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No 1: Orpheus weeps for Eurydice Air de danse Dance of the Angel of Death Interlude
No 2: Pas des Furies Air de danse Interlude Air de danse, conclusion Pas d'action Pas de deux Interlude Pas d'action
No 3: Apotheosis of Orpheus
Stravinsky's score, though routinely included among his neoclassical works, is in many ways quite unlike anything that precedes it. During the war, in America (of which he became a citizen in 1945), he had fulfilled a string of more or less openly commercial commissions, while working quietly on two eventual masterpieces—the Symphony in three movements and the Mass—which he found for a long time hard to crystallize in his mind, perhaps for lack of any likely performance. The uncertainties of war seem to have drawn him back to the church, and also to his Russian inheritance. And at some point he became intrigued by medieval music, especially the Ars Nova of the fourteenth century, with its decorative lines and intricate polyphonic techniques.
All these influences left their mark, however obliquely, on Orpheus, while to some extent directing it towards the more esoteric aspects of the next and final phase of his music. Orpheus is not in any sense a serial work; but it does hint at a new austerity and intensity that might suggest a breaking away from the more mechanical aspects of neoclassicism. Compared with the brusque, almost hearty opening of Jeu de cartes, the introduction of Orpheus has a repressed, secretive quality. Orpheus weeps for Eurydice, the stage direction informs us, but with his back to the audience, standing motionless. Later, as the Angel of Death leads him into Hades, the Furies protest, it seems, mainly in undertones (and soon submit to the beauty of Orpheus' playing). Then, at the crucial moment when Orpheus tears the bandage from his eyes and looks back at Eurydice, who at once falls dead, the music responds with a bar of complete silence. Even the Pas d'action in which the Bacchantes tear Orpheus to pieces is truly violent only for a few bars, which must have seemed a very strange turn of style for the composer of The Rite of Spring.
from notes by Stephen Walsh © 2009


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Greg Wooden, John's grandson, cheers on UConn
Updated: December 22, 2010, 12:31 PM ETBy Lynn Hoppes
HARTFORD, Conn. -- As fans were exiting the XL Center on Tuesday night, University of Connecticut women's basketball coach Geno Auriemma took time to walk over and talk with Greg Wooden, grandson of late UCLA men's basketball coach John Wooden.
"This is a very special day for you," Wooden said to Auriemma, whose team had just won its 89th consecutive game, breaking UCLA's record set in 1971-74.
"I've always been an admirer of your grandfather," Auriemma said, ...
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Have you heard his first solo album 'Candleland' pd? My ex hubby liked them - and I loved that album though I wasn't a fan of the band. Might have to buy it again as I had it on a cassette....aah happy days!
The Stones were great in their heyday. Brown Sugar was my favourite I'd say. Much preferred them to the Beatles - you were in one camp or or the other then! Watched a Ray Davies thing last night - loved early Kinks stuff. You Really Got me and All Day and All of the Night. Superb.
I don't grow a lot from seed but I can vouch for sweet peas, lettuce and basil lasting well, and I've had free packets of seed which have been past their sell by and have germinated well - sunflowers and nigella were two that my daughter sowed. I make sure they're closed up tightly after opening and stored in a cool place out of sunlight as KEF stated.
Oh doc -
Dove - when you strike oil - just remember I'm your favourite on here....
Got some sweet peas in the post yesterday. 2 whites and 2 creams to go with the dark ones I got locally. Good service - English Sweet Peas. Smaller packets -12 seeds - but similar value to the normal sized packs, and free delivery over a fiver. Looking forward to getting them in
Pinching out encourages bushiness by producing more stems/shoots which in turn carry more buds/flowers. Leaving them just means (generally speaking) they'll grow up rather than out and will just have slightly fewer flowers. As Verd says, make sure the plants are healthy though.
I don't grow many annuals other than sweet peas but I pinch them out to give more flower power!
Morning all. Cloudy here but dry for now so will try and get the last bits of hedge put in along the front path. Did some yesterday afternoon once it cleared up. Grass needs cutting but it's still very soggy so will leave it for a while. Picked a few daffs for the house. They're still popping up in the bit I lifted last spring so I didn't get them all!
Off for a look round the forum. Have a good day all
Sounds good Bilje. The little Irises which are in flower just now (Reticulata)are bulbs and these are the ones which tend to disappear eventually. The tall, rhizomatous kind we see in summer, will diminish with age and should be split to regenerate them. The central part gradually becomes less productive and new shoots appear around it. Remove the centre, divide the new sections up and replant them to maintain vigour and a good display.
Yum Yum Dove
Just had a baked potato with daughter's leftover meatball sauce though....
Steady on Dove...you'll need a lie down and a cake
I've done very little today as well, other than some light h****work and buying some compost. Well, it's Friday
Well no muffins for you Verd. You're off my Christmas card list...
Haven't made them anyway. Went out for a coffee with a friend
If you take the 'ones' off those temps Stacey, that's about right for here!
I've put a silly reply on the advertising/spammer...I'm in one of those daft moods again..
Raining now and very windy
The 'sun baking' is for the larger summer flowering types Bilje. They're rhizomes as opposed to bulbs, and are planted on the surface of the ground rather than being buried like a bulb is.


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Posts tagged xblm-holiday-deal-of-the-week
XBLM Deal of the Week: Halo
The fourth and final of Microsoft's weekly Xbox Live Marketplace holiday deals is none other than the original Halo: Combat Evolved. Major Nelson has announced that the franchise-launching, system-selling FPS will be available for 600 ($7.50) starting tomorrow, Tuesday, December 23 for Xbox Live G...
XBLM Deal of the Week: Bomberman Live
Larry Hryb – Xbox Live's Major Nelson – has dropped a bomb via his blog. Er, rather, he delivered news of a dropping in price of the Avatar-supporting Bomberman Live. As of Tuesday, December 16, the game will be on sale for 600 ($7.50), down from 800 ($10). The special price will be in...
XBLM Deal of the Week: Foo's 'Colour and the Shape' Rock Band DLC
Following up on last week's discounted Legendary Map Pack for Halo 3, Major Nelson recently announced the latest Xbox Live Marketplace "Holiday Deal of the Week". For the next seven days, you can pick up one of the best pieces of Rock Band DLC to drop this year (in our most humble, Grohl-worshipin...
XBLM Holiday Deal of the Week: Legendary Map Pack for 400 points
Major Nelson sends word that Microsoft is beginning a new promotion this week, the Xbox Live Marketplace Holiday Deal of the Week. Throughout December, Microsoft will announce a new deal of the week for Gold members only on a certain content from the Xbox Live Marketplace. This week's deal is the L...
Halo 3 Legendary maps kick off 'Holiday Deal of the Week' on XBLM
Following a tepid Black Friday sale this past weekend, Major Nelson has announced the Xbox Live Marketplace "Holiday Deal of the Week." Launching tomorrow, the weekly promotion will feature one discounted item for, well, one week during this month of December. First up, the Halo 3 "Legendary Map Pa...
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This Deal Has Been Marked as Spam
The HUKD community has flagged this deal as reeking of spam, therefore the deal has been incinerated. While you're here, you may find another deal you like. Check out some related deals below.
Recent Hot Deals
HP Stream 7 Signature Edition Tablet on MS Store UK at £99.00, incl. free delivery and return
Check this out: the new HP Stream 7 is now available on pre-order on the MS Store UK with free delivery and free return, at £99 incl. one-year subscription to Office 365 Personal with Skype. . Edit: the tablet is available to order since yesterday, apparently they launched earlier as planned! :) 45 Commentstokkimade hot 2 hours, 13 minutes ago
Receive a £20 voucher when you buy an IKEA real Christmas tree for £25.00
Got this e-mail a few moments ago, looks like a good deal, you purchase the xmas tree from IKEA between 28th November and the 21st December 2014 and then you get a £20 voucher that you can spend instore from 19th January until 8th February 2015. . Technically the christmas tree costs you a fiver. . terms. . *Christmas tree offer available when you buy a real Christmas tree for £25 between 28/11/14 to 21/12/14. Voucher can be redeemed between the 19/01/15 to 08/02/15. While stocks last. This offer is not available at IKEA Cardiff and Croydon. - eslick 18 Commentspugsportmade hot 2 hours, 29 minutes ago
Zavvi - 2 DVDs for £6.00 - including Epic, Life of Pi, Ice Age 4, Rise of the Guardians
2 DVDs for £6 at Zavvi. . Similar deal (2 for £5 I think) was live a couple of weeks ago and stock went quickly. . Not a huge choice but some good stocking fillers 6 Commentsaing69made hot 2 hours, 30 minutes ago
Corsair AF120-Red Led (Dual Pack) case fan £13.00 @ amazon (Purple led £12.50 dispatched within 1 to 2 mths - See Other Sellers on Amazon)
Translucent, frosted fan blades. Nine ultra-thin fan blades, a sleeved bearing system for ultra low noise 7 Commentsstealthxmade hot 2 hours, 31 minutes ago
Monster high nail polish set £1.00 @ Poundworld
Monster high clawlicious nail polish set for only £1 at pound world at Intu in derby. It's currently £8.12 from Amazon. I picked one up for my daughter. Plenty left. Think it's national as they usually have the same stock. It was in the Xmas aisle. Not the toy aisle. Will make a nice stocking filler. . http://www.amazon.co.uk/gp/aw/d/B00CMLF07S?pc_redir=1414193304&robot_redir=1 3 Commentsblackberry786made hot 2 hours, 33 minutes ago
Disney A Christmas carol (2009) BLU-RAY+DVD £4.99 at play/linkent
New and free delivery. Next cheapest is £7.99. . Disney's 'A Christmas Carol', a multi-sensory thrill ride re-envisioned by Academy Award®-winning filmmaker Robert Zemeckis (Forrest Gump, Back to the Future, The Polar Express, Beowulf), captures the fantastical essence of the classic Dickens tale in a groundbreaking 3-D motion picture event. Ebenezer Scrooge (Jim Carrey) begins the Christmas holiday with his usual miserly contempt, barking at his faithful clerk (Gary Oldman) and his cheery nephew (Colin Firth). But when the ghosts of Christmas Past, Present and Yet to Come take him on an eye-opening journey revealing truths Old Scrooge is reluctant to face, he must open his heart to undo years of ill will before it's too late 7 Commentsthebaron69made hot 2 hours, 34 minutes ago
The Hobbit Trilogy 3D Triple Bill @ Cineworld From Just £14.00
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Hisense 8, 16GB Tablet £62.10 - Asda Direct
Similar to a few previous posts. However, Asda are currently adding a 10% discount to the £69.00 price at checkout making it just £62.10. . Reviews. http://the-digital-reader.com/2014/07/20/review-hisense-sero-8/#.VHEHemdybIU. https://www.youtube.com/results?search_query=hisense+8. . . . http://www.youtube.com/watch?v=kIRFFB38DX0. - rejector1 11 CommentsGordon Bellmade hot 2 hours, 49 minutes ago
Lara Croft & The Temple Of Osiris (Steam) £9.99 @ GamesPlanet
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anti tip flatscreen tv strap was £5.99 now £2.99 from argos
Prevents the tv from tipping over if pulled by a child, fittings fit most popular tvs, £2.99 is a better price to pay than paying for a new tv. . Shared Via The HUKD App For Android. 27 CommentsR4CH4EL88made hot 3 hours, 2 minutes ago


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Jeff and Brandi have gone to extra lengths to make sure I'm a happy customer. If I want to buy something they don't sell, they go out and find it for...Read More ->
The service from this company was very professional and they keep you up to date on the status of your order. Thanks a lot!Read More ->


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Big Al's Big Ass Comedy Show: Stand-Up Showcase at Cobb's
* Additional fees apply.
The last date listed for Big Al's Big Ass Comedy Show was Thursday December 22, 2011 / 8:00pm.
Currently at Cobb's Comedy Club
Cobb's Comedy Showcase: Top Bay Area Comics Perform
- Full Price:
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For some 30 years, Cobb's Comedy Club has been one of the hottest venues on the stand-up circuit,...Learn More
Venue Details
I saw him at Cobb's in SF, so between tough parking and long line to get in, give yourself extra time.Comedian Gary Owen info • Oct 30 2014 star this tip starred
If you sit near the front know that you will be targeted for material towards the end of the show :)Comedian Aries Spears info • Sep 25 2014 star this tip starred
Reviews & Ratings
Featured review from Goldstar Member
view more less of this review
The comedians were all good. The silly "band" at the end was horrible. I know they were trying to be funny but just how many times can one say "butt" until it gets really (really) old. Bad way to end the show. Stick to comedians for a comedy club. Leave those guys to... no idea what they would be appropriate for.
star this review starred report as inappropriate
Comedians were not that funny and there was little audience intraction.
star this review starred report as inappropriate
Great evening of comedy.
star this review starred report as inappropriate
Great local comedian showcase, and especially loved the "musical guest" Saw Dem Eyez. Wouldnt get that any other place!
Big Al has performed throughout the States, entertaining audiences with his smooth and easy going delivery. Big Al lives in L.A. but is originally from Silicon Valley, where he studied stage acting and improvisational comedy. He toured with Big City Improv in the early 2000's, and the group won "Best Of The Bay" in 2002. You may have seen Big Al on TV in the national PSA for "Click It Or Ticket" where he played a "happy passenger". Big Al has also toured with The Chicano Comedy All-Stars, The Burger Kings Of Comedy, and The Three Jarheads.


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More than a dozen successful restaurants -- including Marc Forgione's American Cut, four Jose Garces concepts, and the House of Blues -- shuttered on the Atlantic City boardwalk last weekend as Revel and Showboat casinos closed their doors. But, in their shadow, Tony Boloney's has only thrived over ... More >>
If you haven't yet learned to navigate your way to Jersey City, let us help you out: Take the PATH during the week and a newly discounted weekend ferry service from the World Financial Center. If it's been awhile since you've been there, you're missing out on a crop of good restaurants. Here, we've ... More >>
When a towering figure like Nelson Mandela dies -- a revolutionary, an emancipator, South Africa's first black president, one of the most beloved figures ever in the struggle against not just apartheid, but racism and oppression in all its forms -- the most important thing to do is immediately estab ... More >>
Here's another way pro athletes are like cops: most of them live out of town. amNY reports that "only 35 of the 200 players" on our local teams live in New York City. One of them is the Mets' David Wright, who maintains "a spacious $6 million penthouse suite with outdoor space at Infinity Flats, a n ... More >>
Comedian Mike Birbiglia loves to milk his underdog status for laughs as a 30-year-old stand-up (today's his birthday!) who gets no respect (his routine includes the humiliating story of the time he was introduced as a "middle-tier" comedian at Trump Plaza in Atlantic City). But now that he's ... More >>
Comedy Central comes to Central Park
Candidate whisperer: The ex-bartender who's got the ear of the GOP's top contender
New Yorkers Tell Their Tales of Botched Raids
Lifelong Partnership Dissolved


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The NYTimes just ran "Hard-Nosed Advice From Veteran Lobbyist: 'Win Ugly or Lose Pretty' - Richard Berman Energy Industry Talk Secretly Taped". Rick Berman has long been the architect of "public charities" for any client willing to pay. Berman's Center for Consumer Freedom (CCF, EIN 26-0006579) evolved...read more
IPCC Reviewers Peer Into Moral Abyss of Censorship
IPCC Reviewers Peer Into Moral Abyss of Censorship
Previous Comments
I keep seeing code pop out on the top of your post. Hard to tell that you are quoting Phil Jones' in one of the hacked private emails, saying the inverse of what he and others are accused of: trying to censor denialists from peer-review journals.
Indeed, the entire thread, for those of us who have bothered to read it, has them really demanding that the peer review be as stringent as it is supposed to be.
Indeed. As I noted in an earlier post the whole thing seemed to have been triggered off by a case in which the peer review process had notably failed, viz. the publication of Willy Soon and Sally Balunias's paper on global warming in which solar output was declared to be the main culprit.
After this paper was published the chief editor-to-be (Hans Von Storch) and half the editorial board quitted the offending journal in protest
Heres a good video that summarizes some of the key players in climategate. Maybe the desmog team can use some of their investigative reporting skills to discover how these people funneled a bunch of the cash they received.


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Think You're Having A Shitty Day? Try Getting Hit By a Car... Twice
Byon January 9, 2014
Oh, sorry, did you think you were having a shitty day? We get it. After all, Monday was officially the most depressing day of the year. Well, to give you a break and get those seratonin receptors firing in a way that can only be stimulated by marvelling at the misfortune of others, check this scenario out:
Some poor schlep in Hewitt, Texas was driving home from work, minding his own business, and was smoked by another car operated by, not surprisingly, a drunken asshole. Luckily, our hero escaped unscathed, if a little dusted up, and after watching his assumedly unwitting assailant get carted off to instant-karma jail, resumed his homeward trek, this time on foot.
Because, apparently, the cops in Hewitt are too busy doing, er, somethin' else other than sympathetically driving accident victims home. Bear in mind: This is a 6.9-square-mile township.
So our unnamed hero was strolling resolutely, hands no doubt shoved deep in his pockets and visions of a well-deserved adult soda guiding him like a proverbial carrot on a stick when, guess what? Yep.
Hit again.
This time by a black Pontiac Vibe, to be exact, although that detail really doesn't add much but some visual accents to our story. And, alas, Lady Luck, fickle wench that she is, had abandoned our sorry sap victim, and he wound up not home, but in the hospital, with serious injuries. We wish him the best.
Buck up, campers. The polar vortex has danced away and temps are rising, it's almost the weekend, and you (probably) haven't been hit by a car today. Twice.
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Town Square
Arrest in decades-old Mtn. View murder case
Original post made on Jan 8, 2013
Read the full story here Web Link posted Tuesday, January 8, 2013, 9:09 AM
Comments (2)
a resident of Los Altos
on Jan 8, 2013 at 11:49 am
Kudos to the police for closing a cold case. I admire the tenacity to bring someone to justice who probably thought on some level they got away with it (at least in this frequency).
a resident of Crescent Park
on Jan 10, 2013 at 2:27 am
bru is a registered user.
Got away with that crime, but what else has Daniel Garcia been doing in 28 years? There is no way people would give up their guns with people like this around.
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Grab a Bowl of Heaven soon in Mountain View
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The Household's Most Wanted Wow. It's like this family wants you to lay around on the internet and munch on junk drowning in television. So much for "quality time", right? Really. And I so much as put a little dedication in the one sport I picked up and I'm suddenly "The Household's Most Wanted".
It's like we're in some kind of Looney Toon and they're all a bunch of Elmer Fudd's chasing me down with their imaginary rifles. One thing I'd like to point out: Elmer Fudd never actually pins Bugs Bunny down. Maybe I'm alot like Bugs, always on the go, threatened, but never stopped for good.
But the only difference that will remain is this:
I don't plan on living my life as a rerun. I look forward to making every episode more suspenseful and interesting than the previous.
Afterall, you know what they say:
Life is too short.
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Cutting the acid in fresh tomatoes?
This is sad: I've finally had a successful year growing my own tomatoes. I love nothing better than a tomato sandwich. But I've now developed quite a problem with acid reflux at night. Figured I'd have them for breakfast and lunch. (Yes, I can eat a tomato sandwich for breakfast in tomato season.) But after several days of multiple tomato sandwiches, I'm now developing canker sores in my mouth. Argh!
I should have tried to grow the less acidic varieties, but I went with ones that were pretty much guaranteed to grow for incompetent gardeners.
Anyway, with several tomatoes waiting on the kitchen counter and more waiting to be picked outside, does anyone have any ideas on what I can do with all these beauties that will be less injurious to my mouth and esophagus?
I feel like the guy in the Twilight Zone story who finally had all the books in the world and all the time to read them--and then he breaks his glasses.
re: scubadoo97
I wasn't writing for medical advice but for recipes and/or preparation tips. I already know what to do for my reflux and canker sores.
Let the canker sore heal, campho phenique works miracles
Wait until the tomatoes are super super ripe and don't eat them on an empty stomach- have a few bites of something basic like the bread or a cracker first. And cut waaay back- like one tomato a day after your canker sore heals.
Otherwise you'll need to take OTC meds like millions of others do. Everything says people with reflux should generally avoid tomatoes...
Use the excess tomatoes to make a sauce to freeze or bring gazpacho for a party.
re: Susangria
Sugar doesn't neutralize, it just balances (masks) the acidity.
Reminds me of the Twilight Zone where an old man gives a young man a stopwatch, saying "I don't want this thing anymore." Young man realizes that when he clicks it, the whole world freezes. Eventually, he freezes everyone and robs a bank, but drops and breaks that stopwatch on the way out. Ends with a pan out of him screaming.
Anyhow, peel the tomatoes with the standard method - it is easy. I poked around, and that is a suggested solution because the skins seem to cause most of the problem.
I had a reflux problem for a while. I cut out wine and it went away, seemingly for good. I can eat anything now. One weird thing: when i would have a bout of relux, a sour pickle seem to alleviate it. This was counterintuitive to me. I got off the prilosec and just used pickles. When I realized that it was the wine and potentially alcohol in general, I was fine. It was pretty bad, but I've had no reflux for two years.
I know you don't want advice, but have you tried sauerkraut juice for the mouth sores? I know people who swear by it. Lots of info on the net. I think that Kimchi is good, too. Must be the probiotics. Saw one article where a scientist is trying to isolate the compound(s) responsible.
re: Puffin3
I guess I'll have to do that--thanks. We have a long growing season here so I may actually still have to plant a low acid tomato. So glad I planted a cherry tomato called "Chocolate Cherry" which doesn't bother me at all. Think I'll go get another...
re: NotSoHot
Choc. cherry are delicious! Black cherry, too. :)
I use a tiny pinch of baking soda in tomato sauce that is acidic (1/4 teaspoon at a time, it doesn't take much at all!) I don't know if you could sprinkle a tiny amount of a tomato slice and have it work but if all else fails, make sauce with the tomatoes and pick a different variety for next year.
Grow cucumbers. You can't take the acid out of a tomato. But there are lots of other delicious things to grow and to eat.


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Theory of surface plasmons and surface-plasmon polaritons
ABSTRACT Collective electronic excitations at metal surfaces are well known to play a key role in a wide spectrum of science, ranging from physics and materials science to biology. Here we focus on a theoretical description of the many-body dynamical electronic response of solids, which underlines the existence of various collective electronic excitations at metal surfaces, such as the conventional surface plasmon, multipole plasmons, and the recently predicted acoustic surface plasmon. We also review existing calculations, experimental measurements, and applications. Comment: 54 pages, 33 figures, to appear in Rep. Prog. Phys
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ABSTRACT: A large amplitude surface plasma wave (SPW) propagating over a conductor–vaccum interface with Gaussian intensity profile transverse to the direction of propagation ( $ \widehat{z} $ ) and surface normal ( $ \widehat{x} $ ) is shown to undergo periodic self-focusing due to ponderomotive nonlinearity. The ponderomotive force on electrons arises due to the rapid decline in surface wave amplitude with the depth inside the conductor. In case of plasma, this leads to ambipolar diffusion of plasma, whereas in metals, only electron displacement occurs until the space charge balances the ponderomotive force on electrons. For a surface plasma wave, having Gaussian amplitude profile in y, the maximum electron density depression occurs on the axis (y = 0) and the effect weakens as |y| increases. The axial portion of SPW thus travels with slower phase velocity than the nonaxial portion leading to self-focusing.Plasmonics 01/2013; 8(2). · 2.43 Impact Factor
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ABSTRACT: Bimetallic Ag/Au nanoparticles (NPs) with alloyed and core–shell structures were synthesized at a low temperature (75°C) from the preformed gold and silver NPs in aqueous solution. The transmission electron microscope (TEM) mapping images, as well as the evolution of UV/vis spectroscopy, revealed that both the digestive ripening and Ostwald ripening were responsible for the formation of above structures during the heating process. Higher Ag ratios led to the growth of silver shell on the surface of Ag/Au alloy core, while lower Ag ratios tended to the formation of alloy particles. Interparticle diffusion, ligands and chemical potential were presumed to play important roles in the formation of alloyed and core–shell bimetallic Ag/Au NPs.Colloids and Surfaces A-physicochemical and Engineering Aspects - COLLOID SURFACE A. 12/2010;
Article: Impact of C60 adsorption on surface plasmon polaritons on self-assembled Ag(111) islands on Si(111)[Show abstract] [Hide abstract]
ABSTRACT: The influence of C60 adsorption on the properties of surface plasmon polaritons on small Ag islands is discussed. Under illumination with UV light as well as under illumination with femtosecond laser pulses, a decrease of the photoemission yield with increasing C60 coverage is observed. With angular resolved measurements, changes of the band structure during deposition are studied. Based on these experiments, an increase of the work function with increasing coverage is measured. In two photon photoemission, the surface plasmons are imaged as a periodic moiré pattern, the wavelength of which changes because of a modified effective surface dielectric function. Our findings imply that the wavelength of the plasmon wave becomes shorter as a result. Finally, a decrease of the intensity of the moiré pattern maxima compared with the intensity of the first maximum with increasing C60 coverage has been observed. Accordingly, the damping of the plasmon wave becomes stronger.Plasmonics 01/2012; 7(2):229. · 2.43 Impact Factor
Page 1
arXiv:cond-mat/0611257v1 [cond-mat.mtrl-sci] 9 Nov 2006
Theory of surface plasmons and surface-plasmon polaritons
J. M. Pitarke1,2, V. M. Silkin3, E. V. Chulkov2,3, and P. M. Echenique2,3
1Materia Kondentsatuaren Fisika Saila, Zientzi Fakultatea, Euskal Herriko Unibertsitatea,
644 Posta kutxatila, E-48080 Bilbo, Basque Country
2Donostia International Physics Center (DIPC) and Unidad de F´ ısica de Materiales CSIC-UPV/EHU,
Manuel de Lardizabal Pasealekua, E-20018 Donostia, Basque Country
3Materialen Fisika Saila, Kimika Fakultatea, Euskal Herriko Unibertsitatea,
1072 Posta kutxatila, E-20080 Donostia, Basque Country
(Dated: February 6, 2008)
Collective electronic excitations at metal surfaces are well known to play a key role in a wide
spectrum of science, ranging from physics and materials science to biology. Here we focus on a
theoretical description of the many-body dynamical electronic response of solids, which underlines
the existence of various collective electronic excitations at metal surfaces, such as the conventional
surface plasmon, multipole plasmons, and the recently predicted acoustic surface plasmon. We also
review existing calculations, experimental measurements, and applications.
PACS numbers: 71.45.Gm, 78.68.+m, 78.70.-g
I. Introduction
II. Surface-plasmon polariton: Classical
A. Semi-infinite system
1. The surface-plasmon condition
2. Energy dispersion
3. Skin depth
B. Thin films
III. Nonretarded surface plasmon: Simplified
A. Planar surface plasmon
1. Classical model
2. Nonlocal corrections
3. Hydrodynamic approximation
B. Localized surface plasmons: classical
1. Simple geometries
2. Boundary-charge method
3. Composite systems: effective-medium
4. Periodic structures
5. Sum rules
IV. Dynamical structure factor
V. Density-response function
A. Random-phase approximation (RPA)
B. Time-dependent density-functional theory
1. The XC kernel
VI. Inverse dielectric function
VII. Screened interaction
A. Classical model
1. Planar surface
2. Spheres
3. Cylinders
B. Nonlocal models: planar surface
1. Hydrodynamic model
2. Specular-reflection model (SRM)
3. Self-consistent scheme
VIII. Surface-response function
A. Generation-rate of electronic excitations
B. Inelastic electron scattering
C. Surface plasmons: jellium surface
1. Simple models
2. Self-consistent calculations: long
3. Self-consistent calculations: arbitrary
4. Surface-plasmon linewidth
5. Multipole surface plasmons
D. Surface plasmons: real surfaces
1. Stabilized jellium model
2. Occupied d-bands: simple models
3. First-principles calculations
E. Acoustic surface plasmons
1. A simple model
2. 1D model calculations
3. First-principles calculations
4. Excitation of acoustic surface plasmons
IX. Applications
A. Particle-surface interactions: energy loss
1. Planar surface
B. STEM: Valence EELS
1. Planar surface
2. Spheres
3. Cylinders
C. Plasmonics
X. Acknowledgments
Page 2
In his pioneering treatment of characteristic energy
losses of fast electrons passing through thin metal films,
Ritchie predicted the existence of self-sustained collec-
tive excitations at metal surfaces [1].
been pointed out by Pines and Bohm [2, 3] that the long-
range nature of the Coulomb interaction between valence
electrons in metals yields collective plasma oscillations
similar to the electron-density oscillations observed by
Tonks and Langmuir in electrical discharges in gases [4],
thereby explaining early experiments by Ruthemann [5]
and Lang [6] on the bombardment of thin metallic films
by fast electrons. Ritchie investigated the impact of the
film boundaries on the production of collective excita-
tions and found that the boundary effect is to cause the
appearance of a new lowered loss due to the excitation
of surface collective oscillations [1]. Two years later, in
a series of electron energy-loss experiments Powell and
Swan [7] demonstrated the existence of these collective
excitations, the quanta of which Stern and Ferrell called
the surface plasmon [8].
Since then, there has been a significant advance in
both theoretical and experimental investigations of sur-
face plasmons, which for researches in the field of con-
densed matter and surface physics have played a key
role in the interpretation of a great variety of exper-
iments and the understanding of various fundamental
properties of solids. These include the nature of Van
der Waals forces [9, 10, 11], the classical image potential
acting between a point classical charge and a metal sur-
face [12, 13, 14, 15], the energy transfer in gas-surface in-
teractions [16], surface energies [17, 18, 19], the damping
of surface vibrational modes [20, 21], the energy loss of
charged particles moving outside a metal surface [22, 23],
and the de-excitation of adsorbed molecules [24]. Sur-
face plasmons have also been employed in a wide spec-
trum of studies ranging from electrochemistry [25], wet-
ting [26], and biosensing [27, 28, 29], to scanning tun-
neling microscopy [30], the ejection of ions from sur-
faces [31], nanoparticle growth [32, 33], surface-plasmon
microscopy [34, 35], and surface-plasmon resonance tech-
nology [36, 37, 38, 39, 40, 41, 42]. Renewed interest in
surface plasmons has come from recent advances in the
investigation of the electromagnetic properties of nanos-
tructured materials [43, 44], one of the most attractive
aspects of these collective excitations now being their
use to concentrate light in subwavelength structures and
to enhance transmission through periodic arrays of sub-
wavelength holes in optically thick metallic films [45, 46].
The so-called field of plasmonics represents an excit-
ing new area for the application of surface and interface
plasmons, and area in which surface-plasmon based cir-
cuits merge the fields of photonics and electronics at the
nanoscale [47]. Indeed, surface-plasmon polaritons can
serve as a basis for constructing nanoscale photonic cir-
cuits that will be able to carry optical signals and electric
currents [48, 49]. Surface plasmons can also serve as a
It had already
basis for the design, fabrication, and characterization of
subwavelength waveguide components [50, 51, 52, 53, 54,
55, 56, 57, 58, 59, 60, 61, 62, 63, 64]. In the framework
of plasmonics, modulators and switches have also been
investigated [65, 66], as well as the use of surface plas-
mons as mediators in the transfer of energy from donor to
acceptors molecules on opposite sides of metal films [67].
According to the work of Pines and Bohm, the
quantum energy collective plasma oscillations in a free
electron gas with equilibrium density n is ¯ hωp
¯ h(4πne2/me)1/2, ωp being the so-called plasmon fre-
quency [68]. In the presence of a planar boundary, there
is a new mode (the surface plasmon), the frequency of
which equals in the nonretarded region (where the speed
of light can be taken to be infinitely large) Ritchie's
frequency ωs = ωp/√2 at wave vectors q in the range
ωs/c << q << qF (qF being the magnitude of the Fermi
wave vector) and exhibits some dispersion as the wave
vector is increased. In the retarded region, where the
phase velocity ωs/q of the surface plasmon is comparable
to the velocity of light, surface plasmons couple with the
free electromagnetic field. These surface-plasmon polari-
tons propagate along the metal surface with frequencies
ranging from zero (at q = 0) towards the asymptotic
value ωs= ωp/√2, the dispersion relation ω(q) lying to
the right of the light line and the propagating vector be-
ing, therefore, larger than that of bare light waves of the
same energy. Hence, surface-plasmon polaritons in an
ideal semi-infinite medium are nonradiative in nature,
i.e., cannot decay by emitting a photon and, conversely,
light incident on an ideal surface cannot excite surface
In the case of thin films, the electric fields of both
surfaces interact. As a result, there are (i) tangential
oscillations characterized by a symmetric disposition of
charge deficiency or excess at opposing points on the two
surfaces and (ii) normal oscillations in which an excess
of charge density at a point on one surface is accompa-
nied by a deficiency at the point directly across the thin
film. The phase velocity of the tangential surface plas-
mon is always less than the speed of light, as occurs in
the case of a semi-infinite electron system. However, the
phase velocity of normal oscillations may surpass that
of light, thereby becoming a radiative surface plasmon
that should be responsible for the emission of light [69].
This radiation was detected using electron beam bom-
bardment of thin films of Ag, Mg, and Al with thick-
nesses ranging between 500 and 1000˚ A [70, 71]. More re-
cently, light emission was observed in the ultraviolet from
a metal-oxide-metal tunnel diode and was attributed to
the excitation of the radiative surface plasmon [72].
Nonradiative surface plasmons in both thin and thick
films can couple to electromagnetic radiation in the pres-
ence of surface roughness or a grating, as suggested by
Teng and Stern [73]. Alternatively, prism coupling can
be used to enhance the momentum of incident light,
as demonstrated by Otto [74] and by Kretchmann and
Raether [75]. Since then, this so-called attenuated reflec-
Page 3
tion (ATR) method and variations upon it have been
used by several workers in a large variety of applica-
tions [76, 77, 78, 79, 80, 81, 82].
During the last decades, there has also been a signifi-
cant advance in our understanding of surface plasmons in
the nonretarded regime. Ritchie [83] and Kanazawa [84]
were the first to attack the problem of determining the
dispersion ω(q) of the nonretarded surface plasmon. Ben-
nett [85] used a hydrodynamical model with a continu-
ous decrease of the electron density at the metal surface,
and found that a continuous electron-density variation
yields two collective electronic excitations: Ritchie's sur-
face plasmon at ω ∼ ωs, with a negative energy disper-
sion at low wave vectors, and an upper surface plasmon
at higher energies. In the direction normal to the surface,
the distribution of Ritchie's surface plasmon consists of
a single peak, i.e., it has a monopole character; however,
the charge distribution of the upper mode has a node, i.e.,
it has a dipole character and is usually called multipole
surface plasmon.
Bennett's qualitative conclusions were generally con-
firmed by microscopic descriptions of the electron gas.
On the one hand, Feibelman showed that in the long-
wavelength limit the classical result ωs= ωp/√2 is cor-
rect for a semi-infinite plane-bounded electron gas, irre-
spective of the exact variation of the electron density in
the neighborhood of the surface [86]. On the other hand,
explicit expressions for the linear momentum dispersion
of the conventional monopole surface plasmon that are
sensitive to the actual form of the electron-density fluc-
tuation at the surface were derived by Harris and Grif-
fin [87] using the equation of motion for the Wigner dis-
tribution function in the random-phase approximation
(RPA) and by Flores and Garc´ ıa-Moliner [88] solving
Maxwell's equations in combination with an integration
of the field components over the surface region. Quan-
titative RPA calculations of the linear dispersion of the
monopole surface plasmon were carried out by several
authors by using the infinite-barrier model (IBM) of the
surface [89], a step potential [90, 91], and the more realis-
tic Lang-Kohn [92] self-consistent surface potential [93].
Feibelman's calculations showed that for the typical elec-
tron densities in metals (2 < rs< 6) the initial slope of
the momentum dispersion of monopole surface plasmons
of jellium surfaces is negative [93], as anticipated by Ben-
nett [85].
Negative values of the momentum dispersion had been
observed by high-energy electron transmission on unchar-
acterized Mg surfaces [94] and later by inelastic low-
energy electron diffraction on the (100) and (111) sur-
faces of Al [95, 96]. Nevertheless, Klos and Raether [97]
and Krane and Raether [98] did not observe a negative
dispersion for Mg and Al films. Conclusive experimental
confirmation of the negative surface plasmon dispersion
of a variety of simple metals (Li, Na, K, Cs, Al, and Mg)
did not come until several years later [99, 100, 101, 102],
in a series of experiments based on angle-resolved low-
energy inelastic electron scattering [103].These ex-
periments showed good agreement with self-consistent
dynamical-response calculations carried out for a jellium
surface [104] in a time-dependent adiabatic extension
of the density-functional theory (DFT) of Hohenberg,
Kohn, and Sham [105]. Furthermore, these experiments
also showed that the multipole surface plasmon was ob-
servable, its energy and dispersion being in quantitative
agreement with the self-consistent jellium calculations
that had been reported by Liebsch [106].
Significant deviations from the dispersion of surface
plasmons at jellium surfaces occur on Ag [107, 108, 109,
110] and Hg [111], due to the presence of filled 4d and
5d bands, respectively, which in the case of Ag yields
an anomalous positive dispersion. In order to describe
the observed features of Ag surface plasmons, various
simplified models for the screening of d electrons have
been developed [112, 113, 114, 115, 116]. Most recently,
calculations have been found to yield a qualitative un-
derstanding of the existing electron energy-loss measure-
ments by combining a self-consistent jellium model for
valence 5s electrons with a so-called dipolium model in
which the occupied 4d bands are represented in terms of
polarizable spheres located at the sites of a semi-infinite
face-cubic-centered (fcc) lattice [117].
Ab initio bulk calculations of the dynamical response
and plasmon dispersions of noble metals with occu-
pied d bands have been carried out recently [118, 119,
120]. However, first-principles calculations of the surface-
plasmon energy and linewidth dispersion of real solids
have been carried out only in the case of the simple-metal
prototype surfaces Mg(0001) and Al(111) [121, 122].
These calculations lead to an accurate description of the
measured surface-plasmon energy dispersion that is supe-
rior to that obtained in the jellium model, and they show
that the band structure is of paramount importance for
a correct description of the surface-plasmon linewidth.
The multipole surface plasmon, which is originated
in the selvage electronic structure at the surface, has
been observed in a variety of simple metals at ω ∼
0.8ωp [99, 100, 101, 102], in agreement with theoreti-
cal predictions. Nevertheless, electron energy-loss spec-
troscopy (EELS) measurements of Ag, Hg, and Li re-
vealed no clear evidence of the multipole surface plas-
mon. In the case of Ag, high-resolution energy-loss spec-
troscopy low-energy electron diffraction (ELS-LEED)
measurements indicated that a peak was obtained at
3.72eV by subtracting the data for two different impact
energies [123], which was interpreted to be the Ag mul-
tipole plasmon. However, Liebsch argued that the fre-
quency of the Ag multipole surface plasmon should be
in the 6 − 8eV range above rather than below the bulk
plasma frequency, and suggested that the observed peak
at 3.72eV might not be associated with a multipole sur-
face plasmon [124].
An alternative spectroscopy technique to investigate
multipole surface plasmons is provided by angle- and
energy-resolved photoyield experiments (AERPY) [125].
In fact, AERPY is more suitable than electron energy-
Page 4
loss spectroscopy to identify the multipole surface plas-
mon, since the monopole surface plasmon of clean flat
surfaces (which is the dominant feature in electron-
energy loss spectra) is not excited by photons and thus
the weaker multipole surface mode (which intersects the
radiation line in the retardation regime) can be observed.
A large increase in the surface photoyield was observed
at ω = 0.8ωpfrom Al(100) [125] and Al(111) [126]. Re-
cently, the surface electronic structure and optical re-
sponse of Ag has been studied using this technique [127].
In these experiments, the Ag multipole surface plasmon
is observed at 3.7eV, while no signature of the multipole
surface plasmon is observed above the plasma frequency
(ωp= 3.8eV) in disagreement with the existing theoret-
ical prediction [124]. Hence, further theoretical work is
needed on the surface electronic response of Ag that go
beyond the s-d polarization model described in Ref. [124].
Another collective electronic excitation at metal sur-
faces is the so-called acoustic surface plasmon that has
been predicted to exist at solid surfaces where a par-
tially occupied quasi-two-dimensional surface-state band
coexists with the underlying three-dimensional contin-
uum [128, 129]. This new low-energy collective excitation
exhibits linear dispersion at low wave vectors, and might
therefore affect electron-hole (e-h) and phonon dynamics
near the Fermi level [130]. It has been demonstrated that
it is a combination of the nonlocality of the 3D dynamical
screening and the spill out of the 3D electron density into
the vacuum which allows the formation of 2D electron-
density acoustic oscillations at metal surfaces, since these
oscillations would otherwise be completely screened by
the surrounding 3D substrate [131]. This novel surface-
plasmon mode has been observed recently at the (0001)
surface of Be, showing a linear energy dispersion that
is in very good agreement with first-principles calcula-
tions [132].
Finally, we note that metal-dielectric interfaces of ar-
bitrary geometries also support charge density oscilla-
tions similar to the surface plasmons characteristic of
planar interfaces. These are localized Mie plasmons oc-
curring at frequencies which are characteristic of the
interface geometry [133].
plasmons on small particles has attracted great inter-
est over the years in scanning transmission electron mi-
croscopy [134, 135, 136, 137, 138, 139] and near-field
optical spectroscopy [140].
structuring and manipulating on the nanometer scale
have rekindled interest this field [141]. In nanostructured
metals and carbon-based structures, such as fullerenes
and carbon nanotubes, localized plasmons can be ex-
cited by light and can therefore be easily detected as
pronounced optical resonances [142, 143, 144]. Further-
more, very localized dipole and multipole modes in the
vicinity of highly coupled structures are responsible for
surface-enhanced Raman scattering [145, 146] and other
striking properties like, e.g., the blackness of colloidal
silver [147].
Collective electronic excitations in thin adsorbed over-
The excitation of localized
Recently, new advances in
layers, semiconductor heterostructures, and parabolic
quantum wells have also attracted attention over the last
years. The adsorption of thin films is important, because
of the drastic changes that they produce in the electronic
properties of the substrate and also because of related
phenomena such as catalytic promotion [148]; however,
the understanding of adsorbate-induced collective exci-
tations is still incomplete [149, 150, 151, 152, 153, 154,
155, 156]. The excitation spectrum of collective modes in
semiconductor quantum wells has been described by sev-
eral authors [157, 158, 159, 160, 161]. These systems,
which have been grown in semiconductor heterostruc-
tures with the aid of Molecular Beam Epitaxy [162], form
a nearly ideal free-electron gas and have been, therefore,
a playground on which to test existing many-body theo-
ries [163, 164].
Major reviews on the theory of collective elec-
tronic excitations at metal surfaces have been given by
Ritchie [165], Feibelman [166], and Liebsch [167]. Ex-
perimental reviews are also available, which focus on
high-energy EELS experiments [168], surface plasmons
on smooth and rough surfaces and on gratings [169], and
angle-resolved low-energy EELS investigations [170, 171].
An extensive review on plasmons and magnetoplasmons
in semiconductor heterostructures has been given re-
cently by Kushwaha [172].
This review will focus on a unified theoretical descrip-
tion of the many-body dynamical electronic response of
solids, which underlines the existence of various collective
electronic excitations at metal surfaces, such as the con-
ventional surface plasmon, multipole plasmons, and the
acoustic surface plasmon. We also review existing cal-
culations, experimental measurements, and some of the
most recent applications including particle-solid inter-
actions, scanning transmission electron microscopy, and
surface-plasmon based photonics, i.e., plasmonics.
A.Semi-infinite system
1.The surface-plasmon condition
We consider a classical model consisting of two
semi-infinite nonmagnetic media with local (frequency-
dependent) dielectric functions ǫ1and ǫ2separated by a
planar interface at z = 0 (see Fig. 1). The full set of
Maxwell's equations in the absence of external sources
can be expressed as follows [173]
∇ × Hi= ǫi1
∇ × Ei= −1
Page 5
ε 1
ε 2
FIG. 1: Two semi-infinite media with dielectric functions ǫ1
and ǫ2 separated by a planar interface at z = 0.
∇ · (ǫiEi) = 0, (2.3)
∇ · Hi= 0,(2.4)
where the index i describes the media: i = 1 at z < 0,
and i = 2 at z > 0.
Solutions of Eqs. (2.1)-(2.4) can generally be classified
into s-polarized and p-polarized electromagnetic modes,
the electric field E and the magnetic field H being paral-
lel to the interface, respectively. For an ideal surface, if
waves are to be formed that propagate along the interface
there must necessarily be a component of the electric field
normal to the surface. Hence, s-polarized surface oscil-
lations (whose electric field E is parallel to the interface)
do not exist; instead, we seek conditions under which a
traveling wave with the magnetic field H parallel to the
interface (p-polarized wave) may propagate along the sur-
face (z = 0), with the fields tailing off into the positive
(z > 0) and negative (z < 0) directions. Choosing the
x-axis along the propagating direction, we write
Ei= (Eix,0,Eiz)e−κi|z|ei(qix−ωt)
Hi= (0,Eiy,0)e−κi|z|ei(qix−ωt),(2.6)
where qirepresents the magnitude of a wave vector that
is parallel to the surface. Introducing Eqs. (2.5) and (2.6)
into Eqs. (2.1)-(2.4), one finds
iκ1H1y= +ω
iκ2H2y= −ω
cǫ2E2x, (2.8)
i− ǫiω2
The boundary conditions imply that the component
of the electric and magnetic fields parallel to the sur-
face must be continuous. Using Eqs. (2.7) and (2.8), one
writes the following system of equations:
H2y= 0 (2.10)
H1y− H2y= 0,(2.11)
which has a solution only if the determinant is zero, i.e.,
= 0.(2.12)
This is the surface-plasmon condition.
From the boundary conditions also follows the conti-
nuity of the 2D wave vector q entering Eq. (2.9), i.e.,
q1 = q2 = q.Hence, the surface-plasmon condition
[Eq. (2.12)] can also be expressed as follows [174],
q(ω) =ω
ǫ1+ ǫ2,(2.13)
where ω/c represents the magnitude of the light wave
vector. For a metal-dielectric interface with the dielectric
characterized by ǫ2, the solution ω(q) of Eq. (2.13) has
slope equal to c/√ǫ2at the point q = 0 and is a mono-
tonic increasing function of q, which is always smaller
than cq/√ǫ2and for large q is asymptotic to the value
given by the solution of
ǫ1+ ǫ2= 0. (2.14)
This is the nonretarded surface-plasmon condition
[Eq. (2.12) with κ1 = κ2 = q], which is valid as long
as the phase velocity ω/q is much smaller than the speed
of light.
2.Energy dispersion
In the case of a Drude semi-infinite metal in vacuum,
one has ǫ2= 1 and [175]
ǫ1= 1 −
ω(ω + iη),(2.15)
η being a positive infinitesimal.
Eq. (2.13) yields
Hence, in this case
q(ω) =ω
ω2− ω2
2ω2− ω2
We have represented in Fig. 2 by solid lines the dis-
persion relation of Eq. (2.16), together with the light line
ω = cq (dotted line). The upper solid line represents the
Page 6
q (A -1)
ω (eV)
FIG. 2: The solid lines represent the solutions of Eq. (2.16)
with ωp = 15eV: the dispersion of light in the solid (upper
line) and the surface-plasmon polariton (lower line). In the
retarded region (q < ωs/c), the surface-plasmon polariton
dispersion curve approaches the light line ω = cq (dotted
line). At short wave lengths (q >> ωs/c), the surface-plasmon
polariton approaches asymptotically the nonretarded surface-
plasmon frequency ωs = ωp/√2 (dashed line).
field associated with a surface-plasmon polariton propagating
along a metal-dielectric interface. The field strength Ei [see
Eq. (2.5)] decreases exponentially with the distance |z| from
the surface, the decay constant κi being given by Eq. (2.18).
+ and - represent the regions with lower and higher electron
density, respectively.
Schematic representation of the electromagnetic
dispersion of light in the solid. The lower solid line is the
surface-plasmon polariton
ω2(q) = ω2
p/2 + c2q2−
p/4 + c4q4,(2.17)
which in the retarded region (where q < ωs/c) cou-
ples with the free electromagnetic field and in the non-
retarded limit (q >> ωs/c) yields the classical nondis-
persive surface-plasmon frequency ωs= ωp/√2.
We note that the wave vector q entering the dispersion
relation of Eq. (2.17) (lower solid line of Fig. 2) is a 2D
wave vector in the plane of the surface. Hence, if light
hits the surface in an arbitrary direction the external ra-
diation dispersion line will always lie somewhere between
the light line cq and the vertical line, in such a way that it
will not intersect the surface-plasmon polariton line, i.e.,
light incident on an ideal surface cannot excite surface
plasmons. Nevertheless, there are two mechanisms that
q (A-1)
li (A)
FIG. 4: Attenuation length li = 1/κi, versus q, as obtained
from Eq. (2.18) at the surface-plasmon polariton condition
[Eq. (2.17)] for a Drude metal in vacuum. ǫ1 has been taken
to be of the form of Eq. (2.15) with ωp = 15eV and ǫ2 has
been set up to unity. The dotted line represents the large-q
limit of both l1 and l2, i.e., 1/q.
allow external radiation to be coupled to surface-plasmon
polaritons: surface roughness or gratings, which can pro-
vide the requisite momentum via umklapp processes [73],
and attenuated total reflection (ATR) which provides the
external radiation with an imaginary wave vector in the
direction perpendicular to the surface [74, 75].
3.Skin depth
Finally, we look at the spatial extension of the electro-
magnetic field associated with the surface-plasmon po-
lariton (see Fig. 3). Introducing the surface-plasmon con-
dition of Eq. (2.13) into Eq. (2.9) (with q1= q2= q), one
finds the following expression for the surface-plasmon de-
cay constant κiperpendicular to the interface:
ǫ1+ ǫ2,
which allows to define the attenuation length li= 1/κiat
which the electromagnetic field falls to 1/e. Fig. 4 shows
lias a function of the magnitude q of the surface-plasmon
polariton wave vector for a Drude metal [ǫ1of Eq. (2.15)]
in vacuum (ǫ2= 0). In the vacuum side of the interface,
the attenuation length is over the wavelength involved
(l2> 1/q), whereas the attenuation length into the metal
is determined at long-wavelengths (q → 0) by the so-
called skin-depth.At large q [where the nonretarded
surface-plasmon condition of Eq. (2.14) is fulfilled], the
skin depth is li∼ 1/q thereby leading to a strong concen-
tration of the electromagnetic surface-plasmon field near
the interface.
Page 7
B.Thin films
Thin films are also known to support surface collective
oscillations. For this geometry, the electromagnetic fields
of both surfaces interact in such a way that the retarded
surface-plasmon condition of Eq. (2.12) splits into two
new conditions (we only consider nonradiative surface
plasmons), depending on whether electrons in the two
surfaces oscillate in phase or not. In the case of a thin
film of thickness a and dielectric function ǫ1in a medium
of dielectric function ǫ2, one finds [169]:
= 0(2.19)
= 0.(2.20)
Instead, if the film is surrounded by dielectric layers of
dielectric constant ǫ0and equal thickness t on either side,
one finds
κ1ν tanh(κ1a/2)+ǫ0
= 0(2.21)
κ1ν coth(κ1a/2)+ǫ0
= 0, (2.22)
ν =1 − ∆e−2κ0t
1 + ∆e−2κ0t, (2.23)
∆ =κ2ǫ0− κ0ǫ2
κ2ǫ0+ κ0ǫ2
q2− ǫ0ω2
c2. (2.25)
Electron spectrometry measurements of the dispersion
of the surface-plasmon polariton in oxidized Al films were
reported by Pettit, Silcox, and Vincent [176], spanning
the energy range from the short-wavelength limit where
ω ∼ ωp/√2 all the way to the long-wavelength limit
where ω ∼ cq. The agreement between the experimen-
tal measurements and the prediction of Eqs. (2.21)-(2.25)
(with a Drude dielectric function for the Al film and a
dielectric constant ǫ0 = 4 for the surrounding oxide) is
found to be very good, as shown in Fig. 5.
In the nonretarded regime (q >> ωs/c), where κ1 =
κ2= q, Eqs. (2.19) and (2.20) take the form
ǫ1+ ǫ2
ǫ1− ǫ2
= ∓e−qa, (2.26)
00.01 0.020.03
q (A -1)
ω (eV)
FIG. 5: Dispersion ω(q) of the surface-plasmon polariton of
an Al film of thickness a = 120˚ A surrounded by dielectric lay-
ers of equal thickness t = 40˚ A. The solid lines represent the
result obtained from Eqs. (2.21)-(2.25) with ǫ2 = 1, ǫ0 = 4,
and a frequency-dependent Drude dielectric function ǫ1 [see
Eq. (2.15)] with ωp = 15eV and η = 0.75eV [177]. The solid
circles represent the electron spectrometry measurements re-
ported by Petit, Silcox, and Vincent [176]. The dashed line
represents the nonretarded surface-plasmon frequency ωp/√5,
which is the solution of Eq. (2.14) with ǫ2 = 4 and a Drude
dielectric function ǫ1. The dotted line represents the light line
ω = cq.
which for a Drude thin slab [ǫ1of Eq. (2.15)] in vacuum
(ǫ2= 1) yields [1]:
ω =ωp
?1 ± e−qa?1/2.(2.27)
This equation has two limiting cases, as discussed by
Ferrell [69]. At short wavelengths (qa >> 1), the sur-
face waves become decoupled and each surface sustains
independent oscillations at the reduced frequency ωs =
ωp/√2 characteristic of a semi-infinite electron gas with
a single plane boundary. At long wavelengths (qa << 1),
there are normal oscillations at ωpand tangential 2D os-
cillations at
ω2D= (2πnaq)1/2,(2.28)
which were later discussed by Stern [178] and observed
in artificially structured semiconductors [179] and more
recently in a metallic surface-state band on a silicon sur-
face [180].
The classical picture leading to the retarded Eq. (2.12)
and nonretarded Eq. (2.14) ignores both the nonlocality
of the electronic response of the system and the micro-
scopic spatial distribution of the electron density near
Page 8
the surface. This microscopic effects can generally be ig-
nored at long wavelengths where q << qF; however, as
the excitation wavelength approaches atomic dimensions
nonlocal effects can be important.
As nonlocal effects can generally be ignored in the re-
tarded region where q < ωs/c (since ωs/c << qF), here
we focus our attention on the nonretarded regime where
ωs/c < q. In this regime and in the absence of external
sources, the ω-components of the time-dependent electric
and displacement fields associated with collective oscilla-
tions at a metal surface satisfy the quasi-static Maxwell's
∇ · E(r,ω) = −4πδn(r,ω), (3.1)
or, equivalently,
∇2φ(r,ω) = 4π δn(r,ω), (3.2)
∇ · D(r,ω) = 0,(3.3)
δn(r,ω) being the fluctuating electron density associated
with the surface plasmon, and φ(r,ω) being the ω com-
ponent of the time-dependent scalar potential.
A.Planar surface plasmon
1.Classical model
In the classical limit, we consider two semi-infinite me-
dia with local (frequency-dependent) dielectric functions
ǫ1 and ǫ2 separated by a planar interface at z = 0, as
in Section IIA (see Fig. 1). In this case, the fluctuating
electron density δn(r,ω) corresponds to a delta-function
sheet at z = 0:
δn(r,ω) = δn(r?,ω) δ(z),(3.4)
where r?defines the position vector in the surface plane,
and the displacement field D(r,ω) takes the following
D(r,ω) =
z < 0,
z > 0.
Introducing Eq. (3.4) into Eq. (3.2), one finds that self-
sustained solutions of Poisson's equation take the form
φ(r,ω) = φ0eq·r?e−q|z|, (3.6)
where q is a 2D wave vector in the plane of the surface,
and q = |q|. A combination of Eqs. (3.3), (3.5), and
(3.6) with E(r,ω) = −∇φ(r,ω) yields the nonretarded
surface-plasmon condition of Eq. (2.14), i.e.,
ǫ1+ ǫ2= 0. (3.7)
2.Nonlocal corrections
Now we consider a more realistic jellium model of
the solid surface consisting of a fixed semi-infinite uni-
form positive background at z ≤ 0 plus a neutralizing
nonuniform cloud of interacting electrons. Within this
model, there is translational invariance in the plane of
the surface; hence, we can define 2D Fourier transforms
E(z;q,ω) and D(z;q,ω), the most general linear relation
between them being
D(z;q,ω) =
dz′ǫ(z,z′;q,ω) · E(z′;q,ω),(3.8)
where the tensor ǫ(z,z′;q,ω) represents the dielectric
function of the medium.
In order to avoid an explicit calculation of ǫ(z,z′;q,ω),
one can assume that far from the surface and at low wave
vectors (but still in the nonretarded regime, i.e., ωs/c <
q < qF) Eq. (3.8) reduces to an expression of the form of
Eq. (3.5):
D(z;q,ω) =
z < z1,
z > z2,
where z1<< 0 and z2>> 0. Eqs. (3.2) and (3.3) with
E(r,ω) = −∇φ(r,ω) then yield the following integration
of the field components Ez and Dx in terms of the po-
tential φ(z) at z1and z2[where it reduces to the classical
potential of Eq. (3.6)]:
dz Ez(z;q,ω) = φ(z2;q,ω) − φ(z1;q,ω) (3.10)
− i
dz Dx(z;q,ω) = ǫ2φ(z2;q,ω) − ǫ1φ(z1;q,ω).
Neglecting quadratic and higher-order terms in the
wave vector, Eqs. (3.10) and (3.11) are found to be com-
patible under the surface-plasmon condition [88]
ǫ1+ ǫ2
ǫ1− ǫ2
= q?d⊥(ω) − d?(ω)?,(3.12)
d⊥(ω) and d?(ω) being the so-called d-parameters intro-
duced by Feibelman [166]:
d⊥(ω) =
dz zd
dz δn(z,ω)
=dz z δn(z,ω)/
d?(ω) =
dz zd
dzDx(z,ω), (3.14)
where Ez(z,ω), Dx(z,ω), and δn(z,ω) represent the
fields and the induced density evaluated in the q → 0
Page 9
For a Drude semi-infinite metal in vacuum [ǫ2 = 1
and Eq. (2.15) for ǫ1], the nonretarded surface-plasmon
condition of Eq. (3.12) yields the nonretarded dispersion
ω = ωs
where ωsis Ritchie's frequency: ωs= ωp/√2. For neu-
tral jellium surfaces, d?(ω) coincides with the jellium edge
and the linear coefficient of the surface-plasmon disper-
sion ω(q), therefore, only depends on the position d⊥(ωs)
of the centroid of the induced electron density at ωs[see
Eq. (3.13)] with respect to the jellium edge.
?1 − qRe?d⊥(ωs) − d?(ωs)?/2 + ...?, (3.15)
3. Hydrodynamic approximation
In a hydrodynamic model, the collective motion of elec-
trons in an arbitrary inhomogeneous system is expressed
in terms of the electron density n(r,t) and the hydrody-
namical velocity v(r,t), which assuming irrotational flow
we express as the gradient of a velocity potential ψ(r,t)
such that v(r,t) = −∇ψ(r,t). First of all, one writes
the basic hydrodynamic Bloch's equations (the continu-
ity equation and the Bernoulli's equation) in the absence
of external sources [181]:
dtn(r,t) = ∇ · [n(r,t) ∇ψ(r,t)] (3.16)
dtψ(r,t) =1
+ φ(r,t),(3.17)
and Poisson's equation:
∇2φ(r,t) = 4π n(r,t),(3.18)
where G[n] is the internal kinetic energy, which is typi-
cally approximated by the Thomas-Fermi functional
G[n] =
The hydrodynamic equations [Eqs. (3.16)-(3.18)] are
nonlinear equations, difficult to solve.
typically uses perturbation theory to expand the electron
density and the velocity potential as follows
Therefore, one
n(r,t) = n0(r) + n1(r,t) + ...(3.20)
ψ(r,t) = 0 + ψ1(r,t)...,(3.21)
so that Eqs. (3.16)-(3.18) yield the linearized hydrody-
namic equations
dtn1(r,t) = ∇ · [n0(r) ∇ψ1(r,t)], (3.22)
dtψ1(r,t) = [β(r)]2n1(r,t)
+ φ1(r,t), (3.23)
∇2φ1(r,t) = 4π n1(r,t), (3.24)
where n0(r) is the unperturbed electron density and
β(r) =
?1/3?3π2n0(r)?1/3represents the speed of prop-
tem [182].
We now consider a semi-infinite metal in vacuum con-
sisting of an abrupt step of the unperturbed electron den-
sity at the interface, which we choose to be located at
z = 0:
?¯ n,
agation of hydrodynamic disturbances in the electron sys-
n0(z) =
z ≤ 0,
z > 0.
Hence, within this model n0(r) and β(r) are constant at
z ≤ 0 and vanish at z > 0.
Introducing Fourier transforms, Eqs. (3.22)-(3.25)
yield the basic differential equation for the plasma nor-
mal modes at z ≤ 0:
∇2(ω2− ω2
and Laplace's equation at z > 0:
∇2φ1(r,ω) = 0
where both n1(r,ω) and ψ1(r,ω) vanish. Furthermore,
translational invariance in the plane of the surface allows
to introduce the 2D Fourier transform ψ1(z;q,ω), which
according to Eq. (3.26) must satisfy the following equa-
tion at z ≤ 0:
(−q2+d2/dz2)?ω2− ω2
where q represents a 2D wave vector in the plane of the
Now we need to specify the boundary conditions. Rul-
ing out exponential increase at z → ∞ and noting that
the normal component of the hydrodynamical velocity
should vanish at the interface, for each value of q one
finds solutions to Eq. (3.28) with frequencies [183, 184]
ω2≥ ω2
Eqs. (3.29) and (3.30) represent a continuum of bulk nor-
mal modes and a surface normal mode, respectively. At
long wavelengths, where β q/ωp << 1 (but still in the
nonretarded regime where ωs/c < q), Eq. (3.30) yields
the surface-plasmon dispersion relation
ω = ωp/√2 + β q/2,
p+ β2∇2)ψ1(r,ω)(z ≤ 0), (3.26)
(z > 0),(3.27)
p− β(−q2+ d2/dz2)?ψ1(z;q,ω),
p+ β2q2
p+ β2q2+ βq
p+ β2q2?
. (3.30)
which was first derived by Ritchie [83] using Bloch's
equations, and later by Wagner [185] and by Ritchie
and Marusak [186] by assuming, within a Boltzmann
transport-equation approach, specular reflection at the
Page 10
B.Localized surface plasmons: classical approach
Metal-dielectric interfaces of arbitrary geometries also
support charge density oscillations similar to the surface
plasmons characteristic of planar interfaces. In the long-
wavelength (or classical) limit, in which the interface sep-
arates two media with local (frequency-dependent) di-
electric functions ǫ1and ǫ2, one writes
Di(r,ω) = ǫiEi(r,ω),(3.32)
where the index i refers to the media 1 and 2 separated
by the interface. In the case of simple geometries, such as
spherical and cylindrical interfaces, Eqs. (3.1)-(3.3) can
be solved explicitly with the aid of Eq. (3.32) to find
explicit expressions for the nonretarded surface-plasmon
1.Simple geometries
a.Spherical interface.
electric function ǫ1in a host medium of dielectric func-
tion ǫ2, the classical (long-wavelength) planar surface-
plasmon condition of Eq. (3.7) is easily found to be re-
placed by [133]
In the case of a sphere of di-
lǫ1+ (l + 1)ǫ2= 0,l = 1,2,...,(3.33)
which in the case of a Drude metal sphere [ǫ1 of
Eq. (2.15)] in vacuum (ǫ2= 1) yields the Mie plasmons
at frequencies
ωl= ωp
2l + 1. (3.34)
b.Cylindrical interface.
long cylinder of dielectric function ǫ1in a host medium
of dielectric function ǫ2, the classical (long-wavelength)
surface-plasmon condition depends on the direction of
the electric field.For electromagnetic waves with the
electric field normal to the interface (p-polarization),
the corresponding long-wavelength (and nonretarded)
surface-plasmon condition coincides with that of a planar
surface, i.e., [187, 188, 189]
In the case of an infinitely
ǫ1+ ǫ2= 0, (3.35)
which for Drude cylinders [ǫ1 of Eq. (2.15)] in vacuum
(ǫ2 = 1) yields the planar surface-plasmon frequency
ωs= ωp/√2.
For electromagnetic waves with the electric field par-
allel to the axis of the cylinder (s-polarization), the pres-
ence of the interface does not modify the electric field
and one easily finds that only the bulk mode of the host
medium is present, i.e., one finds the plasmon condition
ǫ2= 0.(3.36)
In some situations, instead of having one single cylinder
in a host medium, an array of parallel cylinders may be
present with a filling fraction f. In this case and for
electromagnetic waves polarized along the cylinders (s-
polarization), the plasmon condition of Eq. (3.36) must
be replaced by [190]
f ǫ1+ (1 − f)ǫ2= 0, (3.37)
which for Drude cylinders [ǫ1 of Eq. (2.15)] in vac-
uum (ǫ2 = 1) yields the reduced plasmon frequency
ω =√f ωp.
2. Boundary-charge method
In the case of more complex interfaces, a so-called
boundary-charge method (BCM) has been used by sev-
eral authors to determine numerically the classical (long-
wavelength) frequencies of localized surface plasmons. In
this approach, one first considers the ω-component of the
time-dependent surface charge density arising from the
difference between the normal components of the electric
fields inside and outside the surface:
σs(r,ω) =
4π[E(r,ω) · n|r=r− + E(r,ω) · n|r=r+],
which noting that the normal component of the displace-
ment vector [see Eq. (3.32)] must be continuous yields
the following expression:
σs(r,ω) =
ǫ1− ǫ2
E(r,ω) · n|r=r+,(3.39)
where n represents a unit vector in the direction perpen-
dicular to the interface.
An explicit expression for the normal component of the
electric field at a point of medium 2 that is infinitely close
to the interface (r = r+) can be obtained with the use of
Gauss' theorem. One finds:
E(r,ω) · n|r=r+ = −n · ∇φ(r,ω) + 2πσs(r,ω),
where φ(r,ω) represents the scalar potential. In the ab-
sence of external sources, this potential is entirely due to
the surface charge density itself:
φ(r,ω) =
|r − r′|.(3.41)
Combining Eqs. (3.39)-(3.41), one finds the following in-
tegral equation:
2πǫ1+ ǫ2
ǫ1− ǫ2σs(r,ω) −
d2r′ r − r′
|r − r′|3· nσs(r′,ω) = 0,
which describes the self-sustained oscillations of the sys-
The boundary-charge method has been used by sev-
eral authors to determine the normal-mode frequen-
cies of a cube [191, 192] and of bodies of arbitrary
shape [193, 194]. More recent applications of this method
Page 11
include investigations of the surface modes of channels
cut on planar surfaces [195], the surface modes of coupled
parallel wires [196], and the electron energy loss near in-
homogeneous dielectrics [197, 198]. A generalization of
this procedure that includes relativistic corrections has
been reported as well [199].
3.Composite systems: effective-medium approach
Composite systems with a large number of inter-
faces can often be replaced by an effective homogeneous
medium that in the long-wavelength limit is charac-
terized by a local effective dielectric function ǫeff(ω).
Bergman [200] and Milton [201] showed that in the
case of a two-component system with local (frequency-
dependent) dielectric functions ǫ1 and ǫ2 and volume
fractions f and 1−f, respectively, the long-wavelengthef-
fective dielectric function of the system can be expressed
as a sum of simple poles that only depend on the mi-
crogeometry of the composite material and not on the
dielectric functions of the components:
ǫeff(ω) = ǫ2
1 − f
u − mν
where u is the spectral variable
u = [1 − ǫ1/ǫ2]−1, (3.44)
mν are depolarization factors, and Bν are the strengths
of the corresponding normal modes, which all add up to
Bν= 1.(3.45)
eff(ω) = ǫ−1
1 + f
u − nν
Cν= 1.(3.47)
The optical absorption and the long-wavelength energy
loss of moving charged particles are known to be dic-
tated by the poles of the local effective dielectric function
ǫeff(ω) and inverse dielectric function ǫ−1
tively. If there is one single interface, these poles are
known to coincide.
In particular, in the case of a two-component isotropic
system composed of identical inclusions of dielectric func-
tion ǫ1 in a host medium of dielectric function ǫ2, the
effective dielectric function ǫeff(ω) can be obtained from
the following relation:
eff(ω), respec-
(ǫeff− ǫ2)E = f(ǫ1− ǫ2)Ein, (3.48)
where E is the macroscopic electric field averaged over
the composite:
E = fEin+ (1 − f)Eout,(3.49)
Einand Eoutrepresenting the average electric field inside
and outside the inclusions, respectively [202].
a.Simple geometries.If there is only one mode with
strength different from zero, as occurs (in the long-
wavelength limit) in the case of one single sphere or cylin-
der in a host medium, Eqs. (3.43) and (3.46) yield
ǫeff(ω) = ǫ2
1 − f
u − m
eff(ω) = ǫ−1
1 + f
u − n
, (3.51)
normal modes occurring, therefore, at the frequencies
dictated by the following conditions:
mǫ1+ (1 − m)ǫ2= 0 (3.52)
nǫ1+ (1 − n)ǫ2= 0.(3.53)
For Drude particles [ǫ1of Eq. (2.15)] in vacuum (ǫ2= 1),
these frequencies are easily found to be ω =√mωpand
ω =√nωp, respectively.
Indeed, for a single 3D spherical or 2D circular [203]
inclusion in a host medium, an elementary analysis shows
that the electric field Einin the interior of the inclusion
u − mE,(3.54)
where m = 1/D, D representing the dimensionality of
the inclusions, i.e., D = 3 for spheres and D = 2 for
cylinders.Introduction of Eq. (3.54) into Eq. (3.48)
leads to an effective dielectric function of the form of
Eq. (3.50) with m = 1/D, which yields [see Eq. (3.52)]
the surface-plasmoncondition dictated by Eq. (3.33) with
l = 1 in the case of spheres (D = 3) and the surface-
plasmon condition of Eq. (3.35) in the case of cylinders
(D = 2). This result indicates that in the nonretarded
long-wavelength limit (which holds for wave vectors q
such that ωsa/c < q a << 1, a being the radius of the in-
clusions) both the absorption of light and the energy-loss
spectrum of a single 3D spherical or 2D circular inclusion
exhibit one single strong maximum at the dipole reso-
nance where ǫ1+ 2ǫ2= 0 and ǫ1+ ǫ2= 0, respectively,
which for a Drude sphere and cylinder [ǫ1of Eq. (2.15)]
in vacuum (ǫ2= 1) yield ω = ωp/√3 and ω = ωp/√2.
In the case of electromagnetic waves polarized along
one single cylinder or array of parallel cylinders (s-
polarization), the effective dielectric function of the com-
posite is simply the average of the dielectric functions of
its constituents, i.e.:
ǫeff= f ǫ1+ (1 − f)ǫ2, (3.55)
Page 12
which can also be written in the form of Eqs. (3.50)
and (3.51), but now with m = 0 and n = f, respec-
tively.Hence, for this polarization the absorption of
light exhibits no maxima (in the case of a dielectric host
medium with constant dielectric function ǫ2) and the
long-wavelength energy-loss spectrum exhibits a strong
maximum at frequencies dictated by the plasmon condi-
tion of Eq. (3.37), which in the case of Drude cylinders
[ǫ1of Eq. (2.15)] in vacuum (ǫ2= 1) yields the reduced
plasmon frequency ω =√f ωp.
b.Maxwell-Garnett approximation.
among spherical (or circular) inclusions in a host medium
can be introduced approximately in the framework of the
well-known Maxwell-Garnett (MG) approximation [133].
The basic assumption of this approach is that the av-
erage electric field Einwithin a particle located in a sys-
tem of identical particles is related to the average field
Eout in the medium outside as in the case of a single
isolated (noninteracting) particle, thereby only dipole in-
teractions being taken into account. Hence, in this ap-
proach the electric field Einis taken to be of the form of
Eq. (3.54) but with the macroscopic electric field E re-
placed by the electric field Eoutoutside, which together
with Eqs. (3.48) and (3.49) yields the effective dielec-
tric function and effective inverse dielectric function of
Eqs. (3.50) and (3.51) with the depolarization factors
m = n = 1/D (corresponding to the dilute limit, where
f → 0) replaced by
D(1 − f)
The interaction
m =
n =1
D[1 + (D − 1)f].(3.57)
4.Periodic structures
Over the years, theoretical studies of the normal modes
of complex composite systems had been generally re-
stricted to mean-field theories of the Maxwell-Garnett
type, which approximately account for the behavior of
localized dipole plasmons [133].
ber of methods have been developed recently for a
full solution of Maxwell's equations in periodic struc-
tures [204, 205, 206, 207, 208, 209]. The transfer matrix
method has been used to determine the normal-mode
frequencies of a lattice of metallic cylinders [210] and
rods [211], a so-called on-shell method has been employed
by Yannopapas et al. to investigate the plasmon modes
of a lattice of metallic spheres in the low filling fraction
regime [212], and a finite difference time domain (FDTD)
scheme has been adapted to extract the effective response
of metallic structures [209].
Most recently, an embedding method [207] has been
employed to solve Maxwell's equations, which has al-
lowed to calculate the photonic band structure of three-
and two-dimensional lattices of nanoscale metal spheres
Nevertheless, a num-
plasma and vacuum are interchanged. The top panel rep-
resents the general situation. The bottom panel represents a
half-space filled with metal and interfaced with vacuum. The
surface-mode frequencies ωs1and ωs2of these systems fulfill
the sum rule of Eq. (3.58).
Complementary systems in which the regions of
and cylinders in the frequency range of the Mie plas-
mons [147]. For small filling fractions, there is a surface-
plasmon polariton which in the non-retarded region
yields the non-dispersive Mie plasmon with frequency
ωp/√D. As the filling fraction increases, a continuum of
plasmon modes is found to exist between zero frequency
and the bulk metal plasmon frequency [147], which yield
strong absorption of incident light and whose energies
can be tuned according to the particle-particle separa-
tion [213].
5.Sum rules
Sum rules have played a key role in providing insight
in the investigation of a variety of physical situations. A
useful sum rule for the surface modes in complementary
media with arbitrary geometry was introduced by Apell
et al. [214], which in the special case of a metal/vacuum
interface implies that [215]
s1+ ω2
s2= ω2
p, (3.58)
where ωs1is the surface-mode frequency of a given sys-
tem, and ωs2represents the surface mode of a second
complementary system in which the regions of plasma
and vacuum are interchanged (see Fig. 6).
For example, a half-space filled with a metal of bulk
plasma frequency ωp and interfaced with vacuum maps
into itself (see bottom panel of Fig. 6), and therefore
Eq. (3.58) yields
ωs1= ωs2= ωp/√2, (3.59)
which is Ritchie's frequency of plasma oscillations at a
metal/vacuum planar interface.
Other examples are a Drude metal sphere in vac-
uum, which sustains localized Mie plasmons at frequen-
cies given by Eq. (3.34), and a spherical void in a Drude
Page 13
metal, which shows Mie plasmons at frequencies
ωl= ωp
l + 1
2l + 1.(3.60)
The squared surface-mode frequencies of the sphere
[Eq. (3.34)] and the void [(Eq. (3.60)] add up to ω2
all l, as required by Eq. (3.58).
The splitting of surface modes that occurs in thin films
due to the coupling of the electromagnetic fields in the
two surfaces [see Eq. (2.26)] also occurs in the case of
localized modes. Apell et al. [214] proved a second sum
rule, which relates the surface modes corresponding to
the in-phase and out-of-phase linear combinations of the
screening charge densities at the interfaces. In the case
of metal/vacuum interfaces this sum rule takes the form
of Eq. (3.58), but now ωs1and ωs2being in-phase and
out-of-phase modes of the same system.
For a Drude metal film with equal and abrupt planar
surfaces, the actual values of the nonretarded ωs1and ωs2
are those given by Eq. (2.27), which fulfill the sum rule
dictated by Eq. (3.58). For a spherical fullerene molecule
described by assigning a Drude dielectric function to ev-
ery point between the inner and outer surfaces of radii
r1and r2, one finds the following frequencies for the in-
phase and out-of-phase surface modes [216]:
1 ±
2l + 1
1 + 4l(l + 1)(r1/r2)2l+1
also fulfilling the sum rule of Eq. (3.58).
Another sum rule has been reported recently [210, 211],
which relates the frequencies of the modes that can be
excited by light [as dictated by the poles of the effec-
tive dielectric function of Eq. (3.43)] and those modes
that can be excited by moving charged particles [as dic-
tated by the poles of the effective inverse dielectric func-
tion of Eq. (3.46)]. Numerical calculations for various
geometries have shown that the depolarization factors
mν and nν entering Eqs. (3.43) and (3.46) satisfy the
relation [210, 211]
nν= 1 − (D − 1)mν, (3.62)
where D represents the dimensionality of the inclusions.
Furthermore, combining Eqs. (3.50) and (3.51) (and
assuming, therefore, that only dipole interactions are
present) with the sum rule of Eq. (3.62) yields Eqs. (3.56)
and (3.57), i.e., the MG approximation. Conversely, as
long as multipolar modes contribute to the spectral rep-
resentation of the effective response [see Eqs. (3.43) and
(3.46)], the strength of the dipolar modes decreases [see
Eqs. (3.45) and (3.47)] and a combination of Eqs. (3.43)
and (3.46) with Eq. (3.62) leads to the conclusion that the
dipolar resonances must necessarily deviate from their
MG counterparts dictated by Eqs. (3.56) and (3.57).
That a nonvanishing contribution from multipolar modes
appears together with a deviation of the frequencies of
the dipolar modes with respect to their MG counterparts
was shown explicitly in Ref. [210].
The dynamical structure factor S(r,r′;ω) represents a
key quantity in the description of both single-particle and
collective electronic excitations in a many-electron sys-
tem [217]. The rate for the generation of electronic exci-
tations by an external potential, the inelastic differential
cross section for external particles to scatter in a given di-
rection, the inelastic lifetime of excited hot electrons, the
so-called stopping power of a many-electron system for
moving charged particles, and the ground-state energy
of an arbitrary many-electron system (which is involved
in, e.g., the surface energy and the understanding of Van
der Waals interactions) are all related to the dynamical
structure factor of the system.
The dynamical structure factor, which accounts for the
particle-density fluctuations of the system, is defined as
S(r,r′;ω) =
δˆ ρ0n(r1)δˆ ρn0(r2)δ(ω−En+E0). (4.1)
Here, δˆ ρn0(r) represent matrix elements, taken between
the many-particle ground state |Ψ0? of energy E0 and
the many-particle excited state |Ψn? of energy En, of the
operator ˆ ρ(r) − n0(r), where ˆ ρ(r) is the electron-density
operator [218]
ˆ ρ(r) =
ˆδ(r − ri),(4.2)
withˆδ and ri describing the Dirac-delta operator and
electron coordinates, respectively, and n0(r) represents
the ground-state electron density, i.e.:
n0(r) =< Ψ0|ˆ ρ(r)|Ψ0> . (4.3)
The many-body ground and excited states of a many-
electron system are unknown and the dynamical struc-
ture factor is, therefore, difficult to calculate.
theless, one can use the zero-temperature limit of the
fluctuation-dissipation theorem [219], which relates the
dynamical structure factor S(r,r′;ω) to the dynami-
cal density-response function χ(r,r′;ω) of linear-response
theory. One writes,
S(r,r′;ω) = −Ω
where Ω represents the normalization volume and θ(x) is
the Heaviside step function.
Take a system of N interacting electrons exposed to a
frequency-dependent external potential φext(r,ω). Keep-
ing terms of first order in the external perturbation and
Page 14
neglecting retardation effects, time-dependent perturba-
tion theory yields the following expression for the induced
electron density [218]:
δn(r,ω) =
where χ(r,r′;ω) represents the so-called density-response
function of the many-electron system:
χ(r,r′;ω) =
E0− En+ ¯ h(ω + iη)
E0+ En+ ¯ h(ω + iη)
η being a positive infinitesimal.
The imaginary part of the true density-response func-
tion of Eq. (5.2), which accounts for the creation of both
collective and single-particle excitations in the many-
electron system, is known to satisfy the so-called f-sum
dωω Imχ(r,r′;ω) = −π∇·∇′[n0(r)δ(r,r′)], (5.3)
with n0(r) being the unperturbed ground-state electron
density of Eq. (4.3).
A. Random-phase approximation (RPA)
In the so-called random-phase or, equivalently, time-
dependent Hartree approximation, the electron density
δn(r,ω) induced in an interacting electron system by a
small external potential φext(r,ω) is obtained as the elec-
tron density induced in a noninteracting Hartree system
(of electrons moving in a self-consistent Hartree poten-
tial) by both the external potential φext(r,ω) and the
induced potential
δφH(r,ω) =
dr′v(r,r′) δn(r′,ω),(5.4)
with v(r,r′) representing the bare Coulomb interaction.
Hence, in this approximation one writes
δn(r,ω) =
φext(r′,ω) +
, (5.5)
which together with Eq. (5.1) yields the following Dyson-
type equation for the interacting density-response func-
χ(r,r′;ω) = χ0(r,r′;ω) +
× v(r1,r2)χ(r2,r′;ω),(5.6)
where χ0(r,r′;ω) denotes the density-response function
of noninteracting Hartree electrons:
χ0(r,r′;ω) =
(fi− fj)
ω − εj+ εi+ iη
Here, fi are Fermi-Dirac occupation factors, which at
zero temperature take the form fi= θ(εF−εi), εF being
the Fermi energy, and the single-particle states and ener-
gies ψi(r) and εiare the eigenfunctions and eigenvalues
of a Hartree Hamiltonian, i.e.:
2∇2+ vH[n0](r)
ψi(r) = εiψi(r),(5.8)
vH[n0](r) = v0(r) +
with v0(r) denoting a static external potential and n0(r)
being the unperturbed Hartree electron density:
n0(r) =
B.Time-dependent density-functional theory
In the framework of time-dependent density-functional
theory (TDDFT) [220], the exact density-response func-
tion of an interacting many-electron system is found to
obey the following Dyson-type equation:
χ(r,r′;ω) = χ0(r,r′;ω) +
×{v(r1,r2) + fxc[n0](r1,r2;ω)}χ(r2,r′;ω).
Here,the noninteracting
χ0(r,r′;ω) is of the form of Eq. (5.7), but with the single-
particle states and energies ψi(r) and εi being now the
eigenfunctions and eigenvalues of the Kohn-Sham Hamil-
tonian of DFT, i.e:
2∇2+ vKS[n0](r)
ψi(r) = εiψi(r), (5.12)
vKS[n0](r) = vH[n0](r) + vxc[n0](r),(5.13)
vxc[n0](r) =δExc[n]
Exc[n] represents the unknown XC energy functional and
n0(r) denotes the exact unperturbed electron density of
Page 15
Eq. (4.3), which DFT shows to coincide with that of
Eq. (5.10) but with the Hartree eigenfunctions ψi(r) of
Eq. (5.8) being replaced by their Kohn-Sham counter-
parts of Eq. (5.12). The xc kernel fxc[n0](r,r′;ω) denotes
the Fourier transform of
fxc[n0](r,t;r′;t′) =δvxc[n](r,t)
with vxc[n](r,t) being the exact time-dependent xc po-
tential of TDDFT.
If short-range XC effects are ignored altogether by set-
ting the unknown XC potential vxc[n0](r) and XC ker-
nel fxc[n0](r,r′;ω) equal to zero, the TDDFT density-
response function of Eq. (5.11) reduces to the RPA
Eq. (5.6).
1.The XC kernel
Along the years, several approximations have been
used to evaluate the unknown XC kernel of Eq. (5.15).
a.Random-phase approximation
days, one usually refers to the RPA as the result of
simply setting the XC kernel fxc[n0](r,r′;ω) equal to
[n0](r,r′;ω) = 0, (5.16)
but still using in Eqs. (5.7) and (5.10) the full single-
particle states and energies ψi(r) and εiof DFT [i.e., the
solutions of Eq. (5.12)] with vxc[n0](r) set different from
zero. This is sometimes called DFT-based RPA.
b. Adiabatic local-density approximation (ALDA).
In this approximation, also called time-dependent local-
density approximation (TDLDA) [221], one assumes that
both the unperturbed n0(r) and the induced δn(r,ω)
electron densities vary slowly in space and time and,
therefore, one replaces the dynamical XC kernel by the
long-wavelength (Q → 0) limit of the static XC kernel of
a homogeneous electron gas at the local density:
[n0](r,r′;ω) =d2[nεxc(n)]
δ(r − r′),
where εxc(n) is the XC energy per particle of a homoge-
neous electron gas of density n.
c. PGG and BPG. In the spirit of the optimized
effective-potential method [222], Petersilka, Gossmann,
and Gross (PGG) [223] derived the following frequency-
independent exchange-only approximation for inhomoge-
neous systems:
[n0](r,r′;ω) = −
|r − r′|
where ψi(r) denote the solutions of the Kohn-Sham
Eq. (5.12).
(BPG) [224] devised a hybrid formula for the XC
kernel, which combines expressions for symmetric and
antisymmetric spin orientations from the exchange-only
PGG scheme and the ALDA. For an unpolarized
many-electron system, one writes [224]:
recently, Burke, Petersilka,andGross
[n0](r,r′;ω) =1
+ f↑↓,LDA
?, (5.19)
where f↑↑
with parallel and antiparallel spin, respectively.
short-range XC effects in solids has been focused in a
great extent onto the simplest possible many-electron
system, which is the homogeneous electron gas. Hence,
recent attempts to account for XC effects in inhomo-
geneous systems have adopted the following approxima-
tion [225, 226]:
xcand f↑↓
xcrepresent the XC kernel for electrons
xc[n0](r,r′;ω) = fhom
(˜ n;|r − r′|;ω), (5.20)
where ˜ n represents a function of the electron densities at
points r and r′, typically the arithmetical average
˜ n =1
2[n0(r) + n0(r′)],(5.21)
and fhom
mogeneous electron gas of density ˜ n, whose 3D Fourier
transform fhom
(˜ n;Q,ω) is directly connected to the so-
called local-field factor G(˜ n;Q,ω):
(˜ n;|r − r′|;ω) denotes the XC kernel of a ho-
(˜ n;Q,ω) = −4π
Q2G(˜ n;Q,ω).(5.22)
In the ALDA, one writes
GALDA(˜ n;Q,ω) = G(˜ n;Q → 0,ω = 0)
n=˜ n
which in combination with Eqs. (5.20)-(5.22) yields the
ALDA XC kernel of Eq. (5.17). However, more accu-
rate nonlocal dynamical expressions for the local-field
factor G(˜ n;Q,ω) are available nowdays, which together
with Eqs. (5.20)-(5.22) should yield an accurate (beyond
the ALDA) representation of the XC kernel of inhomo-
geneous systems.
During the last decades,
into the determination of the static local-field factor
Gstatic(˜ n;Q) = G(˜ n;Q,ω = 0) [227, 228, 229, 230, 231,
232, 233, 234], the most recent works including diffu-
sion Monte Carlo (DMC) calculations [235, 236] and the
parametrization of the DMC data of Ref. [236] given by
Corradini et al. [237]:
much effort has gone
Gstatic(˜ n;Q) = CˆQ2+ BˆQ2/(g +ˆQ2) + αˆQ4e−βˆ Q2,


text:
Pharoahe Monch - Get Down
Artist: Pharoahe Monch
Featuring: DJ Revolution
Producer: M-Phazes
Album: Unearthed (Mixed by DJ Revolution)
Avg Rating: 4.0 ( 3 votes )
"I don't want to hear anymore new Pharoahe Monch tracks," said no one ever. Fresh off the release of his new album, P.T.S.D., the "walking over water and drinking 40's of acid" emcee is back with a feature on the upcoming Unearthed compilation album (out 5/13) from Brooklyn-based indie label Coalmine Records. The signature boom bap production from M-Phazes is just what the doctor ordered with 500 cc's of Pharoahe's unique, metered knowledge. You can purchase Get Down on a limited edition 45 vinyl, which was part of Record Store Day from Fat Beats, UGHH, or Turntable Lab. Today's scratches were brought to you by DJ Revolution of The World Famous Wake Up Show.
Pharoahe MonchDJ Revolution IndieLyrical
Written by Bryan H on 04/22/14
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First DJ Booth Appearance:
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