# Electrons getting stuck

Kink in waveguide (red lines): the Green function is calculated for region I, the square with grey regions II removed by embedding.

It’s the last day of my month in San Sebastían, and I am still writing new programs for the way that electrons propagate round kinks in waveguides, using embedding to confine the electrons to the kink. In my last post I described the treatment of the kink using the box pictured at the right, with the grey regions II replaced by an embedding potential. This confines the electrons to region I, giving the kink in the waveguide, shown by the red lines. In the work I described in my previous post I calculated the Green function for region I, with basis functions given by
$\chi_{mn}=\cos(m\pi x/a)\sin(n\pi y/a)$
satisfying zero amplitude at y = 0 and a, and zero derivative at x = 0 and a; in other words the Green function is forced to have a zero derivative boundary condition at the entrance and exit to the kink. From this we can calculate the transmission and reflection properties of the kink, using a method in which the wave-functions in the left and right-hand waveguides are written in terms of incident + reflected waves, and transmitted waves, and then using Green’s theorem to match the reflection and transmission coefficients (Dix and Inglesfield, J. Phys.: Condens. Matter 10 5923-5941 (1998)).

However I realised last week that we can do the problem using a different, simpler approach, based on the fact that embedding allows us to treat mixed boundary conditions, by which I mean that we can calculate the wave-functions or Green function for a system with a combination of zero amplitude and zero derivative boundary conditions. (This has been implicit in our previous publications.)

Kink treated as a rectangular box. Grey area is region I, with confinement at the blue lines, and open boundary conditions at the dotted lines.

We can then treat the kink by considering the rectangular box shown on the left, with confinement – zero amplitude – at the blue lines, and open boundary conditions – zero derivative – at the dashed lines, to which the straight sections of waveguide are joined. Region I is now a very simple shape, and we add the confinement embedding potential over the blue lines, and zero embedding potential at the dashed lines to give the required mixed boundary conditions (at the top and bottom, the choice of basis function ensures zero amplitude). The resulting embedded Green function with zero derivative at the waveguide entrance and exit can then be used, once again, to find the transmission and reflection of the kink.

Transmission through the kink vs. electron energy

Results for the transmission of electrons through the kink are shown in the figure on the right, as a function of electron energy (I should say that the incident electrons are in the first channel, but transmission is into all the open channels). These results are just the same as the transmission results I obtain using the geometry shown at the top of this page – both are describing the same kink. (Actually not quite the same if we haven’t got fully converged results, which we never have in the real world.) First point to notice – for several energies the electrons go unimpeded through the kink, with 100% transmission. There are also energies at which transmission is zero – at such energies, the current density in the kink is as shown in the previous post, with electrons circulating in the kink and getting nowhere. But what is very interesting is all the structure, for example at an energy E = 7.5 a.u. there is a curious sharp feature, and at E = 8.7 there is a sharp dip.

Density of state of kink joined on to waveguides.

To understand these features we calculate the density of states of the kink, that is, how many electronic states there are at a particular energy. Again we can use the rectangular geometry to describe the kink, but this time as well as the confining embedding potentials along the blue lines we must add embedding potentials on the dashed lines to take account of the waveguides, the fact that electrons can move out of the kink into the waveguides. The results are shown in this figure, and we see numerous very sharp features, as well as a continuous background which starts at an energy of E = 0.55 a.u., the energy at which electrons can start to move through the waveguide (which has width 3 a.u.). Just below this energy there is a very sharp peak, which is in fact a bound state, in which an electron is trapped in the kink. The other sharp peaks come from states which are trapped for a long time, but can ultimately leak out – resonances – the width of the peak corresponding to 1/lifetime. If we compare the transmission with the density of states, we see that all the sharp features, peaks and dips, correspond to resonances in the density of states, in other words electrons getting stuck in the kink.

I think I’ve got all the programs working, which relate to the confinement of electrons – famous last words – and there are lots of results and figures to put in chapter 6, which isn’t quite finished after all. I’ve also made progress in relating embedding to scattering theory, and the work which I’m most pleased with is (yet another) calculation of the transmission through the kink using equation 33 from Inglesfield, Crampin and Ishida (Phys. Rev. B 71 155120 (2005)), $\chi(r)=2i\int_S dr_s\int_s dr'_s G(r,r_s)\Im\mathrm{m}\Sigma(r_s,r'_s)\psi(r'_s)$,
which gives the wave-function inside the kink, $\chi$, in terms of the incident wave-function in the left-hand waveguide, $\psi$, the embedding potential $\Sigma$ for the left-hand waveguide, and the full kink Green function $G$. This gives the same results for the transmission as in the figure above, as it should, but in a more straightforward way. So that’s it, for the time being, in San Sebastían. Back to Cumbria, and quite a lot of work in the garden, no doubt, catching up with lawn-mowing, cutting plants back for the winter etc.

# Electrons going round corners

San Sebastián

Kink in a waveguide – the walls of the waveguide are indicated by red lines. Box bounded by dashed lines is treated by embedding.

Halfway through my stay in San Sebastián, and I’m making good progress with the chapter on using embedding to solve electron confinement problems. But I haven’t finished the chapter yet, mainly because I decided to do the calculations on electrons going round corners in waveguides, which I described in my last post. The system which I decided to calculate was the kink in the waveguide, shown in this figure (a bit more clear than the figure in the previous post): the waveguide itself is indicated by the red lines, and we calculate the properties of the electrons moving through the square indicated by dashed lines. The shaded grey areas are removed by embedding, giving the kink. Anyway, I programmed this all up over the last week, and eventually got the programs – and the gnuplot plotting routines – working, giving some very nice results which I can use in chapter 6.

Third eigenstate of kink with zero gradient boundary conditions at entrance to waveguides.

The first calculation I did was of the eigenstates of the system, electrons confined to the kink, but with wave-functions satisfying a zero-derivative boundary condition at the open ends of the box, to which the straight sections of waveguide are joined on. Results are shown in this figure for the third eigenstate, and we see very clearly how well the boundary conditions are satisfied. Note that I’ve plotted the wave-function throughout the square, including the “forbidden” regions II where it is in fact meaningless – what’s important is that the wave-function goes to zero at the edges of these regions, and the zero contour follows these edges very accurately. I’m quite proud of this picture, not only because it shows how well embedding works, but also because the graphics are quite good! (I relied on help from friends, and also from a Google search to get the right gnuplot commands). Then I repeated the calculations done with Elizabeth Dix on the current density, when electrons are directed towards the kink through the left-hand straight section of waveguide.

Current density in the waveguide kink, E=2.1 a.u.

This figure shows the current density when the incident electrons have an energy at which they are almost 100% reflected, and we see that a complicated standing wave pattern is set up in the kink. Actually, when I first got the program going I had the current going backwards – needless to say a minus-sign was wrong. And my first attempts at plotting vector fields were disastrous.

Autumn in the Aude – a wild vine

This weekend I’m spending in my house in France, a beautiful train journey on Thursday from the French border via Toulouse to Carcassonne. The views of the Pyrenees were wonderful, especially between Pau and Lourdes. The weather here in France is lovely, fairly warm, though distinctly autumnal. The wine harvest – the vendange – is still in progress, so it tends to be rather slow driving, when one gets behind a little tractor pulling a large truck full of grapes. On Saturday afternoon I had a very satisfying afternoon helping friends get the logs in for the winter.

When I get back to San Sebastián I hope to explore the transmission properties of the waveguide kink using the well-known expression for the total transmission at energy E,
$T_{lr}(E)=4\mathrm{Trace}[G_{lr}\mathrm{Im}\Sigma_r G^*_{rl}\mathrm{Im}\Sigma_l]$
where $\Sigma_{r/l}$ is the self-energy which couples the kink (in our case) to the right or left waveguides, and $G_{lr}$ is the Green function of the kink, connecting the left-hand end to the right-hand. But in fact the self-energy – this is how it’s usually referred to in the literature – is nothing but an embedding potential, and a few years ago Simon Crampin, Hiroshi Ishida and I re-derived this using the embedding method (Phys. Rev. B 71, 155120 (2005)). We derived some other useful results on the way, which I intend to demonstrate in the waveguide kink. This will contribute to a chapter on embedding in transport – and possibly scattering theory. One of the reasons I wanted to write the book on embedding was to explore connections with other aspects of physics, but how this works out remains to be seen.

Quick note, Tuesday 22 October – I’ve just finished the chapter on the confinement problem with embedding. Next chapter? Embedding and Tight-Binding.

# A month in the Basque Country

I arrived a week ago in San Sebastián/Donostia, in the Basque Country of Spain, for a 4-week visit to DIPC, the Donostia International Physics Centre. During this stay I hope to make real progress with the book – going to seminars, talking to colleagues, using the library, it provides an excellent environment for getting on with the work! San Sebastián is a really beautiful city, with magnificent beaches, and surrounded by green hills (the Basque country gets plenty of rain). And the food here is great – San Sebastián is famous for its tapas.
I’ve brought my Spanish dictionary and grammar with me, as I also want to improve my Spanish – at the moment it’s extremely basic. Going to the supermarket and shopping will help me to build up a bit of vocabulary. Also, watching television helps, with the (Spanish) subtitles turned on. It is hopeless to try to learn any Basque (amazingly, a non-Indo-European language), though you see it everywhere on notices and shop-fronts.

Electrons are confined to region I by an infinite barrier at S.

Since my last post I’ve started the chapter on using the embedding method to solve the problem of electron confinement. This was a most unlikely application of embedding, but one which has proved useful, and has surely many more applications. The basic problem is shown in the figure – we want to solve the electronic Schrödinger equation in region I, which is bounded by an infinite barrier potential over S. (This could just as well be a three-dimensional problem as the two-dimensional problem in the figure.) This type of problem crops up in nano-physics, where semiconductor nano-structures of different shapes can be constructed, confining the electrons. We also find confined electrons on surfaces, where STM techniques can be used to build up “quantum corrals”, and self-assembled islands of atoms occur. To solve the problem using embedding we replace the infinite barrier by a finite potential barrier of height V, where the height is large, and I mean very large. This can be replaced by a local, energy-independent embedding potential over S, with the simple form

$\Sigma(\mathbf{r}_S,\mathbf{r}'_S)=\sqrt{\frac{V}{2}}\delta(\mathbf{r}_S-\mathbf{r}'_S)$

where r_S is a coordinate over S. Simon Crampin, Maziar Nekovee and I derived the method in a paper published nearly 20 years ago, and the first application was to the energy of a hydrogen atom in an impenetrable spherical barrier – off-centre, otherwise the problem would be easy to solve. This seems a fairly artificial problem, but it’s one which has been studied many times, and it has a genuine application to impurities in quantum dots. We replaced the impenetrable barrier by a finite barrier potential, of height V=5 x 10^10 electron volts (almost an infinite barrier, by any standard!). This in turn can be replaced by the embedding potential given above, and the method converged beautifully to give an energy of the ground state of the enclosed atom of -11.26 electron volts, in precise agreement with other calculations of the same problem (H-atom displaced by 0.5 Bohr radii in a sphere of radius 3 Bohr radii, in case you’re interested). Just a glance at the literature convinces one that our method has the virtue of simplicity, as well as accuracy.

Corner and kink in an electron wave-guide.

I still haven’t finished the confinement chapter, as I am now writing about an application made with Liz Dix, about 15 years ago, to electron transport round corners and through kinks in electron wave-guides. The idea was to calculate the Green function for region I (see the diagram), confined by a very high potential in regions II. From this Green function we can then calculate the reflection and transmission properties of the corner or kink, and also the current density profile. This works very well, and we wrote some nice papers on the topic, but I’ve decided to repeat the calculations: I want some new diagrams, and also want to explore the effect of different basis sets. So I’m about to begin some more programming, as I’ve no idea where the original programs have got to (yet again). The new programs can also form the basis for additional material for the book (assuming I can get the programs in a nice elegant form). I also want to explore transport theory à la embedding a bit more. So writing stops (briefly) and I shall explore the physics.

In retrospect we should have pushed this embedding method for confined systems, as there is still a lot of interest in this type of problem. But why bother at this stage?

The weather has turned cooler, after a week of sunshine. It was in fact warm enough to go swimming in the sea several times. Amazing to have beautiful beaches right in the middle of the city. I would attach a picture, but I cannot transfer it from my camera till I get back home!

# Holidays over and back to writing

On the way to the Grotta di Nettuno, Sardinia

After 11 glorious days in Sardinia, it’s back to Cumbria, dull, wet weather, and ….writing! The holiday in Sardinia, actually just north of Alghero in the north-west of the island, was spent either on the beach, or walking in the lovely countryside. Lots of swimming, and the sea was crystal clear and warm. I’m trying to learn Italian – about time, after having had holidays in Italy nearly every year – so I’ve joined an Italian conversation class with the U3A (for those younger readers, this is the self-help “University of the Third Age”).

How hard it is to start writing again after a couple of weeks off. It’s just the same getting back to work after a holiday of course, or starting term again, but those are now distant memories for me. Fortunately I’m still on with the chapter on applications of embedding to surface physics, something which should almost write itself. I finished the section on embedding calculations of the screening of an electric field at a metal surface, work which I still think is pretty good, as it really showed how the screening charge sits right on top of the surface atoms. I’ve included the very interesting work by Ian Merrick on screening at surfaces with steps – vicinal surfaces, and surfaces with islands on top. This shows how the screening charge is concentrated at the step edges, just as one would expect from macroscopic electrostatics.

The Schrödinger equation is solved for the near-surface region, with embedding potentials added to Sm and Sv to replace the metal substrate and vacuum.

The next section, which I’ve just started, is the calculation of image states using embedding. The image states are electron states bound to the surface of a metal by the attractive image potential; if there is a band gap in the band structure the electron cannot leak into the bulk and is localised outside the surface. Embedding is a perfect way to calculate these states – we solve the Schrödinger equation in the “near-surface” region extending 10-20 Å outside the surface, with an embedding potential on one side to replace the metal surface and on the other side an embedding potential to replace the Coulomb tail of the image potential. In this near-surface region the potential is smooth, so the wave-functions or Green function can be expanded in terms of plane waves. The potential which an electron feels in the near-surface region is taken as an interpolation between the local density exchange-correlation potential, and the Coulombic image potential (in reality it feels a dynamic self-energy). The results which I’m going to include in this section are the image states on the magnetic Fe(110) surface – work done by Maziar Nekovee, when we worked together in Nijmegen. I wondered about including the work on the behaviour of the image states near the vacuum threshold – the fact that the spectral density of image states in the near-surface region below the vacuum level is continuous with the density of states above the vacuum zero of energy. In other words, one cannot distinguish the vacuum threshold – a feature of the Coulomb image potential, which occurs in other contexts. But one has to broaden the discrete image states, and this would get me into a long discussion about many-body effects, which I don’t really want to get into at this stage. The magnetic image states are certainly worth including, as not only is their physics very interesting, but the predicted magnetic energy splitting was confirmed after our paper.

In just over a week I’m off for the whole of October to a physics centre in San Sebastián, in the Basque Country. With its good library facilities, and lots of people to talk to, I should get quite a lot done on the book. I’ve written 70 pages so far (including figures), so I have a long way to go before I finish. The first two books are already out in this new Institute of Physics series!

# A lesson to us all…..

Lake District from Cross Fell

I’m enjoying being back in Cumbria, and the weather has been excellent since returning from France. A week ago, on Bank Holiday Monday, I climbed Cross Fell in the Pennines – and hardly met a soul. It’s also been lovely weather for gardening, mostly chopping stuff back and tidying up – but not too tidy – I cannot stand an excessively neat garden.

The title for the book is chosen (for the moment), as is the cover, and I’m writing chapter 5 on applications of the embedding method. What more could I wish for? Well, I wish I’d kept some old results! I’ve just finished the section on calculations of the screening charge at a surface in an electric field, work which was done nearly 25 years ago with Geof Aers. Unfortunately the diagrams in the article on screening at Ag(001) published in Surface Science (217, 367 (1989)) have the surface normal in the -z direction, whereas the convention I’m using in the book is that the normal is in the +z direction. Just a matter, you may think, of making a mirror image of the diagrams in the article (\reflectbox in LaTeX?) but unfortunately these have labels, tic marks etc, all of which I would have to redraw with TikZ. Much easier to plot the figures again from scratch. But to make matters worse, I didn’t keep the computer output – all I have are the hand-drawn graphs in my “good” notes (I do keep all my notes, or at least the final version of my notes).

Planar averaged screening charge at Ag(001) as a function of distance from surface

Here is the figure from my notes, showing the planar average of the screening charge at Ag(001) as a function of z, and of course what I’ve had to do is to read values off this graph and re-plot them with gnuplot etc. I suppose I threw the output away in one of my job moves, after a reasonable length of time when I didn’t think I would need the work again. It had all been published, I’d included the work in several review articles, and I wasn’t counting on writing the book. The moral of this is to hang on to the output, in some form or another. You never know when you will need it again. Of course it’s all much easier nowadays than when we did the work – at least that’s my excuse.

Contours of screening charge at Ag(001) with applied field

Here’s another picture from my good notes, showing contours of the screening charge. Plotting has come on a long way since those days, but you can just about see the contours, and there is a much better version in the paper which I can use almost straightaway (but I cannot use in the blog till I’ve got the publisher’s permission). In this calculation the electric field is positive, repelling electrons from the surface, and the dashed lines give contours of reduced electron density, and the solid lines contours of increased electron density. This screening calculation, with an earlier embedded calculation on screening at Al(001), was the first to include the atoms (there was of course the classic work of Lang and Kohn on screening of an external field at the surface of jellium). The important point to notice is that practically all the screening charge sits on top of the surface atoms – screening at a metal surface is amazingly effective. From these results we can calculate the plane from which the distance between the plates of a capacitor should be measured – and much else besides. Anyway, I’ve finished the sub-section on screening at the Ag(001) surface, and the next sub-section is screening at a stepped surface.

One of the problems with writing the book is to know how much detail to put in. What seems interesting to me, and what I would certainly include in an article in a journal, isn’t necessarily appropriate for the book. I don’t want too many appendices or footnotes, just something which reads well and which other people will find interesting.

Late summer garden

Tomorrow I’m off on holiday to Sardinia, to sun-bathe and swim. Unlike during my stay in France, I’m not going to do any writing at all – though I’ll take my iPad just in case!

# Back to Blighty!

Sunset over the vineyards

Friday 23 August
After spending nearly two months at my holiday home in the South of France, it’s back to Cumbria tomorrow. Conveniently I’ve almost finished chapter 4, on deriving the full-potential embedding potential to replace the semi-infinite bulk in surface electronic structure calculations. By “full-potential” I mean that that the embedding potential is derived from a fully self-consistent bulk calculation, rather than a simplified description of the bulk, such as a muffin-tin potential. This embedding potential gives a perfect description (in principle – there are always numerical inaccuracies) of the effect of the bulk material on electrons in the surface region. The only piece I’ve still got to add to chapter 4 is a sub-section on examples of full-potential embedding in action, and there is no shortage of good examples here.

It’s been quite a difficult chapter to write, especially the sections on the use of transfer matrices to evaluate the embedding potential. These methods represent a real advance, giving embedding potentials for surface and interface calculations defined over a convenient plane which doesn’t intersect atomic spheres. This leads to much greater stability in embedded surface calculations, and the ability to treat open surfaces in which the atomic layers are close together.

Wednesday 28 August
I’ve just finished chapter 4, including some very impressive examples of full-potential embedding from the work of Ishida. One is a calculation of the Al(111) surface from about 15 years ago, in which 3 atomic layers of Al are embedded on to bulk Al: the Al atoms are replaced by pseudopotentials and a plane-wave basis set is used. This is compared with an LAPW treatment of the same surface, 3 Al layers embedded on to bulk Al, by Benesh and Liyanage, and the surface density of states in the two very different calculations is almost identical. Both give almost the same work-function, 4.25 eV and 4.22 eV, compared with the experimental value of 4.24 eV (there may be a more recent experimental value in the literature – that’s something I must check). One of the most impressive features of full-potential embedding is the way that the charge density is beautifully continuous across the embedding plane, the plane which separates the surface calculation from the bulk.

I’ve provisionally called chapter 5 “Surface electronic structure results”, but I’m not particularly happy with this title – not quite snappy enough for the what I’m going to put in. I’m going to describe what I think are the most interesting results from embedding calculations of surfaces and interfaces, but I can hardly put all that in the title. It’s also going to be a very personal selection of work, so I hope that I don’t offend too many people by what I leave out. Apart from getting the title right, this is one of the chapters I’m really looking forward to writing, real physics, where embedding has made a significant contribution. Of course like all methods, everything which is calculated by embedding can be calculated in other ways, but I think that embedding does it better.

Up to now I’ve written 60-odd pages and created well over 30 figures. Well, “created” is a slight exaggeration, as some of the figures are taken from the literature, adapted for the book using TikZ. (All must be acknowledged, and it is important get permission to use them in the book. I cannot simply use these adapted figures in the blog unless I have permission – only the ones that I’ve created from scratch, and this is why this blog is rather lacking in diagrams.) How many pages and figures to go? I have no idea – it all depends how I get on in the autumn, especially as I’m going to spend the whole of October at a theoretical physics institute.

The 7 weeks I spent in France this summer was the longest I’ve been down there at a stretch (well, I was back in England for a week in the middle). It was a great boost for the book, as well as my swimming, and I’m sorry the summer is nearly over. But what a lot to do in the garden now I’m back in Cumbria – chopping back the vigorous summer growth, and generally tidying up. Something I enjoy doing very much, rather like tidying up an article or even a book chapter.

# A week in the Aude

Monday 5 August
Today started another beautiful day, promised to be very hot, but it turned cooler and cloudy in the course of the afternoon. But still nice for a good swim. In the morning I started writing about the transfer matrix, and this went quite well. I got as far as describing the eigenvalues of the layer transfer matrix – these are nothing other than the Bloch phase factors in going from layer to layer in a bulk solid – and this brings me nicely to the idea of the complex band structure, where the Bloch wave-vector is in general complex. Of course the allowed wave-functions in an infinite solid correspond to a real Bloch wave-vector, the usual band structure. States with complex wave-vector increase or decrease exponentially as we move into the solid, and the decaying states are allowed in a semi-infinite solid. So they are relevant to surface states, electronic states lying in a band gap which are localised on the surface. I’ve already discussed surface states in applications of the embedding method in chapter 3, so I must tie this in with the complex energy bands in the section I’m writing. It’s quite hard to tie things together – just remembering what I’ve written is bad enough! Anyway, I got a page written this morning, with a nice figure showing a schematic layer of atoms, and that’s not bad for an August morning.

Tuesday 6 August
I need a figure showing the complex band structure of a metal, Cu or Al would do very nicely. My first thought is to generate this myself, using the old embedding potential program – this uses scattering theory rather than the transfer matrix, but that doesn’t matter, all I need is the band structure, any band structure! The versions of the embedding potential program which I have are in Fortran 77, and have been much adapted over time. Well, I got the program compiled without any difficulty, and got it running for a Cu input potential. It even gave me a band structure. But only one band – where are the other bands? So I insert a “write” statement, and the program stops – a segmentation fault – how vividly I remember those! Overwriting somewhere, presumably, but trying to understand this old program is becoming too difficult and time-consuming. So I shall ask one friend for another version of the program (one which works) and another friend for pictures of calculations of complex bands. As I remarked in an earlier post, it’s only when you go back to old programs that you realise how much better modern Fortran is (and I suppose other programming languages).

Today it was supposed to turn showery and thundery. Not yet here in the Aude, but mustn’t speak too soon: it’s now 8.30 in the evening and there is thunder in the distance.

Wednesday 7 August
My visitor arrived today, so it will be a week with plenty of walking, if the weather stays fine. In fact today turned out much better than forecast, with plenty of sunshine and no rain. But what a rainy and stormy night last night! Awful pictures on the news of houses and cars damaged by massive hailstones, not far from here.

Close-packed peaches

This morning, before I went to the airport, I got quite a lot of writing done, with various blanks for the complex band structure figure. I covered the concept of complex bands, why these are relevant to the general solution of the Schrödinger equation in semi-infinite crystals – in other words a crystal with a surface – hence electron surface states. I’ve reached the calculation of the embedding potential from the eigenstates of the layer transfer matrix. The next stage will be to explain how transfer matrices can be used to transfer the embedding potential from the “natural” embedding surface, which weaves its way between atoms, to a more convenient flat surface. Not bad for a day’s writing, and for once no particular worries about consistency of notation etc.

With my visitor we went this afternoon to the café in the next village for an excellent coffee and ice-cream. I seem to have spent the whole day eating, but my favourite at the moment are the locally grown peaches – incredible.

Thursday 8 August
A beautiful day, perfect for a walk, and we started at Rennes-le-Château for a round walk over a viewpoint called La Pique, not very high but magnificent views towards the Pech (Pic) de Bugarach and over the Aude valley towards the Pyrenees.

Aude view

Rennes-le-Château is a very odd place, new-age, I suppose, associated with all sorts of mad stories about the lost treasure of the Visigoths etc. Having been there several times we avoided it this time, but it was the starting point for the walk – very well marked, but almost no other walkers. They must all have stayed in Rennes.

In the evening we went to the Welsh café in a neighbouring village for their monthly fish and chips evening. Superb – I’ve never tasted better. As well as the expats, quite a lot of French from the village came for this typically British supper.

No physics today!

Friday 9 August
Another beautiful day of walking, this time actually reaching the top of a mountain, Pech Cardou. Not very high at 795 m, but a fine walk with wonderful views from the summit. No physics again!

Saturday 10 August
The fine weather continues, and we went to a lovely valley above the pretty village of Sougraigne. This walk took us to the source of the river Sals, which, as its name suggests, is saline. In fact salt used to be extracted at the source, as the Sals at the source is allegedly several times more salty than the Mediterranean. I’m a bit dubious, as the water didn’t taste particularly salty, but apparently this varies with the rainfall.

Sunday 11 August
The highlight of our walks! Another fine day, so we set off for the Col de Jau, on the border between the departments of Aude and Pyrénées-Orientales. Very narrow roads, after going through the even narrower (and very spectacular) Aude gorges. But so worthwhile – from the col there were amazing views of Canigou, the most easterly (big) peak of the Pyrenees and a symbol of Catalonia, and down to the Mediterranean near Perpignan. From the col, at 1506 m, our goal was Madrès, 2469 m, which in fact I can see from my house. On the map a refuge was marked, quite close to the col, and this seemed an ideal stopping-off point for a cup of coffee, something to really set us up for the long walk ahead. Unfortunately the refuge (a former barracks for workers at a long-abandoned talc quarry) was pretty well derelict! So we set off without the coffee, and after a long and difficult path through forests we eventually reached la Balmette at just over 2000 m, a ridge with an incredible view towards Madrès itself. Still patches of snow around. The mountain flowers were wonderful, and we saw a herd of the Pyrenean chamois, the isard.

By this time it was mid-afternoon, and it was clear that we couldn’t reach the summit that day, so reluctantly we turned back. A hard walk, but immensely enjoyable. Next time we’ll have an earlier start.

Monday 12 August
The weather continues excellent, so we set off for one of our favourite walks, above the village of Valmigère. This is high limestone country, reminiscent of parts of South Cumbria. Again, fabulous views, the Mediterranean in the distance to the east, and to the south the Pyrenees, with the ever-present Pech de Bugarach in the foreground.

Pech de Bugarach

Tuesday 13 August
Today started off wet, but it soon cleared up and got out another beautiful day. Just the day for another favourite, starting off from the village of Bugarach, and following the Sentier Cathare – a marked footpath commemorating the Cathars, who were active and persecuted in this area in the 13th and 14th centuries. Bugarach came into prominence in December 2012 – even making the BBC news – as the village which would be saved at the end of the world on 21 December 2012! Apparently flying saucers would emerge from caves in the Pech de Bugarach (the caves do exist, not so sure about the flying saucers however), and rescue the people in the village.

Wednesday 14 August
My visitor left today, so it’s back to physics in the morning, and swimming in the afternoon!

I’d better get this blog post out – rather longer then a week in Aude.