Estimation of the electric field in atom probe tomography experiments using charge state ratios

(Kingham 1982) provided equations for the probability of observing higher charge states in atom probe tomography (APT) experiments. These"Kingham curves"have wide application in APT, but cannot be analytically transformed to provide the electric field in terms of the easily-measured charge state ratio (CSR). Here we provide a numerical scheme for the calculation of Kingham curves and the variation in electric field with CSR. We find the variation in electric field with CSR is well-described by a simple two or three-parameter equation, and the model is accurate to most elements and charge states. The model is applied to experimental APT data of pure aluminium and a microalloyed steel, demonstrating that the methods described in this work can be easily applied to a variety of APT problems to understand electric field variations.


Introduction
In atom probe tomography (APT), atoms at the apex of a needle-shaped specimen are both fieldevaporated and accelerated towards a position-sensitive detector using strong electric fields.The rate of field evaporation is controlled by varying the applied voltage between 0.5 kV and 10 kV, and with brief pulses of energy provided by a laser or variations in the voltage.By measuring the time between the pulse and the ion reaching the detector, a spectrum of mass-to-charge ratios (or "mass spectrum") can be produced (Gault et al. 2012).Although only a single ionisation (i.e., atom A to ion A + ) is required for field-evaporation to occur, higher charge states (such as A ++ and A +++ ) are commonly observed in mass spectra.(Kingham 1982) provided a model for why higher charge states are observed using post-evaporation ionisation (or "post ionisation").In that work, expressions were derived for the probability of an electron tunnelling from the ion to the surface,  t , in terms of the local electric field at the apex of the specimen, .Manipulating  t provides the probability   of observing a charge state , and (Kingham 1982) plots curves of   () for 24 elements.The importance of these "Kingham curves" to the atom probe community is indicated by their reproduction in APT textbooks (Miller et al. 1996;Gault et al. 2012;Miller and Forbes 2014) and reference in several hundred journal articles.Knowledge of the local electric field is valuable in studies of hydrogen in steels and alloys (Felfer et al. 2012;Chang et al. 2018;Breen et al. 2020), carbides (Thuvander et al. 2011), semiconductors (Mancini et al. 2014;Hans and Schneider 2020;Cuduvally et al. 2022), and into the fundamentals of field evaporation and atom probe instrumentation (Lam and Needs 1992;Vella et al. 2021;Breen et al. 2023;Tegg et al. 2023).Variations in electric field can result in distortions to microstructural features (known as "local magnification") (Larson et al. 2013a;Lawitzki, Stender, and Schmitz 2021), changes to multiple-ion evaporation (Saxey 2011;Peng et al. 2018), and can affect complex ion evaporation (Gault et al. 2012;Larson et al. 2013b), so knowledge of the electric field in a specimen can significantly aid in data interpretation.
The curves in (Kingham 1982) give   (): the probability of observing a certain charge state based on the electric field.However, atom probe researchers more commonly seek  (  ): the electric field calculated using the measured charge state, or the charge state ratio (CSR).This is because several phenomena in atom probe data can be explained by variations in local electric field (Felfer et al. 2012;Chang et al. 2018;Breen et al. 2023).The expressions for  t () cannot be analytically transformed to provide  ( t ).This means that atom probe researchers wanting to calculate electric fields must measure their CSR and either read  graphically from published Kingham curves (Kingham 1982;Gault et al. 2012;Miller and Forbes 2014), or calculate   () themselves.
Here, we present a numerical method for calculating Kingham curves for 60 elements.We also calculate  (CSR), and find that a simple approximate expression describes for  (CSR) across 10 −4 ≤ CSR ≤ 10 4 for each element and combination of charge states up to 0 ≤  ≤ 70 V/nm.This expression can be used to quickly estimate the electric field to within ¡1% of the values predicted from Kingham curves , and is easily applied to experimental data.We provide graphs of  (CSR), a table of coefficients, and the Python 3.11 code used to calculate   () and  (CSR).Ethis work serves as a pratical starting point for estimating surface electric field in order to better understand various phneomna in experimental atom probe data.

Kingham's model
Here we will partially reproduce the model of (Kingham 1982) in order to introduce our computational method.We also include some of the modifications or clarifications made by (Andrén, Henjered, and Kingham 1984;Lam and Needs 1992;Cuduvally et al. 2022).Expressions are introduced in the order they are calculated in the associated Python code, and some have been re-arranged from that shown in the source material.As in (Kingham 1982), the expressions below are given in Hartree atomic units (au).
The reduced Planck constant, the mass of the electron, and the electronic charge are all ℏ =  e =  = 1,

Sur face
Figure 1: A schematic of the model in (Kingham 1982).An ion A n in charge state  ≥ 1 is fieldevaporated and travels along (or very close to) electric field lines (" lines →") at velocity ( 0 ).
Post-ionisation to charge state  + 1 is possible at ion-tip distances  0 that are greater than a critical distance,  c .The electric field  penetrates up to a distance  below the tip surface at  = 0.
In (Kingham 1982)'s model, post-ionisation occurs when an electron tunnels from the field-evaporated ion A n to an unoccupied electronic state on the surface of the tip.The ion initially evaporates in charge state  i but is observed at the detector with charge state  ≥  i .The probability of tunnelling  t from a given charge state  to  + 1 depends on the distance between the tip surface and the ion ( 0 ) and the ionisation energy of the the  + 1 state ( +1 , hereafter ). Figure 1 shows a schematic of the model system.
Tunnelling becomes likely at a critical distance,  c , where the energy of the least-tightly bound electron on A n is of a greater energy than the Fermi level at the tip surface.The ionisation energy of the outermost electron of the ion is less than that of same ion would be in a field-free environment.(Cuduvally et al. 2022) calculate this energy as Here,  0 is the zero-field work function,  = 0.8 is the distance the field penetrates into the tip surface in  0 units (Lang and Kohn 1973), and  image is the image potential, given by at  c .Here we follow (Cuduvally et al. 2022) by assuming the Stark shift is negligible.Equation 1 is a quadratic equation in  c , and the real solution with the positive root is The rate constant of post-ionisation, , is found from the electron probability flux of the outermost electron orbital through a surface  perpendicular to the field direction, where (, ) is an s-type electron wavefunction in polar co-ordinates, and   is the electron velocity normal to the metal surface.(Kingham 1982) provides the approximate solution when the angle between the ion trajectory and the field, , is small: (5) Here, exp [] = e  is the exponential function,  2  is given by and  is introduced here to simplify the typesetting of equation 5: is found by (Kingham 1982) by fitting the calculated  t to experimental values for post-ionisation from Rh + to Rh ++ from (Ernst 1979): Lastly, the ion velocity  is found from where  ion is the mass of the ion (Kingham 1982;Andrén, Henjered, and Kingham 1984).Equations 5 and 9 are used to calculate the probability of post-ionisation at each field and charge state, These probabilities are more commonly expressed as the probability of observing a certain change state at the detector,   , which is found from for the  = 1 (+) charge state, and for the  ≥ 2 (++, +++, and ++++) charge states.The CSR can be found from equations 11 and 12, or from an experimental mass spectrum, by where  is the ranged counts in the higher ( + 1) or lower () charge state for a given element or isotope ().Alternatively,  is the ionic concentration of a given  at  in reconstructed APT data, providing each  and  are sampled from the same reconstructed volume.

Numerical implementation
In this work,  t is calculated using the equations given in the previous section.The main calculation loop is performed in the Hartree atomic units but the input parameters and the output plots are expressed in eV, nm and V/nm.Ionisation energies are taken from (Kramida et al. 2009) and work functions from (Michaelson 1977).The work functions used are those for polycrystalline solids in their room temperature phases, and work function for C is that of polycrystalline graphite.Electric fields are calculated across 0.01 ≤  ≤ 100 V/nm in 1000 steps with equal spacing, charge states 1 ≤  ≤ 4, and distances 10 −2 ≤  0 ≤ 10 6 nm in 5000 steps with exponentially-increasing spacing.The main calculation loop consists of a loop over elements, then electric fields, then charge states, with  t calculated for each.  and CSR curves are then calculated for each element.(Kingham 1982) does not provide equation 3, and (Cuduvally et al. 2022) does not indicate which root to use, or how to handle a complex solution.We find agreement with their   () curves if we use the positive root and set  c = 0 when a complex solution would be obtained, as this would represent a  c inside the metal.Once  c (, ) has been found, subsequent calculations involving  0 are only performed for the domain  0 >  c .Like (Cuduvally et al. 2022), we applied the correction to the exponent in equation 5 given by (Lam and Needs 1992), but do not use their expressions for .Equation 9 is not fully defined in (Kingham 1982), and not expressed for  > 1 in (Cuduvally et al. 2022).(Andrén, Henjered, and Kingham 1984) clarifies that the two sums presented in equation 3.39 of (Kingham 1982) should be combined into one sum, as shown here in equation 9. We have assumed the initial charge state of the field-evaporated ion is always  i = 1.Like (Kingham 1982), we do not observe significant differences in the results for  i ≥ 2. The integral in equation 10 is calculated using the trapezoidal method, and thus the result is sensitive to the array of  0 used.We use an exponentially-spaced array to ensure accurate  c calculation at low , while ensuring sufficiently high  0 are included to suit the infinite upper limit.
The Python code used to perform these calculations is provided as supplementary information to this manuscript.The code was written for Python 3.11.5 with numpy version 1.19.2 (Harris et al. 2020), scipy version 1.5.2 (Virtanen et al. 2020), and matplotlib version 3.3.2(Hunter 2007).Key equations from section 2.1 are expressed as functions, allowing for a modular approach to the calculations of  t and CSR.Atomic and material parameters are also supplied in a comma-separated variable file.The same code was used to prepare the figures shown in this manuscript.

Expression for the electric field
Plots of  (CSR) were produced from the specified arrays of  and the calculated arrays of CSR.We found that equation 14 described plots of  (CSR) with reasonable accuracy: The coefficients  and  were found for each element and CSR using the curve fit function from the scipy.optimizelibrary (Virtanen et al. 2020).The fit was performed over 10 −4 ≤ CSR ≤ 10 4 as it was felt this is the maximum range where CSR can be accurately calculated in typical atom probe data.
This assumption is covered in the discussion section.

Application to experiment
Equation 14 was used to estimate electric field using CSRs in experimental atom probe data.Data from a pure Al specimen was collected using a Cameca Invizo 6000 (Tegg et al. 2023) by using voltage-pulsed acquisition with 20% pulse fraction, 200 kHz pulse rate, 2% target detection rate and at 50 K temperature.
Data from a microalloyed martensitic steel (Lin et al. 2020) was collected using a Cameca Invizo 6000 by using laser-pulsed acquisition with 400 pJ laser pulse energy, 200 kHz pulse rate, 4% target detection rate and at 50 K temperature.Data reconstruction and analysis was performed using the IVAS module within AP Suite 6.3.The Al dataset was reconstructed in detector space to allow for easy calculation of isotope-specific field evaporation images.The microalloyed steel was reconstructed by calculating the sample radius from the standing voltage and the evaporation field of Fe (Tsong 1978).The image compression factor and field factor determined by inspection of crystallographic poles (Gault et al. 2012).
Further details of the analysis methods are provided in section 3.2.

Calculated fields and CSRs
Figure 2 illustrates the method described in sections 2.1-2.3.Figure 2(a) shows Kingham curves for W between 0 ≤  ≤ 70 V/nm, 10 −4 ≤   ≤ 10 4 , and for charge states + to ++++.As the field increases, the probability of observing the + charge state falls, and the probabilities of observing higher charge state rise and fall successively.As such, we felt there was no need to include additional free parameters to equation 14.Note that this residual and percentage error refers to the difference between the  (CSR) calculated from Kingham curves and from the model in equation 14.Other sources of uncertainty are not quantified here and are covered in the discussion section.The annotations on the right vertical axis of (b) indicate the charge states expected for different ranges in .The ranges where a single charge state is observed are relatively narrow, for example the +++ state is expected only between ≈ 40 V/nm and 43 V/nm.This idea is also discussed further in later sections.
Figure 3 shows the  (CSR) curves for 16 elements commonly studied using APT.As with figure 2, hollow circles denote  (CSR) data calculated from successive pairs of Kingham curves, and solid lines denote fits using equation 14.The fit parameters are listed in table 1 and plotted in figure 4. Equation 140 Field, F (V/nm)  4 accurately describes the ++/+ and +++/++  (CSR) curves for all elements studied.There are small variations in the shape of the ++++/+++  (CSR) curves at CSR ≤ 10 −3 for some elements, such as Si shown in figure 3(l), which make the model less accurate in these regions.represents the electric field at infinite CSR and generally increases along a block, e.g., the fourth period d-block transition metals between Ti and Cu.The parameter  is related to the  (CSR → 0), by Figure 4(c) shows the relationship between  and .It was found that  can be described by  with an where  is the lower charge state of the CSR (i.e.,  = 1 for ++/+).Equation 16was fit to the data in figure 4(c) and is shown as a dashed line.The fit parameters are  = 0.0036(3),  = −0.025(3), = 0.11(2),  = 0.21(3),  = −0.012(2)and ℎ = 0.04(2), where the parentheses indicates the uncertainty in the least significant digit (i.e. = 0.0036 ± 0.0003).The fit is better for elements with higher ( = 1), making this expression less accurate for group IIa and f-block metals.A physical interpretation of the constants in equation 16 is beyond the scope of this work.However, equations 14 and 16 show that  can be expressed solely in terms of the CSR,  =  (CSR → ∞), and : Though interesting, we expect that researchers will find equation 14 more useful than equation 17 for calculating  from CSRs in APT data due to its simplicity, and its greater accuracy for ++/+ CSRs and group IIa and f-block elements.

Application to APT data
Figure 5 shows application of the model  (CSR) curves to an atom probe dataset of pure Al collected with voltage-pulsed acquisition on a Cameca Invizo 6000.The reconstruction was performed in detectorspace co-ordinates.The [001] pole is near the centre of the detector, and three ⟨111⟩ poles are visible at the detector periphery.The Al + and Al ++ peaks were ranged separately and binned into 0.25 nm × 0.25 nm pixels, producing the field evaporation images shown in figure 5 (a,b).The ratio of these two images is the ++/+ CSR and is shown in (c).Equation 14with the model parameters from table 1 produced the  (CSR) map shown in (d).The average CSR for this dataset is 5.64 × 10 −2 , corresponding to  (CSR) = 22.7 V/nm. Figure 5 (d) shows the  varies between ≈ 21 V/nm and ≈ 24 V/nm across the detector field-of-view, which is ≈ 10% variation around the average value.As expected (Larson et al. 2013a), the electric field is greater around crystallographic poles.We have demonstrated that electric fields can be easily calculated from experimental data, and represented in a highly visual way that is simple to interpret.Figure 6 shows the result of applying the method described here to Cu precipitates in a microalloyed martensitic steel (Lin et al. 2020).An iso-concentration surface (isosurface) of 15 at.%Cu was calculated on a grid of 0.5 nm spacing with 2 nm of delocalisation.A proximity histogram (proxigram) of Cu, Fe and Mn is shown in (a).Only Fe and Mn were observed in both + and ++ charge states, and their CSRs are shown in (b).The electric field was calculated using equation 14 and table 1 and is shown in (c).As a relatively low-field metal (Tsong 1978), the Cu precipitates have a lower evaporation field than the martensite matrix.This is reflected in the CSR and  curves for both metals, and their values drop in the precipitate.However, the fields predicted by Fe and Mn differ by 1.1 V/nm in the matrix and up to 1.9 V/nm in the precipitate.This difference is within the range of uncertainty in the  curve for Mn, but this ultimately results from uncertainty in the proxigram and the low concentration of Mn in the matrix.Larger datasets would reduce the uncertainty in , but it's likely there would still be a discrepancy between the  calculated from the Fe and Mn CSRs.

Discussion
The results shown here have been calculated over 10 −4 ≤ CSR ≤ 10 4 , as this approximates the maximum range which can reasonably be measured in a typical APT experiment.To justify this, we will consider the relative uncertainty across CSRs and dataset sizes.As weak peaks in mass spectra can be described by a Poisson distribution (Larson et al. 2013b), the uncertainty in the measured counts are, at worst, and similar for the  ,+1 case.Considering equations 13 and 14, and assuming Δ , is not strongly correlated with Δ ,+1 , the relative uncertainty in CSR is where the middle term describes variation with  , , and the right term when when considering  ,+1 .
Figure 7 shows the relative uncertainty in CSR for (a) 10  Like Kingham curves, the  (CSR) curves here can be used to estimate the range of electric fields measurable using CSRs with a given element.Figure 8 shows the ranges in  accessible for each 10 −4 ≤ CSR ≤ 10 4 and element studied in this work.As with figure 4, elements are sorted by atomic number and separated into blocks.The right vertical axis shows elements with overlapping ranges where  (CSR) will be continuously measurable over a wide range of .For example, it should be possible to measure  using Al ++/+ and +++/++ between ≈ 18 ≤  ≤ 45 V/nm, providing the evaporation rate be managed.Most of the elements with wide ranges for  measurements are s-or p-block metals, with the addition of Os and Au.Similarly, most elements have ranges in  where only a single charge [Xe] 4f 14 6s 2 5d 2 ) (Mann, Meek, and Allen 2000;Mann et al. 2000).The low ionisation energies of the f-block metals may also challenge the accuracy of the model, as these were not considered in (Kingham 1982) or any subsequent literature on post-ionisation.There is value in repeating the Rh + → Rh ++ field evaporation experiment from (Ernst 1979) on other metals, as equation 8 underpins (Kingham 1982) and all subsequent studies into post-ionisation, including this one.
Section 3.2 highlighted some applications to the method described in this work: variation in electric field around poles, and reduction in field within low-field precipitates.The importance of the electric field to atom probe means there are many other potential applications.Variations in electric field across precipitates or microstructural features leads to local magnification (Larson et al. 2013a), and knowledge of the electric field allows more sophisticated reconstruction methods (Lawitzki, Stender, and Schmitz 2021;Fletcher et al. 2022).The electric field affects the relative populations of 1 H + and 1 H 2 + seen in mass spectra (Mouton et al. 2019), and controlling the field may allow for discrimination between 1 H 2 + and 2 H + = D + peaks at 2 Da.Acqusition parameters such as laser pulse energy affect the specimen temperature, which in turn affects the electric field needed to induce field evaporation (Larson et al. 2013b).This work aids researchers in thoughtfully choosing acquisition parameters to avoid data issues related to electric field, such as peak overlap or evaporation of metal-hydride species.

Conclusion
In this work we provide a numerical method for calculating Kingham curves: plots of the probability   of observing a charge state  in terms of the electric field  in atom probe tomography (APT) experiments.Using these data we plot  in terms of the charge state ratio (CSR), and find that simple 2or 3-parameter expressions can describe  (CSR) across the first three CSRs and 8 orders-of-magnitude in CSR.We fit this equation to almost all solid elements on the periodic table and provide a table of constants that allows researchers to calculate  using CSR in their reconstructed APT data.We illustrate this application using a field evaporation map of pure Al, where we find the evaporation field increases by ≈ 10% from the average value around crystallographic poles.We also show how the method can estimate the evaporation field inside low-field precipitates in a microalloyed martensitic steel, though calculated values for  differ for the matrix and solute elements.
In general we find our models are most accurate for the ++/+ and +++/++ CSRs, and the p and dblock metals.Aside from common APT considerations such as maximising mass resolving power and consistently ranging mass spectrum peaks, accurate measurement of CSR and  requires relatively large datasets.An uncertainty in CSR of ≤ 1% is only obtainable when there are ≥ 10 4 counts in the smaller of the two peaks used in the CSR calculation.The simplicity of the models reported here allow researchers to more easily estimate the , allowing for more sophisticated interpretation of phenomena in experimental APT datasets.

Figure 2 :
figure2(d)  shows the residual relative to K to provide the percentage error between K and T. Across

Figure 4
Figure 4(a,b) shows the variation in model parameters  and  with atomic number.The shading indicates the s, p, d and f blocks of the elements.With reference to equation 14, the parameter  =  (CSR → ∞)

Figure 5 :
Figure 5: (a) Al + and (b) Al ++ field evaporation images from a voltage-pulsed APT experiment of pure Al.(c) The ++/+ CSR for these field evaporation maps.(d) The  (CSR) calculated using the CSR in (c), equation 14, and the Al ++/+ parameters in table 1.

Figure 6 :
Figure6: (a) Concentration ( , ) of selected elements () and charge states () from a proximity histogram (proxigram) over an iso-concentration surface (isosurface) of Cu ions at 15 at.% in a microalloyed steel(Lin et al. 2020).The + and ++ charge states were observed only for Fe and Mn.Positive distances are inside the Cu precipitates, negative distances in the martensitic matrix.(b) The ++/+ CSR for Fe and Mn.(c) The electric field calculated using equation 14 and the CSRs from (b).The shaded ribbons indicate uncertainties, all propagated from the measurement uncertainty in  , .The arrows indicate the difference between the  for Fe and Mn in the matrix and the precipitate.

Figure 7 :
Figure 7: Relative uncertainty in CSR (ΔCSR/CSR) in terms of (a)  , for CSR ≤ 10 0 and (b)  ,+1 for CSR ≥ 10 0 .The dashed horizontal line indicates a 1% relative uncertainty, and the dashed vertical lines indicate the counts required to ensure 1% relative uncertainty.

Table 1 :
Model parameters  and  from equation 14 for the elements and CSRs explored in this study.
Figure 4: Variation in model parameters (a)  and (b)  with atomic number for the (purple) ++/+, (pink) +++/++ and (orange) ++++/+++ CSRs.Horizontal axis ticks and shading indicate blocks from the periodic table of the elements.(c) Relationship between model parameters  and  for all elements.Dashed lines indicate a fit using equation 16. equation of the form (Tegg et al. 2023)th  , and (b) 10 0 ≤ CSR ≤ 10 4 with  ,+1 .Each line indicates a CSR = 10  for integer −4 ≤  ≤ 4 and text labels indicate the limits of  for that plot.The dashed horizontal line indicates the arbitrary uncertainty threshold used in this section, ΔCSR/CSR = 1%.The dashed vertical lines indicate the counts needed in order to have 1%uncertainty in CSR.To measure CSR = 10 0 at 1% uncertainty,  , =  ,+1 ≈ 10 4 counts are needed in both peaks.To measure CSR = 10 −4 at 1% uncertainty,  , ≈ 10 8 counts are needed in the  peak and thus  ,+1 ≈ 10 4 in the  + 1 peak.In general, ≥ 10 4 counts are needed in the smaller peak in order to measure any CSR with better than 1% accuracy.Consistently recording datasets of ≫ 100 million ions without fracture is not routine, even for the newest generation of atom probe instruments(Tegg et al. 2023), so accurate measurement of CSR ≥ 10 4 or ≤ 10 −4 will not be performed frequently.Additionally, figure6illustrates how modest uncertainties in CSR propagate to large uncertainties in .This is due to the low gradient d/dCSR, particularly for ++/+ CSRs.Thus when measuring , it's important to minimise uncertainty in CSR by maximising the sample size used in a calculation.