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Q: Conversion from strange template type I have a number of vectors (the maths type) implemented as structs. There is a base Vector, which is a template, and then multiple implementations of the this template for vectors of different dimensions:
template<unsigned int D>
struct Vector
{
    float values[D];

    inline Vector<D> operator*(const Vector<D>& v) const
    {
        Vector<D> result;

        for (unsigned int i = 0; i < D; i++)
            result[i] = values[i] * v[i];

        return result;
    }

    float operator[](unsigned int i) const
    {
        return values[i];
    }
};

I have just included a single operator, but there are obviously others and various methods for taking the dot product etc. I then have implementations like so:
struct Vector2 : Vector<2>
{
    Vector2(float x = 0, float y = 0) : Vector<2>()
    { }
};

I then have two Vector2s that I attempt to use the multiplication operator on like: Vector2 textureOffsetPixels = textureOffset * textureSampleSize;
And it throws the error: conversion from Vector<2u> to non-scalar type 'Vector2' requested. What is up with the Vector<2u> and why does this not work? Both variables are of explicit type Vector2.

A: In order to be able to convert the result of textureOffset * textureSampleSize, which is a Vector<2>, into a Vector2, you need a conversion function. A constructor like this will suffice:
Vector2(const Vector<2> &base) : Vector<2>(base) {}

The error mentions Vector<2u> because the template parameter is an unsigned int, so 2 is implicitly converted to 2u.

A: Your operator * returns Vector<2>, not Vector2. The following will work:
Vector<2> a, b, c;
c = a * b;

redpajama_set_name: RedPajamaStackExchange


DELLA Mounted BTU Mini Air Conditioner 115V Remote Control Up NOW! TAX prepared to take the heat with the 8,000 BTU air conditioner. for cooling a to 350 square coveted cooling speeds, 3 fan speeds, and an AC for rooms up to 350 square feet Remote to the temperature and from Display monitor settings added visibility.Antibacterial with side, access cleans air, removing fan and compressor off temporarily when room cool saving time moneyLow power start up conserves and will help down the keeps you cool keeping you awake.Built in Dry Mode.


When thinking in tours or things to do in Cancun you got to consider Tulum archeological ruins. Tulum pyramid is especially unique since it's located on a 12 meter (36 ft.) cliff with the sea at it's feet. All of our Tulum Tours includes air conditioned transportation, entrance fee, guided tour, buffet meal and of course free time to take pictures and hang around. Discover why this tour is now one of Cancun's main attractions. Book now!
If you want to visit Tulum but you have few time, this is for you. Get to know one of the greatest Mayan temples with Tulum Express, this tour will take you from your hotel to the magestic Mayan ruins of Tulum in an air conditioned coach to make your trip very pleasant. A certified bilingual guide will explain you about the ruins while you enjoy the only Mayan settlement next to the sea. Free time for pictures and to take a dip on Tulum beach. Bookn now!
Explore the mythical Tulum mayan ruins and visit the most trendy city in Riviera Maya, Playa del Carmen on this incredible 1 day tour. Discover the cultural richness of Tulum on a guided tour by our english speaking certified guide and walking tour on the iconic Playa del Carmen's Quinta Avenida (full of boutique stores,restaurants, cofee shops, handicraft stores, etc) the best place for shopping and ideal for fashion lovers. Of course roundtrip transportation from your hotel is inlcuded, book now!
Roundtrip air conditioned transportation from your hotels lobby.
Entrance ticket fee on Tulum archeological site.
Guided tour in Tulum mayan ruins.
Free time for photos and dip in Tulum's beautiful beach.
Freetime for shopping, photos, etc.
Please show printed voucher at pick up time.


text:
ACCEPTED

#### According to
International Plant Names Index

#### Published in
null

#### Original name
Acacia loretensis J.F.Macbr.

### Remarks
null

redpajama_set_name: RedPajamaGithub


text:
M5 horror crash: Picture shows air ambulance landing at scene of fatal crash near Wellington
Two air ambulances have landed on the motorway between J26 A38 (Wellington) and J25 A358 (Taunton)
James WoodReporter
Two air ambulances have landed at the scene of a horror crash on the M5 in which a person is said to have died.
Somerset Live digital editor Thomas Cock is live at the scene. He's photographed an air ambulance landing at the scene of the crash.
The motorway has been closed between J26 A38 (Wellington) and J25 A358 (Taunton) after a crash reportedly involving four cars.
One person is confirmed to have died in the horror crash and reports suggest that two air ambulances have landed at the scene.
An air ambulance lands at the scene of a horror crash near Wellington
Great Western Air Ambulance Charity has landed at the scene.
A spokeswoman told Somerset Live: "Great Western Air Ambulance Charity attended an incident on the M5 this morning. The patient was treated by our critical care team and taken in the helicopter to Southmead Hospital."
A large number of emergency service vehicles can be seen on the motorway and huge lines of traffic can also be seen along the route.
M5 traffic delays after crash leads to death of Elaina Beard
Woman killed was 'passionate' carer
Motorway reopens
Police finish investigation
Why closures last so long
Details of horror crash
Tributes to Elaina
Balloon tribute planned
Police think Elaina got out of car
Thomas Cock, who is live at the scene, said: "The crash is about a mile south of J25. There is a large presence by emergency services and highways England.
"Both carriageways are closed.
"About half a mile of stuck traffic on the northbound carriageway is being turned around from the back of the queue and being sent the wrong way down the northbound carriageway towards junction 26.
"Traffic on local roads around Taunton is extremely heavy and difficult.
"A van load of screening barriers has just driven up the road towards the crash scene."
M5 horror crash: Pictures show scene of crash in which person died
South Western Ambulance Service has confirmed it is at the scene.
A spokesman said: "Just before 6.30am we were called to reports of a serious RTC on the M5 near Wellington.
"We responded with 15 resources including six ambulances, four rapid response vehicles, two officers, two air ambulances and one BASICS doctor. There are 10 patients in total.
"The carriageway is likely to be closed whilst emergency services are at the scene, please avoid the area."
A bridge over the M5 near Wellington has also been closed as a result of the crash
Avon and Somerset Constabulary is also at the scene.
A spokesman said: "The M5 remains closed in both directions between junctions 25 and 26 following a fatal collision this morning.
"Sadly, one person has died in the collision, which happened at just before 6.20am today.
"Several vehicles were involved and the northbound and southbound carriageways were both affected.
"Four people have suffered injuries and have been taken to hospital.
"We expect the motorway to be closed for a significant amount of time so would urge motorists to avoid the area and make sure they follow diversions put in place by Highways England. If your journey isn't necessary, please consider delaying it until later in the day."
Are you at the scene? Can you see anything? Let us know. Get in touch using james.wood@westgaz.co.uk , call 07870 505 906 or tweet me @JamesWood24

redpajama_set_name: RedPajamaCommonCrawl


Rolling Hills offers outstanding junior activities and programs for all ages. We host a camping overnight, movie night under the stars, end of school year pool party, teen pizza nights, holiday "splashtaculars", parent/child dances, a fall festival, brunch with Santa, gingerbread house decorating party, and much, much more! The club has a very popular summer camp for children ages 3-11 that offers golf, tennis, paddle and swimming lessons as well as crafts, guest speakers and lunch for 8 weeks.


Chaos is so 2018. Get professionally organized now!
Hello! I hope 2019 is treating you well so far! Here at the Wicker Homes Group, one of our goals for 2019 is to GET ORGANIZED both professionally and personally. Perhaps you have the same goal? In case you do, we wanted to share some information with you.
Recently, at a Keller Williams meeting, we were introduced to Organizing Maniacs. They are a group of "Certified Professional Organizers (CPO®), Certified Professional Organizers in Chronic Disorganization (CPO-CD®) and Productivity Consultants".
What does this mean? They help you sort your possessions, make decisions, organize what is being kept, and environmentally discard the unwanted items. The goal of Organizing Maniacs is to help clients "finally get organized and to feel more confident in their homes so they can stop surviving and start thriving instead."
We hope this was helpful for you! Our team at the Wicker Homes Group loves to be a resource for our community both in life and in "all things Real Estate". Let us know if you or someone you know is looking to buy or sell a home this year. With over 30 years of experience, we are experts in the area and would love to be your chosen professional.


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Q: Custom table or array with multiple columns exported to HTML I am having a hard time creating a table that has multiple columns. The columns will be populated using various cmdlets. At the very end of a script, I want to run a final check and write the hostname to a file, along with various other columns. Here is what I was thinking for the headings:
"HOSTNAME", "FILES REPLACED?", "SCCM UNINSTALLED?"
The rows below these headings will show computer name, TRUE or FALSE, and TRUE or FALSE. So far, I am able to export the data to an HTM file, but it does not show up in separate columns. This is what I have:
$hostname | ConvertTo-Html -Property @{Expression={ $hostname};Label="Hostname"} |
            Out-File C:\Temp\Test.htm -Append -NoClobber
If (Test-Path -Path $FileCheck) {
Test-path -Path $FileCheck | ConvertTo-Html -Property  @{Expression=`
    { [string]"TRUE"};Label="Windows files replace"} | Out-File C:\Temp\Test.htm `
                                                        -Append -NoClobber
}
Else {
Test-path -Path $FileCheck | ConvertTo-Html -Property @{Expression=`
    { [string]"FALSE"};Label="Windows files replaced"} | Out-File C:\Temp\Test.htm `
                                                         -Append -NoClobber
}

I basically just need to understand how to create a table or array with 3 or 4 columns and how to populate the rows for those columns. Each computer will have a different row and show something like (COMPUTER1,TRUE,FALSE,TRUE). The way I currently have it set up, it is creating new rows when I need the headings in new columns.

A: Ok, what you really need is an array of objects. Those objects will have three properties, one for each column in your table. To make an object in PS3 you can simply use the [pscustomobject] type followed by a hashtable containing the values that you want. Something like this:
[pscustomobject]@{
    "HOSTNAME" = $hostname
    "FILES REPLACED" = If (Test-Path -Path $FileCheck) {"TRUE"}else{"FALSE"}
    "SCCM UNINSTALLED" = <some test to see if SCCM is installed>
}

You could then create a loop for the computers/servers in question, create that object once for each machine, add all of the objects to an array, and then output that array to your HTML.
$Results = @()
ForEach($hostname in $machinelist){

    $filecheck = "\\$hostname\c$\path\to\files"
    $Results += [pscustomobject]@{
        "HOSTNAME" = $hostname
        "FILES REPLACED" = If (Test-Path -Path $FileCheck) {"TRUE"}else{"FALSE"}
        "SCCM UNINSTALLED" = <some test to see if SCCM is installed>
    }
}

$Results | ConvertTo-HTML -Property HOSTNAME,'FILES REPLACED','SCCM UNINSTALLED' | Set-Content C:\Temp\Test.htm

Obviously you will need to do some editing there, but that should give you something to build off of.


March's father is Alfie McQuin, a world-famous jewelry thief and a pretty good dad. Then something goes wrong during a heist in Amsterdam. Alfie falls from a rooftop and dies.
March is alone in the world.
He's waiting at the airport with an official he calls Mr. Soda. They're about to board a flight for New York so March can begin his new lonely life in a group home. A different official hurries toward them with a girl his age.
She's the twin sister he never knew he had.
Loot is a fast-paced mystery about mastermind kids who are very good at outwitting adults—which is a good thing, because their lives are at stake.


I am pleased as punch (but not Judy?) to announce that I am now represented by the fabulous Elizabeth Bewley at Sterling Lord Literistic! I snagged this wonderful agent through the Sun vs. Snow contest on Twitter with my Young Adult historical action/adventure HOOD, about the daughter of Robin Hood under the tyrannical rule of King John. I am so excited to be working with her to bring Isabelle to a bookstore near you!
Next My story is going to be a book!


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Partnership Programs Helps Farmers Increase Their Livelihoods
By Digital TeamJanuary 19, 2018No Comments
Independent smallholder farmers are often trapped in a cycle: they need better training and equipment in order to increase their yields and thereby their income, but lack the means to pay for this investment.
It is for this reason that Asian Agri launched its Desa Sawit Lestari Program in 2015. The program identifies communities struggling with low yields, then works with farmers over a three-year period to turn their operations around.
The idea for Desa Sawit Lestari Program, which is part of Asian Agri's Corporate Social Responsibility commitment, came in 2015 when falling palm oil prices hit farmers relying on outdated practices particularly hard.
Such farmers often rely on poor quality seeds and fertilizers, and have limited knowledge of sustainable farming practices and little access to modern equipment.
"Many of them do not understand how to manage plantations well, so we are here to educate them," said Benjamin Hutagalung, Corporate Social Responsibility Coordinator for Asian Agri in Riau Province.
"We want the village communities to be independent so they do not continue to rely on others."
The Desa Sawit Lestari Program was launched initially in two villages: Ukui Dua in Pelalawan Regency, and Petapahan in Kampar Regency.
The community sets aside land to be planted with Asian Agri's Topaz seeds, which are developed by the company's in-house R&D team and are renowned for their quality.
It is managed in a joint project between Asian Agri and the local community, so that farmers get hands-on experience of best practices, and access to quality fertilizers. At the end of three years, when the oil palms are mature, the land is fully handed over to the villagers.
"We provide sufficient knowledge to the villagers until they are eventually able to manage their plantations on their own," Benjamin says.
"In addition, farmers are no longer lost about how to sell their produce, as the palm oil is immediately sold to Asian Agri directly after harvest," he explains.
Under the program, the company is currently managing 11.6ha of community-owned plantations (8.9ha in Ukui Dua Village and 2.7ha in Petapahan Village). Although it has only been running for two years, the program is considered a success as it has resulted in increased palm oil sales for participating villagers.
"Their incomes have increased in line with their plantations' productivity," Benjamin says.
By turning villages into stakeholders in the sustainable palm oil production process in this way, the Desa Sawit Lestari Program aims to help improve the overall economies of villages.
All income gained through the sales of palm oil goes into a village treasury overseen by a cooperative village unit, and the money goes toward public facilities for the village. In this way Asian Agri is able to help improve the welfare of communities in its areas of operation.


The Dennis Moreland Tack HV41 Soft 4 Plait Loping Hackamore has a noseband made of tan 5/16 inch nylon solid braid rope. At 5 inches in width between the cheeks this hackamore is 10 inches in length from top to bottom. I can make additional sizes but you will need to call to order: 817-312-5305.
The hackamore comes with a browband headstall, fiador and with your choice of hackamore rein. The rein you choose will be tied to the hackamore before I send it. Shown here with the CHR1 Hand Braided Grass Hackamore Rein.
Soft 4 Plait Loping HackamoreHow it works: This hackamore is perfect for loping and is one of my very best selling hackamores. It's very soft and comfortable on the horse's nose and is good for everything from trail or pleasure riding to warming up in the show pen. A great tool to help stressed or tense horses relax.
Click here to see more information on this loping hackamore.


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https://www.spahotel.co.uk/
reservations@spahotel.co.uk
TN4 8XJ
Ant Scragg
The Spa Hotel
"My father was always in the industry, he built up a chain of hotels called Marston Hotels for which he was chairman and Managing Director, from one hotel to 22 hotels at the most, they sold that in 2006 to Q hotels and then he bought The Spa Hotel in May 2007.
I left school quite early and was playing Pro Tennis until I was 20, I quit that and then naturally had to quickly find an income, I was put in touch with a hotel manager that he knew and I worked on the leisure desk. Obviously I'd been around hotels my whole life, listening to various management conversations in the car when Dad would be on the phone to work, listening to the stories at dinner after he got home from work etc.
Apparently the General Manager of this particular hotel saw something in me, then he put me on a management training course which was basically 3 months in every department, housekeeping, reception, kitchen, waiting, just about everything and then I went to work for another hotel in Rochester, earned a couple of promotions and a then had a brief stint abroad.
When he, and the Marston Family bought this place he asked me to come and work for him. I worked at duty manager level in 2007 when we started and then made my way up the ranks, in 2013 I was made Deputy General Manager.
I took over in January 2016 and then started to build my own team, generation two! From there, we've just grown and grown, moved with the times, changed the image of the Hotel and the quality of product we are offering has improved a lot also. I've started to network a lot more, which I think we'd lacked before and managed to put the right people in the right roles. We have big plans for the future so hopefully we can continue on this path!"
Set in 14 acres of picturesque grounds close to the centre of Royal Tunbridge Wells, the spa town from which its name is derived, our hotel first opened its doors to guests in 1880 and has since built a reputation for style, sophistication, relaxation and fine dining.
Originally built in 1766 as a country mansion, The Spa Hotel has a long-established history in the town, and with its recent refurbishment it offers a stunning location to stay, dine and for weddings and events. With 70 rooms, an award-winning restaurant as well as our new, contemporary Zagatos Bar and Brasserie, there is plenty for guests to enjoy whilst taking advantage of easy access to historic Tunbridge Wells and the beautiful Kent countryside.


text:
Blog: January 2021
Reflection: The Greatest Compliment
Years ago, my friend and long-time Camden priest, Father Michael Doyle, asked me a question I've not forgotten.
"What's the greatest compliment Jesus gave humanity?"
I confess I was stumped—never pondering the thought. Jesus shared amazing parables and taught with wise, pithy truisms...healed the sick ....performed some incredible miracles...but compliments?
"I'm stumped Father," I deflected, trying to disguise my biblical ignorance.
The aging Irish priest looked at me with his deep penetrating eyes and then responded with his mirthful chuckle, "Love your enemies!"
"That's not a compliment," I countered with a hint of vindication, thinking this Baptist boy had just outsmarted a veteran Catholic priest. "Sounds more like a command to me."
After all, who would interpret 'love your enemy' as a compliment? A near impossible command, maybe. A radical counter-cultural teaching, absolutely. But a compliment? I was feeling pretty good about my reply.
"It was a compliment," assured my friend with unwavering gentleness. "You see, Jesus believed humans could do something incredibly extraordinary and difficult. And that's a great compliment."
Father Doyle had a point. After an undergraduate degree in theology and a few years in seminary, never had I viewed "love thy enemy" as a compliment. It was one of those verses you'd rather erase from the Bible, pretending it was never written—an inconvenient truth. But reflecting more deeply, it made sense.
Jesus is always calling us to a higher place, seeing greater capacity and potential in ourselves than we can even begin to imagine.
And that's an aspirational truth we need these days. The current gravitational force of our culture seems to be pulling us toward our lowest selves and base instincts—not a higher vision of what it means to be fully human and made in God's image. Social media baits us, feeding our insatiable appetite for reasons to distrust those outside our camps and tribes. Jesus believes we have the capacity to do better.
You've probably noticed a little word getting the limelight this past week. Unity. It's found its way into speeches, songs, poems, talk shows. When shared it's often associated with acts of healing, putting aside differences, starting fresh. All sounds great, but ask 100 people what unity means—you'll get 100 different answers. Ask 300 million people the same question...you guessed it. Unity is complicated. Easy to speak about, much harder to implement. So what can we do?
This past week I asked our leadership team what unity means to them—especially in the wake of a very contentious, highly partisan, low-trust, violent, and divisive moment in our nation's history. Some shared Bible verses, others more theoretical definitions. One shared a personal story.
"There's a former staff worker who's been posting a lot on Facebook recently," began my colleague. "I vehemently disagree with his views. My first inclination was to join the group opposing his positions, pointing out why he is so wrong. But I decided to not pile on the criticism."
He continued: "Instead I sent a personal email, telling him I missed him and loved him. I thought it might be a way to preserve the relationship, opening a door to one day have a deeper conversation." Preserving a relationship rather than trying to win an argument....is that unity?
Some might say my colleague's response was a cop out. The duty to speak his true convictions was neglected. Yet if we're honest, building unity with political foes using terse Facebook postings is a low probability proposition. Behavioral data agrees, suggesting "conversion" rates through arguing with adversaries seldom happens. True unity ultimately needs relational roots.
Community might be a more accurate description of what our country needs.
Community calls us to work out differences within the context of geographical and social arrangements—neighborhoods, churches, workplaces and civic organizations. Derived from the Latin 'communitas' (fellowship), the prefix "com" signifies "with, together, in conjunction." So true unity is built in conjunction with others—not in the anonymity of cyberspace.
I wish this stuff was easy. It's not. Difficult days are still ahead. But I'm hopeful. For the past 35 years I've had a front row seat watching an eclectic group of God's people unify across political, theological, racial, economic and social differences to create UrbanPromise—a community where Christ's love and hope is shared and lived each day. It's possible. We just need to believe the compliment.
UrbanPromise
Blog tags:
UrbanPromise Food Co-Op Director to receive prestigious award
Caroline Mitchell coordinates and directs the UrbanPromise Food Co-Op—a program that engages our community in creative ways to alleviate food scarcity, while empowering local citizens to share food and resources with their neighbors. Under Caroline's leadership, Food Co-Op membership has grown from 100 to 400 registered families.
Recently Caroline was selected by the YWCA of Princeton, New Jersey for The Tribute to Women Award—a prestigious award honoring women who have demonstrated sustained leadership and exceptional talent and who have made significant contributions to their professions.
Caroline shared, "Getting this Award has special meaning for me because I connect deeply with and believe in the mission of eliminating racism and empowering women."
At UrbanPromise Caroline is known for her kindness, humility and servant leadership. Our Chief People Officer, Pam Foxx says, "Caroline's level of expertise and empathy for those in need of care and support is nothing short of amazing."
"I have thought a lot about my advocacy work over the years and the people I have worked and served with to bring about change in our community." Caroline adds, "I stay committed to the words of Margaret Mead, "Never doubt that a small group of thoughtful, committed citizens can change the world; indeed, it's the only thing that ever has."
UrbanPromise President Bruce Main adds, "It's exciting to see Caroline get recognized for her decades of community service. Her compassion, commitment and vision to make a difference in our world is inspiring. It's an honor to have Caroline as part of the UrbanPromise community."
Congratulations Caroline!
Reflection: Drum Major Instinct
"Whoever wants to become great among you must be your servant..." Mark 10:43
I've been part of a few contentious staff meetings over the years—you know the kind. Walk into the conference room: arms are crossed, no laughter, no engagement, no eye contact between the participants. Anger can't be measured or seen, but it's in the room. Thick. Palpable. Trust has calcified, losing its limbering quality of loosening social relations. And then there's the pit in your stomach. Uninvited anxiety enters your system. Adrenaline and cortisol flow. Deep breath. You're the leader. Your job—bring the team back from the brink of collapse.
To be a fly on the wall at Jesus' Monday morning staff meeting—the meeting after 10 of his disciples discover their two social climbing colleagues, James and John, secretly lobbied Jesus for prominent cabinet posts in the new regime—would have been fascinating. "Indignant" (10:41) is the adjective used by the writer Mark to describe the feelings of the angered 10. Egos apparently bruised, trust broken, emotions running high. Jesus, it's time to watch a TED talk on "How to Have Tough Conversations."
Martin Luther King, Jr. preached about this very moment in one of his lesser-known sermons called, The Drum Major Instinct—delivered at his home church Ebenezer Baptist on February 4th, 1968. For those who appreciate homiletics as an art form, it's an oratorical masterpiece. The antiquated pre-digital recording (with disruptions, crackling static, parishioners commenting, babies crying) only adds to the drama and authenticity. Oh, and 53 years later, the content is still relevant. Truth never seems to go out of style.
King notes that Jesus doesn't judge James and John harshly for their selfish request. He acknowledges that we all battle the same impulse—our primal need to be recognized, appreciated, "first in line".... It's the "Drum Major Instinct", says King, which beckons us to lead the parade. Deep down we all secretly yearn for attention, approval and status. Psychotherapist Alfred Adler argues it's the dominant impulse of the human condition.
The Drum Major Instinct, however, is only problematic if not controlled—becoming destructive not only to ourselves, but those around us. "It's vitamin A to our ego," shares King with his congregation. "It's what advertisers tap into, compelling us to buy products that elevate our social position, while dragging us into personal debt by living beyond our means." Unchecked, The Drum Major Instinct leads to "snobbish exclusion, classism, racism"—why else would we put others down so we can elevate our self-importance?
So how does Jesus facilitate his contentious Monday morning staff meeting? Brilliantly. According to King, Jesus doesn't condemn John and James or even call them selfish. He doesn't belittle them for raising their ego-driven question. Instead Jesus seems to say: "You want to be great? Wonderful! You want to be important? Fantastic! You want to be significant? Terrific!" It might surprise you, but the "Drum Major Instinct" is NOT condemned. Jesus simply reorders priorities and redirects our energy.
You want to be first? Be first in love! You want to be significant? Be significant in generosity! You want to be great? Be great in humility and service! Jesus provides a whole new definition of greatness—a definition counterintuitive to the human condition. And this leads to one of my favorite lines in King's sermon—words so appropriate for this moment in history.
"Everybody can be great...because everyone can serve. You don't have to have a college degree to serve. You don't have to make your subject and verb agree to serve. You only need a heart full of grace. A soul generated by love."
As King reaches the climax of his sermon, he reflects on his own mortality. Not in a morbid way, but in a way that calls his listeners to reflect on their own brief lives. King shares how he'd like to be eulogized at his funeral:
"I'd like somebody to mention that day that Martin Luther King, Jr. tried to give his life serving others. I'd like for somebody to say that day Martin Luther King, Jr. tried to love somebody...tried to be right on the war question ...tried to feed the hungry....tried to clothe those who were naked...tried to love and served humanity...."
Eerily the recording goes silent midsentence during King's crescendo—it just cuts off, serendipitously leaving a silent pause long enough for the listeners to place themselves in the eulogy.
And that's a good space to place ourselves on this important and commemorative day. A space to reflect on our lives. A space to reflect on the meaning of true greatness. A space to ask ourselves, are we using our Drum Major Instinct for the right things -- like justice, righteousness, peace, humility and service.
Find joy in your service today—
PS. Here's a link to the original sermon. You might enjoy listening.
King's Sermon
Thank you for investing in us!
2020 was a crazy year; there is no doubt about it. But difficult circumstances are often the moments God uses to show up and makes things good, despite our very human concerns that we will never be able to survive.
Did UrbanPromise survive 2020? You bet we did!
Today, we offer up an enormous THANK YOU to all of you who believe in us, believe in what God is doing here in Camden. To all of you who donated in 2020 and those of you who dug extra deep in December to help us reach our match goal - you did it! We did it!
We hit our goal in 2020 despite a crazy global pandemic. What a faith-building moment for us. Today we begin the work of fundraising for 2021 and we do so joyfully knowing that God is bigger than our circumstances and that all of you are on this journey with us.
With love and gratitude,
The UrbanPromise faculty and staff
Reflection: A Time to Improvise
"She placed the child in the basket and put it among the reeds along the bank of the Nile." Exodus 2:3
"Really nice car," I complimented. The young man in the driver's seat smiled.
"I own it," he echoed with a sense of pride. "I bought it with money generated from my business."
I was heading to the airport after visiting some of our ministries in Malawi—a small subsaharan country in east Africa. One of our directors arranged my transportation. A young man named Peter—evidently in his early 20's—picked me up in a shiny, new Toyota hatch back. I was intrigued. With so few jobs in the country—especially for young people—and 62% of the country living in less than $2 a day, curiosity got the best of me.
"What kind of business?"
"I raise chickens and sell them to local restaurants," he added.
"How'd you get started?"
Keeping his eyes on the road, Peter shared his remarkable story. "One day I visited a small restaurant in the city of Zomba," he began. "I ordered chicken and chips."
Peter chuckled. "Ten minutes later the waiter comes back to me and asks if I wanted a thigh, leg or breast with my chips."
A thigh was his request. Ten minutes passed—the waiter returned. "Do you want your chicken fried, baked or charbroiled?"
Peter, now getting hungry and a little annoyed, replied with a hint of impatience, "Fried is just fine." But to his surprise the waiter returned a third time. "Excuse me. One last question. Do want the chips and chicken served together or separately?" Peter gave the waiter a surprised glance, "Together would be nice."
At this point I wasn't sure how the story connected to my initial question—but being a captive audience listened intently. "So the waiter comes back a fourth time," concluded Peter, "He tells me they have NO chicken—but he'd be willing to drive 45 minutes to the city of Blantyre and get my chicken." Peter politely declined, paid for his soda and left the restaurant.
"I decided to do a little research," continued my new friend as he navigated the bumpy road. "I started asking restaurant owners in Zomba how they acquired their chickens. You believe it? They all drove 45 minutes to Blantyre to make their purchases."
So Peter decided to seize the opportunity. One small problem. Having recently graduated from college, he had no money. "I was broke. No job. No prospects," he recalled. "I only owned the cell phone my father gave me as a graduation gift."
Peter then made a difficult decision—especially for a twenty year old. "I asked myself, 'Why do I need a cell phone if I have no job and no money?" His new phone was sold for $30 US dollars and he bought his first 100 chicks. Three years later Peter owns a thriving chicken business in Zomba—making enough to buy a car, support himself, payroll a small staff and run a small non-profit ministry teaching entrepreneurial classes to men coming out of Malawian prisons. "God's using my story to inspire these young men."
It intrigues me how people respond to adversity. Many would walk out of that restaurant, utter a few complaints and post a negative review on Yelp. Yet Peter sells his cell phone to seed a business venture. One person has a horrible dining experience. Another sees opportunity.
One of the reasons I'm drawn to the Bible is it's filled with people like Peter—people who look at bleak, desperate and hopeless situations and see another path forward. Faith inspires their imagination.
Case in point, Moses' mother in the book of Exodus. She's one of the great improvisers of the Old Testament. You may remember her story. Egypt's top political leader, Pharaoh, issued an edict to murder all the Hebrew male babies. Pharaoh, threatened by a growing population of outsiders in his country, creates a policy of infanticide to thwart any threats to his power.
But Moses' mother isn't allowing this dark and hideous situation to negate action. Even as a slave to Egyptian masters, her faith preserves and informs her quest to save her son.
Much like my new friend Peter, Moses' mother takes inventory of what was happening around her...and conceives a brilliant plan. You see, Pharaoh's daughter comes to the Nile each day. Pharaoh's daughter interacts with her helpers, thus revealing her character and personality. Moses' mother watches—her instincts are informed. So her baby Moses is placed in a woven basket, floated to an appropriate position on the river and Moses' sister Miriam stands in a strategic position. Nothing is left to chance. A hopeful outcome is imagined—and ultimately achieved. "Pharaoh's daughter took him as her son and named him Moses, meaning, 'I drew him out of the water.' A baby's life is spared. The tragic consequences of losing a future leader averted.
Exodus reminds me that we are not the first people in history to suffer, to be caught in the crosshairs of political upheaval and live in a moment riddled with anxiety and uncertainty. These stories teach me about courageous people whose faith nurtures improvisation and ignites dreams of redemptive solutions to seemingly intractable and impossible problems.
Yet...sacrificing a cell phone leads to a thriving chicken business and a spin-off ministry. An ancient mother's vision and a few sticks woven into a floating basket liberates a community of oppressed people to flourish. It's the very best of God's people on display.
This is the kind of holy improvisation needed right now, friends. Don't give up. Keep dreaming. Keep scheming of ways to overcome evil with good. Remember Paul's words: "Let's not grow weary in doing good, for at the proper time we will reap a harvest if we do not give up." Galatians 6:9
Onward in 2021—


text:
LOCAL LORE
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Made in the MOV: Gift Guide
COVID-19 Vaccine Update for Washington County
Parkersburg Area Community Foundation Announces $190,522 in Fall Grants
Press Release December 10, 2020
The Parkersburg Area Community Foundation and Regional Affiliates (PACF) announced support for nonprofit organizations and charitable projects throughout the region through its fall Community Action Grant program. The PACF awards grants from a variety of funds created by individuals, organizations, and area businesses to support critical community needs. The PACF's grant awards this fall total $183,272. The Foundation's three affiliates awarded an additional $7,250 in grants, including $1,000 from the Doddridge County Community Foundation (DCCF), $3,000 from the Ritchie County Community Foundation (RCCF), and $3,250 from the Little Kanawha Area Community Foundation (LKACF). Grant awards will support facility renovations, equipment purchases, new program development, and the provision of services to individuals.
"Several grant awards in this cycle address food access, which continues to be a critical need in our region due to the COVID-19 pandemic," said PACF Associate Director for Community Leadership Marian Clowes. "These include grants for food pantries, backpack feeding programs, community and school gardening projects, and a new community kitchen in Calhoun County. Other grants provide funds for organizations to undertake needed facility renovations, develop new services, purchase equipment, and address important community educational and human service needs."
To be considered for a Community Action Grant, an applicant must be a private, nonprofit organization or a public institution. Either the applicant or program to be funded must be located in the Foundation's eleven-county geographic service area (Calhoun, Doddridge, Gilmer, Jackson, Mason, Pleasants, Ritchie, Roane, Wirt, and Wood counties in West Virginia, and Washington County, Ohio).
The fall 2020 Parkersburg Area Community Foundation & Regional Affiliates grant recipients are:
American Red Cross of the Ohio River Valley – $8,000 to purchase a new community emergency vehicle.
Arnoldsburg Elementary Community Sports Complex – $9,000 to support the development of recreation fields for Calhoun County youth.
Arnoldsburg Volunteer Fire Department – $8,700 for HVAC updates and firehouse renovations.
Auburn Community Association – $2,000 from the PACF and $1,000 from RCCF to install vinyl siding on the Auburn Community Building.
Blennerhassett Middle School – $6,990 to purchase a Glowforge system for STEM and entrepreneurship education.
Cairo Crafty Quilters – $1,000 from PACF and $1,000 from RCCF for the creation of a Ritchie County quilt trail.
Calhoun County Park – $7,591 from PACF and $2,700 from LKACF for the installation of a wheelchair lift in the park's two-story community building.
Catholic Charities of West Virginia – $6,000 from PACF and $1,000 from DCCF to support the operation of the Mobile Wellness Works Food Pantry at sites in Calhoun, Doddridge, Ritchie, Roane, and Wirt Counties.
Gabriel Project of West Virginia, Northern Programs Chapter – $5,000 to reestablish client services in Pleasants County and to expand services in Calhoun, Doddridge, Ritchie, and Wood Counties.
GFWC Parkersburg Woman's Club – $6,085 to support the Brown Bag project, which provides nutritious food for the weekends for students at Jefferson and Emerson Elementary Schools.
Gilmer County Elementary School – $1,775 from PACF and $550 from LKACF to purchase materials and equipment for the school's Square Foot Gardening project.
Glenville State College Foundation – $4,000 to support the Smart Classroom Renovation project, designed to add interactive technology in classrooms to improve student achievement.
Jackson County Developmental Center, Inc. – $3,826 to add a kitchenette to the Wood County program site to facilitate independent living skill development programs.
Mountaineer Food Bank – $10,000 to purchase and install interactive white board technology to improve virtual communications.
Normantown Historical Community Center – $10,000 to replace the roof on the building that houses the community clothing closet.
Parkersburg Area Coalition for the Homeless/House to Home – $5,950 to renovate the client bathroom and repair the facility's sidewalk.
Pennsboro Volunteer Fire Department – $5,000 from PACF and $1,000 from RCCF to upgrade and replace outdated fire suppression equipment.
Roane County Commission/Roane County High School – $3,600 to renovate the high tunnel at Roane County High School for use in its agricultural education programs.
Ronald McDonald House Charities of Pittsburgh and Morgantown – $2,500 to support the Jacob Johnson Family Lodging Fund, to defray the costs of stays at RMHC houses for families from the PACF service area.
The Hope Recovery Manor – $4,665 to support training for clients on life and relationship building skills.
The Salvation Army – $15,000 for kitchen and dining room renovations.
Voices of the Streets/Essentially Yours – $1,500 to purchase kitchen supplies to provide to those transitioning to a home after experiencing homelessness.
West Virginia Food and Farm Coalition – $15,000 to develop a community canning and manufacturing kitchen in Grantsville.
Westbrook Health Services – $9,000 to renovate the community meeting room and public restrooms.
Wirt County Missional Group – $6,600 to purchase a new freezer, a double-door cooler, and food for the food pantry program.
Wood County 4-H Camp – $10,400 to support capital improvements at the Wood County 4-H Camp.
Wood County Parks and Recreation Commission/Mountwood Park – $13,000 to replace roofs on cabins in the White Oak Village Section of the park, which houses a substance use disorder treatment facility.
In addition, through the support of the Tamara K. Brown Helping to Heal Fund and the Good Samaritan Clinic Endowment Fund, the PACF provided support to school nurses at the following schools to purchase equipment to help meet student health needs:
Blennerhassett Middle School – $290 to purchase an otoscope.
Parkesburg South High School – $500 to purchase a mobile trauma/first aid kit.
Lubeck Elementary School – $300 to purchase supplies and personal protective equipment.
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Buckeye Hills Connecting Seniors to COVID-19 Vaccine Providers
MCF Approves School District Proposals From $60K For 6 Campaign
Marietta College Announced ILR Winter Courses
Press Release January 9, 2021
Washington County COVID-19 Update
WSCC Part of Inaugural Building Rural Innovation Cohort
Indoor Triathlon Parkersburg YMCA 1800 30th Street
27feb(feb 27)9:00 am(feb 27)9:00 amFeaturedIndoor TriathlonTime to let out your inner athlete and show the world what you've got!
The YMCA of Parkersburg's Indoor Triathlon is a great event for newbies and seasoned veterans alike to compete in swimming, biking, and running (in our pool and on our stationary
The YMCA of Parkersburg's Indoor Triathlon is a great event for newbies and seasoned veterans alike to compete in swimming, biking, and running (in our pool and on our stationary bikes and treadmills).
Triathlon Details
Junior and Adult Divisions will be available for competitors. The Triathlon will measure the distance traveled for each competitor within a 1-HR time span:
● 10 min of lap swims in our 25-yard indoor pool.
● 30 min of cycling on our stationary bikes.
● 20 min of running on our treadmills.
Gold, Silver, and Bronze medals will be awarded to both males and females for each of the 2 divisions.
Participants have the option to select the day and time they'd like to participate through online or in-person registration. To maintain social distancing, each start time below will be limited to 4 people.
February 27th:
9:00 AM, 10:00 AM, 11:00 AM and 1:00 PM, 2:00 PM, 3:00 PM
1:00 PM, 2:00 PM, 3:00 PM
March 6th:
INTERESTED in training for the Indoor Triathlon?
Beginning December 1st, in-person training for the Indoor Triathlon will be held on Tuesdays at 8:00 AM and Thursdays at 5:00 PM. All sessions are led by our Personal Trainer, Michaela Piendel. The trainings will last until the Indoor Triathlon's launch on February 27th.
Division Ages and Fees
Junior Division: (Ages 12-17)………….. $24/members, $30/non-members
Adult Division (Ages 18 )……..…….. $24/members, $30/non-members
*Cost includes a T-Shirt and the Training sessions above.
(Note: If not pre-registered for Triathlon, the training is $5 per class.)
Online or in-person registration is available! Visit www.parkersburgymca.org/yevents/indoortriathlon to learn more.
(Saturday) 9:00 am - 3:00 pm
Parkersburg YMCA
(Sunday) 9:00 am - 3:00 pm
Never miss out. Subscribe to our ridiculously good-looking email newsletter.
Emily Nestor, Mile Marker 181
Where to Find Fireworks this 4th of July
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text:
\section{Introduction}

The electronic structure and Fermi surfaces of conventional metals
have been studied in great detail by ARPES \cite{HUEFNER}.
The question of the determination of the Fermi surface
\cite{CAMPUZANO,OLSON,SHEN,BILAYER}
by ARPES in the normal state of the high $T_c$
superconductors is of great interest, especially since other
Fermi surface probes (like de Haas-van Alphen and positrons) have not
yet yielded useful information on the Fermi surface of the
planar Cu-O states. However, this question is not a trivial one,
since these materials are strongly correlated, and likely not
Fermi liquids, exhibiting very broad, ill-defined electronic
excitations\cite {BILAYER,KAMINSKI}.

The determination of the Fermi surface by ARPES in these systems
is further complicated by the very small dispersion in
the vicinity of the $(\pi,0)$ point of the Brillouin zone
\cite{BILAYER,GOFRON,DESSAU93}, and by
strongly ${\bf k}$-dependent photoemission
matrix elements which lead to intensity variations which have nothing
to do with Fermi crossings. Bi2212 has an additional complication:
final state diffraction of photoelectrons by the ${\bf Q}=(0.21\pi,0.21\pi)$
structural modulation in the BiO layers \cite{BILAYER,AEBI94,COMMENT}.
The combination of all these effects,
if not treated correctly, can be a source of confusion and lead to
apparently contradictory conclusions
\cite{DESSAU93,SAINI97,DESSAU99,SHEN99},
even though the data between various groups are completely consistent
with one another.

It is therefore important to establish the
criteria for unambiguously extracting the Fermi surface from ARPES data.
This is the main goal of the work presented here, in which we
study near-optimal and overdoped samples of
$Bi_{2}Sr_{2}CaCu_{2}O_{8+\delta}$ (Bi2212) and
$Bi_{1.6}Pb_{0.4}Sr_{2}CuO_{6}$ (Bi2201) using a range of
incident photon energies from 17 - 60 eV.
Our extensive study leads us to the same conclusion as our previous
work\cite{BILAYER,Bi2201}, namely that the Fermi surface consists of a
single
hole barrel centered around $(\pi,\pi)$, the most antibonding point in the
Brillouin zone. Furthermore, the Fermi surface is consistent with the
Luttinger
count with its volume scaling as one plus the number of doped holes.

The paper is organized as follows. In Section II, we
describe the samples and experimental details. We begin with ARPES
data over a wide (6eV) scale describing the full valence band in Section
III, and then turn to low energy, near $E_F$ features in the rest of the
paper. Section IV contains a brief description of polarization
selection rules and their experimental implications.
We then discuss in some detail criteria for determining the Fermi surface
in Section V. In Section VI we present data on Bi2212 and Bi2201 and show
how
the various criteria proposed in Section V fare in determining
Fermi crossings. We find that the symmetrization method for inferring
when the spectral function peak goes through the chemical potential is
a very powerful tool and works even when the energy distribution curves
(EDCs) are broad and weakly dispersive. We discuss in Section VII and VIII
the usefulness and limitations of using the integrated intensity
to determine ${\bf k}_F$, emphasizing the importance of the photon energy
and ${\bf k}$-dependence of the ARPES matrix elements. By analyzing
data obtained at different incident photon energies, we explicitly show
how one can experimentally separate matrix element effects from those
due to changes in the momentum distribution.
In Section IX, we finally turn to the Fermi surface in Bi2212
where, in addition
to all the issues discussed above for Bi2201, one also needs to
be careful about BiO superlattice effects. Polarization selection rules
are exploited to disentangle superlattice effects from the intrinsic
CuO$_2$ electronic structure. We conclude in Section X.

An Appendix contains some further technical details related
to the symmetrization procedure.

\section{Experimental details}

Our experiments are on very high quality single crystals of Bi2212
and Bi2201 grown by the traveling solvent floating zone method with
an infrared mirror furnace, with low defect densities, as it can be
appreciated from the high resolution x-ray diffraction rocking
curve shown in Fig.~1. The samples are labeled by their doping levels
(OPT for optimal doped and OD for overdoped) together with their
onset $T_{c}$.

The as-grown Bi2212 samples are slightly overdoped and have $T_c = 87$K
with a transition width of 1K as determined by a SQUID magnetometer. These
samples are most stable in terms of their photoemission
characteristics. We look at Bi2201 samples in the doping range from
OD23K to heavily OD0K.
The samples are cleaved \textit{in situ},
and have optically flat surfaces as measured by specular laser
reflections. It is absolutely essential to characterize the flatness
of the surface on which ARPES experiments are done.
 Another measure of the sample quality, within ARPES,
is the observation of ``umklapp" bands \cite{BILAYER}
in the electronic structure of Bi2212 samples,
due to the presence of a structural superlattice
modulation in the BiO layer. Since the structural superlattice has a
periodicity of $\sim 5$ times the unit cell, very good long range order is
required for its observation.

The experiments were performed at the Synchrotron Radiation Center,
Wisconsin, using a high resolution 4-m normal incidence monochromator
with a resolving power of $10^4$ at $10^{11}$ photons/s.
The samples are carefully oriented in the sample holder to an
accuracy of 1$^\circ$ by Laue diffraction, and the orientation is
further confirmed by the observed symmetry of sharp ARPES features
around high symmetry points, as described below.

Some of the data analyzed below (in particular Figures 5, 11, 14 and
16)
were obtained using a Scienta analyzer,
at a variety of incident photon energies,
with an energy resolution of 16 meV and
a high ${\bf k}$-resolution better than 0.01 $\AA^{-1}$ at 22 eV photon
energy.
The detailed analysis presented in this paper
leads to conclusions which are completely consistent with the recent
high ${\bf k}$-resolution results of
our group \cite{HELEN} (emphasizing low temperature data on Bi2212),
as well as that of Borisenko et al. \cite{BORISENKO}.

For the Brillouin zone of Bi2212 and Bi2201, we use a square lattice
notation
with $\Gamma {\bar M}$ along the CuO bond direction, as shown in
the insets of Fig.~2. $\Gamma = (0,0)$, ${\bar M} = (\pi,0)$,
$X=(\pi,-\pi)$ and $Y=(\pi,\pi)$ in units of $1/a^*$, where $a^* =
3.83\AA$
is the separation between near neighbor Cu ions.
(The orthorhombic $a$ axis is along $X$ and $b$ axis along $Y$).

An example of how ARPES is used in sample alignment is shown in Fig.~2,
where spectra are shown along the
$(0,0)-(2\pi,0)$ and the $(\pi,\pi)-(\pi,-\pi)$
high symmetry lines of the Brillouin zone. This symmetry is reflected
in the position of the peak in the spectra in Fig.~2,
and allows us to accurately find the surface normal, and the angle of
the sample about this normal, completely
determining the momentum ${\bf k}$ of the outgoing electron. Note that
this
alignment procedure only makes use of the symmetry properties of the
peak positions \cite{INTENSITY}, and does not require
a knowledge of the Fermi surface
(indicated by the curves in the top panels
of Fig.~2).

\section{The valence band}

Our main focus will be on near $E_F$ electronic structure, but we
begin with a brief discussion of
angle-resolved photoemission from the complete valence band of Bi2212.
This covers an
energy range of approximately 6 eV from the Fermi energy (the small peak
near zero) to the bottom of the valence band. The electronic
structure can be divided into three groups, as indicated in Fig.~3a:
the most bonding CuO$_{2}$ state is at the bottom of the valence band
(the peak at 6 eV), the antibonding state is at the Fermi energy,
and the non-bonding states are in between.
The ``lump" in the middle also includes states
from elements in the structure other than Cu and O.
But since these layers are insulating, the corresponding states do
not cross the Fermi energy.

By varying the in-plane momentum ${\bf k}$, one can map the complete
electronic structure of the valence band, as shown in Fig.~3a. These
curves were obtained without fitting peaks to the data. Instead, the
second derivative of the observed spectra, as shown in Fig.~3b, was
taken and plotted as a grey scale without any modifications
\cite{TAKAHASHI}.
Two considerations apply: the energy step in these spectra
is only 30 meV, and therefore the details of the dispersion near
the Fermi energy are not clear, and the spectra
were obtained with a particular polarization of the photons,
so that not all states show optimal intensity. Nonetheless, the most
noteworthy features are the most bonding and antibonding states,
highlighted by thick dark lines. In the remainder of the paper,
we will focus exclusively on
the antibonding states in the region near the Fermi energy.

\section{Selection rules}

We now focus on the states crossing the Fermi energy in Fig.~3a and
show how one can determine the symmetry of the initial state in ARPES.
The ARPES intensity is governed by the (square of the) dipole matrix
element
$M_{fi}$ connecting the initial state $\left|{\psi_i}\right\rangle$ to the
final state $\left|{\psi_f}\right\rangle$, given by
$\left|\left\langle{\psi_f}\right|{\bf A}\cdot{\bf p}
\left|{\psi_i}\right\rangle\right|^2$, where $\bf A$ is the vector
potential of the (linearly polarized) incident photon, and ${\bf p}$
is the momentum operator.

We use a simple version of the selection rules proposed by Hermanson
\cite{HERMANSON}. Let the photon beam be incident along a plane of
mirror symmetry of the sample (${\cal M}$).
If the detector is placed in the same mirror plane as shown in Fig.~4a,
then the final state $\psi_f$ must be even with respect to reflection
in ${\cal M}$, because if it were odd the wave function would vanish at
the detector.
The dipole transition is allowed if the entire matrix element
has an overall even symmetry. Thus two possibilities arise \cite{EXPT}.
First, if the initial state $\psi_i$
is even with respect to ${\cal M}$, then the light polarization ${\bf A}$
must also be even, i.e. parallel to ${\cal M}$. Second, if the initial
state is odd with respect to ${\cal M}$, then ${\bf A}$ must also be odd,
i.e.
perpendicular to ${\cal M}$. This can be summarized as:
\begin{equation}
\left\langle{\psi_f}\right|{\bf A}\cdot\hat{\bf
p}\left|{\psi_i}\right\rangle
\left\{{\matrix{{\psi_i\,\,even\,\,\left\langle + \right|+\left| +
\right\rangle
\,\,\Rightarrow \,\,{\bf A}\,\,even}\cr
{\psi_i\,\,odd\,\,\left\langle + \right|-\left| - \right\rangle
\,\,\Rightarrow \,\,{\bf A}\,\,odd}\cr
}}\right.
\end{equation}

Consider hybridized $Cu3d-O2p$ initial states, as shown in Fig.~4b,
which have a $d_{x^2 - y^2}$ symmetry about a Cu site.
These states are even with respect to $(0,0)-(\pi,0)$
(i.e. the plane defined by this symmetry axis and the z-axis)
and odd with respect to $(0,0) - (\pi,\pi)$.
Therefore, measurement along the $(0,0)-(\pi,0)$ direction will be
dipole-allowed (forbidden) if the polarization vector $\textbf{A}$ is
parallel
(perpendicular) to this axis.  Fig.~4c shows that,
consistent with an initial state which is even about $(0,0)-(\pi,0)$,
the signal is maximized when ${\bf A}$ lies in the mirror plane
and minimized when ${\bf A}$ is perpendicular to this plane.
(The reasons for non-zero intensity in the dipole forbidden
geometry are the small, but finite, ${\bf k}$-window of the experiment
and the possibility of a small misalignment of the sample).
Similarly, we have checked experimentally that (for Bi2212 in the
Y-quadrant
where there are no superlattice complications) the initial state is
consistent with odd symmetry about $(0,0) - (\pi,\pi)$.

While the dipole matrix elements are strongly photon energy dependent,
the selection rules are, of course, independent of photon energy.
This has been checked by measurements at 22 eV and 34 eV.
All of these results are consistent with the fact that we are probing
$Cu3d-O2p$ initial states with $d_{x^2 - y^2}$ symmetry.
In addition, as we shall emphasize below,
the selection rules can be exploited to one's great advantage
in disentangling the main CuO$_2$ ``band'' from its umklapp
images due to the superlattice in Bi2212.

\section{Fermi surface criteria}

Many criteria have been used for determining the Fermi surface
in the past without a clear discussion of
the conditions under which they are applicable.
We will present three criteria here: (A) one based on dispersion of the EDC
spectral peaks through the chemical potential, (B) a second one based on
the peak of the spectral function inferred from symmetrized data,
and (C) a third one based on rapid changes in the momentum distribution.
In the following Sections we will show how these criteria fare
when applied to experimental data.
Other criteria, not discussed in this paper, will be briefly
alluded to at the end of this Section.

The ARPES intensity is given by \cite{NK}
\begin{equation}
I({\bf k},\omega)= I_0({\bf k};\nu;{\bf A}) f(\omega)A({\bf k},\omega)
\label{intensity}
\end{equation}
for a quasi-two-dimensional system, assuming validity of
the impulse approximation. Here ${\bf k}$ is the in-plane momentum,
$\omega$ is the energy of the initial state measured relative to
the chemical potential, $f(\omega) = 1/[\exp(\omega/T) + 1]$ is
the Fermi function, and the one-particle spectral function
$A({\bf k},\omega) = (-1 / \pi)\Im mG({\bf k},\omega + i0^{+})$.
The prefactor $I_0$ is proportional
to the dipole matrix element $\left| M_{fi} \right|^2$ and thus
a function of ${\bf k}$ and of the incident photon energy $h\nu$ and
polarization ${\bf A}$.
It is also important to remember that the experimentally observed
EDC involves a convolution of the intensity of Eq.~(\ref{intensity})
with the energy resolution function and a sum over the momentum
resolution window.  There is also an additive (extrinsic) background
contribution to the EDC, however this has little effect on Fermi
surface determination since the background is negligible at the
chemical potential \cite{BACKGROUND}.

We first discuss the simplest case of non-interacting electrons
which have infinitely sharp energy levels leading to a spectral function
$A({\bf k},\omega) = \delta(\omega -\epsilon_{\bf k})$.
A Fermi surface crossing ${\bf k}_F$ is then defined by
the location in ${\bf k}$-space where the sharp peak of
the spectral function crosses
the chemical potential ($\omega = 0$), i.e., $\epsilon_{{\bf k}_F} = 0$.
It is also useful to look at the momentum distribution
\begin{equation}
n({\bf k})=\int_{-\infty}^{+\infty}d\omega f(\omega)A({\bf k},\omega).
\label{nofk}
\end{equation}
For non-interacting electrons $n({\bf k}) = f(\epsilon_{\bf k})$,
the Fermi function. At $T=0$ then, the momentum distribution
shows a jump discontinuity at ${\bf k}_F$. At low temperatures
there is no singularity in $n({\bf k})$ but only a rapid variation
in the vicinity of ${\bf k}_F$.

The case of interacting electrons at finite temperatures is
much more interesting. The energy levels are now broadened and
shifted by the self-energy
$\Sigma = \Sigma^{\prime}+i\Sigma^{\prime\prime}$, with
$G^{-1}({\bf k},\omega) = \omega - \epsilon_{\bf k} - \Sigma({\bf
k},\omega)$.
Thus the spectral function is given by
\begin{equation}
A({\bf k},\omega) = {1 \over \pi}
{{\left| {\Sigma^{\prime\prime}({\bf k},\omega)} \right|} \over
{\left[{\omega-\epsilon_{{\bf k}}-\Sigma^{\prime}({\bf k},\omega)}\right]^2
+ \left[{\Sigma^{\prime\prime}({\bf k},\omega})\right]^2}}.
\label{spectral}
\end{equation}
The electronic dispersion is now given by tracking the peak of the
spectral function.
We define a Fermi surface crossing by the ${\bf k}$-point at which
the spectral function peak crosses the chemical potential ($\omega = 0$)
\begin{equation}
\Re eG^{-1}({\bf k}_F,\omega=0)=0.
\end{equation}

This definition agrees with the standard definition of the Fermi surface
at $T=0$ in an interacting Fermi system which can be described by
Landau's Fermi liquid theory \cite{AGD}.
For this case, there is an additional (equivalent)
characterization of the Fermi surface in terms of a discontinuity in
$n({\bf k})$ at $T=0$.

However, it should be stressed that the discussion above in terms
of the spectral function is very general, and not necessarily
limited to Fermi liquids.
We propose to use the above definition in terms of the peak of the
spectral function, which is valid even at finite temperatures, and
use it to define the Fermi surface for high $T_c$ superconductors
even though the spectral peaks above $T_c$ are too broad for the system
to qualify as a Landau Fermi liquid \cite{KAMINSKI}.
The question of whether the $T=0$ momentum distribution shows any
singularity or not cannot be addressed experimentally, since at $T=0$
one is not in a normal state but rather a broken symmetry state.
We do find, however, that the Fermi surface we experimentally
determine above $T_c$ encloses a number of electrons which are
consistent with the Luttinger count \cite{AGD} of $(1+x)$,
where $x$ is the hole doping.

Let us now discuss in detail how these characterizations of the Fermi
surface
will be used in practice to determine ${\bf k}_F$. The first method
(A) is simply to look at the dispersion of the peaks of the measured
EDCs and determine from this when the peak position crosses the
chemical potential. There are two caveats to this method. First, the
peak of the EDC does {\it not} in general coincide with the peak of the
spectral function $A({\bf k},\omega)$. As can be seen from
Eq.~(\ref{intensity}), if there is a broad spectral function $A$ centered
about $\omega = 0$, then the peak of the EDC will be at $\omega < 0$,
produced by the Fermi function chopping off the peak of $A$, in
addition to resolution effects.
This can readily be seen in the data, as will be discussed in
connection with Fig.~5(c) below, and can be corrected for under favorable
circumstances. The second problem with method (A) is that it may
be difficult to use in cases where the dispersion is very weak, as
for instance near the $(\pi,0)$ point in the cuprates.
We should note that with sufficiently fine ${\bf k}$-sampling,
these problems are minimized, as shown recently by us in
ref.~\onlinecite{HELEN}. However, as we show here, even in the
absence of such data it is possible to make progress.

We turn to the symmetrization method (B)
which allows us to overcome both the limitations of the method (A).
This method was originally introduced by us \cite{NORMAN98}
as a means of ``dividing out the Fermi function'' from the EDC and directly
infer the spectral function $A$. In order to determine $A({\bf k}_F,\omega)$
one had to assume particle-hole symmetry on a low energy scale.
However, we now use this idea for a different purpose, namely
Fermi surface determination. As we show below we {\em do not need
any assumptions about p-h symmetry to determine} ${\bf k}_F$.

For an arbitrary ${\bf k}$, we define the symmetrized ARPES intensity by
\begin{equation}
I_{\rm sym}(\textbf{k},\omega)
= I(\textbf{k},\omega) + I(\textbf{k},-\omega)
\label{symm0}
\end{equation}
For simplicity, we will ignore resolution effects here,
and we use eq.~(\ref{intensity}) on the right hand side of
eq.~(\ref{symm0}).
(The effect of energy and momentum resolution
convolutions are discussed in the Appendix.)
Our goal is to use $I_{\rm sym}$ to determine
${\bf k}_F$ at which $A$ has a maximum at $\omega=0$.

Analyzing the symmetrized intensity about $\omega=0$, for any
${\bf k}$, we see that $dI_{\rm sym}/d\omega(\omega=0) = 0$ and
\begin{equation}
\left. \frac{d^2I_{\rm sym}}{d\omega^2}\right|_{\omega=0}=
\left. \frac{d^2A}{d\omega^2}\right|_{\omega=0}
-\left. \frac{1}{T}\frac{dA}{d\omega}\right|_{\omega=0}
\label{symmt}
\end{equation}
For ${\bf k} \ne {\bf k}_F$,
the second term on the right hand side dominates
at sufficiently low temperature.
For an {\em occupied} state, it is easy to see that
$dA/d\omega(\omega=0) < 0$, so that $d^2I_{\rm sym}/d\omega^2(\omega=0)$ is
positive. Thus the symmetrized intensity will exhibit a local minimum,
or a dip, at $\omega$=0 for an occupied k-state.
Conversely, for an {\em unoccupied} state $dA/d\omega(\omega=0) > 0$,
and the left hand side of eq.~(\ref{symmt}) is negative,
leading to a local maximum in the symmetrized intensity at $\omega$=0.
Precisely at ${\bf k} = {\bf k}_F$, the spectral function
has a maximum at $\omega = 0$. Thus
the second term of eq.~(\ref{symmt}) vanishes and the first term
leads to $d^2I_{\rm sym}/d\omega^2(\omega=0) < 0$ yielding a peak
or local maximum in $I_{\rm sym}$ at $\omega$=0.

In practice, symmetrization is used to determine the Fermi crossing
${\bf k}_F$ as follows.
We symmetrize {\it all} EDCs along a cut in ${\bf k}$-space and
identify ${\bf k}_F$ as the boundary in momentum space
between where symmetrized data have a dip (local minimum)
versus a peak (local maximum) at $\omega=0$.
This will be demonstrated in detail below,
where the symmetrization estimate for ${\bf k}_F$
from ARPES data is also compared with other estimates, wherever
possible, and found to agree.

Before turning to the data (in the next Section), we show one example of
a simulation which illustrates symmetrization
with resolution effects included.
In Fig.~16 we plot symmetrized intensities for five ${\bf k}$ points
along a certain cut in ${\bf k}$-space.
We see that for {\it occupied} states, corresponding to the first
two ${\bf k}$ points, the spectral functions peak at $\omega < 0$, and
thus the corresponding symmetrized spectra show dips, or local
minima, at $\omega = 0$. As ${\bf k}_F$ is approached, this minimum gets
shallower. At $k_F$, half the peak is being chopped off in the EDC.
Therefore, upon symmetrization it is completely restored.
(Note we did not build in any matrix element effects in this simulation).
Once you go beyond ${\bf k}_F$, as in the last two curves,
more than half the peak is chopped off in the EDC, so there is an
intensity drop in the symmetrized plot.
In addition, one observes that unoccupied states continue to
exhibit a peak at $\omega=0$ in their symmetrized intensity in Fig.~16,
as can be understood from the preceding discussion.

We note that symmetrization makes precise, or formalizes, a
rough criterion often used by ARPES practitioners:
at ${\bf k}_F$ a vertical
line through the chemical potential intersects the midpoint (half
the maximum intensity) of the leading edge of the EDC \cite{GAPS}.
It should be emphasized a very precise determination of the
chemical potential ($\omega = 0$) is necessary to determine
${\bf k}_F$ via symmetrization.

Finally we turn to another method (C) devised by us\cite{NK,PH}, based on
the sum rule Eq.~(\ref{nofk}) relating the energy-integrated ARPES
intensity
to the momentum distribution $n({\bf k})$.
In principle, the rapid variation of $n({\bf k})$ offers a very direct
probe of the Fermi surface, which is again not restricted to Fermi
liquids.
(The $T=0$ momentum distribution
for known non-Fermi liquid systems, such as Luttinger liquids in
one dimension, shows an inflection at ${\bf k}_F$.)
In several cases we have demonstrated the usefulness of this method in
our earlier work \cite{NK,PH} where ${\bf k}_F$ was estimated from the
location of $\max \left| {\nabla _{\bf k}n({\bf k})} \right|$.
The same method has also been successfully used later by other authors
\cite{HUEFNER2,SCHABEL}.

However, there is an important caveat to keep in mind: one does not measure
$n({\bf k})$, but rather the integrated intensity
\begin{equation}
I(\textbf{k})=\int_{-\infty}^{+\infty}d\omega I(\textbf{k},\omega)
= I_0({\bf k};\nu;{\bf A}) n({\bf k}).
\label{integratedI}
\end{equation}
Potential problems for Fermi surface determination can, and do,
arise from the ${\bf k}$-dependence of prefactor $I_0({\bf k};\nu;{\bf A})$
due to ARPES matrix elements. In Sections VII and VIII below we
discuss in detail how to distinguish ${\bf k}$-dependences of
the integrated intensity coming from $n({\bf k})$ and from the
matrix elements.

To conclude our discussion of Fermi surface criteria, we note that we will
restrict ourselves here to the normal state. We will {\it not} discuss in
this
paper
the notion of the ``minimum gap locus'' \cite{PH,UDFS} in a gapped state
(either below $T_c$ or in the pseudogap regime) which is a measurement of
the underlying Fermi surface that got gapped out.
We also mention, for completeness,
two other methods of Fermi surface determination, which we
will not discuss in this paper. The first one exploits the
approximate sum rule \cite{NK} that $dn({\bf k}_F)/dT = 0$, i.e.,
the integrated intensity at ${\bf k}_F$ is independent of temperature.
The second method uses the constant $\omega$ scan, or momentum
distribution curve (MDC), as a function of ${\bf k}$, at $\omega = 0$;
see ref.~\onlinecite{KAMINSKI2}.

\section{Fermi Crossing from symmetrization}

We will show below that the symmetrization method provides a simple
and general way of determining a Fermi crossing, even when the
dispersion is very small. As discussed above, we will
identify ${\bf k}_F$ as that ${\bf k}$ for
which the symmetrized data {\em first} shows a clear peak at the chemical
potential
($\omega = 0$).

We begin with the simplest Fermi crossing along
the zone diagonal, $(0,0) \to (\pi,\pi)$, where the electronic
structure shows rapid dispersion. The data for a Bi2212-OD88K sample
shown in Fig.~5(a) were obtained with an incident
photon energy of 22 eV and polarization ${\bf A}$
parallel to the $(0,0)-(\pi,-\pi)$ axis. These $T= 180$K spectra are
extremely broad, and not at all consistent with a Fermi liquid
picture \cite{KAMINSKI,KAMINSKI2}.
We first determine the Fermi crossing ${\bf k}_F$ using the
dispersion of the observed peak.
In Fig.~5(c) we plot the observed peak positions of the
data in (a). One can see that in
the narrow ${\bf k}$ interval plotted in Fig.~5(c), the dispersion is
linear over much of the range, deviating from linearity due to
the effect of the Fermi function (as discussed above).
The extrapolation of the linear part crosses the chemical potential at
$\textbf{k}=(0.375\pi,0.375\pi)$, which is then the estimated ${\bf k}_F$
from the dispersion.

In Fig.~5(b) we plot the symmetrized intensities
$I_{\rm sym}(\textbf{k},\omega)$ obtained from the data of Fig.~5(a). From
this we see that the inferred spectral function
is peaked at $\omega=0$ also at $\textbf{k}=(0.37\pi,0.37\pi)$.
We thus find that in this case the dispersion and symmetrization
methods give identical ${\bf k}_F$ estimates.

In the following we will first look at the simpler case of Bi2201.
In the Pb-doped Bi2201 compounds there are no observable
complications arising from umklapp bands as in oxygen-doped Bi2201 or
Bi2212 (which is discussed in more detail in Section IX).
We now move along the Fermi surface in Fig.~6, where the plotted data
were obtained for an OD23K sample with an incident
photon energy of 22 eV and polarization ${\bf A}$
parallel to the $(0,0)-(\pi,0)$ axis. One can
observe a clear trend: as the $(\pi,0)-(\pi,\pi)$ line is approached,
the dispersion becomes very small. In the vicinity of the
$(\pi,0)$ it becomes very difficult to use the dispersion criterion
to determine ${\bf k}_F$. Nevertheless, symmetrized data provide
completely unambiguous results as can be seen from Fig.~7. From the
top panels (a,b, and c) of Fig.~7 one can determine precisely the Fermi
crossing
from the ${\bf k}$-point at which the spectral function inferred from
symmetrized data first peaks at the chemical potential.

Two points should be noted about the ${\bf k}_F$ estimate from
symmetrization. First, just before approaching ${\bf k}_F$ from the occupied
side,
we expect that the symmetrized data will show two peaks and a small dip,
which would be broadened by resolution into a rather flat topped
symmetrized  spectrum.
Second, there should be an intensity drop upon crossing ${\bf k}_F$
in the symmetrized spectrum, assuming that the matrix elements are not
strong functions of ${\bf k}$. Both of these effects are clearly
seen in the data.

It is equally important to be able to determine the {\it absence} of a
Fermi crossing in a cut in ${\bf k}$-space as shown in Fig.~7(d,e).
In this respect, the raw data along $(0,0)-(\pi,0)-(2\pi,0)$ is quite
difficult to interpret, since the ``band'' flattens while approaching
$(\pi,0)$, and remains extremely close to $E_F$.
Nevertheless, it is simple to see that at no point along this cut do
the symmetrized data show a peak centered at $E_{f}$, thus establishing
the absence of a Fermi crossing along this cut.

The important conclusion from this discussion is that for
Pb-doped Bi2201 one can
continuously follow a Fermi surface contour which traces a hole barrel
centered at $(\pi,\pi)$. Analysis of a large set of data using the
symmetrization method allows an unambiguous determination of the Fermi
surface crossing with high accuracy, even in the unfavorable case of
broad peaks with weak dispersion.

\section{Matrix elements}

As discussed at the end of Section V, great care must be used
in determining a Fermi crossing from
the integrated intensity which is the momentum distribution $n({\bf k})$
multiplied by the prefactor $I_0({\bf k};\nu;{\bf A})$. A
loss of integrated intensity as a function of ${\bf k}$ can arise
either from a drop in $n({\bf k})$ related to a Fermi crossing, or
from the ${\bf k}$-dependence of the matrix elements in $I_0$.

One possibility is to have {\it a priori} information about the
matrix elements from electronic structure calculations \cite{BANSIL}.
But as we now show, even in the absence of such information, one
can experimentally separate the effects of a strong ${\bf k}$-variation
of the matrix element from a true Fermi surface crossing.
The basic idea is to exploit the fact that changing the incident
photon energy one only changes the ARPES matrix elements and not
the momentum distribution of the initial states.

In Fig.~8(a) we shows cuts along $(0,0)-(\pi,0)$ and $(\pi,0)-(\pi,\pi)$
obtained at a photon energy of 34 eV (to be contrasted with the data in
the previous two figures for the same sample at 22 eV). The symmetrized data
are shown in
Figs.~8(b) and (c) from which we see results entirely consistent with those
obtained at 22 eV. There is no Fermi crossing along $(0,0)-(\pi,0)$
since the symmetrized data in Fig.~8(b) never show a peak at $\omega=0$.
Turning to the $(\pi,0)-(\pi,\pi)$ direction, the symmetrized
data (Fig.~8(c)) do show a Fermi crossing occurring at k=0.12$\pi$,
in agreement with the data obtained at 22 eV (k=0.14$\pi$).

However, there is an important difference between the
data sets at 22 eV and 34 eV photon energies, which can be
appreciated in Fig.~9, where the integrated intensity along the two
directions is displayed.
While at 22 eV the maximum intensity occurs
close to $(\pi,0)$ and decreases both towards $(0,0)$ and $(\pi,\pi)$,
the data taken at 34 eV show a strong depression of intensity on
approaching $(\pi,0)$, resulting in a shift of the intensity maximum
away from $(\pi,0)$. This loss of intensity cannot be interpreted
as a Fermi crossing, since at no point in the symmetrized data from
$(0,0) \to (\pi,0)$  (Fig.~8) is there a peak centered at $\omega=0$.
(In fact, this loss of intensity close to $(\pi,0)$ is responsible
for the reduced signal-to-noise ratio in the 34 eV data in Fig.~8
compared with the 22 eV data in Figs.~6 and 7).

We would like to attribute this loss in intensity around $(\pi,0)$
at 34 eV, and in fact the entire variation seen in Fig.~9(a), to
strong ${\bf k}$-dependent matrix element effects.
A direct proof is found from the data:
the EDCs at the same point in the Brillouin zone obtained at
the two different photon energies exhibit exactly the
same lineshape, i.e. one can be rescaled onto the other
as shown in Fig.~9b.

We emphasize that the results of Fig.~9 imply that the photon
energy dependence of the ARPES data is {\it not} a $k_z$ dispersion
effect. If this were the case, different incident photon energies would
be probing initial states with different $k_z$ values.
However, the scaling of Fig.~9(b) proves that it is the {\it same}
two dimensional ($k_z$-independent) initial state which is being probed,
and the photon energy dependence arises from the different
final states that the matrix element couples to.

To further illustrate the role of matrix elements, we present a model
calculation and compare it
with the data of Fig.~9. The purpose of this exercise is to determine
whether or not the data are consistent with a matrix element variation with
${\bf k}$.
In Fig.~10(a) we use dotted lines to
show the dispersion of a model ARPES spectrum along $(0,0)$
to $(\pi,0)$ where the ``band'' approaches $E_F$ near $(\pi,0)$
without a Fermi crossing.
(The dispersion is chosen from the tight binding fit
of Ref.~\onlinecite{NORM95} to ARPES data on Bi2212 for illustrative
purposes
even though we will compare it to data on Bi2201).
The full curves in Fig.~10(a) show the effect of the matrix element
variation on the model spectra, simulated by $I_0({\bf k}) =
\sin^{4}(0.6k_{x})$ along $(0,0)$ to $(\pi,0)$.
This is a simple phenomenological matrix element (squared)
which satisfies the following properties:
it vanishes at the $\Gamma$ point, as dictated by
symmetry, and then has non-monotonic behavior along
$\Gamma$-M with a peak away from the M point, similar
to that obtained by detailed band-structure
calculations \cite{BANSIL}. We emphasize that,
beyond this, no deep meaning should be attached to
the simple analytical form used.

Fig.~10(b) shows the momentum dependence of the
intensity integrated over the (large) energy range of -350 to +50
meV as a solid line, and over the (narrow) energy range -50 to +50 meV as
a dashed line.  One can see that this simple example shows remarkable
agreement with the measured intensity using the same integration
ranges, shown in Fig.~10(c). These results are easy to understand. The
data with the large integration range first increase in intensity
simply following the matrix element variation. The data with
integration over a small energy range however shows a different
intensity behavior. They start to increase for k-values higher
than 0.65$\pi$ because only when the peak is closer to the Fermi energy
does it contribute to the integrated intensity. It then again
decreases rapidly because of the strong decrease of the matrix
element, as in the previous case.

However, not recognizing the role of
matrix elements, some authors have ascribed the differences between
the 22 eV and 34 eV photon energy data to
additional ``mysterious'' states around $(\pi,0)$ \cite{DESSAU99}.
We feel that there is no necessity to invoke such states or
to assert changes in Fermi surface topology with photon energy.
In conclusion, all our data on Bi2201 when analyzed using the methods
described above indicate a large Fermi surface centered around $(\pi,\pi)$
\textit{independent of the photon energy}.

\section{The Momentum Distribution}

Despite the matrix element issues discussed above, it is
nevertheless interesting to study the integrated intensity
(Eq.~(\ref{integratedI})) for a dense data set in the entire zone.
The results obtained are shown in Fig.~11 for an OD0K sample.
In the top panels Figs.~11(a) and (c) we show
the integrated intensity $I({\bf k})$ around the $(\pi,0)$ point
obtained at two different photon energies: 22 eV and 28 eV respectively.
In the lower panels Figs~11(b) and (d), we plot the magnitude of
the logarithmic gradient: $|\nabla_{\bf k}I({\bf k})|/I({\bf k})$
which emphasizes the rapid changes in the integrated intensity.
The logarithmic gradient filters out the less abrupt changes in
the matrix elements and helps to focus on the intrinsic variations
in $n({\bf k})$. This can be seen from the fact that the integrated
intensities in the top panels are quite different for 22 eV and 28 eV,
while the logarithmic gradients are much more similar.

The Fermi surface can be clearly seen as two high intensity arcs
curving away from the $(\pi,0)$ point. Modulo matrix element effects,
the results obtained at the two different photon energies are
quite similar. Moreover, the Fermi surface estimated by this method
is in good agreement with the one obtained from the symmetrization
analysis above.

\section{The Fermi surface of B\lowercase{i}2212}

We now turn to Bi2212 where, in addition to all the issues
discussed above for Bi2201, there is
an added complication due to the presence of
umklapp bands arising from the superlattice modulation
with wavevector ${\bf Q} = (0.21\pi,0.21\pi)$ in the BiO layers.
Another difference with Bi2201, which we will not address here, is that
Bi2212 has a CuO$_2$ {\it bilayer}; we only mention that no bilayer
splitting is observed in the ARPES data on Bi2212 as discussed in detail
in Ref.~\onlinecite{BILAYER}.
In this Section we will first review the effect of the superlattice
on the electronic structure probed by ARPES, emphasizing the usefulness of
polarization selection rules.
In an earlier letter \cite{BILAYER}, we had shown data along the
principal axes of Bi2212. Here we present cuts
throughout the Brillouin zone, analyzed using the symmetrization and
integrated intensity methods discussed above, together with a detailed
study of the photon energy dependence of the matrix elements.
All of these new data and their analysis substantially strengthen our
earlier conclusions. (See also ref.~\onlinecite{HELEN}).

In Fig.~12 we show the electronic structure from EDC peaks
(in panel (c)) and Fermi surface crossings (in panel (b)) determined
from data at incident photon energies of 19 and 22 eV for an OD87K
sample\cite{BILAYER}.
The dark lines in the bottom panel of Fig.~12 are a fit of
the intrinsic planar CuO$_2$ electronic
structure, $\epsilon_{\bf k}$, which we call the ``main band''; see
Ref.~\onlinecite{BILAYER,NORM95} for details. The lighter lines
are simply obtained by plotting $\epsilon_{\bf k \pm Q}$,
where ${\bf Q}$ is the superlattice vector, and it is very important
to note that these lines provide an excellent description
of the data points that do not lie on main band. The data strongly
suggest \cite{SLDIFF} that these additional ``umklapp bands''
arise due to diffraction of the outgoing photoelectron through
the BiO superlattice, which leads to ``ghost'' images of the electronic
structure at $\epsilon_{\bf k \pm Q}$.

     From the point of view of the present discussion of the
Fermi crossings, it is very important to establish conclusively
that the crossings U4 and U5 along $(0,0)$ to $(\pi,0)$
shown in the middle panel of Fig.~12 correspond to umklapp
``ghost'' images and {\it not} to the ``main band''.
The case of the Fermi crossing U5, closer to $(0,0)$, is unambiguous
since it is just obtained from following the dispersion of the EDC peaks
which clearly fall on the umklapp band dispersion in Fig.~12(c).

The Fermi crossing U4 requires more care since the umklapp and main
band dispersions are almost degenerate in the vicinity of $(\pi,0)$.
To disentangle these two contributions, we exploit the polarization
selection rules discussed in Section IV.
In Fig.~12(c) we use filled circle symbols to denote data obtained
in an odd geometry, i.e., initial state odd under reflection in the
corresponding mirror plane, and open circles to denote even geometry.
We see from the dispersion plotted in Fig.~12(c) that the main
band signal is seen in the even geometry, since it is a dipole-allowed
transition. (Actually in this polarization both the main and the umklapp
bands should contribute but in the 17 -- 22 eV photon energy range the
main band intensity is much larger than that of the umklapps.)
However in the odd polarization (filled circles along $(0,0)$ to
$(\pi,0)$) the main band is dipole-forbidden and thus the weaker
umklapp band, which does not have any symmetry restrictions here,
dominates in this geometry. From the dispersion, and in particular the
polarization geometry in which it is observed, we clearly see that
the Fermi crossing U4 must be associated with the umklapp
Fermi surface, and {\it not} with the main band.

Fig.~13 shows various cuts at a photon energy of 22 eV for Bi2212-OD87K.
The Fermi crossings are determined using the symmetrization method
and the Fermi surface is found to be a hole barrel
centered at $(\pi,\pi)$.
Notice that in each cut umklapp bands (broken
lines) can be identified. The labels U1 to U3 correspond to particular Fermi
crossings of the umklapp band as shown in Fig. ~12.
In particular along the $(0,0)$ to $(\pi,\pi)$ direction (right-side panels)
the U1 and U2 crossings of the umklapp bands can be
clearly observed.

These umklapp bands are also responsible
for the $\omega=0$ peaks observed in the symmetrized data beyond
${\bf k_F}$ in the other panels.
The new data, obtained at much higher density in the zone, allows us to
directly visualize the main Fermi surface, together with the ghost
Fermi surfaces due to umklapp bands, using the same procedure as in
Fig.~11. The magnitude of logarithmic gradient
$|\nabla_{\bf k}I({\bf k})|/I({\bf k})$ for Bi2212-OPT90K
over part of the Brillouin zone in the Y quadrant (defined
in Fig.~12) is shown in Fig.~14.
From the intensity pattern in this plot, one can clearly see
the main Fermi surface in the middle,
which is a large hole-like barrel,
and also one of the umklapps (U3 in Fig.~12 notation).
The other umklapps are weaker in intensity, but would be visible in the
figure if a log intensity scale had been used.  It is quite satisfying to
see indications of all the features deduced earlier
(using other methods) in the plots obtained using the straightforward
logarithmic gradient method.

The analysis of the photon energy dependence of the ARPES data in
Bi2212 shows a similar trend to the one observed in Bi2201. As an
example of this, we plot in Fig.~15 the
photon energy dependence of the ratio $I(0.7\pi,0)/I(\pi,0)$
of the energy integrated intensity measured at $(0.7\pi,0)$ and $(\pi,0)$
for OD88K.
Both these ${\bf k}$ points are inside the occupied part of the
zone, and we would not expect the momentum distribution $n({\bf k})$
to vary significantly from one point to the other.
Thus, following Eq.~(\ref{integratedI}), any significant deviation of
this ratio from unity as a function of incident photon
energy must be attributed to the matrix elements.
While around 20 eV, $I(0.7\pi,0)/I(\pi,0)$ is close
to unity indicating a small ${\bf k}$-dependence of the matrix elements
in this part of the zone, the ratio peaks to about
2.5 at 38 eV, signaling the suppression of intensity around $(\pi,0)$,
similar to what is seen in Bi2201.
Even more interesting is the observation that around 54
eV the $I(0.7\pi,0)/I(\pi,0)$ becomes much {\it smaller} than one.
Fig.~15
illustrates once more how dangerous it would be to infer a Fermi
crossing from intensity variations alone.

We have further observed \cite{HELEN} that at photon energies close to
30 eV, where the main band is strongly suppressed around $(\pi,0)$,
the superlattice contributions are in fact strongly enhanced in this
region.
The reason is that these superlattice intensities originate from regions
of reciprocal space where the matrix elements are less suppressed.

To summarize, it is very important that
the superlattice contributions be differentiated from the main band
using polarization selection rules, and all Fermi crossings carefully
checked by a combination of symmetrization analysis together with careful
studies of the integrated intensities as shown above.

\section{Conclusions}

In this paper we have carefully enunciated the criteria to
be used in determining the Fermi surface from ARPES data.
We have illustrated these ideas using data on two high $T_c$ copper oxide
based materials, Bi2201 and Bi2212. However we believe that
these methods should prove to be useful for a large class of
quasi 2D materials.

The high $T_c$ materials are hard to analyze because of the
absence of sharp quasiparticle peaks in their normal state,
and anomalously weak dispersion in parts of the zone.
Nevertheless, the symmetrization method discussed in this
paper is able to deal with both these issues. It effectively
removes the Fermi function from the EDC and determines
${\bf k}_F$ as that point in ${\bf k}$-space at which the
spectral function peaks at the chemical potential.

It is very useful to supplement this analysis with
studies of the momentum distribution, however one has
to be very careful about matrix element effects.
We show that by analyzing data at different incident photon energies,
one can unambiguously distinguish between loss of integrated intensity
arising from matrix element variations from that due to genuine structure
in
$n({\bf k})$. In this connection we have also shown the
usefulness of studying the gradient of the logarithm of
the integrated intensity.

We emphasize that not recognizing the role of matrix elements can lead to
paradoxical conclusions like changes in Fermi surface topology with photon
energy, which of course makes no sense. Some authors have ascribed the
differences between the 22 eV and 34 eV photon energy data to
additional states around $(\pi,0)$ \cite{DESSAU99,SHEN99}.
We find that there is no necessity to invoke such states.

Bi2212 has an additional complication arising from the effect
of the BiO superlattice modulation on the ARPES data. However,
as we argue above, the Fermi surface crossings arising from
``ghost'' images (superlattice umklapp bands) can be
clearly differentiated from the intrinsic planar CuO$_2$
Fermi surface by exploiting polarization selection rules.

In conclusion, all our data on Bi2201 and Bi2212
when analyzed using the methods
described above indicate a single large Fermi surface centered around
$(\pi,\pi)$ \textit{independent of the photon energy}.

\acknowledgments

We thank Jim Allen for discussions which led us to sharpen some
of the arguments presented in the Appendix.
This work was supported by the National Science Foundation DMR 9624048,
and DMR 91-20000 through the Science and Technology Center for
Superconductivity, the U. S. Dept. of Energy, Basic Energy Sciences,
under contract W-31-109-ENG-38, the CREST of JST,
the Ministry of Education, Science, and Culture of Japan, and
the Swiss National Science Foundation. MR is supported in
part by the Indian DST through the Swarnajayanti scheme.

redpajama_set_name: RedPajamaArXiv


I've become a bit discouraged with watercolor skies, so I decided to revisit portraits for a bit. (See my other Famous Faces at this link.) It's probably been over two months since I've done one, but I do enjoy them. It was a pleasure staring into this man's eyes, I'll tell you that.
Please feel free to guess, as I'm holding all comments in moderation. If you guess and do not see your comment appear in a few hours or so, then you've guessed correctly! 😀 I'll reveal everyone's comments/guesses later on this afternoon.
I don't want to say too much about this person, and spoil the guessing game. Suffice it to say that he is a person whom I admire deeply, but not (only) for the obvious reasons. I simply admire him as a human being. I'm looking forward to reading a biography about him. If anyone can recommend one, I'd love to hear about it.
I got a big hand up from the raw umber Polychromos colored pencil. I really cannot recommend these pencils highly enough. This was done in a Strathmore spiral-bound drawing sketchbook. The paper is nice, but only nice. The plan was to sketch him there and then do the final in a Stonehenge sketchbook (perfect marriage of paper and medium). However, I liked the sketch so much that I decided to go with it, instead. I may revisit him in a different pose; he was (and is) a joy to stare at, in my opinion.
It is a very good likeness! so that was easy. I envy you, I am hopeless – cannot even capture a likeness for my husband and I see him every day!! lol!
I think people we know are more difficult though! If you're really interested in portraiture, I'd say try someone like Clint Eastwood, who's fairly easy to recognize. I've tried drawing my husband too and it never works out either.
…..it was immediate recognition once I looked into The Holy Father's beautifully-rendered eyes.
Thank you, Lance. You guessed correctly. I'm so glad you enjoyed this.
I don't know who he is but it's a very good portrait – subtle and delicate.
Thanks, Nicola! I appreciate it. I enjoyed doing it. I've been missing portraits (without realizing it).
Wonderful portrait, beautiful warm expression. And you're right, great eyes.
Brown pencil just seems to work well with people.
Thank you! Yes, all of the lighter browns seem to work well. This one is erasable, and I did need that in this case, as I freehanded it. Had to extend the nose quite a bit in the final analysis, so I'm glad I didn't choose a darker brown. I love his eyes. I'm glad you like this one.
LOL, Thank you, CAS! You guessed correctly! He was so much fun to draw, I think because of that sparkle. He has such a good aura around him, I don't know how else to describe it. I'm glad you liked it!
Likewise, I don't recognise the face, but you've captured his character very well indeed. Well done!
Thank you, Margaret! He was a pleasure to draw. I'm looking forward to revisiting him.
Just back from a rainy graduation party. WIne time and chillin' More graduation parties tomorrow….
Thank you, Teresa! As always, you make me feel like a million bucks! You guessed right! I think I'm better at stormy skies, but fair-weather, puffy-cloud skies are a whole other matter. I'll keep trying though. I don't give up quick!
You got it! 😀 He was wonderful to draw.
Nice rendering, especially of his radiant smile and eyes. Is it Pope Francis?
It is! Thank you! I'm looking forward to drawing him again. So much glee behind his eyes; it was a real pleasure.
Thanks, Jennifer! It was fun. I need to do more.
Yes good to move on to something new when bored glad to see some of your portraits!
Great likeness, and I agree, admirable not for the obvious reasons. How much time do you spend on a portrait sketch, approximately? Seems like you whip them out.
Thank you, Jo. Usually, about an hour. I think this one took closer to two, because I've not done one in awhile and also because I kept seeing things I wanted to fix. Other than MLK, I've never cared so much about getting a portrait right. There are still things I could fix about this, but I would've been fussing all night and finally decided to stop fussing. I'd like to revisit him instead. I do plan to revisit MLK as well. I've found the key to a good portrait is to choose someone I really admire. And it helps if staring into their eyes…..you see something inspirational there. If that makes any sense. I'm glad you enjoyed this.
Lovely portrait of the Pope – do like the expression and likeness.
Thanks, Mary! I'm happy you like it.
Nice portrait of Pope Francis, Laura. Your soft marks convey your sense of his serene nature. Good portraits focus on the triangle of the eyes and mouth, and you captured it nicely. They also say the most identifiable feature of a person is the shape of the head – that if you don't get that right none of the details matter. Looks to me like you nailed it. I saw the Pope immediately. Compelling portrait!
Thank you so much, Tom! You are correct, and I learned a few things from your very kind & informative comment! I didn't realize the shape of the head was so important. I'm really glad you mentioned that. I'll try and keep that in mind for future portraits. I do enjoy them. Thanks again!
You got it, Rebecca! Thank you!
Great drawing of the Pope.
Thank you, Jean. I was hoping you would like this one. I never know if a portrait will turn out when I sit down to do one, but I'm really glad this one did. Glad you enjoyed it. Gollum liked it, too! Lol!
Stupid me. I thought it was the drummer for the Stones.
Francis! I like him! He's just what the world needs right now!
Thank you so much, Fabio! You're very kind! He is a wonderful subject.