The Stephens formalism in practice

Here, I am going to look at a couple of exams that use the Stephens formalism in their consideration of data collected from strained samples. The examples will all be orthorhombic, so that the relevant Stephens parameters are S_{400}​, S_{040}​, S_{004}​, S_{220}​, S_{202}​ and S_{022}​.


I will start with a study of the microstructural strain energy in \alpha-uranium, by Manley et al., published in Physical Review B. This is orthorhombic with space group Cmcm. The study was carried out on polycrystalline samples, collecting neutron powder diffraction patterns from 77 K to 300 K. The authors extract the Stephens parameters from their fits, and later give 3D representations of what they call the microstrain, generated from the Stephens parameters – an example is shown below.

Microstrain representation obtained from work by Manley et al.  This image is the top right panel of Figure 4.  This figure is copyright of the American Physical Society; rights and permissions information is available at

Unfortunately, the actual parameters extracted from the fits are not given in the paper. They also found that they the measured parameters were different in different detector banks, pointing to textural changes in the sample.

They then compared the directionality of the strain broadening with both the Young’s modulus data availables and the hydrostatic compressibility, finding that the pattern is closer to the Young’s modulus state, indicating that the observations more closely mimic a uniaxial state rather than a hydrostatic state.

There then follows an interesting discussion on how to evaluate the strain energy stored in the microstructure. As noted previously, it is not possible to extract values for the stiffness constants directly from the Stephens parameters (note that an orthorhombic lattice has 9 independent stiffness constants, three pure normal stresses, three pure shear stresses, and three coupled terms). They then consider two limits. In the first, the strains are statistically isotropic, such that there are no correlations and the three coupled terms are assumed to be zero. The second limit considers the cases where the stresses are statistically isotropic. In this case, there will be contributions to the strain in one direction, and stress in another through the Poisson effect, and characterized by the Poisson ratio. In reality, the result will lie somewhere between these two limits.


This material was being investigated as a colossal magnetoresistive manganite, by Yaicle and co-authors, in a paper published in the Journal of Solid State Chemistry. Some synchrotron X-ray powder diffraction was done, where there was significant broadening of some Bragg peaks, and so the Stephens formalism was used to characterise this. Of the six parameters, four of them were quite large, indicating that the problem could not be simplified to a consideration of only one dimension. However, changes in the strain parameters appeared to correlate with the onset of a charge ordered/orbital ordered state. We can make the same visualisation plots as Manley et al. used to see how the response changes.

This makes clear the changes in the strain parameters between 300 K and 10 K, where the two phases have quite different levels of strain – in the Phase I case, the scale is 5 times larger than the other two, indicating the dramatic difference between Phase I and Phase II.

Observing strain – the Stephens formalism

When a powder diffraction pattern is looked at, the peak linewidths may vary as a function of scattering angle 2\theta in a non-uniform fashion. This could arise from a given internal strain distribution, from anisotropic sample size broadening, or from a given defect pattern (for example, stacking faults). Here, I will concentrate on the broadening due to strain. The simplest method for dealing with this is to consider isotropic or uniform strain, in which case the full-width half-maxima (FWHMs) of the Bragg peaks will be proportional to tan \theta. If the strain is not uniform, the problem is a bit harder. Here, I will follow P. W. Stephens’ approach, outlined in an article in the Journal of Applied Crystallography, published in 1999.

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Distorted hexagonal lattices

When a vortex lattice forms in a Type-II superconductor, it usually forms a hexagonal lattice, although this can be distorted, for example by some sort of coupling to the underlying crystal lattice, or by the directions of any nodes in the superconducting gap.  In some cases this can give rise to a square lattice.  Here, we are going to focus on small distortions of the hexagonal latiice.  In the perfect hexagonal case, the triangles that make up the hexagon are equilateral.  The easiest distortion to consider is that these equilateral triangles become isosceles triangles.  If they become scalene triangles, as in niobium, this is a bit trickier, and we will leave that for later.

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Lessons learned – ONL191

The #ONL191 course has come to an end, and we have been asked to reflect on what we have learned, and what we will do differently in the future, as a result of the course.

What have I learned?

I have seen that online-only courses can work, and I have tried to identify the areas that I consider most important in making them work. In particular, I can now give a better description of the advantages and disadvantages of various online (and offline) methods and tools. In some ways, one point that has come through clearly for me is that there is nothing new under the sun, as many of the key observations were noted and applied by the early distance learning institutions formed in the 1970s, long before the rise of the internet. One key aspect that is general and not related to specific course content is the importance of community building (covered in one of my earlier blog posts). I would also like to add that I think that is important for students to be able to appreciate the effects of their interactions with others and of their own actions. In some sense I am arguing here that it is crucial that the students grasp their own agency.

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Tuning intuition in online teaching

For our group work this time round, we prepared a mock syllabus for a course on Online Collaborative Writing, peppered with roll-over points describing the thinking behind our choices, using a tool called Thinglink.  This helped to concentrate our thoughts on underlying design issues, which have been laid out clearly by my colleague Sebastian Schwede.

In this blogpost, I want to take a step back from the individual course, and think about what it is we are trying to teach in a more global sense.  I teach physics, and so we are trying to impart some facts and information, but there is far more out there than can be crammed into an undergraduate course, so we engage in identifying what we think the most important bits are, and, more importantly, in training students to think “like physicists”.

xkcd: physicists

Physics World (the magazine of the UK Institute of Physics) runs a monthly column called ‘Once a physicist’, interviewing people who studied physics at some point, and then moved into careers in areas that or more (e.g. sales engineer for a scientific instruments company) or less (e.g. opera singer) related.  One thing that is often mentioned in these articles is the transferable skill of being able to develop an approach to solve a problem, or check for feasibility using approximations. The archetypal problem for this latter case is “How many piano tuners are there in Chicago?”  The version I have used in my own small group teaching is usually what is the mass of an aeroplane.

Of course, it is sometimes (but not always) trivial to Google the answers to these questions.  This does not necessarily help with exploiting or understanding the answers.  For example, in the case of the aeroplane, the follow-up question would be: why do the cabin crew sometimes insist that you shouldn’t change seats on a half-full flight?

So, how do we learn (and teach) things like this?  What kind of learning are we aiming for?  There is a a huge amount of literature on this, generally arguing for deep learning as opposed to surface learning – here following the definitions of Marton and Säljö, later expanded by Biggs in his book Teaching for Quality Learning at University.

I now want to illustrate this by using several examples that may seem irrelevant.  Consider juggling.

Made by Koxinga – Wikimedia Commons

This is a learned skill.  I learned how to juggle three balls over a summer by starting with two against a wall, then three against the wall, then three freestanding.  With time and practice, you adapt to the ball trajectories in advance, and can cope with unforeseen obstacles (a distracting friend, a gust of wind, etc).  Or consider snooker.  This can easily be visualised as Newtonian mechanics in action.  Understanding the physics can help (why should I hit the white ball there?) but being a good snooker player requires a lot of practice at the table, until your responses are built on instinct.

Or consider landing a plane.  A pilot has to study a lot of theory, but this is combined with a lot of practice so that, when there is suddenly a strong cross-wind at the landing strip, the pilot can build on their lived experience to respond appropriately.

All of these examples can be nicely described by the very clear work of Daniel Kahnemann and Amos Tversky.  I highly recommend Kahnemann’s excellent book Thinking Fast and SlowThis assigns two systems to the brain.  System 1 is “fast, automatic, frequent, emotional, stereotypic, unconscious”, and System 2 is “slow, effortful, infrequent, logical, calculating, conscious”.

In all of the examples above, the aim is to push as much processing into System 1 as possible.  If you are using System 2 to juggle, you might understand what is going on, but the balls end up all over the floor.

For physics, the aim is not dissimilar.  By means of extensive practice and consideration we want to incorporate a ‘physics intuition’ into the brain.  I want my students to be able to look at the graph they have made from their experimental results and immediately realise that there has been a problem with the error calculation.  But, it takes time and effort.

I have some experience at trying to do this in offline courses, but how do I do this in online or blended courses?  Well, for certain algorithmic questions, it is possible to generate a whole stack of them for the students to plough through, receiving immediate feedback as they work.  There are many open educational resources, providing high quality illustrations, animations, videos and notes.  Eventually, however, consuming such resources is insufficient.

Offline, the peer-to-peer component is vital for cementing learning.  Online, this can be done with discussion boards and other fora for online engagement, but it is harder.  Nonetheless, there must be opportunities (obligations?) to engage with others in some way to jointly construct knowledge.  Obviously this can be done – here are some student reviews from the Open University, UK, for an advanced course on quantum mechanics.  One particular comment I want to highlight is the following:

This is simply the most fascinating course I have done with the OU to date. One where you make a key transition in your study of physics: from relying on your physical intuition to inform the maths; to relying on the mathematics to tell you what your intuition should be!

This is getting right at the question I wanted to address with this post, the retraining of your intuition!  Now, I just need to figure out how I can implement it myself!

Online fika

The third topic in #ONL191 is concerned with learning in communities, obviously with the focus on the networked aspect.  There is a lot of interesting evidence for the advantages of online working, most interestingly to me how this can promote higher-order thinking, meaning that students have more time to reflect and develop their ideas.  A very nice discussion of this is given by Katrina Meyer in an article on threaded discussions.  This was also recognised very early on in the history of online interactions, as illustrated by this article from 1996, which looked at prisoner’s dilemma type games mediated by online interaction.  Elena Rocco found here that online discussions were not as dominated by a small number of participants.  However, this only worked out if the groups had alread established a sense of community prior to participation in the game.  In the article, this was done by prior face-to-face meetings.  Today, there are many tools to try and replace this, with success.

In the specific learning context, one question that arises is the question of collaborative learning.  It is well-documented that students benefit extensively from strong peer networks to support individual knowledge construction.  In the online context, we therefore want to replicate this, building on the strong points of online interaction.  Accordingly, in the course we have looked at online principles for working together, and also discussed a hierarchy of interactions, progressing from discussion, to the simple welding together of individual pieces of work, to actual sharing of ideas and common work together, albeit with each individual having their own purpose, through to the development of a common purpose and identity.  This type of description can be found in many places – I have borrowed from Siemens here, and have deliberately avoided using the words cooperative or collaborative.  One of my colleagues on the course, Elisabeth Öhman, has addressed this issue on her blog.

So, how do we build this community?  There are a lot of guides out there on the internet.  I link to one here, but there are many examples, often talking about netiquette, group contracts, rules of engagement, etc.  One thing that is less well developed is the question of light social contact, or serendipitous exchange.  In professional contexts, the value of this is well recognised including the possibility of forcing random communication.  Legitimising non-work related discussions helps to build community feeling.  The guidelines for online learning communities that I have found tend to focus on purely transactional interactions, with a focus on the importance of an online photo and small profile, consisting of a couple of sentences.  I must admit I fail to understand how this helps in the building of community!

Taking the example of my group in #ONL191, I think that we have established a good rapport, with generally good attendance, and interesting discussions on the various topics at home.  However, our interactions are mainly limited to our scheduled video meeting slots, which have a limited time.  Accordingly, we have tended to be course- and task-oriented in our verbal exchanges, with limited opportunity for one-on-one interaction.  Our sense of community is built on the framework of the assigned tasks; I am not sure if this constitutes the type of community that the literature encourages us to aim for.

Looking through a selection of online information, I have also found some literature from the student’s perspective, in particular a piece by Richard West at Brigham Young University.  To quote,

Many online communities have a designated space for members to socialize about topics not related to course content.  Often these spaces are most beneficial if they are for students only, not instructors.  This allows students to have the kinds of informal discussions that they might normally have in the hallways before an F2F class begins.

The article then gives some examples of this using TappedIn – a US-based project that terminated in 2013, funded originally by the National Science Foundation and then by participating institutions.  I haven’t investigated further as yet to see what descendants of this exist, but I think that it is important that such spaces should not be relegated to the purely commercial sphere (the Facebook group, the Whatsapp group, etc) as that can restrict access.  Just as the physical University provides the physical corridor space, perhaps it should also provide the equivalent online space.

Tapped In Campus – from

Sharing and Openness

The second topic on #ONL191 has been sharing and openness, and the kick-off was a presentation by Alastair Creelman and Kay Oddone.  This has been followed up by various activities to explore advantages, disadvantages, levels and practicalities. My thoughts are this are not well-organized, and run into well-known linguistic issues – does open mean accessible or free, does free mean libre or gratis, etc.  Still, here are some observations…

It is clear that there is a strong case that knowledge/education is a public good that should be freely (openly) available, both for the person searching for education, and for the society that they live in.  However, education has historically required educators, and as resources are finite there is a strong and constant tension over how much education should be freely available.  This tension has played out in different ways in every country.

Where then does open education sit?  Even where higher education is free at the point of delivery, there are often other (academic) requirements that students must meet to be able to attend university.  Open programmes that aim to remove these access limitations have existed for a very long time, one example is the Open University in the UK, founded in 1969, but many others exist.

The internet has removed a lot of resource restrictions, so that it is now easy to disseminate and access freely available structured information to learn about something new.  This is great for auto-didacts – all you need is an internet connection, and you can learn anything.  There are large repositories of open educational resources, e.g. at OER Commons.   Although you will have to navigate various potholes that are not documented, as exist in every trade (although the patent exists, the recipe for original Polaroid film is lost).

However, not everybody is suited to completely self-driven learning, and even for those who are, human engagement is extremely valuable in the learning process.  This is where some MOOC courses are pushing forward, as can be seen by briefly surveying the different options at MOOC course repositories like Class Central.

That means we run right back into the resource problem that we started with, although probably with a shift in the learner/teacher ratio.  Some of these issues are discussed in a study of the effect of internet-drive course delivery on the resilience of higher education institutions by Weller and Andersen.

As more and more teachers make open course materials, eventually some sorting mechanisms will need to be applied, to identify the most useful methods and materials, to avoid potential learners getting lost in a maze of material.  This is bound to be a fraught process, as we see in many articles online about the issues surrounding the algorithms used to manage content in Google, or Facebook, amongst others.  Institutions provide some level of accreditation that helps with this.  As with other networked spheres, there will be big winners and big losers.

However, this is not directly concerned with the issue of openness in my teaching.  As indicated above, I think that having a lot of material freely available can be beneficial.  What about from the student perspective?  Here, I am less convinced that the learning process needs to (or should) be fully open.  bell hooks urged for ‘Radical Openness‘, but for the teacher, and not necessarily the student. My instinctive reaction is that students need to have the opportunity to work out their thinking in private, or inside the classroom.   Some of these questions are also touched on by Boudreau.




Am I what I say?

Having argued in my previous post that to be active successfully in online fora, you should be media literate, the next obvious question is how do you acquire this literacy?  As a part of my group’s #ONL191 work, we have looked at a variety of practical tips as well as more philosophical points.  As with any environment, you need to know the rules of engagement.  There are a lot of guides out there, for example for “academic Twitter”, or for blogging as an educator.  These sorts of guides are very helpful, although often very platform- or tool-specific.

More generally, this kind of information should be included in journalism training programmes, as we can see from the course structure for a Masters programme at the University of Kent, where the impact of online media is discussed, as well as methods of storytelling.  For the moment, my own exploration of the topic is a bit more limited, so I will concentrate on one particular aspect, that of context collapse.  I came across this term in a set of articles by Bonnie Stewart, in particular one called ‘Collapsed publics‘.  The article is primarily concerned with a case study of several academics with a strong online presence, with a focus on Twitter coming to the fore.  The participants in the study have several common ways of interacting that are considered appropriate, including such facets as sharing (interesting) work by other, as well as invitations for congratulation or commiseration. One thing that comes across clearly is that online credibility belongs to those with charisma and good (digital) social skills; this is not necessarily congruent with expertise.  However, was it not ever thus?

So, what is context collapse?  In the simplest sense, it is the reading of texts by multiple audiences, some (most) of whom are not privy to the intentions and larger surrounding discussion.  Stewart frames this using some very interesting work by Ong on secondary orality (broadcast media culture) and literacy; later, Ong ascribes digital communications as secondary literacy – “textualised verbal exchange registers psychologically as having the temporal immediacy of oral exchange”.  The possibility of immediate response online encourages us to act as if we are in a conversation, but our words are recorded forever.  When we speak in person, we are highly conscious of the immediate environment and power relations, but this is lost online after the immediate discussion has taken place.

So, our intention does not carry through.  Instead, as Lanclos and White have put it, content is king, and this leads to an explicit identification of the person with the content, hence the title of this post.  Am I (only) what I have said/written?  Plainly not; the unreliable narrator is a mainstay of fiction.  The learner or student is also in this position; their ideas are not yet fully formed.  But, to pose a rhetorical question, what else can the reader do?  Stewart concludes that:

the dominance of oral-style interaction as the perceived price of admission may be the key factor in keeping academic Twitter a relatively minimal threat to academia’s structure and tenets, among a professional population deeply conditioned to the internalized, distanced register […]

Digital literacies vs digital languages

The first topic on #ONL191 covers online participation and digital literacies.  One of the pieces of suggested reading presents a model by White and Le Cornu, developed in 2011, to categorise users as Visitors or Residents of the digital world.  I liked this description, and think that it has a lot of applicability.  The two roles are also presented without judgement.  A post by one of my colleagues on the course, Sara Ihlman, provides some more background to this description and where it developed from.

One of the drivers for White and Le Cornu to propose this new description was the broader switch from comparing digital skills to language learning (Prensky’s native/immigrant approach), to instead talking about digital literacy, and indeed literacies in the plural.  I find that I don’t have a good grasp on this distinction, and so that is what I will discuss now.

On its own, literacy usually has two meanings, the first being the ability to read and write, and the second referring to competence or knowledge in a particular area.  Language is a communcation system, and it is common to refer to computer programming languages, for example.

One of the first times I came across the term ‘computer literacy’ (which I assumed to be a precursor of the phrase ‘digital literacy’) was in the acronym CLAIT (Computer Literacy and Information Technology).  This was the name for a series of courses in the UK covering basic computing skills such as how to use word processors and spreadsheets.  The computer literacy covered in the initial courses was simply the ability to use these tools in standard ways, for example, writing a letter in a word processor.  The approach promoted the tool over understanding, as you don’t have to know how to program a spreadsheet to be able to use it.  At some point, this approach hits limits, as understanding the kind of things that computers can do can increase dramatically the types of things you can envisage doing with spreadsheets (including building flight simulators!).

A first flight simulator in Excel, by George Lungu (

A first flight simulator in Excel, by George Lungu (

In White and Le Cornu’s paper, they provide a short summary of how digital literacy has been defined and used, which I quote here (without the references included in the original):

  • Mastering ideas, not keystrokes

  • Conceptual definitions as distinct from standardized operational definitions

  • Individuals’ ability to participate in a multimodal culture

This sounds like the primacy of understanding over implementation.  Or perhaps I could rephrase this as the lowering in importance (indeed relevance, in this description) of technical competence, and an increase in the importance of competence in media studies.  These definitions are a long way from the literacy of CLAIT, and its modern replacements. The ability to use digital tools is assumed and is not considered a part of the definition.

In fact, I might go so far as to say that this is a description of a more general media literacy, but now more obviously including the ‘writing’, or active, aspect that had once been limited to journalists in mass media.  In that context, I can see that this is indeed a literacy, rather than a language, with the digital implicitly meaning ‘digital media’.