Formal Relationship Between the Firearm “Memory Effect” and the Decay Time of the GSR Particles Present on the Shooter’s Hands

Abstract

Background/Objectives: The Scanning Electron Microscope (SEM), combined with an Energy Dispersive Spectrometer (EDS), has been, for over fifty years of practical experience and research in the field, the analytical system of choice for the investigation and analysis of Gun Shot Residues (GSRs). However, the interpretation of analytical results has profoundly changed in recent decades. Specifically, the criteria for evaluating particles presumptive of contamination of a possible discharge have evolved, assessments regarding possible primary/secondary transfer phenomena have been refined, and the retention times of particulate matter on various types of surfaces involved during the discharge have been revalued. The purpose of this study is to provide a formal representation that links together the firearm memory effect, namely the formation of composite characteristic GSRs resulting from the use of the same Firearm but with ammunition having different metallic alloy constituents and different primer mixtures, and the decay time. Methods: The deduced mathematical model is based on experimental results reported in the scientific literature listed below, and it has been elaborated with a series of non-contradictory assumptions, each of which plays a specific role in the mathematical formalism used. Results and Conclusions: This model, although not yet validated through rigorous experimentation, represents a valuable tool in investigations related to the firearm memory effect when forensic specialists have collected GSR samples from the hands of the alleged shooter within four hours of the shooting.

1. Introduction

A critical analysis of the scientific literature regarding the classification criteria for GSR particles reveals a significant evolution over the past few decades. In fact, until the early 2000s, the international scientific community believed that there were particulate materials that could be attributed solely to the phenomenon of firearm discharge. However, with scientific progress, this assertion was challenged, leading to a gradual decrease in the evidentiary importance of the mere chemical composition of the recovered particulate matter. Nevertheless, this awareness did not signal the end of the SEM/EDS GSR analysis method; rather, it further highlighted the necessity of employing additional evaluation criteria for identifying GSR particles, as had already been extensively described in the late 1970s [1]. In fact, to determine whether a particle should be considered a GSR or not, it is not sufficient to identify, among the chemical elements that constitute it, those elements or a combination of them that are generally present in the primer mixtures of common ammunition (e.g., lead-barium-antimony in the case of ammunition with SINOXID^® primer mixture or a combination of titanium-zinc-strontium-gadolinium-gallium-copper-tin in the case of ammunition with Heavy Metal Free primer mixture).

Since it is possible that a lot of ammunition is fired in a gun and that cleaning operations are either not carried out or are essentially mild, it is possible to hypothesize that successive shots fired in the same firearm but with ammunition containing a primer mixture of different elemental compositions could produce GSR particles with a mixed composition (known as the “memory effect”) due to chemical–physical factors related to the firing [2,3].

GSR particles disperse from the skin of shooter’s hands solely due to gravity and mechanical actions related to everyday body movements (e.g., hand swinging). For this reason, these particles can be sampled with the stub only if the collection is performed by the forensic specialists within a reasonable time frame after the shooting. The longer the interval between the shooting and the particle collection, the higher the probability of finding particles with very small diameters on the skin. In fact, particles of such small sizes tend to disperse more slowly from the skin, as they are generally embedded in skin folds.

The number of shots fired in the same firearm is not directly proportional to the number of particles that can be found on surfaces near the shooter. This is due to the previously mentioned mechanical actions, which, together with the effect of gravity, cause the loss of particles based on their size.

The purpose of this study is to formalise the finding, within a narrow time window established by the scientific literature, of composite characteristic GSRs (i.e., those particles having a mixed characteristic composition based on different metallic alloy constituents and different primer mixtures) on skin surfaces that were close, at the time of firing, to the firearm employed by the shooter (usually the palms and backs of the hands).

Specifically, the different mechanisms of particle production have been evaluated by various authors both in relation to the type of primer used in the ammunition [4] and to the mechanical functioning of the firearm (revolver [5] and semi-automatic handgun [6]).

The current particle classification system is described in the ASTM E1588 standard [7]. This standard also provides guidelines to determine whether any particles found on the surface of a stub can be considered characteristic, indicative, or commonly associated with GSR.

As for the “memory effect”, it is important that this phenomenon be considered by the forensic specialist when interpreting the results obtained [2].

Particular attention must be paid in the case of gunshot residues sampled in the cartridge chamber of a firearm, as it contains the cartridge case, which, at the moment of firing, has expanded to confine the explosive reaction and thus prevent the escape of the propellant gases. For these reasons, the propellant gases cannot come into direct contact with the GSR particles that form inside the cartridge case. Specifically, in the case of automatic or semi-automatic firearms, complex fluid dynamic phenomena occur that initially cause the expansion of the gas cloud into the surrounding atmosphere and subsequently lead to its partial suction. In fact, the air–air or gas–air interface ends laterally in a vortex due to the shear forces acting around the edge of the jet flow [8].

Within the scope of the proposed mathematical model, it is worth highlighting the results presented in several studies concerning the combined analysis of IGSR and OGSR [9], including those collected using skin swabs [10], as well as the considerations put forward by other authors regarding the factors that influence the deposition, persistence, and collection of OGSR and IGSR [11], particularly from the perspective of contamination [12].

The second scientific assessment that the forensic specialist will have to make concerns the “time factor”, which refers to the knowledge of the retention times of GSR particles on the back and palm of a shooter’s hands. Specifically, Andrasko and Maehly [13] found that the number of GSR particles on a shooter’s hand decreases rapidly over time; the larger particles, about 10 µm, disappeared from the hands within the first hour after the shooting, while after 2 h or more, only particles with a diameter of less than 3 µm were observed.

The short retention times of GSR particles were also observed by personnel at the Metropolitan Police Forensic Science Laboratory in London. Their manual [14], whose fundamental contents have never been denied but rather confirmed by subsequent scientific literature, indicates that the larger particles (easily found during the scanning of the stub’s surface) disperse first, and after three to four hours following the shooting, only small particles will be found. The same authors propose the following GSR particle temporal decay curves both as a function of their diameter (Figure 1a) and as a function of their number (Figure 1b).

Therefore, in a critical reading of the reported curves, most of the large GSR particles will disperse from the shooter’s hands within the first hour after the shot, while the smaller ones tend to remain longer as they are generally embedded in the skin folds. For these reasons, the discovery of GSR particles several hours after the shooting is most often attributable to transfer [15] or redistribution [16], or that the population observed is the result of a legitimate firearm association unrelated to the criminal event under investigation [17].

The graph of the IAMA Newsletter (Figure 2) [18] provides further confirmation of the rapid dispersion times of GSR particles from the skin following shooting, due solely to the mechanical actions of the shooter’s limbs.

The results presented are extremely interesting because the vertical error bars on the decreasing exponential curve indicate that there are cases where the shooter’s hands are found to be free of GSR particles in samples taken within 0.5 h after the shot.

Although the results presented in Figure 1 cannot be attributed high significance due to the lack of important details regarding how the experimental tests were conducted and how the data were collected, it must be clear that the trend of the reported decreasing exponential curves has a well-established heritage in the scientific literature in the forensic field [19] and has formed the basis for the formal elaboration described below.

In relation to the time interval, Wallace [20] also confirms the time limit for collecting GSR particles from a living subject as three hours after the shooting; according to European forensic specialists from ENFSI [21], this limit could even decrease to two hours after the shooting.

To summarize the above, it can be said that the persistence of particulate on the hands depends on the time elapsed between the shooting and the collection, the size of the particles, as well as the activities performed by the shooter with their hands (from a mechanical point of view).

The memory effect and the temporal decay of GSR particles, as is easy to infer, are two phenomena closely related to each other. In this regard, the link between the persistence of particulate on the shooter’s hands and the memory effect have been studied by Jalanti and collaborators [22]. Finally, these phenomena could also influence the primary and secondary transfer of particles [23], both within specific sub-groups of the population and the general population [24].

It is precisely from the above considerations that the authors of this preliminary study wish to further explore the relationship between the formation of composite characteristic GSR particles (memory effect) and their dispersal from the shooter’s hands as a function of the time elapsed since the shot.

In this regard, it is important to clarify that, in most cases, the firearm used to commit the crime under investigation is never recovered. At most, investigators may find one or more spent cartridge cases at the crime scene, which, unlike the firearm, cannot provide forensic specialists with useful information about the memory effect. It may also be possible to sample the victim’s clothing using stubs; however, even in this case, the analytical findings on any GSR particles discovered would provide forensic specialists with only limited information about the memory effect, unless such results are later compared with those obtained from the firearm following its recovery.

2. Materials and Methods

The search and identification of GSR particles on a stub is an activity that requires a deep understanding of both the phenomena related to firearm discharge and particle formation, as well as the relevant international scientific literature on the subject.

Given the above, it is therefore of fundamental importance to distinguish between particles composed of elements attributable exclusively to the fired ammunition and particles composed of elements derived from previous firings and subsequently incorporated. To do this, the forensic specialist must verify the following:

• particles with a pure composition of the first type of primer are present, collected, and analysed inside the barrel of the suspect firearm;
• particles with a pure composition of the second type of primer are present, collected, and analysed inside the barrel of the suspect firearm;
• particles with a mixed composition containing elements present in both the first and second types of primer are present, collected, and analysed inside the barrel of the suspect firearm;
• particles of the three types described are simultaneously present, collected, and analysed on the ballistic evidence (e.g., cartridge cases and bullets).

In particular, the forensic specialist who conducted the analysis must evaluate

• the possibility that the examined particles may or may not have incorporated elements from the ammunition used (e.g., cartridge case, bullet) or from the internal surfaces of the firearm (e.g., rust in the barrel [25], hybrid mixtures based on molybdenum-sulphur-lead [26]), as well as the possibility that contaminations related to the propellants in the ammunition [27] may have occurred or that the evanescence of mercury in mercury fulminate-based primers [28] has taken place;
• the possibility of distinguishing particles derived from two profoundly different primer compositions (e.g., SINTOX^® and SINOXID^®). However, the likelihood of successfully distinguishing particles produced by one primer mixture from another drastically decreases when the compositions of the primer mixtures are very similar to each other (e.g., SINOXID^® with and without the presence of aluminium);
• the possibility that, due to the processes of compression and rarefaction of the air following the expansion of the propellant gases, resulting in a backflow of the gases, adhesion of particles from the barrel to the surface of the cartridge chamber of the firearm occurred after the ejection of the cartridge case.

The aforementioned assumptions play an important role in defining the phenomenon of the memory effect in firearms, as they delineate its formal scope of application.

In the following section, a mathematical model will be developed with the aim of providing the reader with formal guidelines for approaching this phenomenon (not immediately quantitative) observable within the four-hour time frame after the shooting, i.e., from the primary transfer of GSR particles onto the shooter’s hands.

By “mathematical model”, the Authors refer to its current definition, meaning, in general terms, a formal and/or qualitative–quantitative representation of a natural or anthropogenic phenomenon [29].

Specifically, the proposed mathematical model aims to represent, within the time frame of GSR particle decay from the surfaces of the shooter’s hands, the fundamental interactions between particles generated during each individual shot, those adhering to the firearm’s surfaces, and those detached and lost due to fluid dynamic phenomena related to the shooting, which are then transferred to the shooter as primary transfer.

3. Discussion

Before proceeding with the explanation of our formal processing, it is necessary to clarify that it takes into account two distinct variables: the number of characteristic GSR particles found and the time interval that has elapsed since the shot that produced them.

In fact, the evaluation of the mechanisms that influence the deposition, persistence, transfer, and recovery of characteristic GSR particles and their loss rate is complex in nature, and their formalisation from a mathematical perspective would have involved the definition of multidimensional spaces (which require one dimension for each correlated variable) without being able to reference experimental uncertainties and mutual inferences.

The proposed mathematical model is based on the previously stated definition of a composite characteristic GSR particle without introducing additional multidimensional complications due to the various mixed elemental compositions that might originate in reality.

It is important to clarify that this model was not developed using software and/or artificial intelligence but is based on the formalism of differential equations, which relate an unknown function to its derivatives [30].

The following parameters (m, n, o, p), which act as exponents, together with the parameters serving as bases (x, y), indicate the number densities [31]. As intensive quantities [32], they describe the concentration of a substance in space composed of countable objects (in our case, GSR particles).

Based both on the previously outlined decision-making process and on preliminary assumptions concerning only the average number of GSR particles counted on the surface of the sample after sampling from the shooter’s hands (without any data regarding their size or elemental composition but assuming that the elemental profile of the “characteristic of” type is the simplest—i.e., a single allogeneic tracer element—and the most repetitive possible without “indicative” or “commonly associated with” particles), the exponential decay of composite characteristic GSR particles can be formalised using the following first-order linear homogeneous differential equation:

$$x^{\prime }\left(t\right)+kx\left(t\right)=0\hspace{0.33em}\hspace{0.33em}t\in \left\{0,\dots ,4\right\},k\in \mathbb{R}$$

The general integral of our differential equation is of the form

$$x\left(t\right)=e^{-A\left(t\right)}\left(\int f\left(t\right)e^{A\left(t\right)}+k\right),t\in \left\{0,\dots ,4\right\},k\in \mathbb{R}$$

Calculating the integrating factor $e^{A\left(t\right)}$:

$$e^{A\left(t\right)}=e^{\int kdt}=e^{k\int dt}=e^{kt}t\in \left\{0,\dots ,4\right\},k\in \mathbb{R}$$

it is possible to write the following general integral:

$$x\left(t\right)=x_0{e}^{-kt}\hspace{0.33em}t\in \left\{0,\dots ,4\right\},k\in \mathbb{R}$$

where k is a constant that, considering the error bars shown in the graph in Figure 2, must be determined on a case-by-case basis, while ${x\left(0\right)=x}_0$ represents the quantity of particles that deposited on the shooter’s hands at the moment of the shooting, $t=0$.

Therefore, it is possible to define a component $x^n$ related to the number of GSR particles produced as a result of a single shot and a component ($x^{n-p}+y$) related to the number of GSR particles expelled from the firearm (e.g., from the barrel, the ejection port, the gap between the barrel and the cylinder in the revolver) and the mechanical locking system of the bolt carrier (for example, blowback operation, where the bolt carrier moves back during the passage of the bullet through the barrel, or locked breech, where the bolt carrier moves back after the bullet exits the barrel). Specifically, the component ($x^{n-p}$) is represented by the particles originating from the last shot and expelled from the firearm, while the component [$y=\beta \hspace{0.33em}(y^m-y^{m-o})$] is represented by those particles originating from previous shots and liable to detach from the internal surfaces of the barrel.

$$\left(x^{n-p}\right)+\beta (y^m-y^{m-o})$$

This equation represents the GSR particles $x_0$ that were deposited on the shooter’s hands at the moment of the shot $t=0$. Considering the general integral of the differential equation, originally analysed, the above equation becomes

$$x\left(t\right)=\alpha [\left(x^{n-p}\right)+\beta \left(y^m-y^{m-o}\right)]e^{-kt}$$

where $\alpha \left[\left(x^{n-p}\right)+\beta \left(y^m-y^{m-o}\right)\right]$ represents the number of GSR particles that, once expelled from the firearm, have settled on the shooter’s hands.

To better analyse the condition at time $t=0$, given that there are small values of t close to 0, it is possible to use the Taylor–McLaurin series expansion to better understand the behaviour of the GSR particles. Therefore, let us consider the Taylor–McLaurin series expansion of the exponential function in compact form:

$$e^t=\sum _{n=0}^{\mathrm{\infty }}{\displaystyle \frac{t^n}{n!}} \forall t$$

which for ($e^{-kt}$), stopping at the first order, becomes

$$e^{-kt}\approx 1-kt$$

Substituting ($e^{-kt}$) into $\left[x\left(t\right)\right]$ we obtain

$$x\left(t\right)\approx \alpha [\left(x^{n-p}\right)+\beta \left(y^m-y^{m-o}\right)]·(1-kt)$$

$$x\left(t\right)\approx \alpha \left[\left(x^{n-p}\right)+\beta \left(y^m-y^{m-o}\right)\right]-kt{x}^{n-p}\alpha -kt{y}^m\alpha \beta +kt{y}^{m-o}\alpha \beta$$

which, for $t=0$, becomes

$$x\left(0\right)\approx \alpha \left[\left(x^{n-p}\right)+\beta \left(y^m-y^{m-o}\right)\right]$$

Thus, the number of GSR particles that have settled on the shooter’s hands at the moment of the shot $t=0$ evolves according to the law

$$x\left(t\right)=\alpha [\left(x^{n-p}\right)+\beta \left(y^m-y^{m-o}\right)]$$

Meanwhile, close to the fourth hour after the shooting occurred, $t=4$, the number of detectable GSR particles on the shooter’s hands will evolve according to the law

$$x\left(4\right)=0=\alpha [\left(x^{n-p}\right)+\beta \left(y^m-y^{m-o}\right)]e^{-k4}$$

Since the GSR particles that settled on the internal surfaces of the firearm are

$$[x^n-x^{n-p}(<x^n\left)\right],$$

and those that settled as a result of previous shots, susceptible to detachment from the internal surfaces of the firearm y, are in the condition related to the shot that originated them, i.e.,

$$[y<y^m-y^{m-o}(<y^m\left)\right],$$

it follows that the likelihood of finding GSR particles on the shooter’s hands at a generic time t* is a function of $\alpha$, $\beta$, $(n-p)$ and $(m-o)$.

Finally, it can be inferred that the elemental composition of GSR particles, sampled from the shooter’s hands close to the fourth hour after the shooting and originating from shootings that occurred at different times, will contribute only minimally and negligibly to the memory effect if the shootings were carried out with ammunition containing different primer mixtures. Furthermore, this contribution will be entirely insignificant if the shootings were conducted using the same ammunition.

Therefore, this result will only be valid under the conditions underlying the mathematical model (no more than four hours must have passed since the shooting, and the GSR particles must be sampled from the shooter’s hands). However, in the case of timely sampling (close to the time of the shooting), the application of this mathematical model could provide the forensic specialist with different results regarding the firearm’s memory effect, specifically in terms of the numerical density of GSR particles.

4. Conclusions

Scientifically verifying the use of a firearm by a suspect has always been one of the most highly sought-after pieces of investigative data by investigators. However, the search for and identification of particles linked to a gunshot, even if found to be positive, cannot be considered as clear or decisive evidence (precisely because of the possibilities of the transfer phenomena) but as a most important or fundamental indication, which must, with other clues, construct a logical and indisputable proof in the case-by-case approach.

The memory effect also negatively impacts the number of characteristic GSR particles that can be found on the shooter’s hands after the shot (their number decreases within a few hours); additionally, there can be a discrepancy between the elemental composition of the GSR particle population present in the last spent cartridge case and those collected at time t* on the shooter’s hands.

Based on the assumptions outlined above, a mathematical model has been developed, through which the forensic specialist can obtain formal indications regarding the memory effect detectable within four hours of the shooting, that is, from the primary transfer of GSR particles onto the shooter’s hands. Specifically, the model aims to represent, within the decay time of the particles, the basic interactions between those generated by each individual shot, those adhering to the surfaces of the firearm, and those detached and lost from the firearm due to the fluid dynamic phenomena associated with the shot.

The above model was not developed using software and/or artificial intelligence but is based on the formalism of differential equations, which relate an unknown function to its derivatives. Moreover, the boundary conditions, which are the conditions that the solution of a differential equation must satisfy, were chosen to relate the number density to the time instant (t).

In this context, a thorough scientific investigation must necessarily consider various phenomena, such as the “memory effect” and the “time factor”, that is, the decay time. This preliminary study, in the assumptions detailed, aims to provide a formal relationship between composite characteristic GSR particles (using ammunition with a primer mixture of different elemental compositions in the same firearm) and the exponential dispersion of particulate from the shooter’s hands.

In conclusion, it can be stated that the manifestation of the “memory effect” (characterised by a peculiar statistical variability, with data that need to be supplemented based on dedicated experimental evidence) is closely related to the “time factor”, namely the decay time of the GSR particles that have settled on the shooter’s hands, and therefore to the phenomena of primary and secondary transfer.

Although the experimental validation of the model may prove particularly complex, the authors have nonetheless based it on various analytical results extrapolated from the aforementioned scientific literature, formulating a series of non-contradictory hypotheses, each of which plays a specific role within the mathematical formalism employed. Finally, as repeatedly stated throughout the discussion, the developed model should be considered valid only under the conditions on which it is based: a time interval from the shooting not exceeding four hours and the sampling of GSR particles conducted directly from the shooter’s hands.