities may indeed be found in the fast-loading regime in the spatial loading region close to the center of the trap, in particular for large

q. Investigating this possibility further is beyond the scope of this paper, but is a good topic for future research. It is also possible that the electrons, on their way out of the trap, have sufficient energy to cause secondary ionizations by electron-neutral collisions. These events may be exceedingly rare, but may still deserve some further attention, if only to rule out this ionization process.

#### 2.3. Insertion Heating

Whenever a new ion is created in the loading region of the trap, it is inserted into an already existing charge distribution formed by the trapped ions. In pseudo-potential approximation we may assume that the ion charge density is constant [see (

15)]. Thus, inserting an additional ion into this space-charge cloud a distance

r away from the trap’s center costs the insertion energy [

19]

In the loading zone, that is,

$r\ll {R}_{\mathrm{cut}}$, we have

which means that each ion insertion adds an amount of energy

to the trapped ion cloud. Defining

$\lambda $ as the number of ions loaded per rf cycle, the insertion heating rate,

$\eta $, is given by

where we introduced the unit of energy

in order to connect with the unit of energy used in [

17]. The rf heating rate

H in [

17] is dimensionless, expressed in units of

${E}_{0}/{t}_{0}$. In order to compare our insertion heating rate with the dimensionless heating rate

H in [

17], we define the dimensionless insertion heating rate

In [

17],

H is graphed for various ion numbers in the trap. Therefore, it is convenient to express

$\widehat{\eta}$ with the help of the pseudo-potential saturation number

${N}_{s}^{pp}$ [defined in (

18)]:

For the case

$q=0.2$, shown in

Figure 1 of [

17], this can be simplified further to

At

$q=0.2$, the maximal rf heating rate

${H}^{\mathrm{max}}$ in [

17] is well approximated by

resulting in the ratio

of the two heating rates. Assuming

$\lambda =1$, a loading rate shown in [

13] to be located in the plateau region II of the loading curve for

$q=0.2$, we obtain

$\widehat{\eta}/{H}^{\mathrm{max}}=11.1$, 5.5, 2.8, and 1.1 for

${N}_{s}^{pp}=50$, 100, 200, and 500, respectively. We see that in these cases the insertion heating rates are systematically larger than their corresponding rf heating counterparts, more than an order of magnitude on the low ion-number side. We also see that the insertion-heating advantage is reduced with increasing ion number

${N}_{s}^{pp}$. Nevertheless, for the cases studied here, insertion heating is certainly the dominant heating mechanism. For

${N}_{s}=50$, and reducing the loading rate one order of magnitude to

$\lambda \sim 0.1$, the two heating rates are of the same order of magnitude, and rf heating dominates for

$\lambda \sim 0.01$. This range of loading rates is, in fact, part of the range of loading rates studied in detail in the following

Section 2.4.

#### 2.4. Saturation Curves

So far, we studied the number

${N}_{s}$ of ions in the saturation regime in the context of the static pseudo-potential approximation. We now switch on the rf term in (

1) and study the steady-state ion capacity,

${N}_{s}\left(\lambda \right)$, under saturation conditions for the combined time-dependent processes of rf drive and ion loading at a loading rate

$\lambda $, where

$\lambda $ is the number of ions loaded per rf cycle.

Actual hybrid traps contain of the order of

${10}^{5},\dots ,{10}^{6}$ ions [

14]. While it may be possible to simulate the loading and saturation dynamics of such a trap for a single choice of trap control parameters and loading rates, detailed simulations for ranges of parameters, as, for example, presented in this subsection, are impractical to do for more than ∼

${10}^{3}$ stored ions with the available computer equipment. Therefore, we need to keep the number of particles down so that extensive simulations can be performed in a reasonable time. A good compromise was to scale down the actual experimental trap dimensions by about a factor 10 and define our model trap with a loading-zone radius of

${\widehat{R}}_{L}=3$ and a trap size that we vary in steps of 5 from

${\widehat{R}}_{\mathrm{cut}}=15$ to

${\widehat{R}}_{\mathrm{cut}}=30$. In

Figure 1 we show examples of loading curves, similar to the ones obtained in [

13,

14], for

${\widehat{R}}_{\mathrm{cut}}=15$ and loading rates ranging from

$\lambda =0.01$ to

$\lambda =100$. In contrast to the loading curves in [

13,

14], which show ion saturation numbers

${N}_{s}\left(\lambda \right)$ as a function of loading rate

$\lambda $,

Figure 1 shows scaled ion numbers

where

${N}_{s}^{pp}$, the ion number expected in the pseudo-potential approximation, is defined in (

18) and

${N}_{s}\left(\lambda \right)$ is the actual number of ions in the trap according to our simulations. Scaling is convenient since this way we are able to show loading curves for a wide range of trap parameters (

$q=0.1,\dots ,0.4$) in the same frame. The scaling also provides us with the advantage of a direct comparison of the actual ion saturation number with the pseudo-potential prediction.

Figure 1 confirms the universal shape of loading curves predicted in [

13,

14] to occur for all rf traps. Focusing on the loading curve for

$q=0.2$, the four dynamical regimes are marked in

Figure 1 with the Roman numerals I, II, III, IV, respectively. Dynamical regime I corresponds to the slow-loading regime. Ions are loaded from the MOT into the 3DPT with near-zero initial kinetic energy, and are, therefore, initially, trapped deeply in the potential well of the trap. Subsequently, they slowly gain energy due to the combined action of rf- and insertion heating. Having gained enough energy from these two heating mechanisms, they escape from the trap. The saturation ion number in this regime is determined by the balance between the rate of newly loaded ions and the rate of escape dictated by the heating rates. Increasing the loading rate, more ions can simultaneously undergo this process of loading, heating, and escaping, which explains why the region-I loading curve rises monotonically with increasing loading rate. At some point, however, the increase of the number of ions in the trap with increasing loading rate has to stop, and we enter dynamical region II. In this region, the role of heating is diminished, and the role of Coulomb-Coulomb interactions is emphasized, roughly resulting in a situation of a strongly coupled (Coulomb) fluid (plasma), where each new “drop” (ion loaded) overflows the “bucket” (the trap) such that the total number of fluid (ions) contained in the bucket (trap) cannot exceed the maximum capacity of fluid in the bucket (ions in the trap). This situation results in the quasi-stationary behavior (plateau) that characterizes dynamical region II (see

Figure 1). Increasing the loading rate even further corresponds to a situation where fluid (ions) accumulate with a density larger than the average trap density in the loading region (MOT region) of the trap, actively pushing ions out of the trap due to strong Coulomb forces. This results in the dip of the loading curves, marked as dynamical region III in

Figure 1. Further increase of the loading rate piles up so much inertially confined charge in the loading region that a highly charged “core” develops in the trap (see

Section 2.5), that accelerates all charge outside of the core swiftly and ballistically out of the trap. This explains the sharp rise of the loading curve in dynamical region IV (see

Figure 1). Dynamical region IV is currently the best understood region and resulted in an experimentally confirmed behavior of

${\nu}_{s}\sim {\lambda}^{2/3}$ [

13].

In

Figure 1 all four dynamical regimes are present for

$q=0.2$ and

$q=0.3$. However, in

Figure 1 our

$\lambda $ scale does not extend far enough to the left and to the right to see all four dynamical regimes for

$q=0.1$ and

$q=0.4$. But we still see regions II, III, and IV for

$q=0.1$, and regions I, II, and the beginnings of region III for

$q=0.4$. This confirms expectations. More importantly, however, we see that only in the fast-loading regime, not currently used for ion-neutral collision experiments, do the loading curves reach the level of 1 (and beyond) predicted by the pseudo-potential approximation. In all other dynamical regimes,

${\nu}_{s}$ is significantly smaller than 1, indicating that the static pseudo-potential approximation overestimates the number of ions stored in the trap. So, we have a first result: The saturation numbers do not agree with the pseudo-potential prediction in two important ways: (i) Instead of being constant,

${\nu}_{s}\left(\lambda \right)$ shows a non-monotonic structure, already found in [

13,

14]. (ii) Even at the maximum,

${\nu}_{s}\left(\lambda \right)$ is significantly below the pseudo-potential prediction. We also notice that for increasing

q, the maxima of the loading curves in dynamical regime II shift very quickly toward larger loading rates.

The depression of the saturation ion levels with respect to the pseudo-potential prediction is investigated further in

Figure 2. This figure shows the scaled maxima of the loading curves of

Figure 1,

for several values of

q and

${\widehat{R}}_{\mathrm{cut}}$.

Figure 2 confirms that the scaled saturation levels are substantially below 1 and reach down to

$\approx 0.4$, that is, a trap capacity of only about 40% of the pseudo-potential prediction, at

$q=0.4$. Even for small

q, the trap capacity is only about 80% of the pseudo-potential prediction. However, we also see that the ion capacity scales ∼

${\widehat{R}}_{\mathrm{cut}}^{3}$. Therefore, the scaled

${\nu}_{s}^{\mathrm{max}}$ is approximately independent of

${\widehat{R}}_{\mathrm{cut}}$ and depends only on

q. A scaling function may be derived from

Figure 2, which can be used to calibrate

${N}_{s}\left(\lambda \right)$ with respect to the pseudo-potential prediction.

The rapid shift of the saturation plateaus to the right as

q increases is illustrated in

Figure 3, which shows the loading rate

${\lambda}^{\mathrm{max}}$ at which the loading-platau maxima occur (see

Figure 1) for several different

q values. In the region from 0.1 to 0.3 we see an exponential dependence of

${\lambda}^{\mathrm{max}}$ on

q, approximately given by (see straight, blue line in

Figure 3):

In [

13] we found that for

$q=0.2$ the maximum of

${N}_{s}\left(\lambda \right)$ occurs at

$\lambda $∼ 1.

Figure 3 shows that this is not always so. Quite the contrary: Indicated by closely following the straight blue line in

Figure 3, representing the function (

37),

${\lambda}^{\mathrm{max}}$ shows extreme (exponential) sensitivity to the trap control parameter

q in the region

$0.1<q<0.3$. The deviation from exponential behavior in the region

$q>0.3$ may be due to the fact that for increasing

q the maximum in region II turns into a shoulder on the region-IV curve (see

Figure 1), masking a possible continuation of the exponential behavior. However, a clearer understanding of the deviation from exponential behavior has to await further analytical progress. The deviation of the data point at

$q=0.05$ has a simpler explanation. It is due to the fact that our simulations are cut at

$\lambda =0.01$ and thus do not extend far enough to the left to capture the true location of the maximum of the loading curve at

$q=0.05$.

In this subsection we compared actual steady-state saturation ion numbers obtained from molecular-dynamics simulations with the ion numbers expected from a model that assumes that the pseudo-potential is filled with a homogeneously distributed charge. In reality, however, the ions in the trap are localized, discrete charges, which, in analogy to spherical Coulomb crystals [

20] leads to a reduced ion-number capacity in the trap. Therefore, it is possible that the agreement between the pseudo-potential approximation and the simulations may be improved somewhat by taking the discretization of charge into account. However, even in this improved pseudo-potential model, the decrease of

${\nu}_{s}^{\mathrm{max}}$ with increasing

q, observed in

Figure 2, cannot be accounted for.

#### 2.5. Densities

Because of its quadratic nature, the static pseudopotential discussed in

Section 2.1 predicts that the charge density

${\widehat{\rho}}^{pp}$, given in (

16), is constant throughout the trap. A constant density is important for ion-neutral collision experiments, since only this way we may assume homogeneous and isotropic conditions throughout the reaction volume.

In [

13,

14] it was assumed that the best conditions in terms of charge density should occur at the maxima of the loading curves (see

Figure 1). To test this assumption,

Figure 4 shows the charge densities

$\widehat{\rho}\left(\widehat{r}\right)$ (histograms) obtained as a result of full three-dimensional molecular-dynamics simulations (see

Section 3) for

$q=0.1,0.2,0.3,0.4$ as a function of the dimensionless radial position

$\widehat{r}=|\widehat{\overrightarrow{r}}\phantom{\rule{0.166667em}{0ex}}|=\sqrt{{\widehat{x}}^{2}+{\widehat{y}}^{2}+{\widehat{z}}^{2}}$. Also shown, for comparison, is the constant-density pseudo-potential prediction (red, horizontal line). We see that, while, at least in some regions, the actual densities are close to the pseudo-potential prediction, the condition of constant charge density is approximately satisfied over a large range of the trap volume only for

$q=0.1$, the smallest of the

q values shown in

Figure 4. And even in this case, we see a significant deviation from a constant density, especially toward the radius of the trap at

${\widehat{R}}_{\mathrm{cut}}$. In fact, the deviation from a constant density at

$\widehat{r}\approx {\widehat{R}}_{\mathrm{cut}}$ accounts for much of the missing ions (see

Figure 2) as compared to the pseudo-potential prediction. We see this in the following way. In a simple model of the density shown in

Figure 4a, we assume that the density is constant and equal to

${\widehat{\rho}}^{pp}$ in

$0\le \widehat{r}\le {\widehat{R}}_{1}$, decreases linearly in

${\widehat{R}}_{1}\le \widehat{r}\le {\widehat{R}}_{2}$, and reaches 0 at

$\widehat{r}={\widehat{R}}_{2}$. Thus, in this model, the density is given by

The number of particles in this model is

Therefore, if we now normalize the actual number of stored ions,

${N}_{s}$, to the pseudo-potential prediction,

${N}_{s}^{pp}$, we obtain

Inserting

${\widehat{R}}_{1}=20$ and

${\widehat{R}}_{2}=25$, appropriate for the case of

Figure 4a, we obtain

${\nu}_{s}=0.74$, close to the actual observed ratio of

${\nu}_{s}\approx 0.8$, the small hump in the density occurring before the onset of the decline of the density at

${\widehat{R}}_{1}=20$ accounting for much of the difference. So, we now have an explanation for the missing ion numbers: For small

q, such as in

Figure 4a, the missing ions are due to the behavior of

$\widehat{\rho}\left(\widehat{r}\right)$ close to the cut-off at

$\widehat{r}={\widehat{R}}_{\mathrm{cut}}$.

We now turn to the densities corresponding to larger values of

q.

Figure 4b shows the case for

$q=0.2$. Contrary to the case of

$q=0.1$, where the loading-curve maximum corresponding to the density shown in

Figure 4a occurs at a loading rate of

${\lambda}^{\mathrm{max}}=0.01$, the loading rate at the maximum of the loading curve in the case of

$q=0.2$, corresponding to the density shown in

Figure 4b, now occurs at

${\lambda}^{\mathrm{max}}=2.5$. Thus, the loading rate in the case of

Figure 4b is a factor 250 larger than the loading rate in

Figure 4a, although the trap control parameter only doubled. This, of course, is a result of the exponential sensitivity of the loading rate in

q as shown in

Figure 3 and quantified in (

37). Accordingly, reflecting the vastly larger loading rate, we now see a pile-up of density in the loading region

$0\le \widehat{r}\le {\widehat{R}}_{L}=3$. The pile-up of charge in the loading zone may be explained in the following way. Charge created in the region

$\widehat{r}\le {\widehat{R}}_{L}$ needs to be transported away from the loading zone. One mechanism to ensure this is rf heating, which imparts kinetic energy on the ions in the loading zone. However, since rf heating results from the micro-motion, and the micro-motion amplitude of an ion is proportional to its distance

$\widehat{r}$ from the trap’s center [

16], rf heating is not very effective close to the trap’s center, that is, in the loading zone, resulting in the observed pile-up of charge in the loading zone. Consequently, together with its behavior at the edges of the trap, the ion density in the trap is now significantly different from the predicted constant both in the vicinity of

$\widehat{r}\approx 0$ and around

$\widehat{r}\approx {\widehat{R}}_{\mathrm{cut}}=25$. As shown in

Figure 4c,d, this trend continues for larger

q values, corresponding to ever larger loading rates, according to

Figure 3, so that the

$\widehat{r}$-interval of constant density shrinks with increasing

q and is near negligible in the case of

$q=0.4$ [see

Figure 4d]. We also see that with increasing

q the radius

${\widehat{R}}_{1}$ in our model above decreases significantly, explaining the depression of the ion-storage capacity of the trap down to about 40% of the pseudo-potential prediction at

$q=0.4$ (see

Figure 2). Apparently, the overshoot of the density in the loading region

$0\le \widehat{r}\le {\widehat{R}}_{L}=3$ cannot compensate for the ion loss due to the decline of the density toward

$\widehat{r}={\widehat{R}}_{\mathrm{cut}}$.

What causes the decrease of the density close to

${\widehat{R}}_{\mathrm{cut}}$? Since the 3DPT is an rf trap, the micro-motion comes to mind as a possible direct explanation. According to [

16] the micro-motion amplitude is given by

and is proportional to both

q and

$\widehat{r}$. Therefore, if

$\Delta {\widehat{r}}_{mm}$ would be of the order of

${\widehat{R}}_{\mathrm{cut}}-{\widehat{R}}_{1}$, one could immediately explain the missing ions in the vicinity of

${\widehat{R}}_{\mathrm{cut}}$ as being knocked out of the trap directly by the micro-motion. Moreover, since

$\Delta {\widehat{r}}_{mm}$ is proportional to

q, this would even explain why more ions are missing for increasing

q. However, according to the following reasoning, the micro-motion amplitude can be ruled out as the direct cause for the missing ions. At

${\widehat{R}}_{\mathrm{cut}}=25$ and

$q=0.1$, the micro-motion amplitude works out to be

$\Delta {\widehat{r}}_{mm}=1.25$ and increases only to

$\Delta {\widehat{r}}_{mm}=5$ for

$q=0.4$. So, while the micro-motion amplitude is certainly not negligible, and certainly plays a role in carrying ions across the boundary at

$\widehat{r}={\widehat{R}}_{\mathrm{cut}}$, its magnitude is too small to explain the shape of the density in

Figure 4 via this direct knock-out mechanism. However, the relatively large micro-motion amplitude in the vicinity of

${\widehat{R}}_{\mathrm{cut}}$ indicates that the time-dependence of the trap field may play a major role in explaining the missing ions. To nail this down, in

Figure 2 we compare the ion saturation numbers for the time-dependent 3DPT with the ion saturation numbers obtained from fully three-dimensional molecular-dynamics simulations (see

Section 3) where, in addition to ion loading, only the time-independent, static pseudopotential was switched on (stars in

Figure 2). We see that, although even in the dynamically loaded pseudo-oscillator trap the saturation numbers are smaller than predicted by the static pseudo-potential approximation without dynamic ion loading, the densities obtained from the the pseudo-oscillator simulations for large

q are significantly above those for the fully time-dependent 3DPT, and merge with those of the 3DPT only in the small-

q regime, in which the rf heating power is small. This proves conclusively that the main reason for the missing ions is strong rf heating at large values of

$\widehat{r}$ close to

${\widehat{R}}_{\mathrm{cut}}$, and in addition explains, because of the proportionality of

$\Delta \widehat{r}$ to both

$\widehat{r}$ and

q why the effect of the missing ions increases both with increasing

$\widehat{r}$ and increasing

q. Thus, the decrease of the density in

${\widehat{R}}_{1}\le \widehat{r}\le {\widehat{R}}_{2}$ is dynamical in origin, that is, it is caused by the combined effects of rf heating and the flow of ions generated in the loading zone and absorbed at

$\widehat{r}={\widehat{R}}_{\mathrm{cut}}$. That the flow itself is an important mechanism is shown by the fact that inspection of the densities obtained from the simulation of loading the pseudopotential shows the same linear decrease of the density in the region between

${\widehat{R}}_{1}$ and

${\widehat{R}}_{2}={\widehat{R}}_{\mathrm{cut}}$, only that in the pseudo-potential case

${R}_{1}$ is significantly closer to

${\widehat{R}}_{\mathrm{cut}}$ than in the case of the time-dependent trap, accounting for the significantly larger ion saturation numbers in the pseudo-potential case (stars in

Figure 2). Since the flow is important, a combined model consisting of heating and hydrodynamic transport components might explain the different shapes of the charge densities in the different loading regimes. This, however, is beyond the scope of this paper.

The temperature is also thought to be constant throughout the trap at saturation. Again, this may not be true, since the micro-motion amplitude, $\Delta {\widehat{r}}_{mm}$, and therefore the effectiveness of rf heating, depends on the distance $\widehat{r}$ from the trap’s center. Therefore, the local temperature may be a function of distance from the center of the trap. This, too, is a promising topic for future research.