Physics SL
Sample Extended Essays
B
20 mins Read
3,840 Words
English
Free

# How does the interference fringe width (distance between two maxima i.e. two bright spots) depend on the diameter of the slits (behaving as source), refractive index of medium present between the source of the light (double slit) & the screen?

## Rationale

Since childhood, I have been intrigued by the colors of light I saw on the soap bubbles, the sky, and the window screen. Hence, I would always question “How and Why such patterns of light are created?” This led me to performing small experiments based on the topic of light as was taught in the Physics class. Physics is an integral part of life. All that we observe around us can be justified using physical experiments, laws and calculations. I believe that education takes place when there is a complete understanding of the topic along with the perfect amalgamation between theory and its application and the IB has always maintained a balance between the two. IB has groomed me to become an inquirer and this trait has motivated me to gather more information on the topics of physics. Last summer, I had joined a course in physics to know more about the interesting aspects of it. I was especially interested in the topic of wave optics which is an integral part of IB syllabus and asked “Can the colors of light observed in different situations be analyzed under this scope of wave optics?” One of the most interesting and important aspects of physics is optics. It deals with the interaction of light with objects around us. This led to further questioning myself -Can I experimentally obtain the natural phenomenon of light occurring around me?” This interest had driven me to perform a study on the double slit experiment. I learnt further from my in-depth study of Interference of light that it is an interesting domain as it invokes the wave-particle duality of matter and the theoretical field of quantum mechanics.

And when it comes to the point of contact between these physical domains, Young’s double slit experiment emerges as a prime example in the justification of a surprising optical phenomenon using the quantum theory and duality of matter. I thus decided to delve deeper into this field of study by going through several articles and deciding to come up with an original experiment myself that would aim to derive a relation between a few common aspects like refraction, interference. I thus finally stated my research question as:

My exploration is a deeper delve into other fundamental properties of wave optics such as interference, and refraction which I aim to make a part of this original experiment and thus deduce a correlation between these physical tendencies.

## Background information

### Interference of light

The process in which two or more light or electromagnetic waves of the same frequency combine to reinforce or cancel each other, the amplitude of the resulting wave being equal to the sum of the amplitudes of the combining waves. The effect is that of the addition of the amplitudes of the individual waves at each point affected by more than one wave. This results in constructive interference leading to formation of bright spots or destructive interference leading to formation of dark spots.

• David McKay

Ivy league graduate, 4+ yrs of experience, vetted IB Examiner!

• ### Double slit experiment

In this experiment light from a point source of a constant wavelength (laser light of red colour) is allowed to pass through two narrow slits in a dark room. This results in the formation of alternating dark and bright fringes of light (called interference pattern) on screen.

Figure 1 - Interference Pattern Of A Double Slit (Top) Monochromatic Light : Green, (Bottom) Polychromatic Light: White Light.
Figure 2 - Double Slit Experiment

The figure is the schematic diagram of a Double slit experiment, where a monochromatic light acting as the primary source (marked in Red) is incident on the screen with two slits which acts as the two sources of light causing interference pattern. Light from these sources (two slits) behaves as a wave hence interferes constructively and destructively depending on the superposition of waves formed. On the optical screen these interfering patterns of light fall to give an illumination as shown in the front view. This consists of alternating bright and dark fringes which are equally spaced. This distance between the consecutive bright dark fringes are known as fringe width.

### Central maxima

The central point midway between the two slits on the screen where waves coming from either of the slits constructively interfere and reinforce each other to form the brightest fringe in the pattern is called the central maxima. The other fringes on either side of the central maxima have the same intensity but are known as secondary maxima.

Figure 3 - Graph Of Intensity Showing Maximas (Region Of Maxima Intensity) In Double Slit Experiment.

### Interference fringe width

Fringe width is the distance between two consecutive bright spots (maxima, where constructive interference take place) or two consecutive dark spots (minima, where destructive interference take place).

Thus we can also say it is the linear distance between the central maxima and the next consecutive constructive interference

$$B=\frac{\lambda × D}{d}$$

where,

B is fringe width

λ is wavelength

D is distance between source and screen

d is the distance between two slits within which the diameters of slits are made.

Figure 4 - Fringe Width Is Represented As The Distance Between The Central Maxima And Consecutive Bright Fringes Between The Dark Ones

### Diameter of the slit

The diameter of the slit is varied keeping the distance between the two extreme points of the two slits constant. Hence the distance between two slits “d” reduces.

• David McKay

Ivy league graduate, 4+ yrs of experience, vetted IB Examiner!

• ### Refractive index

The absolute refractive index of any given medium is defined as the ratio of the speed of light in vacuum to the speed of light in that medium. Refractive index of medium 1 with respect to medium 2 is the speed of light in medium 2 divided by speed of light in medium 1. It is similarly defined as the ratio of the sine of the angle of incidence to the sine of the angle of refraction at the interface of two transparent optical media.

Figure 5 - Refractive Index Is Calculated By The Ratio Of Sine Of Angle Of Incidence To That Sine Of Angle Of Refraction

### Exploration methodology

In a black paper a fixed distance is marked within which two pinholes are made (S1 and S2) which act as the two slits. This is mounted with a laser of fixed wavelength acting as the primary source. As the light passes through the two slits interference patterns are observed on the white screen (optical screen) placed at a fixed distance D. The fringe width “B” is measured. The Refractive medium between the source and the screen is then varied to observe their effect on the fringe width in the interference pattern. The fringe width obtained is noted. The same thing is repeated thrice. Then the diameter of the slits was varied and the fringe width is measured for each diameter for three trials. This experiment is again repeated thrice and the data thus produced is noted.

• David McKay

Ivy league graduate, 4+ yrs of experience, vetted IB Examiner!

• ### Literature survey

In order to gain insight on the topic, a few previously conducted researches were consulted to identify the nature of experiments with polarizers. The paper titled, “The double slit experiment with polarizers” by M. Holden, D.G.C. McKeon, and T.N. Sherry inspired me a lot. The experiment thus performed helped realise the relationship between the polarizing angles and the refractive index of the medium. It also established that with an increase in the wavelength of the light incident, the fringe width would decrease. It helped me set up the experiment with both slits and variable refractive index. The plausibility of the experiment and the realizability of the aim was thus inferred.

## Hypothesis

Predictions

• It was assumed that with the increase in diameter of the slit the fringe width will increase.
• It was also assumed that with the increase in refractive index of the medium the fringe width will decrease.

Justification

• Since the distance within which the slits were made was kept constant, with the increase in slit diameter the distance between the slits decreased, this further decreased the region of light waves superposing thus leading to greater area of illumination. Hence, the fringe width would increase.
• Since increase in refractive index would reduce the speed of light thus fringe width would decrease.

## Variables

• David McKay

Ivy league graduate, 4+ yrs of experience, vetted IB Examiner!

• ### Independent variable

Refractive index of medium

Three different types of medium – glass, water and glycerin were chosen and the medium were one-by-one replaced and the resultant fringe widths were measured. These materials were used as medium since they were readily available at low cost and unlike plastic, do not cause much harm to the environment. Since the refractive index is a pure ratio hence has no unit.

Diameter of slit

Diameter of the double-slit was changed by using needles of different diameters, specifically 0.05mm., 0.06mm., 0.07mm., 0.08mm. and 0.09mm. For each diameter value, the experiment was performed thrice and the resultant data was obtained. These values were taken since both the slits had to fall inside the spot created by the laser pointer. As the pointer does not create a large spot, the slits could not have larger values for diameter. Also, for a greater diameter the particle nature of optics would be predominant hence fringes will not be observed. And it would not be feasible too, to make the slit of a diameter smaller than 0.05mm as it cannot be measured with the screw gauge at hand.

### Dependent variable

Fringe width

In this exploration, fringe width of the central maxima was taken as the dependent variable. It was measured by screw gauge and recorded thrice during every change in the independent variables. It was measured in millimeters (mm).

### Controlled variables

Distance between the two extreme points of the two slits

The diameter of the slit is varied keeping the distance between the two extreme points of the two slits at 0.2mm always constant. Hence the distance between two slits “d” reduces. This was the motive of our experiment.

Distance between the slit and screen

With an increase in the distance between the screen and slits, the fringe width would increase and hence, the observation of this exploration that the dependence of refractive index and the diameter of the slit would be disputed. To control that, the distance between the slits and the screen “D” has been kept constant at 1m ± 0.01m and measured using a ruler.

Wavelength of light used

As has been seen in the literature survey with an increase in the wavelength of the light incident, the fringe width would decrease and hence, the observation of this exploration that the dependence of refractive index, and the diameter of the slit would be disputed. To control this wavelength of light, a laser torch of the same colour (λ=700nm) was used in all the trials.

• David McKay

Ivy league graduate, 4+ yrs of experience, vetted IB Examiner!

• ## Apparatus

Apparatus Name
Quantity
Capacity
Least Count
Uncertainty (±)
Screw gauge
1
±0.01 mm
0.005mm
Laser Pointer
1
5mW λ=700nm (RED)
-
-
Needle
5
0.05mm, 0.06mm, 0.07mm, 0.08mm, 0.09mm
-
-
Burette Stand
2
-
-
-
Figure 6 - Table On Apparatus

## Materials used

• Black paper was used to make the slit.
• White paper was used to make the screen.
• String was used to mount the black paper on the burette stand.
• Different Medium: water, glass and glycerin

## Procedure

For variation of fringe width with respect to diameter of slit

• An opaque black paper was taken and two points were marked at a difference of 0.2 mm using a pen.
• Two holes were made with a needle of 0.05 mm.
• The black paper was placed and fixed on the experimental table using four strings that are tied at the four ends of the paper by making four holes at the ends. The other ends of the strings are tied to a burette stand.
• A red laser (λ=700nm) torch of 5mW power was taken and made fixed on the experimental table using a vertical clamp at a distance of 5 cm from the paper, such that it projects the light at the position of the holes or slits.
• At a distance of 1 meter from the paper, a white chart paper (optical screen) was set using four strings that are tied at the four ends of the paper by making four holes at the ends. The other ends of the strings are tied on the second burette stand.
• All the doors and windows of the room were closed and covered with black curtains; other lights of the room were turned off. This, ensured total darkness in the room.
• The laser was turned on.
• The pattern of light that was formed on the screen was observed.
• The distance between the central maxima and the first constructive interference was measured using a screw gauge and noted.
• The laser torch was turned off.
• At an interval of one minute, the laser torch was turned on and the distance between the central maxima and the first constructive interference was measured using the screw gauge and noted. This step was repeated three times.
• The paper with slits was then changed and the paper with the slits of diameter 0.05 mm was replaced by the paper with slit of diameter 0.06 mm.
• Steps 2 to 12 were repeated for 0.05mm to 0.09mm of diameter of the slit with a difference of 0.01mm and for each diameter three trials were taken.

For variation of fringe width with respect to refractive index

• An opaque black paper was taken and two points were marked at a difference of 0.2 mm using a pen.
• Two holes were made with a needle of 0.05 mm.
• The black paper was placed and fixed on the experimental table using four strings that are tied at the four ends of the paper by making four holes at the ends. The other ends of the strings are tied to a burette stand.
• A red laser (λ=700nm) torch of 5mW power was taken and made fixed on the experimental table using a vertical clamp at a distance of 5 cm from the paper, such that it projects the light at the position of the holes or slits.
• At a distance of 1 meter from the paper, a white chart paper (optical screen) was set using four strings that are tied at the four ends of the paper by making four holes at the ends. The other ends of the strings are tied on the second burette stand.
• All the doors and windows of the room were closed and covered with black curtains; other lights of the room were turned off. This ensured total darkness in the room.
• The laser was turned on.
• The pattern of light that was formed on the screen was observed.
• The distance between the central maxima and the first constructive interference was measured using a screw gauge and noted.
• The laser torch was turned off.
• At an interval of one minute, the laser torch was turned on and the distance between the central maxima and the first constructive interference was measured using the screw gauge and noted. This step was repeated three times.
• A glass slab was placed in between the laser torch and the white chart paper (screen) on the experimental table such that it covered the distance between the source and screen. Thus, the refractive index of the medium was changed.
• The distance between the central maxima and the first constructive interference was measured using screw gauge and noted and also the lateral shift between position of central maxima with and without medium (glass) was noted.
• Then the whole setup was inserted in a flat basin containing water and steps 2 to 12 were repeated.
• Then the whole setup was inserted in a flat basin containing glycerine and steps 2 to 12 were repeated.
• Thus, the same process was repeated for each material placed between the light source and the screen.
• David McKay

Ivy league graduate, 4+ yrs of experience, vetted IB Examiner!

• ## Considerations

Safety Precautions

• Lab coat was worn for safety reasons during experimentation.
• In this experiment, gloves were worn throughout since it required handling of brittle materials like glass.
• As the exploration required use of laser pointer, safety goggles were worn to prevent harm to the eyes.

Ethical Considerations

• In this experiment, the materials chosen were taken in optimum quantity so that no wastage was done and this doubled up too, as a cost-effective method.
• The materials, like paper, were reused and thus ensured that no harm was done to the surroundings.
• Moreover, plastic materials were avoided and biodegradable products were used.

Environmental Considerations

• No significant harm was observed on the environment, thus there were no environmental considerations.

## Data collection

Figure 7 - Table On Raw Data For Fringe Width (in ± 0.01 mm) Versus Diameter Of The Slit (in ± 0.01 mm) For Water (Refractive Index = 1.33)

Formula used

Mean = $$\frac{Σ(trial \,values)}{number \,of \,trials}$$

Standard deviation (SD) =√ $$\frac{Σ(trial \,values-mean\, value)^2}{number \,of \,trials}$$

• David McKay

Ivy league graduate, 4+ yrs of experience, vetted IB Examiner!

### Video Course

• Figure 8 - Table On Raw Data For Fringe Width (in ± 0.01 mm) Versus Diameter Of The Slit (in ± 0.01 mm) For Glycerin (Refractive Index = 1.46)
Figure 9 - Table On Raw Data For Fringe Width (in ± 0.01 mm) Versus Diameter Of The Slit (in ± 0.01 mm) For Glass (Refractive Index = 1.50)

### Data processing

Figure 10 - Table On Mean Value Of Fringe Width Versus Diameter Of The Slit For Water, Glass And Glycerin

Key: R.I = Refractive Index

Sample calculation

For water,

Average fringe width of the medium

= $$\frac{3.02+3.23+3.41+3.63=3.83}{5}$$= 3.42 ± 0.01 mm

Error propagation

For Row - 1 of Table - 4

Percentage error in mean fringe width for water

= $$\frac{absolute\,uncertainty}{value\, of\,mean\,fringe\,width}$$× 100

=$$\frac{±0.01}{3.02}$$× 100 = ± 0.33

Percentage error in mean fringe width for glass

= $$\frac{absolute\,uncertainty}{value\, of\,mean\,fringe\,width}$$× 100

=$$\frac{±0.01}{3.04}$$× 100 = ± 0.32

Percentage error in mean fringe width for glycerin

$$\frac{absolute\,uncertainty}{value\, of\,mean\,fringe\,width}$$× 100

=$$\frac{±0.01}{3.05}$$× 100 = ± 0.32

Percentage error in diameter

= $$\frac{absolute\,uncertainty}{value\, of\,mean\,fringe\,width}$$× 100

=$$\frac{±0.01}{0.05}$$× 100 = ± 20.00

### Analysis

Figure 11 - Mean Value Of Fringe Width Versus Diameter Of The Slit For Water, Glass And Glycerin
• In this graph, the diameter of the slit (in mm) has been taken along the x-axes since it is the independent variable and fringe width (in mm) has been taken along the y-axes since it is the dependent variable.
• There are nearly no outliers thus the graph shows a perfect positive correlation. This can be justified by the high value of regression coefficient (= 0.999). The fringe width is seen to be increasing with increase in diameter.

The equation of trendline of fringe length with respect to diameter of slit for water is:

y = 20.2x + 2.01 ... ... ... (equation 1)

The equation of trendline of fringe length with respect to diameter of slit for glass is:

y = 14.5x + 2.315 ... ... ... (equation 2)

The equation of trendline of fringe length with respect to diameter of slit for glycerin is:

y = 10.3x + 2.525 ... ... ... (equation 3)

• It is also observed that the slope of the graph decreases as the refractive index increases. The values of gradient are 20.2, 14.5 and 10.3 for water, glass and glycerin respectively. This clearly shows that as the refractive index of the medium increases for 1.33 to 1.50 from water to glycerin, the change of fringe width against diameter of the slit becomes less significant. Thus, more the refractive index of the medium, lesser the change in fringe width with the increase in diameter of the slit.
• David McKay

Ivy league graduate, 4+ yrs of experience, vetted IB Examiner!

### Video Course

• Figure 12 - Change Of Average Fringe Width (±0.01 mm) With The Refractive Index Of The Medium
• In this graph, the refractive index has been taken along the X-Axis since it is the independent variable and fringe width (in mm) has been taken along the Y-Axis since it is the dependent variable.
• The outliers may be attributed to the mediums being of different state of matter, thus optical density variation or the speed of light variation does not have perfect correlation. The graph shows a negative correlation. This can be justified by the high value of regression coefficient (= 0.95). The fringe width is seen to decrease with increase in refractive index.
• The equation of trendline of fringe width with respect to refractive index is obtained as:

y = - 0.8797x + 4.5947 ... ... ... (equation 3).

Scientific justification:

From the first graph we see the fringe width increases with increase in diameter of slit which is justifying the dependency on separation between slits. This is because as the diameter increase in two slits within the given constant length 0.20 mm it leads to decrease in the distance of separation between the slits (d). As more and more waves diffract through the increasing aperture, the fringe width increases too.

Thus, we can infer

• Fringe width diameter of the slit
• diameter of the slit  $$\frac{1}{Distance\, between \,the \,sites}$$

Hence,

Fringe width  $$\frac{1}{Distance\, between \,the \,sites}$$

From the second graph, it is observed that the fringe width decrease with increase in refractive index, which might be due to the fact that light waves travel slower in media of increasing refractive indices as the resistance to flow of waves increases with increase in optical density, thus decreasing the velocity of light waves. Thus, the wavelength decreases too. and conversely the fringe width increases. Thus

$$\lambda=\frac{\lambda}{\mu}$$

Fringe width ∝ λ

Fringe width ∝ $$\frac{\lambda}{\mu}$$

Fringe width ∝ $$\frac{1}{\mu}$$

where,

ƛ is changed wavelength

ƛ is wavelength

μ is refractive index

## Conclusion

How does the interference fringe width (distance between two maximas i.e. two bright spots) depend on the diameter of the slits (behaving as source), refractive index of medium present between the source of the light (double slit) and the screen?

• Through the experimentation, it was observed that the fringe width increases proportionately with increasing diameter of slit but decreases proportionately with the increasing refractive index.
• The equation of trendline of fringe length with respect to diameter of slit for water is:

y = 20.2x + 2.01

The equation of trendline of fringe length with respect to diameter of slit for glass is:

y = 14.5x + 2.315

The equation of trendline of fringe length with respect to diameter of slit for glycerin is:

y = 10.3x + 2.525 ... ... ... (equation 3)

It is thus observed that there is a straight-line graph obtained for all the mediums (of different refractive index) each having an intercept value almost of the same order.

It is also observed that the slope of the graph decreases from 20.2 to 10.3 as the refractive index increases from 1.33 to 1.50 which justifies the equation:

$$B = \frac{\lambda× D}{d}$$ and the relation Fringe width ∝ $$\frac{1}{\mu}$$

There are nearly no outliers thus the graph shows a perfect positive correlation. This can be justified by the high value of regression coefficient (= 0.999). The fringe width is seen to be increasing with increase in diameter.

The equation of trendline of fringe length with respect to refractive index is obtained as:

y = - 0.8797x + 4.5947

The negative slope indicates fringe width is inversely related to refractive, justifying the equation:

Fringe width ∝ $$\frac{1}{\mu}$$

The graph shows a negative correlation. This can be justified by the high value of regression coefficient (= 0.95). The fringe width is seen to decrease with increase in refractive index.

Thus, we finally conclude that the interference fringe width (distance between two maxima i.e. two bright spots) depend directly on the diameter of the slits (behaving as source) and inversely on refractive index of medium present between the source of the light (double slit) and the screen, keeping the length within which, the two slits have to be made constant, distance between the source and screen, and wavelength of source of light also constant.

• David McKay

Ivy league graduate, 4+ yrs of experience, vetted IB Examiner!

• ## Evaluation

### Strengths

• All three states of medium were taken: solid(glass,), liquid (water, glycerin) and gas (air) (used as control which made the experiment more widespread and results were thus more conclusive.
• Each part of the experiment had a graphical representation which made the understanding of this exploration easier.
• There have been three trials in each part to ensure greater accuracy and precision.

### Limitations

Systematic error

• Instrumental error in screw gauge.

It increases the fringe width during measurement since the precision is 0.01mm only. It can be corrected by using a travelling microscope to measure the fringe width.

• Alignment error in laser pointer.

Because of this, the light may shift from the exact position of the slits. It can be corrected by using a properly aligned and efficient laser pointer, mounted in a sturdier tripod.

• The least count of the screw gauge has limited the accuracy to which the fringe widths could be measured. Furthermore, the loss in intensity of fringes has rendered measurement inaccurate when the angle was increased. Moreover, the initial considerations such as an inextensible string and sturdy plane of paper may result in other methodological errors.

Methodological error

• The slits might not be made properly since making pinholes with needles is not an efficient method.
• Secondly, the paper might get disturbed due to external factors when mounted on a burette stand. This would tamper with the empirical data.
• There could have been some space filled with air when solid medium was taken.
• David McKay

Ivy league graduate, 4+ yrs of experience, vetted IB Examiner!

• ### Future scope

This experiment could again be conducted focusing more on the variation in fringe width due to a variation in the angles between the polarizers. Since a slight relation could be ascertained, a deeper and more extensive study would provide better justification for changes in fringe width observed due to change in not just mutually horizontal angles but even mutually vertical angles. The research question could thus be framed as: How does the fringe width between central maxima and consecutive secondary maxima vary with the change in position of polarizers kept mutually horizontal and vertical by turning the two by different angles during a double slit experiment? To perform the exploration, firstly, a black paper could be taken and two pinholes could be made, before mounting it before another black screen and a laser is to be fixed so as to let the light pass through them and four objects (polarizers) to be kept between the source and the screen to observe their effect on the fringe width in the interference pattern thus generated. The polarizers could then be mutually rotated vertically and horizontally and the change in position of pattern and fringe width could be noted.

## References

Rueckner, Wolfgang, and Joseph Peidle. “Young’s Double-Slit Experiment with Single Photons and Quantum Eraser.” American Journal of Physics, vol. 81, no. 12, Dec. 2013, pp. 951–58. DOI.org (Crossref), https://doi.org/10.1119/1.4819882.

Radin, Dean, et al. “Consciousness and the Double-Slit Interference Pattern: Six Experiments.” Physics Essays, vol. 25, no. 2, June 2012, pp. 157–71. DOI.org (Crossref), https://doi.org/10.4006/0836-1398- 25.2.157.

Thomas, Joseph Ivin. “The Classical Double Slit Experiment–a Study of the Distribution of Interference Fringes Formed on Distant Screens of Varied Shapes.” European Journal of Physics, vol. 41, no. 5, Sept. 2020, p. 055305. DOI.org (Crossref), https://doi.org/10.1088/1361-6404/ab9afd.

Kolenderski, Piotr, et al. “Time-Resolved Double-Slit Interference Pattern Measurement with Entangled Photons.” Scientific Reports, vol. 4, no. 1, Apr. 2014, p. 4685. www.nature.com, https://doi.org/10.1038/srep04685.

Tavabi, Amir H., et al. “The Young-Feynman Controlled Double-Slit Electron Interference Experiment.” Scientific Reports, vol. 9, no. 1, July 2019, p. 10458. www.nature.com, https://doi.org/10.1038/s41598-019- 43323-2.

Wolf, Emil. “Young’s Interference Fringes with Narrow-Band Light.” Optics Letters, vol. 8, no. 5, May 1983, pp. 250–52. www.osapublishing.org, doi:10.1364/OL.8.000250.

Song, Guiju, et al. “Effect of Varied Fringe Width on Measured Profile in Structured Line Projection Method.” Interferometry XIII: Techniques and Analysis, vol. 6292, International Society for Optics and Photonics, 2006, p. 62920T. www.spiedigitallibrary.org, https://doi.org/10.1117/12.681577.

Jones, C. E., et al. “Refractive Index Distribution and Optical Properties of the Isolated Human Lens Measured Using Magnetic Resonance Imaging (MRI).” Vision Research, vol. 45, no. 18, Aug. 2005, pp. 2352–66. ScienceDirect, https://doi.org/10.1016/j.visres.2005.03.008.

Holden, M., et al. “The Double Slit Experiment with Polarizers.” Canadian Journal of Physics, vol. 89, no.
11, Nov. 2011, pp. 1079–81. DOI.org (Crossref), https://doi.org/10.1139/p11-122.

• David McKay

Ivy league graduate, 4+ yrs of experience, vetted IB Examiner!