Sometimes I wonder if degrees in academia aren’t necessarily about the research, but are only conferred when you jump through the correct number of hoops in the right manner and at the right time.  This spring, the stars aligned and the MIT Corporation saw fit to grant me a degree that declares to all the world that I have a master’s in hoop-jumping Earth & Planetary Sciences from MIT.  Amazing!

About 1,500 people receive degrees from MIT at Commencement.  Here are some of the graduates sitting in Killian Court as more file in behind them.

There was even a brass band.

The trees in the courtyard are conscripted into holding up canvases. We were lucky to have a day in the mid-70s without rain.

After MIT President Susan Hockfield handed me my degree, we went to celebrate with the rest of the Earth, Atmospheric, and Planetary Sciences department. Here, Dr. Linda Elkins-Tanton is flanked by Stephanie Brown and myself, her two master’s students this semester.

More photos are here. Many thanks to all who made this degree possible—I really appreciate your encouragement and help!

# The best defense…

… is one that you pass! I had my master’s thesis defense this afternoon, and my committee decided to accept my draft and presentation.

After working on this project for so long, it’s nice that all the parts have resolved themselves into a coherent whole. If you’d like to read a draft, it’s available here. Thank you to everyone who helped in this process: writing and finalizing the presentation have been anything but a solitary effort: the process turned out to not only be rather collaborative, but also a good deal of fun.

# Isotope nomenclature: it’s all Greek to me

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I’ll be the first to admit: my coursework during my undergraduate and graduate years has been focused on physics, astrophysics, and planetary astronomy.  I probably couldn’t tell a peridotite from most other mafic rocks unless really pressed (no pun intended), and I certainly don’t have much of an intuition for isotope geochemistry.  This field requires an entirely new alphabet and vocabulary; an appreciation for isotope fractionation lines; and most importantly, an understanding of what that difference in the millions’ place of a decimal means for the evolution of Earth’s mantle.

One way isotope geochemists measure differences between two similar values, say a sample and an established standard, is by taking the ratio of these two values, then subtracting one, since the ratio of these values is usually very close to unity.  Some isotope ratios differ by one part in 1,000, so for those, we use delta notation.  Oxygen isotopes, for example, are measured compared to the stable 16 and 18 isotopes:

$$\delta^{18}O = \left[ \frac{\left ( \frac{^{18}O}{^{16}O} \right )_{sample}}{\left ( \frac{^{18}O}{^{16}O} \right )_{standard} }- 1 \right ] \times 1,000$$

δ18O values can be used to measure changes in sea level and climate in ice cores, and are calculated compared to a standard for water with a specific set of isotopes.  At present, a δ18O of ~0 corresponds to that of equatorial ocean waters on Earth.  This notation has been used since at least the 1950’s, so if you see delta notation in the literature, it should refer to parts in 1,000.

When Günter Lugmair started investigating concentrations of neodymium in Apollo and meteorite samples, most of the interesting variation was less than one part in 1,000, so he started to use epsilon notation to represent parts in 10,000:

$$\epsilon^{143}Nd = \left[ \frac{\left ( \frac{^{143}Nd}{^{144}Nd} \right )_{sample}}{\left ( \frac{^{143}Nd}{^{144}Nd} \right )_{standard} }- 1 \right ] \times 10,000$$

Note that, unlike delta notation, for epsilon the lighter isotope is on top in the ratios.

When Richard Carlson started with osmium isotope analysis, he continued this trend and coined “gamma” as measuring parts in 100:

$$\gamma^{187}Os = \left[ \frac{\left ( \frac{^{187}Os}{^{188}Os} \right )_{sample}}{\left ( \frac{^{187}Os}{^{188}Os} \right )_{standard} }- 1 \right ] \times 100$$

Though there is not yet a consistent unit for parts per million, some have started using μ:

$$\mu^{142}Nd = \left[ \frac{\left ( \frac{^{142}Nd}{^{144}Nd} \right )_{sample}}{\left ( \frac{^{142}Nd}{^{144}Nd} \right )_{standard} }- 1 \right ] \times 1,000,000$$

For ε143Nd and μ142Nd, it’s important to be consistent with what you’re using as the standard in these calculations, as most use a terrestrial standard such as La Jolla, making ε143Nd or μ142Nd equal to 0. However, if you’re using a chondrite standard, your values would change. There’s enough variation in 142Nd in chondrites that sticking with a terrestrial standard is the most safe at the moment.

# Vector plots, pcolor, MATLAB, and you

I make most of my figures in MATLAB, using the print -depsc command. This generates lovely, vectorized figures for most incantations of plot and its cousins, but annoyingly returns bitmap graphics if you use that command on plots involving pcolor. If you want vectorized .eps files with your colormap rendered nicely rather than .eps figures containing bitmaps, use the -painters flag (it’s a type of Renderer) while printing (thanks, MATLAB Central):

print -depsc -painters file.eps

Now that you have a vector image containing your colormap information, it’s time to display your figure. On some systems, your PS/PDF renderer will insert some crisscrossing white lines while rendering your vectorized image. However, you can usually turn off anti-aliasing on your viewer and you should be to witness your glorious figures sans white crosshatching (again, thanks to MATLAB Central).

Sure, I could probably tackle these problems in Adobe Illustrator or Inkscape, but I like having things work properly on the first go when it comes to creating figures and otherwise minimizing the number of non-automated steps in my image-creation workflow.

(PS: a draft of my thesis is available here.)

# Solidifying the Earth layer by layer: What’s the composition of the magma ocean?

I’m ostensibly working on a project which involves doing some bookkeeping on where rare-Earth elements go during magma ocean solidification of Earth’s early mantle.  After happening in fits and spurts for longer than I’d care to admit, there was a bit of a breakthrough tonight, thanks to the efforts of my brilliant chemical engineer housemate.

Our model splits Earth’s mantle into 1000 concentric shells, then solidifies the shells layer by layer from the bottom up (because the adiabatic temperature profile and the solidus intersect at the bottom of the mantle).  Assuming that once a layer has solidified, the rest of the mantle evenly mixes to have the same composition means you can just think about this problem two layers at a time: the layer you’re solidifying (with pre-solidification composition measured in mass percentage liquidn and mass massn), and the layer just above that.  The above layer will have the same composition as the rest of the mantle (liquidn+1), which means you can just concentrate (ha) on those two layers without having to worry about the mass of the rest of the mantle.

We’re also keeping track of the composition (solid) and mass (mass (solid)n) of what fractionates (solidifies) out of the liquid, as well as the volumes of each shells (vol).

With all these parameters in mind, we can calculate the composition of the next layer of liquid (and by proxy, the rest of the mantle) after solidification liquidn+1:

This equation assumes that mixing of the remaining liquids post-solidification in the magma ocean occurs on small timescales compared to that of the solidification process, producing a homogenous liquid mantle.  The magma ocean is assumed to have a bulk silicate mantle of melted material with a composition from Hart and Zindler (1986) and Bertka and Fei (1997), with average chondritic trace elements from Anders and Grevesse (1989). For solidified minerals, mineral phase behavior is based on experimental results (Elkins-Tanton et al., 2003; Trønnes and Frost, 2002; Bertka and Fei, 1997).

Why bother with going to all this trouble for mass balance?  The previous equation for calculating the composition of the next liquid layer (and thus the remainder of the mantle’s magma ocean) would make some of the mass percentages in that layer go to zero, or worse, negative, meaning that the rest of your model was being fed rather bad values.  Now, after much digging and wrangling through MATLAB’s debugging mode (not to mention initially blaming some nuances of clinopyroxene and magnesiowüstite’s density behavior at various temperatures and pressures ), it looks like the problem might have been in how the composition of the next liquid layer was calculated, or at least the problem has shifted to other points in the model.  Although the liquid composition is no longer getting assigned negative values, there’s no certainty that the values being written into it are reasonable, much less physically sane.

Regardless: thank you, generous and insightful housemate!  Next on deck: what common mantle rock-forming minerals have densities greater than 4,000 kg m-3?  What happens to the last 0.03% of the mantle that doesn’t solidify?  What’s the viscosity and density of mantle material below the magma ocean and above the core?  How do you explain that melting starts where the adiabat intersects the solidus?  Stick with us next time for As The Earth Solidifies (And Overturns)!