text:
Home | Programming Resources | * General FAQ * HTML for the Conceptually Challenged * Home Page Builder 2.0 * The CareWare Idea Anti-Spam Electric Duet Excel 7.0 Spreadsheets Home Assistant / Insteon Rescue Home Automation Home Page Builder 2.0 — Help I Dumped Insteon JavaScript Signal Generator / Spectrum Analyzer Perl / CGI Scripts, C++ Programs Phase-locked loops Regular Expression Laboratory Speak & Search Speech to Text TCP/IP Utilities Version 2.0 Where did my files go? Share This Page
Home Assistant / Insteon Rescue

How to configure Home Assistant and survive Insteon's collapse

Most recent update:

(double-click any word to see its definition)

Update
Since writing this article, I have dumped Insteon and installed all Z-Wave equipment instead (a big project). I did this because it looked like no one was going to step up and build new Insteon parts, and I had a few irreplaceable and aging parts that would completely disable my system if they failed. Read the details.
Introduction

In mid-April 2022 the Insteon company abruptly ceased operations. Without warning the Insteon Website went dark and their customers found themselves cut off from some parts of their Insteon-based home automation systems.

In this article I'll show Insteon system owners how to rescue their systems, either by making their Insteon gear work again, or by transitioning to a different home automation technology, or both at once.

The key to this rescue is called "Home Assistant". It's a free, open-source, widely used environment that can integrate home automation elements from different manufacturers into a coherent system that's easy to use and is much better than the original Insteon software.

A little Background

The home automation business is a bleak landscape with many recent casualties: Charter/Spectrum, Connect, Iris, Revolv, SmartThings, Wink — and now Insteon. It may not be obvious why this business is so lethal to its players, but with a little thought, the reasons become apparent:

• Home automation's goal is to drag a pre-technological household into the 21st century — automate and control appliances and lights, make them respond to their owner's whims, and maybe provide security as well.
• This goal would be much better served if different companies' system talked to each other, but that is just not so, in fact it seems the players do all they can to freeze out their competitors, refuse to cooperate by sharing technology or by agreeing on a common communications protocol.
• This refusal to cooperate results from the famously cutthroat American corporate culture, in which stock market investors see a given market segment as a zero-sum game, in which every Insteon customer is lost to Blink or vice versa.
• To meet its customers' needs, home automation should have a shared, cooperative communications protocol, but this necessity collides with the equities market, where competition is everything. To date there's no realistic plan to resolve this conflict.
• This is where Home Assistant steps in. Home Assistant is an application, often run as a separate guest environment running on a host, that allows different technologies to talk to each other in spite of the frankly obstructionist motives of their corporate creators.

My prior Insteon article set

Years ago, when Insteon was the "new thing", I wrote a comprehensive article set that explained how Insteon worked and presented a step-by-step installation tutorial. That earlier article set might be worth reading as this discussion unfolds.

Phase 1: Gather Information

Here's a typical (pre-Home Assistant) Insteon home automation diagram to guide this phase:

Figure 1: Insteon System Diagram

Collect Information from the Insteon Hub

Because Insteon's Internet network has gone offline, many Insteon customers report they can no longer access the administrative console part of their hubs and therefore have lost the ability to reconfigure their systems. In this section I'll describe some remedies, with the goal of preserving the information the Insteon hub contains, not to rely on it long-term. One goal of this project is the complete elimination of the Insteon-compatible hub (from all manufacturers), while at the same time recovering control over the Insteon house-wiring network.

This strategy works because Home Assistant can scan, detect and control your Insteon devices using only your Insteon PowerLinc Modem (PLM) (the green rectangle in Figure 1 above), which remains an essential part of the system.

My Insteon-compatible hub is an ISY994i-pro, (made by Universal Devices) a common hub in an Insteon network. It's my hope that these instructions will apply to similar models including those made by Insteon itself.

Why replace a working hub?

Update: readers online have aked me, since (unlike hubs built by Insteon) my specific hub will continue to work after the Insteon collapse, why not continue using it? My reply is that whenever I can replace hardware with software, I leap at the chance. Here are some of the reasons:

Property Dedicated hardware hub Home Assistant running on a Raspberry Pi
Cost US$250.00 US$42.00 (Home Assistant is free)
Software quality very low excellent
Critical failure modes numerous nonexistent

The reason the failure possibility is rated as nonexistent for the Raspberry Pi running Home Assistant is because (a) reliable backup is child's play, and (b) the Raspberry Pi approach doesn't rely on cloud resources, the vulnerability that doomed all Insteon hubs in mid-April 2022.

Typical hubs in this class provide a local Web page for everyday control and (in the case of the ISY994i-pro) an administrative console that relies on a Java applet for its functioning (for security reasons modern browsers will refuse to display Java applets). The administrative console may be accessible in some cases, but for this process I'll assume it has become inaccessible because of the collapse of Insteon, and as many Insteon users have reported. We can get the information we need just from the Web page, which in most cases remains accessible post-Insteon collapse.

The purpose of this section is to collect the names and addresses of each Insteon control in your network, so you can identify and label those controls in Home Assistant later on. Each Insteon control has a three-part hexadecimal address that looks like this: 0B.CA.55. By collecting a list of these codes in advance, the reader should be able to track and name each control in Home Assistant without guesswork.

For best results, make an organized list of each control and its hexadecimal address, so later on, when Home Assistant detects that control, you can use the hexadecimal address to locate and relabel the control with its everyday name. I recommend creating a spreadsheet with separate columns for everyday name and address, but a plain-text list will work.

Make a list of friendly names and Insteon addresses

Here are step-by-step instructions for this information-collecting task:

• Access the Web page of your Insteon-compatible hub as you have done in the past (this Web page should remain accessible in spite of Insteon's collapse). It should look more or less like this (image from my Universal Devices ISY-994i-pro):

• Click on the "My Devices" link. This will show a list of named devices in your network.

• Now hover your pointing device over each name in the list and notice the information at the lower left of your browser's display — example:

• See the characters "C BF 84 1" in the image above? (This is just an example, your data will be different.) For similar numbers you find on your network, ignore the "1" and add the "C BF 84" part to your list along with the friendly name of the control. Be sure to add a leading zero to address numbers that have only one digit — in this case you would enter "0C BF 84".

• Each list entry should look more or less like this:

 Control Name Address Lower Staircase 1 0C.BF.84
• Do the same for each Insteon control in the Web page list. This list will help you identify your controls in Home Assistant after it automatically scans your network and detects Insteon controls. The point is that Home Assistant can only identify Insteon controls by their addresses, not their names — you need to tell Home Assistant the names you prefer.

• The not-recommended and hard way to get this information is to remove the switch plates from every control in your house and read the original Insteon label that has the control's address. This is a bad idea, lots more work, even risky if you're not familiar with precautions around electrical equipment, and only necessary if for some reason you can no longer access your Insteon hub at all.

Phase 2: Prepare the Insteon PLM

This section's purpose is to eliminate the now-disabled Insteon-compatible hub (the orange rectangle in Figure 1 above) from your home network. The goal is to connect the existing Insteon PLM (the green rectangle in Figure 1 above) directly to a computer running Home Assistant.

For those with an Insteon 2413U:

You're in luck — with little effort, the 2413U model PLM can be directly connected to a computer for use with Home Assistant. The 2413U looks like this:

The connector at the bottom of this PLM is a standard USB connection, so an ordinary USB cable is all that's required to connect to a computer:

For those with an Insteon 2413S:

This task is only necessary if you have the Insteon 2413S, a serial PLM with a network-style RJ45 connector — it looks like this:

The connector at the bottom is a network-style RJ45 with eight conductors. To connect this PLM to a computer without an Insteon-compatible hub, you need to create a somewhat complex cable adaptor that allows connection to a USB serial adaptor. You will need:

• Any convenient length of RJ45 network cable:

• A special RJ45 to DB9 serial adaptor (StarTech GC98FF — Amazon link):

Here's the wiring pinout table for this adaptor:

 RJ45 DB9 Color 1 2 Blue 7 5 Brown 8 3 White

I just built this adaptor setup for my Insteon 2413S for use with Home Assistant, so I know it works.

Insteon Parts Shortage

Some readers may ask, "Wait, is all this engineering necessary? Can't we just buy the much-preferred Insteon 2413U that connects directly to a computer?" The answer, unfortunately, is no — on the news of Insteon's collapse, people quickly bought up all the remaining Insteon PLMs of both kinds, to avoid the parts vacuum we're now facing.

But other players may step in and fill this vacuum ...

Role of Universal Devices

Update: It's been reported that Michel Kohanim, president of Universal Devices (the original maker of my old ISY994i-pro network interface), is in talks with the current Insteon rights holders with the intention of "buying the license or perhaps the whole thing ..." and become a source for Insteon equipment. Kohanim estimates that there are 100,000 Insteon system users out there, a non-trivial marketing oportunity (direct link to the conversation). This is a new event, unfolding at I write, so expect changes and updates.

Phase 3: Install Home Assistant

After Home Assistant has been integrated into the reader's network, it will look more or less like this:

Figure 2: Insteon network plus Home Assistant

The goal in this phase is to create as little disruption as possible. If this phase is carried out correctly, everything in your Insteon network should function as it did before, but with a much better, more comprehensive control system and user interface provided by Home Assistant.

Prepare a tiny computer to host Home Assistant

The best, most reliable way to proceed is this:

• Acquire one of the popular, inexpensive Raspberry Pi tiny computers — should be a Pi 3 or 4 model. It would be wise to also acquire a plastic case for it, for protection. This computer and its case should cost much less than the original Insteon hub it replaces.
• Dowload a copy of the Home Assistant software suitable for your Raspberry Pi, from this page. Be sure to choose a 64-bit version of the software — both the Raspberry Pi 3 and 4 models are 64-bit.
• Using the instructions on this page, flash the software onto a micro USB (preferred) device, or a USB stick, of adequate size (32 GB or larger).
• Plug the micro USB device or USB stick into your Raspberry Pi computer.
• Connect your Insteon PLM to the Pi using the instructions from the prior section.
• Make a network connection:
• Preferred: connect the Pi to your home network using an Ethernet cable (faster and more efficient during the lengthy automatic setup process).
• Alternate: rely on your home wireless network to communicate with the Pi (slower during setup). This approach may require that you connect a keyboard and display to the Pi to be able to enter a WiFi access password if one exists on your network (as it should).
• Power up the Raspberry Pi.
• After 20-30 minutes of automatic setup/configuration, Home Assistant should appear on your home network with the default URL http://homeassistant.local:8123, and you can begin detailed setup and customization. In fact, if all necessary steps have been taken and enough setup time has elapsed, clicking this link should connect you to your local instance of Home Assistant.

Other Options

An alternative install path for Home Assistant is to download a custom virtual machine (VM) version tailored to one of several existing VM manager programs, and install Home Assistant into a user-created VM using VirtualBox, Vmware or Hyper-V, using the instructions on this page. However, the Raspberry Pi install method is easier, more reliable and is much preferred.

Phase 4: Configure Home Assistant

Home Assistant works with home automation products from many different manufacturers, which is one reason I chose it for this project. If parts of your Insteon network fail, if replacement parts become unavailable over time, you can add components from other manufacturers incrementally while retaining some Insteon elements, rather than having to completely dismantle your Insteon network in one go.

At the end of the prior section's activities, plus some time delay for automatic setup activities, your Home Assistant install will become available using this link. This Home Assistant Web page explains the first steps in configuring your Home Assistant. Briefly, the program will ask you to set up a user and password, then establish your location and time zone.

Then the program will list any non-Insteon home automation devices it has automatically detected. If your home network is entirely Insteon-based, the program won't detect any devices yet — for that we need to install a Home Assistant add-on meant for Insteon devices.

Install Insteon integration

Click here for the Home Assistant Insteon instruction page (my instructions below are clearer and easier to follow).

After the preliminaries are complete and you have clicked "Finish", you will see the default Home Assistant home page. To detect your Insteon devices and make them accessible to Home Assistant, you need to install what is called an "integration" meant for Insteon devices:

• Click "Settings", a gear icon at the lower left.
• From the list that will appear, click "Devices & Services" (or click this link).
• You will see a list of currently-installed integrations plus a button at the lower right labeled "Add Integration". Click that button.
• A search dialog will appear that allows you to enter a partial search string to whittle down what will otherwise be a very long list of available integrations. Enter "insteon" as a search term.
• This search should produce a single result named "Insteon". Click that choice and the Insteon integration will be installed.
• Once installed, the Insteon module will search your network for Insteon devices. This is why it's important to have the Insteon PLM connected to the computer running Home Assistant in advance of this step (as shown in Figure 2 above). But don't worry — you can re-scan your network later.

Insteon networks have some quirks. One is that battery-operated Insteon devices (motion devices, remote controls, etc.) need to be put in "pairing mode" (details in each device's literature) in order for the Insteon integration to find and list them. Devices connected to power — switches etc. — don't need this step, but battery-operated devices do.

Force a network re-scan

The Insteon integration will periodically but randomly re-scan your network on an arbitrary schedule, but there's a way to force an immediate rescan. This allows a battery-operated Insteon device to be prepared for detection, followed by an immediate network re-scan. Here's how to do it:

• Click the "Developer Tools" icon (a hammer in recent Home Assistant versions) at the lower left.
• Click the "Services" tab at the top of the display.
• In the "Service" search window, enter "insteon".
• From the resulting list, select "Insteon: Load all link database". This is a manual re-scan that tries to locate any Insteon devices in your network. Warning — this scan can be slow, be prepared to wait a significant amount of time for results.
• Prepare any battery-operated devices for detection, usually by putting them into "linking mode", using the instructions specific to the devices.
• You have the option of deleting and reloading all records, by clicking the "Reload All Records" checkbox. This is useful if a prior scan misidentified one or more Insteon controls.
• Under the "Entity" entry window, type "all". This will scan for any and all Insteon devices — the most productive, but the slowest, scan mode.
• Once you have entered "all" as an entity identifier, the activation button at the lower left labeled "Call Service" will become active.
• Click the "Call Service" button, then walk your dog, or read "War & Peace", something that requires patience.

Give devices friendly names

Once Home Assistant has a list of local network Insteon devices (which Home Assistant called "entities"), you can use them for your home automation activities. But first you should give the entities useful names:

• Click the "Settings" icon (a gear in recent Home Assistant versions) at the lower left.
• Click "Device & Services".
• The Insteon integration will show two links, named "(number) devices" and "(number) entities".
• Click the "(number) entities" link, which will show a list of entities the Insteon integration detected in its searches.
• In this step you will use the list of addresses and friendly names you created earlier, to make the entities easier to recognize and use.
• For each entity in the list, click its row and use your previously compiled cross-reference table to give it a friendly name.
• Each entity will have a device identifier and address, example "switch.on_off_module_2d_33_7a". The device address is the group at the right — in this hypothetical example, "2d_33_7a" would correspond to "2D.33.7A" in your list.
• Find the corresponding address in your list and enter the friendly name you chose in the "Name" entry.

This step is needed because Home Assistant can find Insteon devices, but the devices only have hexadecimal addresses, not the friendly names you gave them in the old Insteon hub software — you must restore that lost information.

At this point your home automation network should be reasonably integrated with the general Home Assistant environment, so most of the methods common to all home automation devices (not specific to Insteon) should work as expected. This means you may benefit from visiting and reading the main Home Assistant documentation pages for configuration tips and methods.

This section describes some Home Assistant add-ons/changes the reader may find useful in advanced configuration and programming activities.

For those who want to customize and control Home Assistant in detail, the most important addition is a way to remotely access its configuration and programming files. For this we need a Secure Shell server, and as it happens, one is available and easily installed.

• Click the "Settings icon (a gear at the lower left).
• At the lower right you will see a button: "Add-On Store". Click it.
• In the search dialog that will appear, enter "terminal & ssh".
• This entry should produce one result: "Terminal & SSH — Allow logging in remotely to Home Assistant using SSH". Click that result.
• A detailed page will open up including an "Install" link. Click the link to install this add-on.
• After the install is complete, an item named "Terminal & SSH" will appear on the "Add-Ons" page. Click the "Terminal & SSH" item to reveal its options.
• Among the selectors on this page will be one called "Show in Sidebar." Select that option.
• At the top of the "Terminal & SSH" display you will see a row of options including "Configuration". Click that option (this gives you easy access to a command-line terminal).
• Among the available configurations are:
• Network/SSH Port: For local network access, enter port number 22 unless you have a compelling reason to use another port.
• Having made these entries, click "Save" at the lower right.
• Now return to the main "Terminal & SSH" access page and click "Restart". This action will make a Secure Shell server available on your local network.
• Let's test this plugin by accessing the Secure Shell server and the Home Assistant instance:
• From Linux (wait ... you're not running Linux?):
• Open a terminal and enter:
• ssh root@homeassistant.local
• (When accessing Home Assistant by Secure Shell, you're always root, so be careful.)
• Enter the password you chose for access.
• At this point you should see a terminal-based Home Assistant welcoming page:
username@servername:~$ssh root@homeassistant.local | | | | /\ (_) | | | | | |__| | ___ _ __ ___ ___ / \ ___ ___ _ ___| |_ __ _ _ __ | |_ | __ |/ _ \| '_ \ _ \ / _ \ / /\ \ / __/ __| / __| __/ _\ | '_ \| __| | | | | (_) | | | | | | __/ / ____ \\__ \__ \ \__ \ || (_| | | | | |_ |_| |_|\___/|_| |_| |_|\___| /_/ \_\___/___/_|___/\__\__,_|_| |_|\__| Welcome to the Home Assistant command line. System information IPv4 addresses for eth0: xxx.xxx.xxx.xxx/24 IPv6 addresses for eth0: xxxx:xxxx:xxxx:xxxx/64 IPv4 addresses for wlan0: OS Version: Home Assistant OS 7.6 Home Assistant Core: 2022.5.1 Home Assistant URL: http://homeassistant.local:8123 Observer URL: http://homeassistant.local:4357 [core-ssh ~]$ |

• To avoid having to enter a password for local logins, log out and issue this command:
• ssh-copy-id root@homeassistant.local
• This command copies your public authorization key onto the server and eliminates the password requirement for future logins.
• From Windows 10/11 (it's okay, you'll figure out why Linux is better):
• Windows doesn't have any built-in Secure Shell abilities or applications. The explanation is simple — Microsoft doesn't own it, it's open-source, so ... forget it.
• But intense user pressure has compelled Microsoft to allow Secure Shell as an optional user-installed component.
• Follow this guide to install Secure Shell on Windows 10. For this specific task, you only need the client component (but a Secure Shell server is a very nice addition to Windows).
• When the install is complete, open a PowerShell instance.
• Enter this Secure Shell login command:
• ssh root@homeassistant.local
• Enter the password you chose for Home Assistant Secure Shell access.
• This login should produce a terminal-based Home Assistant welcoming page in your Windows PowerShell session:
PS C:\Users\username> ssh root@homeassistant.local

| |  | |                          /\           (_)   | |            | |
| |__| | ___  _ __ ___   ___     /  \   ___ ___ _ ___| |_ __ _ _ __ | |_
|  __  |/ _ \| '_ \ _ \ / _ \   / /\ \ / __/ __| / __| __/ _\ | '_ \| __|
| |  | | (_) | | | | | |  __/  / ____ \\__ \__ \ \__ \ || (_| | | | | |_
|_|  |_|\___/|_| |_| |_|\___| /_/    \_\___/___/_|___/\__\__,_|_| |_|\__|

Welcome to the Home Assistant command line.

System information

OS Version:               Home Assistant OS 7.6
Home Assistant Core:      2022.5.1

Home Assistant URL:       http://homeassistant.local:8123
Observer URL:             http://homeassistant.local:4357
[core-ssh ~]\$ |

• If this login should fail, re-check that you have selected a password and specified port 22 for access, and restarted the Home Assistant Secure Shell server.
• Be careful with this root login — learn what purposes it serves and what can go wrong.
• There are many applications for this direct terminal access, to be covered later.

Backing up the Home Assistant configuration

An important advantage to running a Secure Shell server on Home Assistant is that you can make a copy of the current system configuration for safekeeping. Home Assistant has an integrated backup utility, but I prefer to make a separate backup of the configuration against the possibility of a serious failure (example, a failed/failing micro USB storage device on the Raspberry Pi) that might make both the original and any backups inaccessible and require reinstalling Home Assistant from scratch.

As before, there are two approaches to getting GUI file explorer access to Home Assistant, one for Linux, one for Windows:

• From Linux:
• Open a file explorer that has an address bar option for direct user entry of an URL (in Nautilus, a.k.a. Files, press Ctrl+L to show the address bar) and make this entry:
• sftp://homeassistant.local
• If all prior configuration steps have been taken and if Linux has been configured correctly, this should produce a list of the Home Assistant root directory with some identifying directory names: /addons, /backup, /config, and many others, clear indications that you're looking at the Home Assistant filesystem.
• To make a backup of your Home Assistant configuration that will survive even a complete failure of the micro USB card (yes, they do fail), copy the /config directory to another location. This can be accomplished by simply dragging the folder name /config to a suitable backup location.
• Make sure you're making a copy of this important directory, not moving the directory itself or creating a symbolic link, the two other common actions resulting from a drag gesture in a Linux file explorer.
• From Windows 10/11:
• Windows doesn't have any built-in Secure Shell abilities or applications. The explanation is simple — Microsoft doesn't own it, it's open-source, so ... forget it.
• But you can install a third-party Secure Shell-aware file explorer to allow easy file/folder transfers and backups. The best-known is WinSCP — download and install it.
• Having install WinSCP, open the application and enter your login information:
• For host name, enter homeassistant.local.
• For user name, enter root.
• Enter the password you chose for the Terminal & SSH plugin of Home Assistant.
• Click the "Login" button and you should see a list of Home Assistant directories including /config.
• Now you can make a copy of the Home Assistant /config directory to one or more backup locations.

YAML Issues

Most advanced Home Assistant programming is accomplished by way of YAML ("Yet Another Markup Language") files. YAML files are very tricky to get right, even for seasoned programmers, so the Home Assistant automation management module hides the underlying YAML syntax with a "Visual Editor" that allows the user to make simple entries that create many kinds of home automation.

That's fine, until you try to do something that Home Assistant can't do. One of those things is join several house controls into a group that, when any one of the controls is activated, all the others respond in kind. This makes it possible to turn one switch on or off, and the other controls that are members of the group turn on or off also.

Those familiar with Home Assistant may object that Home Assistant can do that — from the Home Assistant user interface. From the user interface, if you activate a control that has been made a member of a group-entity card, all the other controls will respond in kind. That's true, but if you switch one of the controls directly (not by way of the Home Assistant user interface), it won't work.

To address this problem, I wrote the following YAML script. It's the first of several YAML script examples that provide missing abilities and useful functions, things one can't create through the visual editor.

Remember about these script examples that to use them, you need to replace my example entity_id list with entity identifiers acquired from your own Home Automation configuration.

YAML Example 1: Joining Controls

The original Insteon hub administration console had a way to join controls together to make controls function in unison (something called "scenes"), but when I retired my Insteon hub I intentionally erased all links between controls, scenes and all other associations, realizing that if I didn't do it then, I would have to manually clear each control later on in order to change their relationships. I took this step because I knew I would be able to create software associations between controls, which is a much better, more flexible control method than physically programming devices to force a particular relationship.

After some programming and testing I now have a way to join sets of controls using a YAML script that responds to the physical activation of any particular control in a defined group, so the other group members automatically respond in the same way — without needing to do this by way of the Home Assistant user interface. Here's an example automation script that joins controls so they act in concert if any one of them is turned on or off:

alias: join trigger living room
description: ''
trigger:
- platform: state
entity_id:
- switch.switchlinc_relay_0b_ba_1d
- switch.togglelinc_on_off_56_0f_2f
- switch.togglelinc_on_off_58_20_91
action:
- delay: 00:00:01
- service: "switch.turn_{{ trigger.to_state.state }}"
target:
entity_id:
- switch.switchlinc_relay_0b_ba_1d
- switch.togglelinc_on_off_56_0f_2f
- switch.togglelinc_on_off_58_20_91
mode: single

NOTE: After some testing, I edited the above YAML script to include a one-second delay between state detection and activation. Because Insteon networks aren't protected against packet collisions, this delay produced a significant reliability improvement.

Remember again that to use this script you need to replace the entity_id lists with entity identifiers from your own Home Asistant configuration. Also, you need to create a separate automation for each control group you want to act in concert (I have 9 separate control joining scripts in my somewhat complicated home automation setup). And see below for a way to create individual YAML scripts for each automation, which is a way to simplify the management of complex configurations like mine.

This method works and doesn't require the user to activate the control from the Home Assistant user interface (but that remains an option). It responds more slowly than the old Insteon hardware-linking method, but it's a pure software solution, which means no more manual linking of Insteon controls, and easy reconfiguration with a keyboard instead of a screwdriver. It should work with other kinds of home automation controls, not just those from Insteon.

Even though it solves the issue I wanted addressed, this script shows one of the many drawbacks to YAML as a scripting language. Obviously it would be better to list the grouped controls just once and use a variable to refer to them, to avoid having to list them twice, but to date (May 2022) there's nothing like this in Home Assistant.

YAML Example 2: Lights On, Lights Off.

These two scripts turn user-selected lights on around sunset, then turns all lights off at a user-specificed time. This has many practical applications including making your house look occupied when you're traveling.

Lights On

alias: 'Chosen Lights On Near Sunset'
description: Automatically turn on selected lights/devices at a time near sunset
trigger:
- platform: sun
event: sunset
offset: '-00:30:00'
action:
- service: switch.turn_on
entity_id:
- switch.switchlinc_relay_0b_ba_1d
- switch.switchlinc_relay_0b_ca_55
mode: single

A simple script, very reliable. Choose any time offset that suits your needs. Now for its partner:

Lights Off

alias: 'All Lights Off At Defined Time'
description: Automatically turn off all lights/devices at a user-specified time
trigger:
- platform: time
at: '02:00:00'
action:
- service: switch.turn_off
entity_id: all
mode: single

This script uses a special argument 'all' instead of an entity_id list. This special argument applies the defined action to all lights and devices in your network. If all you have are lights and switches that control other similar devices, then 'all' will work, but if you have devices that must run 24/7, you'll need to make a detailed list of entities to which the command applies.

YAML Example 3: Motion Sensor Actions

I have an Insteon motion sensor, battery operated, model 2842-222, looks like this:

Insteon 2842-222 Motion Sensor

This device includes three sensors, separately identified and usable by Home Assistant -- motion, light/dark and low battery. Each of them can be made to trigger a separate action. So you can activate exterior lights for a specified time interval if motion is detected --

alias: Motion Detector Exterior Lights
description: Activate a specified light on detected motion, after sunset
trigger:
- platform: state
entity_id: binary_sensor.motion_sensor_37_b7_82_motion
to: 'on'
condition:
- condition: state
entity_id: sun.sun
state: below_horizon
action:
- service: switch.turn_on
target:
entity_id: switch.switchlinc_relay_dual_band_23_f3_cb
data: {}
- delay: '00:05:00'
- service: switch.turn_off
target:
entity_id: switch.switchlinc_relay_dual_band_23_f3_cb
data: {}
mode: restart

-- and you can alert the Home Assistant operator if the battery is getting weak:

alias: Motion Detector Low Battery Alert
description: Create a persistent alert when the motion detector's battery is low
trigger:
- platform: state
entity_id: binary_sensor.motion_sensor_37_b7_82_battery
to: 'on'
action:
data:
title: 'Home Assistant: Motion Sensor Battery Monitor'
message: 'Low battery detected in motion sensor.'
mode: single

This script creates a Home Assistant alert that would be difficult to miss, upon detecting a low battery condition.

Module Detection Issues

The only problem with the motion sensor module is that it's battery operated, so it's difficult to make Home Assistant detect it. Forcing a detection involves removing the device's battery cover, pressing a button to put the device in pairing mode, then running a Home Assistant Insteon scan for new devices immediately afterward as explained here.

Separate YAML automation scripts

By default Home Assistant uses a single script file for all its automations. This single file is located in the Home Assistant filesystem at /config/automations.yaml. In the file, each automation is separated by a unique identifier number. The file looks more or less like this:

- id: '1651435471354'
alias: Example automation
# remainder of script
- id: '1651520849957'
alias: Another example
# remainder of script

This is acceptable if you have only a few automations and don't need to edit them outside Home Assistant, but as the number of automations increases, and in particular if you have script content that the visual editor can't handle, this single-file scheme quickly becomes unworkable. Here's a way to create one file per automation, stored in a separate directory:

• Log into Home Assistant using the Secure Shell access method described in a prior section.
• Move to the /config subdirectory:  cd /config
• Create a new subdirectory:  mkdir automations
• Now edit the configuration.yaml file:  nano configuration.yaml
• Go to the line that will originally either (a) be not be present or (b) look like this:
automation: !include automations.yaml
• Edit (or add) the line so it looks like this:
automation: !include_dir_list automations
• Save the edited result.
• Create as many individual automation files you want and put them in the new automation directory. You can do this by editing the original automations.yaml file and extracting individual automations (without their ID numbers) and placing them in separate files, each of which should be stored in the new automations directory.
• When you have made these changes and created the new files, reload the automation configuration:
• Click the "Developer Tools" icon (a hammer) at the lower left -- or click this link.
• Under Developer Tools, choose the YAML tab -- or click this link.
• This action reloads the automations configuration and (in this case) will read the files in the new automations directory.
• If while reading your scripts Home Assistant detects any syntax or other errors, it will provide an alert dialog.
• The new automations should be listed here, each with an activation/deactivation control.
• For an installation with a large number of automations these changes represent a better way to organize them, but it prevents use of the automation visual editor. From this point onward you'll be editing the automation files directly using the rules of YAML syntax. But at the point where you need to have individual automation files, the visual editor will likely have become more a hindrance than a help.
Conclusion


text:
Figure 4: Classification results. To quantify the difference in low-level feature statistics across categories (see Figure~\ref{fig:chevrons2}), we used a standard Support Vector Machine (SVM) classifier to measure how each representation affected the classifier's reliability for identifying the image category. For each individual image, we constructed a vector of features as either (FO) the histogram of first-order statistics as the histogram of edges' orientations, (CM) the chevron map subset of the second-order statistics, (i.e., the two-dimensional histogram of relative orientation and azimuth; see Figure 2 ), or (SO) the full, four-dimensional histogram of second-order statistics (i.e., all parameters of the edge co-occurrences). We gathered these vectors for each different class of images and report here the results of the SVM classifier using an F1 score (50\% represents chance level). While it was expected that differences would be clear between non-animal natural images versus laboratory (man-made) images, results are still quite high for classifying animal images versus non-animal natural images, and are in the range reported by~\citet{Serre07} (F1 score of 80\% for human observers and 82\% for their model), even using the CM features alone. We further extend this results to the psychophysical results of Serre et al. (2007) in Figure 5. Go back to manuscript page.
welcome:
• Page.execute = 0.010s
• getACL = 0.003s
• i18n_init = 0.023s
• init = 0.023s

format: plain


text:
# Unit-4: Free electron Fermi Gas

## INTRODUCTION

We can understand a number of important physical properties of metals, particularly the simple metals, in terms of the free electron model. According to this model the most weakly bound electrons of the constituent atoms move about freely through the volume of the metal. The valence electrons of the atoms become the conduction electrons. Forces between conduction electrons and ion cores are neglected in the free electron approximation: all calculations proceed as if the conduction electrons were free to move everywhere within the specimen. The total energy is exclusively the kinetic energy; the potential energy is neglected.

Even in metals for which the free electron model works well, the actual charge distribution of the conduction electrons is known to reflect the strong electrostatic potential of the ion cores. The usefulness of the free electron model is greatest for experiments that depend essentially upon the kinetic properties of the conduction electrons.

The simplest metals are the alkali metals (lithium, sodium, potassium, rubidium, and cesium). Conduction electrons in all metals are observed to act very much like free electrons, except those metals where the d or f shell electrons overlap or fall close in energy to the conduction band. Electrons in d shells tend to be more localized and less mobile than electrons in s and p shells. Other simple metals are Be, Mg, Ca, Sr, Ba, Al, Ga, In, TI, and Pb. Metals that are not simple are the noble metals (copper, silver, gold), the transition metals, and the lanthanide and actinide metals.

Figure 1. Schematic model of a crystal of sodium metal. The atomic cores arc Na + ions; they are immersed in a sea of conduction electrons, the conduction electrons are derived From the 3s valence electrons of the free atoms. The atomic cores contain 10 electrons in the configuration is 1s2 2s2 2p6. In an alkali metal the atomic cores occupy a relatively small part (15 %) of the total volume of the crystal, but in a noble metal (Cu, Ag, Au) the atomic cores are relatively larger and may be in contact with each other. The common crystal structure at room temperature is bcc for the alkali metals and fcc for the noble metals.

Conduction electrons in a simple metal arise from the valence electrons of the constituent atoms. In a sodium atom the valence electron is in a 3s state [Fig. 1.]; in the metal this electron becomes a conduction electron, roving throughout the crystal. A monovalent crystal which contains N atoms will have N conduction electrons and N positive ion cores. The 10 electrons of the Na + ion core fill the 1s, the 2s, and the 2p states in the free ion; the distribution of core electrons is essentially the same in the metal as in the free ion. The ion cores fill only about 15 percent of the volume of a sodium crystal. The interpretation of metallic properties in terms of the motion of free electrons was developed long before the invention of quantum mechanics. The classical theory had several conspicuous successes, notably the derivation of the form of Ohm’s law and the relation between the electrical and thermal conductivity. The classical theory fails to explain the heat capacity and the magnetic susceptibility of the conduction electrons.

There is a further difficulty. From many types of experiments it is clear that a conduction electron in a metal can move freely in a straight path over many atomic distances, undeflected by collisions with other conduction electrons or by collisions with the atom cores. In a very pure specimen at low temperatures the mean free path may be as long as 10 interatomic spacing (more than 1 cm). Why is condensed matter so transparent to conduction electrons? The answer to our question contains two parts:

a. A conduction electron is not deflected by ion cores arranged on a periodic lattice because matter waves propagate freely in a periodic structure. As the propagation of x-rays in periodic lattice.

b. A conduction electron is scattered only infrequently by other conduction electrons. This property we shall show is a consequence of the Pauli Exclusion Principle. By a free electron Fermi gas, we mean a gas of free and non-interacting electrons subject to the Pauli Principle.

## THE FREE-ELECTRON GAS

In the free-electron model, the conduction electrons are assumed to be completely free, except for a potential at the surface (Fig. 2.), which has the effect of confining the electrons to the interior of the specimen. According to this model, the conduction electrons move about inside the specimen without any collisions, except for an occasional reflection from the surface, much like the molecules in an ideal gas. Because of this, we speak of a free-electron gas.

Figure 2. The potential in the free-electron model.

Let us look at the model a little more closely. It is surprising that it should be valid at all, because, at first sight, one expects the conduction electrons to interact with the ions in the background, and also with each other. These interactions are strong, and hence the electrons ought to suffer frequent collisions; a picture of a highly non ideal gas should therefore emerge. Why then does the free-electron model work? The answer to this fundamental question was not known to the workers who first postulated the model. We now know the answer, but since it requires the use of quantum mechanics, hence a brief qualitative statement is offered here.

The reason why the interaction between the ions appears to be weak is as follows. Although the electron does interact with an ion through coulomb attraction, quantum effects introduce an additional repulsive potential, which tends to cancel the coulomb attraction. The net potential known as the pseudopotential turns out to be weak, particularly in the case of alkali metals. Another way of approaching this is to note that, when an electron passes an ion, its velocity increases rather rapidly in the ion's neighborhood (Fig. 3.), due to the decrease in the potential. Because of this, the electron spends only a small fraction of its time near the ion, where the potential is strong. Most of the time, the electron is far away in a region in which the potential is weak, and this is why the electron behaves like a free particle, to a certain approximation.

Figure 3. Variation of the local velocity of electrons in space.

We come now to the interaction between the conduction electrons themselves, and the reason for the weakness of this interaction. There are actually two reasons: First, according to the Pauli Exclusion Principle, electrons of parallel spins tend to stay away from each other. Second, even if their spins are opposite, electrons tend to stay away from each other, in order to minimize the energy of the system. If two electrons come very close to each other, the coulomb potential energy becomes exceedingly large, and this violates the tendency of the electron system to have the lowest possible energy. When these two considerations are carried out mathematically, the following situation results: Each electron is surrounded by a (spherical) region which is deficient of other electrons. This region, called a hole, has a radius of about 1A0 (the exact value depends on the concentration of electrons). As an electron moves, its hole-sometimes known as a Fermi hole moves with it. We see now why the interaction between electrons is weak. If we examine the interaction between two particular electrons, we find that other electrons distribute themselves in such a manner that our two electrons are screened from each other. Consequently there is very little interaction between them. Free-electron gas in metals differs from ordinary gas in some important respects. First, free-electron gas is charged (in ordinary gases the molecules are mostly neutral). Free-electron gas is thus actually similar to plasma. Second, the concentration of electrons in metals is large: $N{\thicksim}$ 1029 electrons m − 3. By contrast, the ordinary gas has about 1025 molecules m − 3. We may thus think of free-electron gas in a metal as dense plasma.

Our model of the electron (sometimes called the jellium model) corresponds to taking metallic positive ions and smearing them uniformly throughout a sample. In this way there is a positive background which is necessary to maintain charge neutrality. But, because of the uniform distribution, the ions exert zero fields on the electrons; the ions form a uniform jelly into which the electrons move.

## THE DRUDE MODEL

The Drude model is essentially based on the classical kinetic theory of gases. According to Drude the metal must have two types of particles as against only one type in the simplest gases. The discovery of electrons bearing negative charge made it mandatory to accept the presence of positively charged entities (or particles) on the requirement of the charge neutrality condition. It is assumed that when metal atoms come together to form a metal, the valence electrons get liberated and move freely within the metal. The remainder of the atom is a positive ion carrying the major portion of the atomic mass. Drude took these particles as heavy and immobile. Under the action of an external electric field the free electrons referred to as conduction electrons in metals move in the background of immobile positive metal ions. In the absence of a relevant theoretical framework to deal with the free electron gas, Drude took recourse to the methods of kinetic theory of ideal gases for examining metallic conduction without getting deterred by the large electron densities (1022 cm − 3). It is assumed that the reader is acquainted with the postulates of kinetic theory of gases through a prior exposure, and therefore assumptions specific to only free electron gas are being given below:

1. The electron-electron and electron-ion interactions are neglected. The approach is identical to working in the independent electron approximation and the free electron approximation in that order. To be exact, the free electron approximation is not adhered to in strict literal sense because the electrons are considered to remain confined within the metal in the Drude Model and this is possible only if the electron-ion attractive force is active.

2. Under the action of an external electric field, electrons move opposite to the field direction and make collisions with immobile and impenetrable ion cores. An electron bumps from ion to ion and between successive collisions moves in a straight line as determined by the Newton's equations of motion.

3. All electrons move with the RMS speed of a Maxwell-Boltzmann distribution, representing the random or thermal velocity of electrons. The average electron velocity immediately after the collision is zero.

## DC ELECTRICAL CONDUCTIVITY OF METALS

Now, we apply Drude theory to derive an expression for dc conductivity. An external static electric field can affect electron velocity during the time interval between two successive collisions. But the gain in velocity is destroyed each time a collision occurs since the average velocity immediately after the collision is zero. A large influence of electric field is reflected in a larger mean free time or relaxation time τ . The probability of collision per unit time is defined as $\frac{1}{\tau}$ . Therefore, the probability that a collision occurs in a small time interval dt is simply $\frac{dt}{\tau}$ .

Taking the acceleration of an electron of mass as $\frac{e F}{m}$ , the mean drift velocity can be written as

$v_d=\left(-\frac{e F}{m}\right)\tau$ ----------------------------------(1)

Figure 4. (a) An electric field applied to metallic wire. (b) Random versus drift motion of electrons. Cirles represent scattering centers.

If there are a total of n electrons per m3 in the metal, all with constant drift velocity vd , we have the following relation for the net electric current density j

$j=-n e v_d=\left(\frac{ne^2\tau}{m}\right)=\sigma E$ --------------------------------(2)

where the positive scalar quantity

$\sigma=\frac{ne^2\tau}{m}$ ------------------------------------(3)

is defined as the electrical conductivity (the reciprocal of the electrical resistivity $\boldsymbol{\rho}$ ). The form of the expression (3) remains the same in all models including those based on quantum physics. The difference lies only in the way n,m , and τ are defined.

We are now beginning to accomplish our aim of highlighting the gains of the Drude model. The first and the foremost of these is the derivation of the Ohm's law contained in (2). The law was initially set up purely on an empirical basis.

It must be, however, stated that the electrical conductivity σ is not universally scalar since in some complicated situations j becomes nonlinear with respect to E making σ to behave as a tensor. The electrical conductivity is often expressed in terms of the drift mobility of electrons

μ as

σ = neμ ---------------------------------(4)

Where

$\mu=\frac{v_d}{E}$ ----------------------------------------------(5)

According to the kinetic theory thermal velocity vth, expressed by the following relations:

$\tau=\frac{\land}{v_{th}}$ --------------------------------------(6)

where $\land$ is the electron mean free path. We also know that

$\frac{3}{2}k_BT=\frac{1}{2}m v^2_{rms}=\frac{1}{2}m v^2_{th}$ -------------------------------(7)

Using these relations, (3) can be written as

$\sigma=\frac{n e^2\land}{(3 m k_BT)^{1/2}}$ ----------------------------------(8)

The relation (8) is another achievement of the Drude theory since it gives the right magnitude of electrical conductivity and correctly describes that conductivity increases with decrease in temperature. But the dependence on $T^{-\frac{1}{2}}$ is not in agreement with the observed T − 1 dependence in most of the common metals. The modern theory that resolves this issue attributes this inconsistency to the unrealistic assumption of electron-ion elastic collisions and the use of classical statistics to describe the electrons in the Drude model. The Drude model is further plagued by its inability to account for the low electron heat capacity and the temperature independent behavior of paramagnetic susceptibility of conduction electrons.

By feeding the measured value of conductivity in (3), we obtain the value of relaxation time τ . The estimated value of τ is in the range 10 − 15 to 10 − 14 s. The thermal velocities estimated from (7) are around 105 m s − 1. Using (6), we find that the typical values of mean free path $\land$ are of the order of a few angstroms which compare with the interatomic separations. This in a way supports the electron-ion collisions in Drude theory. But the low temperature measurements on some of the purest and least imperfect crystals show the mean free path as large as a few centimeters. This strongly contradicts the Drude picture in which the electron bumps along from ion to ion. On the other hand, the modern theory tells that electron waves cannot be scattered by a perfectly periodic potential leading to an infinite mean free path in an ideal crystal. The experimental data in no way casts aspersion on the above theoretical view, simply because nobody has ever produced any perfect and ideal crystal. Since every real crystal has deviations from periodicity because of the presence of imperfections and even impurities, the value of the mean free path is limited by the scattering from these centers.

## THERMAL CONDUCTIVITY OF METALS

In the Drude model it is assumed that the major contribution to thermal conductivity of metals comes from conduction electrons. This assumption has basis in the general observation that metals are far better conductors of heat than insulators. The participation of metal ions is reflected in the phonon contribution which is neglected in Drude theory on account of its relatively much small measure.

Consider a metal bar whose two ends are maintained at a constant difference of temperature. This situation refers to the steady state when the whole of thermal energy being fed at the hot end is received at the cold end without any net absorption in the bar. Let the temperature gradient along the length of the bar be defined as $-\frac{\partial T}{\partial x}$ meaning thereby that the temperature decreases as length increases by x . For a small temperature gradient, the thermal current j , which is a measure of the thermal energy flowing per unit sectional-area of the bar per unit time, is found to be proportional to the temperature gradient and thus,

$j=K_{el}\left(\frac{\partial T}{\partial x}\right)$ -------------------------------------(9)

which shows that the thermal energy flows in a direction opposite to that of the gradient.

Let us confine ourselves to the one-dimensional flow of heat energy in which electrons move parallel to the length of the bar. The number of heat carriers in this case (electrons) is conserved while calculating j. Let us concentrate on a group of n electrons per unit volume. At a point x along the length, half of them $\left(\frac{n}{2}\right)$ arrive from the hot end and the rest $\left(\frac{n}{2}\right)$ from the cold end. The electron flowing from the hot end should have met its last collision at (xvxτ) whereas the one from the cold end would have its last collision at (x + vxτ) If the thermal energy per electron in thermal equilibrium at T be denoted by $u\left[T(x)\right]$ , the net thermal current can be written as

$j=\frac{1}{2}n v_x\left[u(T(x-v_x\tau))-u(T(x+ v_x\tau))\right]$ -------------------------------------(10)

where T is the temperature at x and vx is the x -component of the average electron velocity. In the approximation that the variation in temperature over a mean free path length $\land$ is small, (6.10) can be expanded about x giving,

$j=n e^2_x\tau\frac{\partial u}{\partial T}\left(-\frac{\partial T}{\partial x}\right)$ ---------------------------------------(11)

The factor $n\left(\frac{\partial u}{\partial T}\right)$ may be replaced by the electron heat capacity per unit volume Cel and $v^2_x$

by $\frac{1}{3}v^2$ in (11) to get

$j=\frac{1}{3}C_{el}v^2\tau\left(-\frac{\partial T}{\partial x}\right)$ ---------------------------------(12)

Comparing (12) with (9), we get .

$K_{el}=\frac{1}{3}C_{el}v^2\tau\frac{1}{3}C_{el}v\land$ --------------------------------(13)

where v is the RMS speed of an electrons.

The calculation of Cel is made by taking the thermal energy per electron at temperature T as $\frac{3}{2}k_B T$ , in accordance with the law of equipartition of energy. This gives the heat capacity per electron as $\frac{3}{2}k_B$ , and $C_{el}=\frac{3}{2}n k_B$ . Furthermore, the value of v2 is obtained from (7). Making these substitutions in (13), we get

$k_{el}=\frac{3}{2}\left(\frac{n k^2_B\tau}{m}\right)T$ ---------------------------(14)

Even before the electrons were known, in 1835 Wiedemann and Franz were able to notice that all good conductors of electricity are good conductors of heat too. They established that the ratio of thermal conductivity to electrical conductivity is proportional to the temperature for a large number of metals. Later in 1881 Lorenz (different from Lorentz) observed that the proportionality constant equaling $\frac{K}{\sigma T}$ has almost the same value for most of the common metals. This constant is known as the Lorenz number and the empirical law became famous as the Wiedemann-Franz law. From (3) and (14), Drude obtained

$\frac{K}{\sigma}=\frac{3}{2}\left(\frac{k^2_B}{e}\right)^2 T$ ---------------------------(15)

The coefficient of T in (15) is a constant. Drude was thus able to derive the Wiedemann-Franz law which is acknowledged as the biggest success of the Drude model. The Lorenz number L as defined in the theory is given by

$L=\frac{K}{\sigma T}=\frac{3}{2}\left(\frac{k^2_B}{e}\right)^2$ -------------------------------------------(16)

## LORENTZ MODIFICATION OF THE DRUDE MODEL

Lorentz took up the task of modifying the oversimplified Drude model and constructed his theory on the basis of following points:

1. The assumption that all electrons move with the same thermal velocity is abandoned.

2. The classical Maxwell-Boltzmann velocity distribution is perturbed by the presence of an electric field or a thermal gradient. Both of these tend to displace the equilibrium velocity distribution and distort its symmetry.

3. The approach of Boltzmann transport equation is followed to describe the transport of charge and kinetic energy of electrons by a statistical distribution of mobile electrons constituting the electron gas.

To avoid duplication, the derivation of the expression for electrical conductivity will be taken up at a proper stage for the quantum model of the free electron gas, i.e. for the free electron Fermi gas. Nevertheless, we give below the relation as obtained by Lorentz;

$\sigma_L=\left(\frac{8}{3\pi}\right)^{\frac{1}{2}}\frac{n e^2\tau}{(3m k_B T)^{1/2}}$ --------------------------------------(17)

If we denote the Drude conductivity of (8) as σD then

$\sigma_D=\left(\frac{3\pi}{8}\right)^{\frac{1}{2}}\sigma_L=1.09\sigma_L$ --------------------------------------(18)

From (17) and (18) we learn that the Lorentz modifications provide neither any noteworthy change in the quantitative measure nor any change in the temperature dependence of electrical conductivity. Lorentz, however, went on to analyze the effect of a uniform magnetic field on current carrying conductors. This effect is known as the Hall Effect, The net force on an electron moving with velocity ν under the action of a static electric field E and a uniform magnetic field B is

$F=-e\left[E+(v\times B)\right]$ ------------------------------------------(19)

where F is called the Lorentz force.

Lorentz solved the Boltzmann transport equation under the above conditions and obtained the following expression for the Hall coefficient.

$R_H=-\left(\frac{3\pi}{8}\right)\frac{1}{ne}$ ------------------------------------(20)

The electron density n can be estimated from (20) by feeding the measured value of Hall coefficient in it. The estimate is consistent with the contribution of one conduction electron per atom in sodium. The result is not so good for silver and is worse for other metals. This points to an otherwise established fact that the free electron approximation is best applicable to alkali metals with the lone loosely bound outermost 3s1 electron. The subject is effectively treated by the band theory that also accounts for the observed positive Hall coefficient in certain metals.

## ENERGY LEVELS AND DENSITY OF ORBITALS IN ONE DIMENSION

Consider a free electron gas in one dimension, an electron of massm is confined to a length L by infinite barrier (Fig. 5). The wave function Ψn(x) of the electron is a solution of the Schrodinger equation HΨ = εψ ; with the kinetic energy $H=\frac{P^2}{2m}$ , where P is the momentum. So that, the Schrӧdinger equation is given by

$i \hbar \frac{\Psi_n}{dt}=-\frac{\hbar^2}{2m}\frac{d^2\Psi_n}{dx^2}=\epsilon_n\Psi_n$ ------------------------------(21)

Where εn is the energy of the electron in the orbital. We use the term orbital to denote a solution of the wave equation for a system of only one electron. The term allows us to distinguish between an exact quantum state of the wave equation of a system of N electrons and an approximate quantum state which we construct by assigning the N electrons to N different orbitals, where each orbital is a solution of a wave equation for one electron. The orbital model is exact only if there are no interactions between electrons.

The boundary conditions are Ψn(0) = 0  ;Ψn(L) = 0 , as imposed by the infinite potential energy barrier; they are satisfied if the wave function is sine like with an integral number a of half-wavelengths between 0 and L: ;

Figure 5. First three energy levels and wave functions of a free electron of mass m confined to a line of length L. The energy the levels are labeled according to the quantum number n which gives the number of half-wavelengths in the wave function. The wavelengths are indicated on the wave functions.

$\Psi_n=A sin\left(\frac{2\pi}{\lambda_n}x\right);\frac{1}{2}n\lambda_n=L$ --------------------------------(22)

where A is a constant. We see that (22) is a solution of (21), because

$\frac{d\Psi_n}{dx}=A\left(\frac{n\pi}{L}\right)cos\left(\frac{n\pi}{L}x\right)$; and
$\frac{d^2\Psi_n}{dx^2}=-A\left(\frac{n\pi}{L}\right)^2 sin\left(\frac{n\pi}{L}x\right)$ -------------------------------------(23)

Where the energy εn, is given by

$\epsilon_n=\frac{\hbar^2}{2m}\left(\frac{n\pi}{L}\right)^2$ ----------------------------------------(24)

We want to accommodate N electrons on the line. According to the Pauli Exclusion Principle no two electrons can have all their quantum numbers identical. That is, each orbital can be occupied by at most one electron. This applies to electrons in atoms, molecules, or solids. In a linear solid the quantum numbers of a conduction electron orbital are n and ms, where n is any positive integer and $m_s=\pm\frac{1}{2}$ , according to the spin orientation. A pair of orbitals labeled by the quantum number n can accommodate two electrons, one with spin up and one with spin down. If there are six electrons, then in the ground state of the system the filled orbitals are those given in the following table:

More than one orbital may have the same energy. The number of orbitals with the same energy is called the degeneracy.

Let nF denote the topmost filled energy level, where we start filling the levels from the bottom ( n = 1) and continue filling the higher levels with electrons until allN electrons are accommodated. It is convenient to suppose that N is an even number. The condition 2nF = N determinesnf , the value of n for the uppermost filled level. The Fermi energy εF is defined as the energy of the topmost filled level in the ground state. By (24) with n = nF we have in one dimension:

$\epsilon_F=\frac{\hbar^2}{2m}\left(\frac{n_F\pi}{L}\right)^2=\frac{\hbar^2}{2m}\left(\frac{N\pi}{2L}\right)^2$ --------------------------------------(25)

## EFFECT OF TEMPERATURE ON THE FERMI-DIRAC DISTRIBUTION

The ground state is the state of the system at absolute zero. What happens as the temperature is increased? This is a standard problem in elementary statistical mechanics and the solution is given by the Fermi-Dirac distribution function. The kinetic energy of the electron gas increases as the temperature is increased: some energy levels are occupied which were vacant at absolute zero, and some levels are vacant which were occupied at absolute zero (Fig. 6). The Fermi-Dirac distribution gives the probability that an orbital at energy ε will be occupied in an ideal electron gas in thermal equilibrium;

$f(\epsilon)=\frac{1}{\left(e^{\frac{(\epsilon-\epsilon_F)}{K_B T}}+1\right)}$ ----------------------------(26)

Figure 6 Fermi-Dirac distribution function at various temperatures.

The quantity εF is a function of the temperature;εF is to be chosen for the particular problem in such a way that the total number of particles in the system comes out correctly — that is, equal to N . At absolute zero εF = μ, because in the limit $T\rightarrow 0$ the function f(ε) changes discontinuously from the value 1 (filled) to the value 0 (empty) at ε = εF = μ . At all temperatures f(ε) equal to$\frac{1}{2}$ when ε = εF = μ, for then the denominator of (26) has the value 2. The quantity μ is called the chemical potential, and we see that at absolute zero the chemical potential is equal to the Fermi energy. The Fermi energy is the energy of the topmost filled orbital at absolute zero.

The high energy tail of the distribution is that part for which $\epsilon-\mu\gg k_B T$; here the exponential term is dominant in the denominator of (26), so that$f(\epsilon)\cong e^{(\epsilon-\epsilon_F)lk_B T}$ . This is the Boltzmann distribution.

## FREE ELECTRON GAS IN THREE DIMENSIONS

The free-particle Schrödinger equation in three dimensions is

$i\hbar=\frac{d\Psi_k}{dt}=-\frac{\hbar^2}{2m}\left(\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}+\frac{\partial^2}{\partial z^2}\right)\Psi_k(r)=\epsilon_k \Psi_k(r)$ -------------------------------------------(27)

If the electrons are confined to a cube of edge L , the wave function is the standing wave

$\Psi_n(r)=A sin\left(\frac{\pi n_x}{L}x\right)sin\left(\frac{\pi n_y}{L}y\right)sin\left(\frac{\pi n_z}{L}z\right)$ ----------------------------------------(28)

where nx,ny,nz are positive integers.

It is convenient to introduce wave functions that satisfy periodic boundary conditions. We now require the wave functions to be periodic in x,y,z with period L, thus

Ψ(x + L,y,z) = Ψ(x,y,,z) --------------------------------------(29)

and similarly for the y and z coordinates. Wave functions satisfying the free-particle Schrodinger equation and the periodicity condition are of the form of a traveling plane wave:

Ψk(r) = exp(ik.r) = e(ik.r) ---------------------------------------(30)

provided that the components of the wave vector k satisfy

$k_x=0;\pm\frac{2\pi}{L};\pm\frac{4\pi}{L};\pm\frac{6\pi}{L};...$ -------------------------------(31)

and similarly forky and kz . That is, any component of k is of the form 2nπ / L , where n is a positive or negative integer. The components of k are the quantum numbers of the problem, along with the quantum numberms for the spin direction. We confirm that these values of k satisfy (29), for

$exp(ik_x(x+L))=exp(i2n\pi(x+L)/L)=exp(i2n\pi x/L)=e^{(ik_x.x)}$ --------------------------------(32)

On substituting (30) in (27) we have the energy εF of the orbital with wave vector k:

$\epsilon_k=\frac{\hbar^2}{2m}k^2=\frac{\hbar^2}{2m}(k^2_x+k^2_y+k^2_z)$ ----------------------------------(33)

The magnitude of the wavevector is related to the wavelength by k = 2π / λ .

The linear momentum p may be represented in quantum mechanics by the

operator $p=-i \hbar \triangledown$, whence for the orbital(30)

$p\Psi_k(r)=-i \hbar \triangledown\Psi-k(r)=\hbar k\Psi_k(r)$ --------------------------------------(34)

so that the plane wave Ψk , is an eigenfunotion of the linear momentum with the eigenvalue $\hbar k$. The particle velocity in the orbital k is given by $v=\frac{\hbar k}{m}$ .

Figure 7. In the ground state of a system of N free electrons the occupied orbitals of the system fill a sphere of radiuskF , where $\epsilon_F=\frac{\hbar^2 k^2_F}{2m}$ is the energy of an electron having a wave vector kF.

In the ground state of a system of N free electrons the occupied orbitals may be represented as points inside a sphere in k space. The energy at the surface of the sphere is the Fermi energy; the wave vectors at the Fermi surface have a magnitude kF such that (Fig. 7):

$\epsilon_F=\frac{\hbar^2}{2m}k^2_F$ --------------------------------(35)

From (31) we see that there is one allowed wave vector that is, one distinct triplet of quantum numbers kx,ky,kz — for the volume element $\left(\frac{2\pi}{L}\right)^3$ of k space. Thus in the sphere of volume $\frac{4}{3}2\pi k^3_F$ the total number of orbitals is

$2\frac{\frac{4}{3}\pi k^3_F}{\left(\frac{2\pi}{L}\right)^3}=\frac{V}{3\pi^2}k^3_F=N$ ----------------------------(36)

where the factor 2 on the left comes from the two allowed values of ms , the spin quantum number, for each allowed value of kF . Then

$k_F=\left(\frac{3\pi^2 N}{V}\right)^{\frac{1}{3}}$ --------------------------------(37)

which depends only on the particle concentration and not on the mass. Using (35),

$\epsilon_F=\frac{\hbar^2}{2m}\left(\frac{3\pi^3 N}{V}\right)^{\frac{2}{3}}$ --------------------------------(39)

This relates the Fermi energy to the electron concentrationN / V . The electron velocity vF at the Fermi surface is

$v_F=\left(\frac{\hbar k_F}{m}\right)=\left(\frac{\hbar}{m}\right)\left(\frac{3\pi^2 N}{V}\right)^{\frac{1}{3}}$ ---------------------------------(40)

We now find an expression for the number of orbitals per unit energy range, D(ε) often called the density of states. We use (39) for the total number of orbitals of energy$\le\epsilon$ :

$N=\left(\frac{V}{3\pi^2}\right)\left(\frac{2m\epsilon}{\hbar^2 }\right)^{\frac{3}{2}}$ ---------------------------------(41)

so that the density of orbitals (Fig. 8) is

$D(\epsilon)=\frac{dn}{d\epsilon}=\left(\frac{V}{2\pi^2}\right)\left(\frac{2m}{\hbar^2}\right)\epsilon^{\frac{1}{2}}$ ---------------------------------(42)

This result may be obtained and expressed most simply by writing (4) as

$ln N=\frac{3}{2}ln \epsilon+ cons tan t$ ;   $\frac{dN}{N}=\frac{3}{2}\frac{d\epsilon}{\epsilon}$ ----------------------------------------(43)

Whence

$D(\epsilon)\equiv\frac{dN}{d\epsilon}=\frac{3N}{2\epsilon}$ -----------------------------(44)

Within a factor of the order of unity, the number of orbitals per unit energy range at the Fermi energy is just the total number of conduction electrons divided by the Fermi energy.

Figure 8. Density of single-particle states as a function of energy, for a free electron gas in three dimensions. The dashed curve represents the density f(ε,T)D(ε) of filled orbitals at a finite temperature, but such that kBT is small in comparison with εF . The shaded area represents the filled orbitals at absolute zero. The average energy is increased when the temperature is increased from 0 to T, for electrons are thermally excited from region 1 to region 2.

## HEAT CAPACETY OF THE ELECTRON GAS

Classical statistical mechanics predicts that a free particle should have a heat capacity of $\frac{3}{2}k_B$ , where kB is the Boltzmann constant. If N atoms each give one valence electron to the electron gas, and the electrons are freely mobile, then the electronic contribution to the heat capacity should be $\frac{3}{2}N k_B$ . But the observed electronic contribution at room temperature is usually less than 0.01 of this value. This discrepancy distracted till the discovery of the Pauli Exclusion Principle and the Fermi-Dirac distribution function. Fermi found the correct equation, and he wrote, “One recognizes that the specific heat vanishes at absolute zero and that at low temperatures it is proportional to the absolute temperature.”

When we heat the specimen from absolute zero not every electron gains an energy kBT as expected classically, but only those electrons in orbitals within an energy range kBT of the Fermi level are excited thermally; these electrons gain an energy which is itself of the order of kBT , as in Fig. 8. This gives an immediate qualitative solution to the problem of the heat capacity of the conduction electron gas. If N is the total number of electrons, only a fraction of the order of T / TF can be excited thermally at temperature T , because only these lie within an energy range of the order of kBT of the top of the energy distribution. Each of these NT / TF electrons has a thermal energy of the order of kBT, and so the total electronic thermal energy U is of the order of

$U\approx N \left(\frac{T}{T_F}\right)k_B T$ -------------------------------(45)

The electronic heat capacity is given by $C_{el}=\frac{dU}{dT}\approx N K_F\left(\frac{T}{T_F}\right)$ and is directly proportional toT , in agreement with the experimental results. At room temperature Cel , is smaller than the classical value $\frac{3}{2} N k_B$ by a factor of the order of 0.01 or less, for $T\approx 5\times 10^4 K$ .

At the low temperatures $\frac{k_B T}{\epsilon_F}<0.01$ the electronic specific heat can be given as,

$C_{el}=\int\limits_{0}^{\infty}(\epsilon-\epsilon_F)D(\epsilon)\frac{\partial f}{\partial T}d\epsilon$ --------------------------------------(46)

$\therefore C_{el}\approx D(\epsilon)\int\limits_{0}^{\infty}(\epsilon-\epsilon_F)\frac{\partial f}{\partial T}d\epsilon$ --------------------------------------------(47)

But $f(\epsilon)=\frac{1}{\left(e^{(\epsilon-\epsilon_F)/k_B T}+1\right)}$ so $\frac{\partial f(\epsilon)}{\partial T}=\frac{(\epsilon-\epsilon_F)}{k_B T^2}\frac{e^{(\epsilon-\epsilon_F)/k_B T}}{(e^{(\epsilon-\epsilon_F)/k_B T}+1)^2}$

Let $\frac{\left(\epsilon-\epsilon_F\right)}{k_B T}=x \rArr d\epsilon=k_B T d x$

$\therefore C_{el}\approx k^2_B T D(\epsilon)\int\limits_{-\epsilon_F/k_B T}^{\infty}\frac{x^2 e^x}{(e^x+1)^2}dx$ ---------------------------------------(48)

The ex factor in the integrand is negligible for $x\le-\frac{\epsilon_F}{k_B T}$ i.e., at low temperatures. Therefore, the lower limit of integration may be safely extended to $-\infty$ transforming the integral into a standard form:

$\int\limits_{-\infty}^{\infty}\frac{x^2 e^x}{(e^x+1)^2}dx=\frac{\pi^2}{3}$ --------------------------(49)
$C_{el}\approx k^2_B T D(\epsilon)\frac{\pi^2}{3}$ ----------------------------(50)

But we have $D(\epsilon)=\frac{3N}{2\epsilon_F}$

$C_{el}\approx k^2_B T \left(\frac{3N}{2\epsilon_F}\right)\frac{\pi^2}{3}=\frac{\pi^2}{2}N k_B\left(\frac{T}{T_F}\right)$ ---------------------------------(51)

The electronic heat capacity of one and two-dimensional electron gases are also linearly proportional T to T at temperatures $T\le T_F$ . The dimensionality of the gas does not enter the argument except by a numerical factor.

## EXPERIMENTAL HEAT CAPACITY OF METALS

At temperatures much below both the Debye temperature and the Fermi temperature, the heat capacity of metals may be written as the sum of electronic and lattice contributions: C = λT + αT3 , where λ and α are constants characteristic of the material. The electronic term is linear in T and is dominant at sufficiently low temperatures. It is convenient to exhibit the experimental values of C as a plot of C / T versus T2 ,

$\frac{C}{T}=\lambda+\alpha T^2$ -------------------------------------(52)

Figure 9. Experimental heat capacity values for Cu, plotted as C/T versus T2.

Where the first term represent the electron contribution and the second term is the Debye term contributed by phonons. The phonon or ionic contribution dominates at high temperatures but drops spectacularly at low temperatures where it becomes even smaller than the electron contribution.

The observed values of the coefficient λ are of the expected magnitude, but often do not agree very closely with the value calculated for free electrons of mass m by use of the relation (50). It is common practice to express the ratio of the observed to the free electron values of the electronic heat capacity as a ratio of a thermal effective mass mth to the electron mass m , where mth is defined by the relation

$\frac{m_{th}}{m}\equiv \frac{\lambda(observed)}{\lambda(free)}$ --------------------------------(53)

This form arises in a natural way because εF, in the expression (50) is inversely proportional to the mass of the electron, whence λαm . The departure from unity involves three separate effects

1. The interaction of the conduction electrons with the periodic potential of the rigid crystal lattice. The effective mass of an electron in this potential is called the band effective mass and is treated later.

2. The interaction of the conduction electrons with phonons. An electron tends to polarize or distort the lattice in its neighborhood, so that the moving electron tries to drag nearby ions along, thereby increasing the effective mass of the electron.

3. The interaction of the conduction electrons with themselves. A moving electron causes an inertial reaction in the surrounding electron gas, thereby increasing the effective mass of the electron.

## ELECTRICAL CONDUCTIVITY AND OHM’S LAW

The momentum of a free electron is related to the wave vector by $p=m v=\hbar k$. In an electric field E and magnetic field B the force F on an electron of charge e is $-e\left[E+\frac{1}{c}(v\times B)\right]$ , so that Newton’s second law of motion becomes (CGS)

$F=m\frac{dv}{dt}=\hbar\frac{dk}{dt}=-e\left[E+\frac{1}{c}(v\times B)\right]$ ----------------------------------(54)

In the absence of collisions the Fermi sphere (Fig. 10) in k space is displaced at a uniform rate by a constant applied electric field, we integrate with B = 0 to obtain

$k(t)-k(0)=-\frac{e E t}{\hbar}$ -----------------------------------(55)

If the field is applied at time t = 0 to an electron gas that fills the Fermi sphere centered at the origin of k space, then at a later time t the sphere will be displaced to a new center at

$\delta k=-\frac{e E t}{\hbar}$ -----------------------------------------(56)

Because of collisions of electrons with impurities, lattice imperfections, and phonons, the displaced sphere may be maintained in a steady state in an electric field. If collision time is τ , the displacement of the Fermi sphere in the steady state is given by (56). The incremental velocity is $v=-\frac{e E \tau}{m}$ . If in a constant electric field E there are n electrons of charge q = − e per unit volume, the electric current density is

$j=n q v=\frac{n e^2\tau E}{m}$ ------------------------------------------(57)
a
b

Figure 10. (a) The Fermi sphere encloses the occupied electron orhitals in space in the ground state of the electron gas. The net momentum is zero, because for every orbital there is an occupied orbital at - . (b) Under the influence of a constant force acting for a time interval every orbital has its vector increased by . This is equivalent to a displacement of the whole Fermi sphere by . The total momentum is , if there are electrons present. The application of the force increases the energy of the system by .

This is in the form of Ohm’s law. The electrical conductivity σ is defined by j = σE , so that

$\sigma=\frac{n e^2\tau}{m}$ ----------------------------------(58)

The electrical resistivity $\boldsymbol{\rho}$ is defined as the reciprocal of the conductivity, so that

$\boldsymbol{\rho}=\frac{m}{n e^2 \tau}$ ---------------------------------(59)

It is easy to understand the result (58) for the conductivity. We expect the charge transported to be proportional to the charge density ne ; the factor e / m enters because the acceleration in a given electric field is proportional to e and inversely proportional to the mass m ; and the time describes the free time during which the field acts on the carrier.

Mean free paths as long as 10 cm have been observed in very pure metals in the liquid helium temperature range.

## EXPERIMENTAL ELECTRICAL RESISTIVITY OF METALS

a
b

Figure 11. Electrical resistivity in most metals arises from collisions of electrons with irregularities in the lattice, as in (a) by phonons and in (b) by impurities and vacant lattice sites.

The electrical resistivity of most metals is dominated at room temperature (300 K) by collisions of the conduction electrons with lattice phonons and at liquid helium temperature (4 K) by collisions with impurity atoms and mechanical imperfections in the lattice (Fig. 11). The rates of these collisions are often independent to a good approximation, so that if the electric field were switched off the momentum distribution would relax back to its ground state with the net relaxation time

$\frac{1}{\tau}=\frac{1}{\tau_L}+\frac{1}{\tau_i}$ ------------------------------(60)

where τL and τ − i are the collision times for scattering by phonons and by imperfections, respectively. The net resistivity is given by

$\boldsymbol{\rho}=\boldsymbol{\rho}_L+\boldsymbol{\rho}_i$ ----------------------------(61)

where $\boldsymbol{\rho}_L$ is the resistivity caused by the thermal phonons, and $\boldsymbol{\rho}_i$ the resistivity caused by scattering of the electron waves by static defects that disturb the periodicity of the lattice. Often $\boldsymbol{\rho}_L$ is independent of the number of defects when their concentration is small, and often $\boldsymbol{\rho}_i$ is independent of temperature. This empirical observation expresses Matthiessen’s rule.

Although the most characteristic feature of a metal, the electrical resistivity is not at all easy to treat theoretically. The collision time depends on the details of the electron-phonon interaction. One feature that emerges is that τL and τi are different functions of the electron wavevector k . Thus the contributions to the net relaxation time τ(k) are no longer simply additive as in (60). Matthiessen’s rule is not always valid. Calculations of the phonon resistivity $\boldsymbol{\rho}_L$ , have been quite successful for some metals.

## MOTION OF CHARGE PARTICLE IN MAGNETIC FIELDS (Hall Effect)

When a magnetic field is applied perpendicular to a conductor carrying current, a voltage is developed across the specimen in the direction perpendicular to both current and the magnetic field. This phenomenon is known as Hall Effect.

Figure 12. The standard geometry for the Hall effect a rod-shaped specimen of rectangular cross-section is placed in a magnetic field , as in (a). An electric field applied across the end electrodes causes an electric current density to flow down the rod. The drift velocity of the electrons immediately after the electric field is applied as shown in (b). The deflection in the direction is caused by the magnetic field. Electrons accumulate on one face of the rod and a positive ion excess is established on the opposite face until, as in (c) the transverse electric field (Hall field) just cancels the force due to the magnetic field.

The magnetic field gives rise to an electric field in a direction mutually orthogonal to the direction of current and magnetic field. The reason for this effect is apparent, when the forces on the current carriers are considered. The electric field Ex which produces the current I , causes a force, of magnitude eEx to act on electron. In the presence of magnetic field, a magnetic force proportional to the magnetic field strength Bz and the electrons average velocity vx also acts on the electron. This force is at right angle to the direction of Bz and vx , and therefore each electron is deflected towards one side of the conductor. When the electron reach the surface of the conductor, an electric charge is built up there, this in turn produces an additional electric field. Under the equilibrium conditions, the sideway force on the moving carriers due to this field is just balance that arising from the magnetic field and the electrons can again move freely down the conductor.

Now the force on the electron in the electromagnetic field in the present condition is given as,

$F_y=-e\left[E_y+\frac{1}{c}(v_x\times B_z)\right]$ ------------------------------(62)

## Hall Voltage and Hall coefficient

In the steady state, the force due to accumulation of electrons becomes equal to the magnetic force and so the flow of electron stops i.e.,Fy = 0

$F_y=0=-e\left[E_y+\frac{1}{c}(v_x\times B_z)\right]$ -----------------------------(63)

$E_y=\frac{1}{c}v_x B_z$ ----------------------------------------(64)

Where Ey is called Hall voltage and vx is average drift velocity, therefore, the current density per unit volume can be written as

j = nevx ---------------------------------------(65)
$v_x=-\frac{j_x}{ne}$ ------------------------------------(66)

Substituting this value of vx in the Ey we have

$E_y=-\frac{1}{c}\frac{j_x}{ne}B_z$ --------------------------------(67)

Rearranging the terms in the above equation we have,

$\frac{E_y}{j_x B_z}=R_H=-\frac{1}{nec}$ ---------------------------------(68)

Where RH is known as Hall coefficient in e.s.u and in e.m.u it will be

$R_H=-\frac{1}{ne}$ ------------------------------------(69)

The Hall coefficient is determined essentially by the sign of the charge carriers. Means it is to be negative if the conduction is by electron and it is positive if conduction is by holes.

## Mobility and Hall angle

When current carrying particles acquire velocity per unit electric field, the velocity is known as mobility and is represented by μ i.e.

$\mu=\frac{v_x}{E_x}$ ------------------------------------(70)
v = μEx -----------------------------------------(71)

Substituting this value in equation (64) we have

$E_y=\frac{1}{c}\mu E_x B_z=\mu E_x B_z$ (in emu)

Rearranging the above equation we have

$\mu=\frac{E_y}{E_x}\frac{1}{B_z}=\phi \frac{1}{B_z}$ ----------------------------(72)

Where $\frac{E_y}{E_x}=\phi$ is the Hall angle, it is also given as,

φ = μBz --------------------------------------(73)

## Importance of Hall Effect:

1. The sign of the charge carriers can be determined

2. The number of charge carriers per unit volume can be calculated from the Hall Coefficient RH .

3. The mobility is measured directly.

4. It can be used to determine the electronic structure of the substance i.e., whether these are metals, insulator or semiconductors.

5. From the Hall voltage we can calculate the applied unknown magnetic field on the substance.

6. It gives the concept of negative mass.


text:
Two plates are connected by filler welds of size $10$ mm and subjected to tension, as shown in the sketch. The thickness of each plate is $12$ mm. The yield stress and the ultimate tensile stress of steel are $250$ MPa and $410$ MPa, respectively. The welding is done in the workshop $(Y_{mw}=1.25)$. As per the Limit State Method of $IS \: 800:2007$, the minimum length (rounded off to the nearest higher higher multiple of $5$ mm) of each weld to transmit a force $P$ equal to $270$ kN is

1. $100$ mm
2. $105$ mm
3. $110$ mm
4. $115$ mm


text:
# What is 0.218 As A Fraction?

0.218 as a fraction is a mathematical representation of a decimal number. It is one of the most common ways of expressing numbers in mathematics. Decimals are used to represent numbers that are between whole numbers, such as 0.218. A fraction is a way of expressing a number as a ratio of two numbers. To convert a decimal number to a fraction, you need to express the decimal number as a fraction with a denominator of 1. This means that 0.218 can be expressed as the fraction 21/100.

## How to Simplify 0.218 As A Fraction

Simplifying a fraction is the process of reducing a fraction to its lowest terms. This is done by finding the greatest common factor of the numerator and the denominator and dividing both the numerator and denominator by it. In the case of 0.218 as a fraction, the greatest common factor is 7. Therefore, the fraction can be simplified to 3/25. This fraction, 3/25, is the simplest form of 0.218 as a fraction.

## How to Convert 0.218 As A Fraction to a Decimal

Converting a fraction to a decimal is also a common operation in mathematics. To convert a fraction to a decimal, divide the numerator by the denominator. In the case of 0.218 as a fraction, the numerator is 3 and the denominator is 25. Therefore, when 3 is divided by 25, the answer is 0.12. This means that 0.218 as a fraction can be expressed as 0.12 as a decimal.

## How to Convert 0.218 As A Fraction to a Percent

To convert a fraction to a percent, multiply the fraction by 100. In the case of 0.218 as a fraction, the fraction is 3/25. When multiplied by 100, the answer is 12%. This means that 0.218 as a fraction can be expressed as 12% as a percent.

## How to Convert 0.218 As A Fraction to a Mixed Number

A mixed number is a number that is composed of a whole number and a fraction. To convert a fraction to a mixed number, divide the numerator by the denominator and then write the result as a mixed number. In the case of 0.218 as a fraction, the numerator is 3 and the denominator is 25. When 3 is divided by 25, the answer is 0.12. This means that 0.218 as a fraction can be expressed as 0 and 12/25 as a mixed number.

##

0.218 as a fraction is a mathematical representation of a decimal number. It can be expressed as 21/100 or simplified to 3/25. It can also be converted to a decimal, a percent, and a mixed number. Understanding how to convert decimals to fractions and fractions to decimals is an important skill in mathematics. Knowing how to simplify fractions and convert them to other forms is also important for solving mathematical problems.