Date of experiment: 16/03/2012Date of report: 30/03/2012

Investigation of Interference Patterns and Newton’s Rings

Experiment conducted by Sultan LeMarc.

Abstract

This experiment looks to investigate the physical optics of the interference pattern called Newton’s rings and calculate the wavelengths of various filters as well as the radius of curvature of a plano-convex lens. Through the consideration of the wavelength dependence of the diameter of the rings, mathematical relationships are derived and applied on the measured data to verify the known wavelength values. The results produced by the data successfully show the relationship between diameters of the rings, their order, the refractive index of the medium causing the interference pattern and the wavelength of the incident light. Graphical analysis confirms that the diameters of the rings are larger at higher orders, and larger still for longer incident wavelengths.

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Introduc tion

The aim of this experiment is to gain an understanding of the wave behaviour of light through the examination of interference patterns of light, more specifically through the production of Newton’s rings. The investigation will look to demonstrate the dependence of interference patterns on the wavelength of incident light through the manipulation of lenses and filters on an optical bench. The objective is to gain additional quantitative insight through the measurement of the Newton’s rings in order to calculate the radius of curvature of the lens and the incoming wavelength of the light.

In order to understand the interference of light, the model of light’s wave nature must be considered instead of the model of rays. Optical effects that depend on behaviour of this nature are considered to be part of the physical optics field.

Interference of light, as an electromagnetic wave, arises by the combination of the electromagnetic fields that constitute the individual light waves. This phenomenon is governed by the principle of superposition, which states that when two or more waves travelling in the same medium at the same time overlap, the resultant displacement at any point in space or time is the vector sum of the individual instantaneous wave displacements.

Interference in light waves from two sources was first demonstrated by Thomas Young with an experiment now called Young’s double-slit experiment. This showed that light can display characteristics of both waves and particles by passing light through two apertures. The light waves passing through the apertures interfered and thus superimposed, producing interference patterns of bright and dark parallel bands called fringes.

Figure 1: Interference pattern produced by Young’s double-slit experiment.

This established the principle of wave-particle duality and the distribution of the brightness is explained by the interference of the wavefronts. This behaviour of light can be modelled using classical wave theory. The Huygens’s principle states that each point on a wavefront generates a secondary spherical wavelet which then superimpose individually during the interference.

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The phase of a wave is the fraction of its period that a given point completes. When two or more components of a wave are of the same frequency and phase, they superimpose such that their amplitudes are reinforced and constructive interference occurs. This is a result of the crest of one wave coinciding with the crest of the other, producing the bright fringes.

When the waves are out of phase by half a period, where one is a minimum when the other is a maximum, they superimpose such that the resultant amplitude is the difference between the individual amplitudes. If the waves are of equal amplitude they cancel each other producing complete annulment, producing the dark fringes.

Figure 2: Sinusoidal representation of constructive and destructive interference.

Sinusoidal waves are characteristic of monochromatic light, which is light of a single wavelength. Monochromatic light is effective in observing interference patterns as it has waves of the same wavelength and same amplitude. In addition, all waves from a monochromatic source are permanently in phase, and this constant phase relationship for the same frequency makes it coherent.

Lasers are an example of monochromatic light however ordinary light sources can be made monochromatic with the use of optical filters. These are devices which selectively modify the component wavelengths of mixed light in order to transmit light of a specific wavelength corresponding to its colour. Filters may be made of coloured glass, plastic or gelatin. An alternative is to use a monochromatic filter, also known as a monochromator, which mechanically selects a narrow band of wavelengths of light.

Interference effects are commonly observed in thin films as a result of wave reflection from the two surfaces of the film. Light waves are reflected from the two boundaries of thin films such that interference, both constructive and destructive, takes place between the reflected waves.

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Incident light wave shining on the first boundary of a thin film with thickness t is partly reflected before it enters the film, whilst the remaining wave which are transmitted through to the second boundary are again partly reflected back out.

The reflected waves interfere with each other and the type of interference is determined by the phase relationship and the various wavelengths of different colours. This is why colours fringes or rings are produced by oil on water or soap films. This difference in phase of the waves, and thus the degree of constructive and destructive interference, is determined by the angle of incidence of the incident wave, the thickness of the film and the refractive index of the film.

Figure 3: Interference pattern of rings produced when white light is incident on a thin oil film.

The refractive index, also known as the index of fraction, commonly denoted by n, is a measure of the refraction of light when passing from one medium to another. It is related to the angles of incidence θi and refraction θr by Snell’s law; n = sinθi/sinθr.

It also equates to the ratio of the speed of a given wavelength in a vacuum to its speed in a medium. They are characteristic values unique to each medium. The refractive index value of a vacuum is 1, whilst that of air is 1.002 ± 2.5 × 10 -9

(Quoc, 2009).

The phase relationships which determine the degree of constructive and destructive interference are dependent on how the refractive indices of the film and the medium adjacent to it compare. As for the case of an oil film and air in Figure 3, the reflected waves from the upper boundary undergo a polarity inversion; a phase change of 180o (π) with respect to their respective incident waves.

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The second set of reflected waves from the lower boundary of the film undergo no phase change because they are reflected from a medium with a lower refractive index. These waves travel an extra distance of approximately double the thickness of the film, 2t, if close to the normal before recombining with the reflected waves of the first boundary. For the waves that are not close to the normal, the path difference is larger than 2t.

Figure 4: Reflected and refracted light paths through a thin film.(Serway, 2010)

As the two sets of waves are out of phase by 180o, the path difference is equivalent to λn/2, where λn is the wavelength of the light in the film. When 2t = λn/2, the waves combine in phase and constructive interference takes place. The general condition for constructive interference in thin films for waves that are 180o out of phase is given by:

2 t=(m+ 12 ) λn m=0 , 1 , 2, … (1)

(Young, 2010)

If the extra distance travelled by the secondary reflected waves corresponds to a multiple of λn, the waves combine out of phase and destructive interference takes place. The general condition for destructive interference in thin films is given by:

2 nt=m λn m=0 , 1 ,2 ,…(2)(Young, 2010)

If neither or both waves have a 180o phase shift, the conditions are reversed.

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The production of Newton’s rings is fundamentally based on such thin-film interferences. When a plano-convex lens is placed on a plane glass surface, with its convex surface facing the glass, an air film of increasing thickness is formed between the lower surface of the lens and the upper surface of the glass surface. A film is considered to be thin when its thickness is of the order of the wavelength of light.

Figure 5: Plano-convex lens on plane glass sheet. Size of air film exaggerated.

The thickness of the film at the point of contact is zero and increases from the centre towards the edge of the lens. The point at which the thickness of the air film is constant is at the central point of contact.

As a result of this air film, when parallel light is normally incident on the flat side of the convex lens, it is partially reflected from the lower surface of the lens and the upper surface of the glass. The reflections of interest are those involving the surfaces in contact, reflections from top of the lens and bottom of the glass do not contribute to the pattern.

The reflected waves do not experience a change of phase at the lens-air boundary as the light is propagating through a higher refractive index medium to a lower refractive index medium. In contrast, the light that is transmitted past to the air-glass boundary is reflected with a change of phase of 180o as the air has a lower refractive index than the glass. As a result, both of the reflected waves from the contact boundary travel in the same direction and are coherent because they are derived from the same incident wave by division of amplitude.

The interference between the reflections produces a concentric ring pattern of spectrum colours for white light, or alternate dark and light rings in monochromatic light. The interference rings can be observed both in reflection and in transmitted light. These concentric rings are called Newton’s rings. The distances between the interference rings are not constant as the convex boundary is curved.

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Figure 6: Diagram for formation of Newton’s rings through the transmission of light.

The radius of curvature of the convex lens, R, is related to the radius of the flat surface r and the thickness of the air film t by:

R= r2

2 t(3)

(Harvard.edu)

Figure 7: Image of Newton’s rings interference pattern.

The interference pattern shows a central dark spot because at the point of contact of the glass and the lens the thickness of the air film is effectively zero relative to the wavelength of light. Therefore there is a zero path difference between the interfering waves at this point and out of phase by 180o, resulting to destructive interference and producing a dark spot. In transmitted light the interference pattern is complementary to the reflected interference pattern but the central spot is bright due to constructive interference. The type of interference is determined by the conditions of equation (1) and (2). The conditions for interference in transmitted light are the opposite to the conditions for reflected light.

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The fringes are circular with equal thickness throughout the circumference due to each fringe being a locus of constant film thickness at a particular point. The rings in the pattern get closer as the order m increases because the diameter does not increase in the same proportions.

The radius of the mth bright fringe is given by:

rm=√R (2m+1) λ2(4)

(Harvard.edu)

The radius of the mth dark fringe is given by:

rm=√mR λ (5)(Harvard.edu)

The radius of a dark ring is proportional to the square root of the radius of the curvature of the lens.

The diameter, Dm, of the mth fringe from the centre is related to the radius of curvature, the refractive index, nx, of the medium between the lens and glass, and the wavelength λ of the light by:

D2m=

4 R λmnx

(6)

(Yusaf, 2010)

This equation governing the interference pattern shows that the diameter is inversely proportional to the refractive index. This means that the greater the refractive index of the medium between the lens and the glass, the smaller the diameter of the rings. The radius of any given ring will be less with for example a liquid in place than with air. This effect can be used to the measure the refractive indices of various mediums.

Furthermore, equation (6) also shows that the greater the wavelength of a given light, the greater the diameter of the rings. This means that red light would produce the largest rings whilst blue light would produce the smallest. Subsequently, this can be used to measure the wavelength and frequency of a given light.

Method

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The optical set up used to produce a Newton’s rings interference pattern, through transmission of light, to find the radius of curvature and the wavelength of the incoming monochromatic light is shown by the optical bench below:

Figure 8: Diagram of set up on optical bench.

The light source is connected to the power supply and fixed on the optical bench, aligned with the three lenses. The plano-convex lens with the plane glass surface was placed in between two collimator lenses. This lens produced the interference pattern, whilst the collimator lenses either side of it aligned the incident light in order to replicate the source at an infinite distance. This made the incident light parallel, first into the plano-convex lens and then again for the projection screen. The plano-convex lens has a ruled scale of 20mm printed on it which is also projected.

The projection screen allowed the interference pattern to be observed by viewers. A larger projection screen, such as a plain wall, was not required as rings up to the 10th order fitted within the range of the screen. The experiment was conducted in full covers from external light in order to prevent light attenuation, allowing the projection of light to be as bright as possible.

The experiment used three different coloured filters, blue, green and red, with stated wavelengths of 437nm, 546nm and 578nm respectively. The radius of curvature of the plano-convex lens was stated to be 12.13 ± 0.005m.

First, the experiment looked to confirm these claimed filter wavelengths with the known radius of curvature value of the lens and the refractive index value of air. Equation (6) was used with λ as the subject:

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λ=nx D2

m

4 Rm(7)

Each of the three filters was individually tested. They were attached to the first collimating lens by inserting them into the lens bracket. The optical bench allowed the lenses to be moved at varying distances until the projection of the interference pattern was in focus.

The distances had to be adjusted for different filters to achieve a focused projection. However, once the vertical position settings of the lenses were fixed to align them with the optical axis, they were not adjusted in any trials. The screws that tightened the lenses in place were fixed to a limit such that they did not provide excessive contact pressure and risk deformation of the lenses. The filters were allowed to cool from the heat of the light before being handled.

Measurements of the diameters of the dark rings up to the 10th order were recorded for each filter using the ruled scale. The maximum order of rings was measured to 10th as the rings of higher orders were not well defined and thus could not be distinguished adequately. The diameters were measured as opposed to the radii because they are the larger lengths; this reduced the relative instrumental error of the measuring scale. The measurement was taken from the middle of the ring’s thickness.

Due to the limited resolution of the scale, a plain sheet was attached to the projection screen and the edges of each ring were marked on to it along with the ruled scale. This allowed the marked diameters to be measured more accurately by a ruler with consideration of the scale conversion. This was repeated for all three filters.

Lastly, the data recorded was then used to calculate the radius of curvature of the lens by assuming that the stated wavelengths of the filters are correct using equation (6) with R as the subject:

R=nx D 2

m

4 λ m(8)

Results

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Blue Filter (λ: 436nm)

Orderm

DiameterDm

± 0.25 (mm)

Theoretical Diameter

DmT

(mm)

Wavelengthλ (nm)

Error in wavelength

∆λ (nm)

Radius of curvature

R (m)

Error in

radius ∆λ (m)

1 5.5 4.6 620.7 31.2 17.38 0.872 7.5 6.5 580.8 29.0 16.16 0.813 9.0 8.0 560.6 28.8 15.51 0.784 10.3 9.2 540.4 27.1 15.09 0.755 11.3 10.3 520.7 26.1 14.54 0.736 12.3 11.3 520.5 25.8 14.37 0.727 13.3 12.2 520.9 25.9 14.41 0.728 14.0 13.0 510.9 26.0 14.08 0.709 14.8 13.8 500.2 25.0 13.89 0.6910 15.8 14.5 510.3 25.6 14.25 0.71

Table 1: Measurements and calculations for blue filter of wavelength 436nm.

The theoretical diameter was calculated by equation (5) to show what the diameter for the given wavelength and order should be and to provide a reference for deducing the deviation of the experimental values.

The wavelength λ was calculated using equation (7) using the literature values of nx = 1.002 ± 2.5×10-9 and R = 12.13 ± 0.005 m. The error in wavelength ∆λ was calculated using:

∆ λ=√( ∆(D2 nx)

D2 nx)

2

+( ∆(4 Rm )4 Rm )

2

(9)

The average wavelength of the blue filter was calculated as λB1 = 538.6 ± 12.0 nm. The error on this average value was calculated at 65% confidence level by:

∆ λB=σ

√N=38.0

√10(10)

The radius of curvature R was calculated by equation (8) using the stated wavelength value λ = 436nm. The error in the radius of curvature ∆R was calculated using:

∆ R=√(∆ (D2 nx )

D2nx)

2

+(∆ (4 λ m)4 λ m )

2

(11)

The average radius of curvature was calculated as RB1 = 14.97 ± 0.35 m . The error on this average was calculated at 65% confidence level with σ = 1.10.

Green Filter (λ: 546nm)Order

mDiameter

Dm

± 0.25 (mm)

Theoretical Diameter

DmT

Wavelengthλ (nm)

Error in wavelength

∆λ (nm)

RadiusR (m)

Error in

radius

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(mm) ∆λ (m)1 6.0 5.1 743.5 37.2 16.52 0.832 8.0 7.3 660.8 33.0 14.68 0.733 10.0 8.9 688.4 34.4 15.29 0.764 11.0 10.3 624.7 31.2 13.88 0.695 12.5 11.5 645.4 32.3 14.34 0.726 13.5 12.6 627.3 31.4 13.94 0.707 14.5 13.6 620.3 31.0 13.78 0.698 15.5 14.6 620.2 31.0 13.78 0.699 16.5 15.4 624.7 31.2 13.88 0.6910 18.5 16.3 706.8 35.3 15.70 0.79

Table 2: Measurements and calculations for green filter of wavelength 546nm.

The average wavelength of the green filter was calculated as λG1 = 656.2 ± 13.6 nm. The error on this average value was calculated at 65% confidence level with σ = 43.2. The average radius of curvature was calculated as RG1 = 14.58 ± 0.30 m. The error on this average was calculated at 65% confidence level with σ = 0.96.

Red Filter (λ: 578nm)

Orderm

DiameterDm

± 0.25 (mm)

Theoretical Diameter

DmT

(mm)

Wavelengthλ (nm)

Error in wavelength

∆λ (nm)

RadiusR (m)

Error in

radius ∆λ (m)

1 6.0 5.3 743.5 37.2 15.60 0.78

2 8.0 7.5 660.8 33.0 13.87 0.69

3 10.0 9.2 688.4 34.4 14.45 0.72

4 11.3 10.6 653.4 32.7 13.71 0.69

5 12.5 11.8 645.5 32.3 13.54 0.68

6 14.0 13.0 674.6 33.7 14.16 0.71

7 15.0 14.0 663.8 33.2 13.93 0.70

8 16.0 15.0 660.8 33.0 13.87 0.69

9 17.0 15.9 663.1 33.2 13.92 0.70

10 18.0 16.7 669.1 33.5 14.04 0.70Table 3: Measurements and calculations for blue filter of wavelength 578nm.

The average wavelength of the red filter was calculated as λR1 = 672.3 ± 8.7 nm . The error on this average value was calculated at 65% confidence level with σ = 27.6.The average radius of curvature was calculated as RR1 = 14.11 ± 0.18 m . The error on this average was calculated at 65% confidence level with σ = 0.58.

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Figure 9: Graph of the results from Table (1), (2) and (3), plotting diameter as a function of order, D2 vs m for all three wavelengths.

The line of best fit for the data points was determined using the method of least squares fit. The equations of least squares fit require the values of Sx, Sy, Sx2 and sxy of the data to be calculated for each of the three tables using:

Sx=∑i=1

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x10 , S y=∑i=1

10

y10 , S x2=∑i=1

10

x210 , Sxy=∑

i=1

10

x10 y10(12)

(Yusaf, 2010)

The plots for all three wavelengths show that the diameter of the rings increases as their order increases, which confirms what is seen in the illustration in Figure 7.The plots also showed that the diameters were largest for the largest wavelength, the red light, and were smallest for the smallest wavelength, the blue light.

The gradients, m, were calculated using the least squares fit formulae and represents:

m=nS xy−Sx Sy

nSx2−( Sx )2

=4 R λnx

(13 )

The errors on the gradients, ∆m, were calculated using:

∆ m :2σm

√ N, where σ m=√ n S

D2

(n−2)(n Sx2−( Sx )2)(14 )

(Yusaf, 2010)

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The gradient of the plot for the blue filter is mB = 23.6 mm2, giving values of λB2

= 487.4 ± 5.8 nm and RB2 = 13.6 ± 0.16 nm. The gradient of the plot for the green filter is mG = 29.5 mm2, giving values of λG2 = 609.2 ± 6.8 nm and RG2 = 13.5 ± 0.09 nm. The gradient of the plot for the red filter is mR = 32.2 mm2, giving values of λR2

= 665.0 ± 7.8 nm and RR2 = 13.9 ± 0.08 nm.

The average radius of curvature values established from Table 1, 2 and 3 is Rav1 = 14.55 ± 0.25 m. The average established from the values of Figure 9 is Rav2 = 13.67 ± 0.11 m.

The errors on these averages are calculated by:

∆ R=∆ R

√N(15)

Analysis

The biggest source of error in the investigation was the measurement of the ring diameters. Although measuring the diameter as opposed to the radius of each ring meant that the relative instrumental error of the scale was reduced, the nature of the measurement itself resulted to the rise of uncertainty in the recordings.

Firstly, the point at which the interference pattern is in focus was difficult to determine as there was a range of arrangements on the optical bench which produced a projection that could be deemed to be in focus. As a result there was an appreciable range on focus in which the ring sizes differed considerably.

In addition, deducing the centre of the each ring’s thickness as the start and end points of the diameter gave rise to further error in the measurements as it introduced the systematic parallax error. This meant that the judgement of the centre was not constant and thus varied for each measurement. Also the limited resolution of the projected scale compromised the measurements as the rings and the scale had to be marked out on a plain sheet and measured with a ruler. Lastly, there was no precaution to ensure that the lengths being measured are of the diameter, exactly perpendicular to the edge of a ring, and not of a chord.

Alongside taking repeat measurements, this source of error can be improved by various ways. To address the issue of determining the focus, an iris diaphragm can be added to the optical set up, placed in front of the projection screen. This will optimize the bright-dark contrast and make the point of focus more distinguishable. This can also be addressed by taking measurements over the

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range of different focus points and averaging the data. A level with a scale rule can be used to ensure that the length being measured is exactly straight.

The results of the investigation show trends that agree with the background physics. The results of the first task, in which the diameters are measured and the known radius of curvature value is used to find the wavelength, show that the diameters of Newton’s rings increase the larger the order gets.

The results also demonstrate the wavelength dependence of the diameters, as the longer wavelength of the red filter had the larger diameter at given order than the shorter wavelength of the blue filter. This relationship is reflected by equation (6) where wavelength is directly proportional to the diameter. This is because the larger wavelength has a larger path difference in the air film.

A similar relationship can be seen between the diameter and the radius of curvature of the lens which is producing the interference pattern. Equation (6) dictates that the radius of curvature is also directly proportional to the diameter.

The two sets of results for each value, calculated by two different approaches, are summarised in Table 4. The percentage differences from the literature values are included below each value.

Blue Wavelength

(nm)

Green Wavelength

(nm)

Red Wavelength

(nm)

Radius of curvature R (m)

Numerical calculation

(Equation 7, 8)

538.6 ± 12.0 656.2 ± 13.6 672.3 ± 8.7 14.55 ± 0.25

Relative error 2.23% 2.07% 1.29% 1.71%

Difference from literature value

23.53% 20.18% 16.31% 19.95%

Graph(Figure 9)

487.4 ± 5.8 609.2 ± 6.8 665.0 ± 7.8 13.67 ± 0.11

Relative error 1.19% 1.12% 1.17% 0.80%

Difference from literature value

11.80% 11.58% 15.05% 12.70%

Table 4: Analysis of calculated values and comparison with literature values.

The analysis shows that graphical approach to establishing the values of the wavelength and radius of curvature was more accurate as all four of the values established have a lower percentage difference from the literature value compared to those from the numerical calculation.

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Moreover, the relative errors of the values from this method are also smaller, making it the more effective method for deducing these quantities.

Although the closest a value has got to its corresponding literature value is by an 11.58% difference, the results for both sets show the wavelengths within the correct range. Also they have the correct relative magnitudes for red, green and blue filters, with blue as the shortest wavelength and red as the largest.

Conclusion

To summarise, the experiment achieved its objective to consider the wave behaviour of light through the investigation of interference patterns and Newton’s rings. The results successfully demonstrated the dependence of interference patterns on the wavelength of the light and how such patterns can be produced through the use of lenses. More specifically, Newton’s rings can be produced by using a plano-convex lens adjacent to a optically flat glass surface in order to exploit the thin-film phenomena.

The observations made and data recorded successfully allowed mathematical relationships to be applied in order to calculate the radius of curvature of the lens and the incident wavelength of three different filters. The results confirmed that the radius of curvature of the lens is within close range of the known value of 12.13 m, as well as verifying the stated wavelengths of the filters. A thorough quantitative analysis showed that the graphical approach of plotting the ring diameter as a function of the order provided the most accurate values.

Extensions of this experiment include the use of different mediums as the film between the convex lens and the glass surface. Example of an alternative medium is a liquid such as water, or particular gases. Furthermore, the thin film’s thickness can be varied in order to examine the relationship between the radii of the rings and the thickness of the film. This can be achieved by using two convex surfaces in contact with each other. Finally, the apparatus can be used to measure the expansion coefficients of crystals.

In conclusion, this experiment was a success as a whole as the entire set of results correctly followed the principles of physics and demonstrated the mathematical relationships by producing hypothesised results. Overall the investigation has been effective in exploring Newton’s rings and considering the wave nature of light in its interaction with other waves.

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Stone, M. (2009). "Mathematics for Physics". Cambridge University Press

Weber, H. (2004). “Mathematic Methods for Physicists”Academic Press

Giancoli, D. (2005). "Physics". Pearson Education Inc.

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